Hindawi Publishing CorporationISRN BiomathematicsVolume 2013 Article ID 637640 12 pageshttpdxdoiorg1011552013637640

Research ArticleGlobal Dynamics of an Exploited Prey-Predator Model withConstant Prey Refuge

Uttam Das1 T K Kar2 and U K Pahari1

1 Department of Mathematics Sree Chaitanya College Habra North 24 Parganas 743268 India2Department of Mathematics Bengal Engineering and Science University Shibpur Howrah 711103 India

Correspondence should be addressed to Uttam Das uttam das76yahooin

Received 27 May 2013 Accepted 8 July 2013

Academic Editors M T Figge M Jose A A Polezhaev and J H Wu

Copyright copy 2013 Uttam Das et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper describes a prey-predator model with Holling type II functional response incorporating constant prey refuge andharvesting to both prey and predator species We have analyzed the boundedness of the system and existence of all possible feasibleequilibria and discussed local as well as global stabilities at interior equilibrium of the system The occurrence of Hopf bifurcationof the system is examined and it was observed that the bifurcation is either supercritical or subcritical Influences of prey refugeand harvesting efforts are also discussed Some numerical simulations are carried out for the validity of theoretical results

1 Introduction

Prey-predator models are of great interest to researchers inmathematics and ecology because they deal with environ-mental problems such as communityrsquos morbidity and how tocontrol it and optimal harvest policy to sustain a communityIn the physical sciences genericmodels can be constructed toexplain a variety of phenomena However in the life sciencesa model only describes a particular situation So a variety ofmodels are needed due to the complexity of the ecosystemTheoretical and numerical studies of these models are able togive us an understanding of the interactions that are takingplace

While investigating biological phenomena there aremany factors which affect dynamical properties of biologicaland mathematical models One of the familiar nonlinearfactors is the functional response In population dynamics afunctional response of the predator to the prey density refersto the change in the density of prey attached per unit time perpredator as the prey density changes [1] Holling [2] suggestedthree different kinds of functional response for different kindsof species tomodel the phenomena of predation whichmadethe classical Lotka-Volterra system more realistic The otherfactor which affects dynamical properties is harvesting Thesubject of harvesting in predator-prey systems has been of

interest to economists ecologists and natural resource man-agers for some time now There are basically three types ofharvesting reported in the literature (i) constant harvestingwhere a constant number of individuals are harvested per unittime (ii) proportional harvesting where the number of indi-viduals harvested per unit time is proportional to the currentpopulation and (iii) nonlinear harvesting Most research hasfocused attention on optimal exploitation guided entirely byprofits from harvesting First of all depending on the natureof the applied harvesting strategy the long-run stationarydensity of population may be significantly smaller than thelong-run stationary density of a population in the absence ofharvesting Therefore while a population can in the absenceof harvesting be free of extinction risk harvesting can leadto the incorporation of a positive extinction probability andtherefore to potential extinction in finite time Secondly ifa population is subjected to a positive extinction rate thenharvesting can drive the population density to a dangerouslylow level at which extinction becomes a sure thing no matterhow the harvester affects the population afterwards

Effects of harvesting on various types of prey-predatormodels have been considered by many researchers Math-ematical modeling with harvesting renewable resourcesstarted with the studies of Clark [3 4] Xiao et al [5]have investigated the dynamics of a system with constant

2 ISRN Biomathematics

PreyPredator

Figure 1 Conceptual diagramof theRosenzweig-MacArthurmodelwith prey refuge Small quadrilateral and oval shapes represent theprey and predator respectively The vertical dashed line representsthe boundary of a refuge where prey species are invulnerable topredators The refuge could protect either a constant number or aconstant fraction of the total prey population

harvesting on the predator whereas Leard et al [6] havestudied the dynamics of ratio-dependent models that includenonconstant harvesting on the prey Xia et al [7] studiedthe combined effects of harvesting and time delay RecentlyPahari and Kar [8] analyzed a prey-predator fishery modelwith harvesting where they applied a tax to regulate thefishery

The existence of prey refuges can clearly have impor-tant effects on the coexistence of predators and their preyAccording to Taylor [9] the different kinds of refuges can bearranged into three types (a) those which provide permanentspatial protection for a small subset of the prey population(b) those which provide temporary spatial protection and (c)those which provide a temporal refuge in numbers that isdecrease the risk of predation by increasing the abundanceof vulnerable prey It is also observed that refuge has astabilizing effect on the equilibrium for a simple Lotka-Volterra model The problem of predator-prey interactionsunder a prey refuge has been studied by some authorsMcNair [10] showed that several kinds of refuges could exerta locally destabilizing effort and create stable large-amplitudeoscillations which would damp out if no refuge was presentMcNair [11] obtained that a prey refuge with legitimate entryexisting dynamics was quite capable of amplifying ratherthan damping predator-prey oscillation Kar [12] proposeda predator-prey model incorporating a prey refuge andindependent harvesting on either species He showed thatusing the harvesting efforts as control it is possible to breakthe cyclic behavior of the system Wang et al [13] considereda prey-predator system where individuals from a prey fishpopulation could hide in holes where predators were unableto enter Ji and Wu [14] considered a predator-prey model

incorporating a constant prey refuge and a constant-rate preyharvesting and showed that the system is controlled by usingconstant harvesting or prey refuge Prepredator interactionswith prey refuge also may be found in the work of Huang etal [15] Wang and Pan [16] Kar et al [17] and so forth

Theobjective of this paper is to study the combined effectsof harvesting and constant prey refuge on the dynamics ofpredator-prey model We also examined the existence of aHopf bifurcation and determined the conditions for whichthe bifurcation is either supercritical or subcritical

Gonzalez-Olivares and Ramos-Jiliberto [18] studied thefollowing prey-predator model with constant number of preyusing refuges as shown in the Figure 1

119889119909

119889119905= 120572119909(1 minus

119909

119896) minus

120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)

119889119910

119889119905= minus119889119910 +

119888120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)

(1)

with 119909(0) gt 119898 119910(0) gt 0 where 119909 and 119910 denote the prey andpredator populations respectively at any time 119905 and 120572 and 119896respectively represent the intrinsic growth rate and carryingcapacity of the prey 119898 is a constant which representsnumber of prey which seeks refuge from predation 119889 is thedeath rate of the predator and 119888 is the conversion factor Theterm 120573119909 | (1 + 119886119909) denotes the functional response of thepredator which is termed Holling type II response function(see [2]) They examined the local stability of equilibria andexistence of limit cycle Chen et al [19] considered the samemodel and examined also global stability and uniqueness oflimit cycle Also Ji and Wu [14] considered the prey-predatormodel with constant prey refuge and a constant-rate preyharvesting ℎ gt 0

119889119909

119889119905= 120572119909(1 minus

119909

119896) minus

120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)minus ℎ

119889119910

119889119905= minus119889119910 +

119888120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)

(2)

and studied instability and global stability of the equilibriaand uniqueness of limit cycle and showed also the influenceof constant prey refuge and constant-rate prey harvesting

Motivated by the paper of Gonzalez-Olivares and Ramos-Jiliberto [18] we considered the following prey-predatorsystem

119889119909

119889119905= 120572119909(1 minus

119909

119896) minus

120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)minus 11990211198641119909

119889119910

119889119905= minus119889119910 +

119888120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)minus 11990221198642119910

(3)

where 1198641ge 0 and 119864

2ge 0 denote the harvesting efforts for the

prey and predator respectively 11990211198641119909 and 119902

21198642119910 represent

the catch of the prey and predator population where 1199021and

1199022represent the catchability coefficients respectively 120572 120573 119896

119889 119886 119888119898 1199021 and 119902

2are positive constants

Our paper is organized in the following way In Section 2we have discussed the boundedness of the solutions of system

ISRN Biomathematics 3

(3) All possible equilibria of the system (8) and the stabilitycriterion at those equilibria and uniqueness of limit cycles atinterior equilibrium are discussed in Section 3 In Section 4we have discussed Hopf bifurcation at interior equilibriumInfluence of refuge parameter 119898 and harvesting efforts 119864

1

and 1198642are discussed in Section 5 Numerical simulations are

given in Section 6 A brief concluding remark is given inSection 7

2 Boundedness of the System

Boundedness of amodel guarantees its validityThe followingtheorem establishes the uniform boundedness of the modelsystem (3)

Theorem 1 All solutions of the system (3) which start in 1198772

+

are uniformly bounded

Proof Let (119909(119905) 119910(119905)) be any solution of the system withpositive initial conditions 119909(0) gt 119898 and 119910(0) gt 0

Now we define the function119882 = 119909 + 119910119888Therefore time derivative gives

119889119882

119889119905=119889119909

119889119905+1

119888

119889119910

119889119905

= 120572119909 minus1205721199092

119896minus 11990211198641119909 minus

119889119910

119888minus11990221198642119910

119888

(4)

Now for each V gt 0 we have

119889119882

119889119905+ V119882 = 120572119909 minus

1205721199092

119896minus 11990211198641119909

+ V119909 +V119910

119888minus119889119910

119888minus11990221198642119910

119888

(5)

We have 119889119882119889119905 + V119882 le (1198964120572)(120572 + V minus 11990211198641)2minus (1119888)(119889 +

11990221198642minus V)

Let us choose V gt 119889 + 11990221198642 then the right-hand side is

positive As we assume that both 1198641and 119864

2are bounded the

right-hand side is bounded for all (119909 119910) isin 1198772+

Thus we choose a 119906 gt 0 such that 119889119882119889119905 + V119882 lt 119906Applying the theory of differential inequality [20] we

obtain

0 lt 119882(119909 119910) lt119906

V(1 minus 119890

minusV119905) +

119882(119909 (0) 119910 (0))

119890V119905

for 119905 997888rarr infin 0 lt 119882 lt119906

V

(6)

Thus all solutions of the system (3) that start in 1198772

+are

confined to the region 119861 where

119861 = (119909 119910) isin 1198772

+ 119882 =

119906

V+ 120576 for any 120576 gt 0 (7)

This completes the theorem

Now for simplicity let us introduce119883 = 119909 minus 119898 then thesystem (3) of equations changes to (still denote119883 = 119909)

119889119909

119889119905= 120572 (119909 + 119898) (1 minus

119909 + 119898

119896)

minus120573119909119910

1 + 119886119909minus 11990211198641(119909 + 119898)

119889119910

119889119905= minus119889119910 +

119888120573119909119910

1 + 119886119909minus 11990221198642119910

(8)

3 The Steady States and Their Stability

We now study the existence and nature of the steady statesParticularly we are interested in the interior equilibrium ofthe system To begin with we list all possible steady states ofthe system (8) as follows

(i) equilibrium in the absence of predator 1198751(1199091 0)

where 1199091= (119896120572 minus 119898120572 minus 119902

11198641119896)120572 exists provided

119896120572 gt 119898120572 + 11990211198641119896

(ii) the interior equilibrium 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast)

minus11990211198641119896 (119909lowast+ 119898))

(9)

If 120573119888 gt 119886119889 + 11988611990221198642holds and any of the following conditions

hold

(H1) 119860 gt 0 119861 gt 0 1198602 ge 4119861 and (119860 minus radic1198602 minus 4119861)2 lt 119898 lt

(119860 + radic1198602 minus 4119861)2

(H2) 119860 gt 0 119861 lt 0 and 0 lt 119898 lt (119860 + radic1198602 minus 4119861)2

(H3) 119860 lt 0 119861 lt 0 and 0 lt 119898 lt (119860 + radic1198602 minus 4119861)2

where119860 = 119896minus2119909lowastminus11990211198641119896120572 and119861 = (119909

lowast)2minus119896119909lowast+11990211198641119896119909lowast120572

then the interior equilibrium 119875lowast(119909lowast 119910lowast) exists

For the stability analysis of the equilibria let the varia-tional matrix of the system (8) at an arbitrary point (119909 119910) be119872(119909 119910)

At 1198751(1199091 0) the eigenvalues of the variational matrix

119872(1199091 0) are 119864

11199021minus 120572 and 119888120573119909

1(1 + 119886119909

1) minus 119889 minus 119902

21198642

Now if (1205721199021)(119896minus119898)119896minus(119889+119902

21198642)119896(119888120573minus119886119889minus119886119902

21198642) lt

1198641lt 120572119902

1and 120573119888 gt 119886119889 + 119886119902

21198642 then both eigenvalues are

negative and hence 1198751(1199091 0) is locally asymptotically stable

To determine the local stability character of the interiorequilibrium 119875

lowast(119909lowast 119910lowast) we compute the variational matrix

119872(119909lowast 119910lowast) at (119909lowast 119910lowast)

The characteristic equation of the variational matrix119872(119909lowast 119910lowast) is given by

1205742+ 120574 (

2119898120572

119896minus1198721) +119872

2= 0 (10)

4 ISRN Biomathematics

where

1198721= 120572 minus 119902

11198641minus

120573119910lowast

(1 + 119886119909)2minus2119909lowast120572

119896

1198722=

1198881205732119909lowast119910lowast

(1 + 119886119909lowast)3gt 0

(11)

Routh-Hurwitz criterion states that all roots of the character-istic equation (10) have negative real parts if (2119898120572119896 minus119872

1) gt

0Therefore interior equilibrium 119875

lowast(119909lowast 119910lowast)will be asymp-

totically stable if119898 gt 11987211198962120572 and unstable if119898 lt 119872

11198962120572

Theorem 2 If 119875lowast(119909lowast 119910lowast) exists with 119898 gt 11987211198962120572 then

119875lowast(119909lowast 119910lowast) is locally asymptotically stable

31 Global Stability for the Interior Equilibrium

Theorem 3 If 119877(119909) gt 0 then the system (8) will be globallyasymptotically stable around the interior equilibrium 119875

lowast(119909lowast

119910lowast)

Proof To show the global stability of system (8) we define aLyapunov function as follows

119881 (119909 119910) = 1198601(119909 minus 119909

lowastminus 119909lowast ln 119909

119909lowast)

+ 1198611(119910 minus 119910

lowastminus 119910lowast ln

119910

119910lowast)

(12)

where 1198601and 119861

1are positive constants to be determined in

the subsequent stepsThe time derivative along the trajectories of (8) is

119889119881

119889119905= 1198601

119909 minus 119909lowast

119909

119889119909

119889119905+ 1198611

119910 minus 119910lowast

119910

119889119910

119889119905

= 1198601(119909 minus 119909

lowast) [120572 (1 +

119898

119909)(1 minus

119909 + 119898

119896)

minus120573119910

1 + 119886119909minus 11990211198641minus11990211198641119898

119909]

+ 1198611(119910 minus 119910

lowast) (minus119889 minus 119902

21198642+

119888120573119909

1 + 119886119909)

= minus1198601

(119909 minus 119909lowast)2

119909lowast[120572119909lowast

119896+120572119898

119909(1 minus

119898

119896) minus

11990211198641119898

119909

minus119886120573119909lowast119910lowast

(1 + 119886119909) (1 + 119886119909lowast)]

+ 120573 [1198881198611minus 1198601(1 + 119886119909

lowast)]

(119909 minus 119909lowast) (119910 minus 119910

lowast)

(1 + 119886119909) (1 + 119886119909lowast)

= minus1198601

(119909 minus 119909lowast)2

119909lowast119877 (119909)

+ 120573 [1198881198611minus 1198601(1 + 119886119909

lowast)]

(119909 minus 119909lowast) (119910 minus 119910

lowast)

(1 + 119886119909) (1 + 119886119909lowast)

(13)

150 200 250 300 350 400 450 5000

002

004

006

008

01

012

014

016

018

02

R(x)

x

Figure 2 Graph of 119877(119909)

where 119877(119909) = [120572119909lowast119896 + (120572119898119909)(1 minus 119898119896) minus 119902

11198641119898119909 minus

119886120573119909lowast119910lowast(1 + 119886119909)(1 + 119886119909

lowast)]

Choosing 1198601= 1 and 119861

1= (1 + 119886119909

lowast)119888 we have

119889119881

119889119905= minus

(119909 minus 119909lowast)2

119909lowast119877 (119909) (14)

Thus if 119877(119909) gt 0 then 119889119881119889119905 lt 0 This completes the proof

In Figure 2 we show that 119877(119909) may be positive for somepositive value of 119909 by using the numerical value of theparameters 120572 = 8 119886 = 05 120573 = 4 119888 = 08 119889 = 2 119896 = 501199021= 06 119902

2= 02 119864

1= 04 119864

2= 02 and119898 = 075

32 Uniqueness of Limit Cycles It is known that for prey-predator systems existence and stability of a limit cycle arerelated to the existence and stability of a positive equilibriumIf the limit cycles do not exist in this case the equilibriumis globally asymptotically stable On the other hand if thepositive equilibrium exists and is unstable there must occurat least one limit cycle

Let us consider system (8) in the form

119889119909

119889119905= 119909119892 (119909) minus 119910ℎ (119909) 119909 (0) gt 0

119889119910

119889119905= 119910 (minus119889 minus 119902

21198642+ 119902 (119909)) 119910 (0) gt 0

(15)

where119892(119909) = 120572(1minus119909119896)minus2120572119898119896minus11990211198641+(1119909)(120572119898minus120572119898

2119896minus

11990211198641119898) ℎ(119909) = 120573119909(1 + 119886119909) and 119902(119909) = 119888120573119909(1 + 119886119909)Nowwe consider the following theorem (see [21]) regard-

ing uniqueness of limit cycles of the previous system

Theorem 4 Suppose for system (15) that

119889

119889119909(1199091198921015840(119909) + 119892 (119909) minus 119909119892 (119909) (ℎ

1015840(119909) ℎ (119909))

minus119889 minus 11990221198642+ 119902 (119909)

) le 0 (16)

ISRN Biomathematics 5

in 0 le 119909 lt 119909lowast and 119909lowast lt 119909 le 119896 Then the previous system has

exactly one limit cycle which is globally asymptotically stablewith respect to the set

(119909 119910) 119909 gt 0 119910 gt 0 119875lowast(119909lowast 119910lowast) (17)

Following Theorem 4 we may state that when 119898 lt 11987211198962120572

system (8) has unique globally stable limit cycle Thus wesee that when the system is unstable there exists a uniqueglobally stable limit cycle

4 Hopf Bifurcation

To discuss Hopf bifurcation of the system (8) we take the helpof the paper [22]

Let us now consider system (8) in the form

119889119909

119889119905= ℎ (119909) (119891 (119909) minus 119910)

119889119910

119889119905= 119888119910 (ℎ (119909) minus 119889

1015840)

(18)

where 119891(119909) = (120572119896120573)(119896 minus 119909)(1 + 119886119909) minus (2120572119898119896120573)(1 +

119886119909) minus (11990211198641120573)(1 + 119886119909) + (1119909120573)(120572119898 minus 120572119898

2119896 minus 119902

11198641119898)(1 +

119886119909) ℎ(119909) = 120573119909(1 + 119886119909) and 1198891015840 = (119889 + 11990221198642)119888

Let (120575 119910120575) be the interior equilibrium of the system (18)

where

120575 =119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910120575=1 + 119886120575

120573119896120575(120572 (120575 + 119898) (119896 minus 119898 minus 120575) minus 119902

11198641119896 (120575 + 119898))

(19)

In this section we examined the Hopf bifurcation occurringat (120575 119910

120575) taking 120575 as a bifurcation parameter

Variational matrix of the system (18) at (120575 119910120575) is

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0) (20)

where 119882(120575) = ℎ(120575)1198911015840(120575) 119884(120575) = minusℎ(120575) and 119885(120575) =

119888119891(120575)ℎ1015840(120575)

The characteristic equation of the variational matrix 119869 isgiven by 1205832 minus 120583119879 + 119863 = 0 where 119879 = tr 119869 = 119882(120575) and 119863 =

det 119869 = minus119884(120575)119885(120575)Now if 119879 lt 0 and 119863 gt 0 then (120575 119910

120575) is locally asymp-

totically stable On the other hand if 119879 gt 0 and 119863 gt 0 then(120575 119910120575) is unstable Also 119879(120575) = 0 and the Jacobian matrix

119869(120575) has a pair of imaginary eigenvalues 120583 = plusmn119894radicminus119884(120575)119885(120575)Let 120583 = 120573(120575) plusmn 119894120596(120575) be the roots of 1205832 minus 120583119879 + 119863 = 0 when 120575is near 120575 then

120573 (120575) =119860 (120575)

2 120596 (120575) =

radicminus4119884 (120575) 119885 (120575) minus 119882 (120575)2

2

1205731015840(120575) =

ℎ (120575) 1198911015840(120575)1015840

2=ℎ1015840(120575) 1198911015840(120575) + ℎ (120575) 119891

10158401015840(120575)

2

(21)

120575 is a root of 1198911015840(120575) = 0 (as ℎ(120575) = 1198891015840= 0) that is 120575 = 120575

satisfies the equation

21205721198861205753+ (2120572119886119898 + 119902

11198641119886119896 + 120572 minus 119886120572119896) 120575

2

+ (120572119898119896 minus 1205721198982minus 11990211198641119896119898) = 0

(22)

and also 11989110158401015840

(120575) = 0Therefore 1205731015840(120575)|

120575=120575= ℎ(120575)119891

10158401015840(120575)2 = 0 as ℎ(120575) = 0

The system (18) undergoes a Hopf bifurcation at (120575 119910120575)

when 120575 = 120575 By further analysis we determined that thebifurcation is either supercritical or subcritical by the firstLyapunov coefficient (see [23ndash25])

119886 (120575) =1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(23)

The computation of 119886(120575) is technical and the detailedcalculations are given in the appendix and only the results arestated in the following theorem

Theorem 5 The system (18) undergoes a Hopf bifurcationat (120575 119910

120575) the Hopf bifurcation is supercritical and backward

(subcritical and forward resp) if 119886(120575) lt 0 (119886(120575) gt 0) where119886(120575) is defined in (23)

5 Influence of the Parameter 119898 andHarvesting Efforts 119864

1and 119864

2

Interior equilibrium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast)

minus11990211198641119896 (119909lowast+ 119898))

(24)

Notice that the scaling from system (3) to system (8) is 119883 =

119909 minus 119898 Let the interior equilibrium of the system (3) be11987510158401015840(11990910158401015840 11991010158401015840) where

11990910158401015840=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

+ 119898

11991010158401015840=1 + 119886 (119909

10158401015840minus 119898)

120573119896 (11990910158401015840 minus 119898)(12057211990910158401015840(119896 minus 119909

10158401015840) minus 1199021119864111989611990910158401015840)

(25)

The model (3) is a one-prey-one-predator model with con-stant prey refuge Now we discuss the influence of the preyrefuge on the model dynamics

Now 11988911990910158401015840119889119898 = 1 gt 0

Therefore 11990910158401015840 is a strictly increasing function of119898

6 ISRN Biomathematics

Table 1

119898 11990910158401015840

11991010158401015840

01 103578 3084702 113578 3375390398969 133578 39500406 153578 45256075 168578 4951751 193578 5655723 393578 11005210 1093578 25775320 2093578 3620882331422lowast 2425002 36898024 2493578 36868525 2593578 36719730 3093578 34093435 3593578 28329740 4093578 19428847 4793578 739071475 4843578 0195172lowastAt119898 = 2331422 the value of 11991010158401015840 is maximum

That is increasing the amount of prey refuge can increaseprey population

To check the effect of 119898 on predator differentiating 11991010158401015840with respect to119898 we get11988911991010158401015840119889119898 = minus(2120572(1+119886119909

lowast)120573119909lowast119896)(119898+

119909lowastminus1198962+119902

111986411198962120572) where119909lowast = (119889+119902

21198642)(120573119888minus119886119889minus119886119902

21198642)

Case 1 If119898 gt 1198962minus119909lowastminus119902111986411198962120572 then11991010158401015840 is amonotonically

decreasing function of119898That is if the value of prey refuge 119898 gradually increases

above threshold value 1198962minus119909lowast minus119902111986411198962120572 then the predator

population gradually decreases

Case 2 If 119898 lt 1198962 minus 119909lowastminus 119902111986411198962120572 then 119910

10158401015840 is a mono-tonically increasing function of119898

That is if the value of prey refuge 119898 gradually decreasesbelow threshold value 1198962minus119909lowastminus119902

111986411198962120572 then the predator

population gradually increases

Case 3 If = 1198962 minus 119909lowastminus 119902111986411198962120572 then11991010158401015840 has a maximum

value

To construct Table 1 and Figure 3 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02

1198641= 04 and119864

2= 02 in appropriate units From the analysis

shown we see that increasing the amount of prey refuge canincrease prey population and that increasing the amount ofprey refuge can increase the density of predator species andthis happened due to predator species still having enoughfood for predation with 119898 being small but if the prey refugeis larger than a threshold that is as the prey refuge becomeslarge enough then the increasing amount of prey refuge candecrease predator species and this happened due to the lossof food for predator species

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

y998400998400

x998400998400

andy998400998400

m

x998400998400

Figure 3 Change of 11990910158401015840 and 11991010158401015840 with the change of the refuge

parameter119898

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

35

4

45

5ylowast

E1

Figure 4 Change of 119910lowast with the change of the harvesting effort 1198641

Influence of the Harvesting Efforts E1and E

2 Interior equilib-

rium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast) minus 11990211198641119896 (119909lowast+ 119898))

(26)

At first let119898 and 1198642be fixed

It is observed that 119909lowast is independent of the parameter 1198641

whereas the value of 119910lowast depends on 1198641 Therefore 119864

1has no

effect on the interior equilibrium level on 119909To check the effect of 119864

1on predator differentiating 119910lowast

with respect to 1198641 we get

119889119910lowast

1198891198641

= minus1199021(1 + 119886119909

lowast) (119898 + 119909

lowast)

120573119909lowastlt 0 (27)

This shows that 119910lowast is a strictly decreasing function of 1198641

ISRN Biomathematics 7

Table 2

1198641

119909lowast

119910lowast

1 093578 3627553 093578 3022485 093578 241746 093578 211487762878 093578 1622196 093578 10257411 093578 060218612 093578 029965125 093578 0148382129 093578 00273671

Table 3

1198642

119909lowast

119910lowast

1 104762 3799685 176471 41513910 333333 52381915 714286 77939417 108 977236194068 226799 12578196 246667 12499320 3000 115087205 406667 611798207 472308 0539829

To construct Table 2 and Figure 4 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02 119898 =

035 and 1198642= 02 in appropriate units From the analysis

shown we see that as the harvesting effort 1198641increases the

prey species remain unchanged but predator species decreaseand this happens due to loss of food for predator species andgoes to extinction when 119864

1is large

Now let119898 and 1198641be fixed then we have

119889119909lowast

1198891198642

=1205731198881199022

(120573119888 minus 119886119889 minus 11988611986421199022)2gt 0 (28)

Therefore 119909lowast is a strictly increasing function of 1198642

Also

119889119910lowast

1198891198642

= [1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572 minus 120572

minus2120572119886119909lowastminus 11990211198641119896119886 minus

119896120572119898

(119909lowast)2]119889119909lowast

1198891198642

(29)

0 2 4 6 8 10 12 14 16 18 20 220

10

20

30

40

50

60

ylowast

xlowast

E2

xlowast

andylowast

Figure 5 Variation of 119909lowast and 119910lowast with the change of the harvestingeffort 119864

2

Since 119889119909lowast1198891198642gt 0 therefore

119889119910lowast

1198891198642

gt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

gt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

119889119910lowast

1198891198642

lt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

lt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

(30)

Thus as the harvesting effort 1198642increases the predator

population increases when 1198642is smaller than the threshold

value But if the harvesting effort 1198642gradually increases

above the threshold value that is as the harvesting effort1198642becomes large enough then the increasing amount of the

harvesting effort 1198642can decrease predator population

To construct Table 3 and Figure 5 we take 120572 = 8 119886 =

05 120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2=

02 1198641= 05 and 119898 = 035 in appropriate units From

the analysis shown we see that as the harvesting effort 1198642

increases the prey and predator species increase But if theharvesting effort 119864

2increases larger than a threshold value

that is as the harvesting effort1198642becomes large enough then

increasing amount of the harvesting effort 1198642can increase

the prey species but will decrease predator species and go toextinction of the predator species when 119864

2is large

6 Numerical Simulation

As the problem is not a case study the real-world data arenot available for this model We therefore take here somehypothetical data with the sole purpose of illustrating theresults that we have established in the previous sections Letus consider the parameters of the system as 120572 = 8 119886 = 05

8 ISRN Biomathematics

0 02 04 06 08 1 12 14 16 18 24

5

6

7

8

9

10

y

x

m = 075 gt 0398969

(093578 495175)

Figure 6 Phase space trajectories corresponding to different initiallevels which shows that (093578 495175) is a global attractor

120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2= 02 119864

1= 04

and 1198642= 02 in appropriate units For these value of para-

meters we get the critical value of 119898 as 119898lowast = 0398969Thus it is easy to verify that for this set of parameters thesystem (8) is locally asymptotically stable around its interiorequilibrium 119875

lowast(119909lowast 119910lowast) for 119898 gt 119898

lowast and is unstable for 119898 lt

119898lowast Thus for 119898 = 119898

lowast= 0398969 the system (8) undergoes

a Hopf bifurcation Now for 119898 = 075 we have interiorequilibrium (093578 495175)which is asymptotically stable(see Figure 6) but 119898 = 02 and the interior equilibrium(093578 337539) is unstable (see Figure 7) Thus taking119898 as a control parameter it is possible to drive the prey-predator system to require equilibrium and to prevent thecycle behaviour of the system FromFigures 8 9 10 and 11 wesee that 119864

1and 119864

2may also be used as controls for the system

(8) Hopf bifurcation occurs when 1198641= 119864lowast

1= 762878 (here

1198642= 02 119898 = 035) and 119864

2= 119864lowast

2= 194068 (here 119864

1= 05

119898 = 035) For the previous values of parameters and 119898 =

0398969 we obtained one value of 120575 say 120575 = 093578 and119886(120575) = minus0332712 lt 0 Thus we may conclude that the Hopfbifurcation around the interior equilibrium is supercriticaland backward

7 Concluding Remarks

This paper deals with a prey-predator model with Hollingtype II functional response incorporating a constant preyrefuge and independent harvesting in either species Oscil-latory behavior and existence of limit cycles in harvestedprey-predator system are common in nature It is notedthat constant prey refuge plays an important role in thedynamics of the proposed model system It is also observedfrom the obtained results that constant prey refuge cancause an unstable equilibrium to become stable and evena simple Hopf bifurcation occurred when the parameter 119898passes through its critical value There exists a threshold

0 5 10 150

5

10

15

20

25

30m = 02 lt 0398969

x

y

Figure 7There is a stable limit cycle surrounding (093578 33754)with119898 = 02

0 05 1 15 2 25 3 35 405

1

15

2

25

3

35

4

45

5

55E1 = 96 gt 762878

x

y

Figure 8 Phase space trajectories corresponding to different initiallevels Here 119864

1= 96 119864

2= 02 and119898 = 035

value of 119898 such that for the prey refuge smaller than thisthreshold increasing the amount of prey refuge can increasethe predator population and if the prey refuge is largerthan the threshold increasing the amount of prey refugecan decrease the predator population We have proved thatexactly one stable limit cycle occurs when the positiveequilibrium is unstableWe also determined the critical valueof 120575 at which Hopf bifurcation occurs and observed that thebifurcation is supercritical and backward It was also foundthat it is possible to control the system in such a way that thesystem approaches a required state using the efforts 119864

1and

1198642as controlsOur analytical results and numerical simulation also indi-

cate that dynamic behavior of the model not only depends onthe prey refuge parameter 119898 but also depends on harvestingefforts 119864

1and 119864

2 Hence it is possible to control the system in

ISRN Biomathematics 9

0 05 1 15 2 25 3 35 41

15

2

25

3

35

4

45

5

55

6E1 = 6 lt 762878

x

y

Figure 9 There is a stable limit cycle surrounding (093578

211487) with 1198641= 6 119864

2= 02 and119898 = 035

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14E2 = 196 gt 194068

x

y

Figure 10 Phase space trajectories corresponding to different initiallevels with 119864

1= 05 119864

2= 196 and119898 = 035

such a way that the system approaches a required state usingthe harvesting efforts 119864

1and 119864

2or prey refuge119898 as controls

In our model we have considered the catch-rate functionbased on catch-per-unit-effort hypothesis that is ℎ = 119902119864119906 (119906and 119864 denote the prey or predator population and harvestingeffort resp) But this type of catch-rate function embodiessome defects in that (i) it assumes random search for fish (ii)it assumes equal likelihood of being captured for every fish(iii) there is unbounded linear increase of ℎ with respect to 119864for fixed 119906 and (iv) there is unbounded linear increase ℎwithrespect to 119906 for a fixed119864These unrealistic features can largelybe removed by adopting the alternative functional form ℎ =

119902119864119906(1198991119864 + 1198992119906) where 119899

1and 1198992are positive constants but

we leave it for our future research work The entire study ofthe paper is mainly based on the deterministic frameworkOn the other hand it will be more realistic if it is possible

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50

Time

Prey

Predator

PreyPredator

xy

E2 = 17 lt 194068

Figure 11 There exist Hopf-bifurcating small amplitude periodicsolutions with 119864

1= 05 119864

2= 17 and119898 = 035

to consider the model system in the stochastic environmentdue to some ecological fluctuations and other factors Thusa future research problem would be considered in stochasticenvironment

Appendix

Detailed Calculation of Formula (23)First we translate the equilibrium (120575 119910

120575) to the origin by

translation 119909 = 119909 minus 120575 119910 = 119910 minus 119910120575 (Still denote 119909 and 119910 by 119909

and 119910 resp) Thus the system (18) becomes

119889119909

119889119905= ℎ (119909 + 120575) (119891 (119909 + 120575) minus (119910 + 119910

120575))

119889119910

119889119905= 119888 (119910 + 119910

120575) (ℎ (119909 + 120575) minus 119889

1015840)

(A1)

We write the system (A1) as follows

119889119909

119889119905= ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

minus (ℎ (119909 + 120575) (119910 + 119910120575) minus ℎ (120575) 119910

120575)

119889119910

119889119905= minus119888119889

1015840119910 + (119888ℎ (119909 + 120575) (119910 + 119910

120575) minus 119888ℎ (120575) 119910

120575)

(A2)

where 119891(120575) = 119910120575and ℎ(120575) = 119889

1015840

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 ISRN Biomathematics

PreyPredator

Figure 1 Conceptual diagramof theRosenzweig-MacArthurmodelwith prey refuge Small quadrilateral and oval shapes represent theprey and predator respectively The vertical dashed line representsthe boundary of a refuge where prey species are invulnerable topredators The refuge could protect either a constant number or aconstant fraction of the total prey population

harvesting on the predator whereas Leard et al [6] havestudied the dynamics of ratio-dependent models that includenonconstant harvesting on the prey Xia et al [7] studiedthe combined effects of harvesting and time delay RecentlyPahari and Kar [8] analyzed a prey-predator fishery modelwith harvesting where they applied a tax to regulate thefishery

The existence of prey refuges can clearly have impor-tant effects on the coexistence of predators and their preyAccording to Taylor [9] the different kinds of refuges can bearranged into three types (a) those which provide permanentspatial protection for a small subset of the prey population(b) those which provide temporary spatial protection and (c)those which provide a temporal refuge in numbers that isdecrease the risk of predation by increasing the abundanceof vulnerable prey It is also observed that refuge has astabilizing effect on the equilibrium for a simple Lotka-Volterra model The problem of predator-prey interactionsunder a prey refuge has been studied by some authorsMcNair [10] showed that several kinds of refuges could exerta locally destabilizing effort and create stable large-amplitudeoscillations which would damp out if no refuge was presentMcNair [11] obtained that a prey refuge with legitimate entryexisting dynamics was quite capable of amplifying ratherthan damping predator-prey oscillation Kar [12] proposeda predator-prey model incorporating a prey refuge andindependent harvesting on either species He showed thatusing the harvesting efforts as control it is possible to breakthe cyclic behavior of the system Wang et al [13] considereda prey-predator system where individuals from a prey fishpopulation could hide in holes where predators were unableto enter Ji and Wu [14] considered a predator-prey model

incorporating a constant prey refuge and a constant-rate preyharvesting and showed that the system is controlled by usingconstant harvesting or prey refuge Prepredator interactionswith prey refuge also may be found in the work of Huang etal [15] Wang and Pan [16] Kar et al [17] and so forth

Theobjective of this paper is to study the combined effectsof harvesting and constant prey refuge on the dynamics ofpredator-prey model We also examined the existence of aHopf bifurcation and determined the conditions for whichthe bifurcation is either supercritical or subcritical

Gonzalez-Olivares and Ramos-Jiliberto [18] studied thefollowing prey-predator model with constant number of preyusing refuges as shown in the Figure 1

119889119909

119889119905= 120572119909(1 minus

119909

119896) minus

120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)

119889119910

119889119905= minus119889119910 +

119888120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)

(1)

with 119909(0) gt 119898 119910(0) gt 0 where 119909 and 119910 denote the prey andpredator populations respectively at any time 119905 and 120572 and 119896respectively represent the intrinsic growth rate and carryingcapacity of the prey 119898 is a constant which representsnumber of prey which seeks refuge from predation 119889 is thedeath rate of the predator and 119888 is the conversion factor Theterm 120573119909 | (1 + 119886119909) denotes the functional response of thepredator which is termed Holling type II response function(see [2]) They examined the local stability of equilibria andexistence of limit cycle Chen et al [19] considered the samemodel and examined also global stability and uniqueness oflimit cycle Also Ji and Wu [14] considered the prey-predatormodel with constant prey refuge and a constant-rate preyharvesting ℎ gt 0

119889119909

119889119905= 120572119909(1 minus

119909

119896) minus

120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)minus ℎ

119889119910

119889119905= minus119889119910 +

119888120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)

(2)

and studied instability and global stability of the equilibriaand uniqueness of limit cycle and showed also the influenceof constant prey refuge and constant-rate prey harvesting

Motivated by the paper of Gonzalez-Olivares and Ramos-Jiliberto [18] we considered the following prey-predatorsystem

119889119909

119889119905= 120572119909(1 minus

119909

119896) minus

120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)minus 11990211198641119909

119889119910

119889119905= minus119889119910 +

119888120573 (119909 minus 119898) 119910

1 + 119886 (119909 minus 119898)minus 11990221198642119910

(3)

where 1198641ge 0 and 119864

2ge 0 denote the harvesting efforts for the

prey and predator respectively 11990211198641119909 and 119902

21198642119910 represent

the catch of the prey and predator population where 1199021and

1199022represent the catchability coefficients respectively 120572 120573 119896

119889 119886 119888119898 1199021 and 119902

2are positive constants

Our paper is organized in the following way In Section 2we have discussed the boundedness of the solutions of system

ISRN Biomathematics 3

(3) All possible equilibria of the system (8) and the stabilitycriterion at those equilibria and uniqueness of limit cycles atinterior equilibrium are discussed in Section 3 In Section 4we have discussed Hopf bifurcation at interior equilibriumInfluence of refuge parameter 119898 and harvesting efforts 119864

1

and 1198642are discussed in Section 5 Numerical simulations are

given in Section 6 A brief concluding remark is given inSection 7

2 Boundedness of the System

Boundedness of amodel guarantees its validityThe followingtheorem establishes the uniform boundedness of the modelsystem (3)

Theorem 1 All solutions of the system (3) which start in 1198772

+

are uniformly bounded

Proof Let (119909(119905) 119910(119905)) be any solution of the system withpositive initial conditions 119909(0) gt 119898 and 119910(0) gt 0

Now we define the function119882 = 119909 + 119910119888Therefore time derivative gives

119889119882

119889119905=119889119909

119889119905+1

119888

119889119910

119889119905

= 120572119909 minus1205721199092

119896minus 11990211198641119909 minus

119889119910

119888minus11990221198642119910

119888

(4)

Now for each V gt 0 we have

119889119882

119889119905+ V119882 = 120572119909 minus

1205721199092

119896minus 11990211198641119909

+ V119909 +V119910

119888minus119889119910

119888minus11990221198642119910

119888

(5)

We have 119889119882119889119905 + V119882 le (1198964120572)(120572 + V minus 11990211198641)2minus (1119888)(119889 +

11990221198642minus V)

Let us choose V gt 119889 + 11990221198642 then the right-hand side is

positive As we assume that both 1198641and 119864

2are bounded the

right-hand side is bounded for all (119909 119910) isin 1198772+

Thus we choose a 119906 gt 0 such that 119889119882119889119905 + V119882 lt 119906Applying the theory of differential inequality [20] we

obtain

0 lt 119882(119909 119910) lt119906

V(1 minus 119890

minusV119905) +

119882(119909 (0) 119910 (0))

119890V119905

for 119905 997888rarr infin 0 lt 119882 lt119906

V

(6)

Thus all solutions of the system (3) that start in 1198772

+are

confined to the region 119861 where

119861 = (119909 119910) isin 1198772

+ 119882 =

119906

V+ 120576 for any 120576 gt 0 (7)

This completes the theorem

Now for simplicity let us introduce119883 = 119909 minus 119898 then thesystem (3) of equations changes to (still denote119883 = 119909)

119889119909

119889119905= 120572 (119909 + 119898) (1 minus

119909 + 119898

119896)

minus120573119909119910

1 + 119886119909minus 11990211198641(119909 + 119898)

119889119910

119889119905= minus119889119910 +

119888120573119909119910

1 + 119886119909minus 11990221198642119910

(8)

3 The Steady States and Their Stability

We now study the existence and nature of the steady statesParticularly we are interested in the interior equilibrium ofthe system To begin with we list all possible steady states ofthe system (8) as follows

(i) equilibrium in the absence of predator 1198751(1199091 0)

where 1199091= (119896120572 minus 119898120572 minus 119902

11198641119896)120572 exists provided

119896120572 gt 119898120572 + 11990211198641119896

(ii) the interior equilibrium 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast)

minus11990211198641119896 (119909lowast+ 119898))

(9)

If 120573119888 gt 119886119889 + 11988611990221198642holds and any of the following conditions

hold

(H1) 119860 gt 0 119861 gt 0 1198602 ge 4119861 and (119860 minus radic1198602 minus 4119861)2 lt 119898 lt

(119860 + radic1198602 minus 4119861)2

(H2) 119860 gt 0 119861 lt 0 and 0 lt 119898 lt (119860 + radic1198602 minus 4119861)2

(H3) 119860 lt 0 119861 lt 0 and 0 lt 119898 lt (119860 + radic1198602 minus 4119861)2

where119860 = 119896minus2119909lowastminus11990211198641119896120572 and119861 = (119909

lowast)2minus119896119909lowast+11990211198641119896119909lowast120572

then the interior equilibrium 119875lowast(119909lowast 119910lowast) exists

For the stability analysis of the equilibria let the varia-tional matrix of the system (8) at an arbitrary point (119909 119910) be119872(119909 119910)

At 1198751(1199091 0) the eigenvalues of the variational matrix

119872(1199091 0) are 119864

11199021minus 120572 and 119888120573119909

1(1 + 119886119909

1) minus 119889 minus 119902

21198642

Now if (1205721199021)(119896minus119898)119896minus(119889+119902

21198642)119896(119888120573minus119886119889minus119886119902

21198642) lt

1198641lt 120572119902

1and 120573119888 gt 119886119889 + 119886119902

21198642 then both eigenvalues are

negative and hence 1198751(1199091 0) is locally asymptotically stable

To determine the local stability character of the interiorequilibrium 119875

lowast(119909lowast 119910lowast) we compute the variational matrix

119872(119909lowast 119910lowast) at (119909lowast 119910lowast)

The characteristic equation of the variational matrix119872(119909lowast 119910lowast) is given by

1205742+ 120574 (

2119898120572

119896minus1198721) +119872

2= 0 (10)

4 ISRN Biomathematics

where

1198721= 120572 minus 119902

11198641minus

120573119910lowast

(1 + 119886119909)2minus2119909lowast120572

119896

1198722=

1198881205732119909lowast119910lowast

(1 + 119886119909lowast)3gt 0

(11)

Routh-Hurwitz criterion states that all roots of the character-istic equation (10) have negative real parts if (2119898120572119896 minus119872

1) gt

0Therefore interior equilibrium 119875

lowast(119909lowast 119910lowast)will be asymp-

totically stable if119898 gt 11987211198962120572 and unstable if119898 lt 119872

11198962120572

Theorem 2 If 119875lowast(119909lowast 119910lowast) exists with 119898 gt 11987211198962120572 then

119875lowast(119909lowast 119910lowast) is locally asymptotically stable

31 Global Stability for the Interior Equilibrium

Theorem 3 If 119877(119909) gt 0 then the system (8) will be globallyasymptotically stable around the interior equilibrium 119875

lowast(119909lowast

119910lowast)

Proof To show the global stability of system (8) we define aLyapunov function as follows

119881 (119909 119910) = 1198601(119909 minus 119909

lowastminus 119909lowast ln 119909

119909lowast)

+ 1198611(119910 minus 119910

lowastminus 119910lowast ln

119910

119910lowast)

(12)

where 1198601and 119861

1are positive constants to be determined in

the subsequent stepsThe time derivative along the trajectories of (8) is

119889119881

119889119905= 1198601

119909 minus 119909lowast

119909

119889119909

119889119905+ 1198611

119910 minus 119910lowast

119910

119889119910

119889119905

= 1198601(119909 minus 119909

lowast) [120572 (1 +

119898

119909)(1 minus

119909 + 119898

119896)

minus120573119910

1 + 119886119909minus 11990211198641minus11990211198641119898

119909]

+ 1198611(119910 minus 119910

lowast) (minus119889 minus 119902

21198642+

119888120573119909

1 + 119886119909)

= minus1198601

(119909 minus 119909lowast)2

119909lowast[120572119909lowast

119896+120572119898

119909(1 minus

119898

119896) minus

11990211198641119898

119909

minus119886120573119909lowast119910lowast

(1 + 119886119909) (1 + 119886119909lowast)]

+ 120573 [1198881198611minus 1198601(1 + 119886119909

lowast)]

(119909 minus 119909lowast) (119910 minus 119910

lowast)

(1 + 119886119909) (1 + 119886119909lowast)

= minus1198601

(119909 minus 119909lowast)2

119909lowast119877 (119909)

+ 120573 [1198881198611minus 1198601(1 + 119886119909

lowast)]

(119909 minus 119909lowast) (119910 minus 119910

lowast)

(1 + 119886119909) (1 + 119886119909lowast)

(13)

150 200 250 300 350 400 450 5000

002

004

006

008

01

012

014

016

018

02

R(x)

x

Figure 2 Graph of 119877(119909)

where 119877(119909) = [120572119909lowast119896 + (120572119898119909)(1 minus 119898119896) minus 119902

11198641119898119909 minus

119886120573119909lowast119910lowast(1 + 119886119909)(1 + 119886119909

lowast)]

Choosing 1198601= 1 and 119861

1= (1 + 119886119909

lowast)119888 we have

119889119881

119889119905= minus

(119909 minus 119909lowast)2

119909lowast119877 (119909) (14)

Thus if 119877(119909) gt 0 then 119889119881119889119905 lt 0 This completes the proof

In Figure 2 we show that 119877(119909) may be positive for somepositive value of 119909 by using the numerical value of theparameters 120572 = 8 119886 = 05 120573 = 4 119888 = 08 119889 = 2 119896 = 501199021= 06 119902

2= 02 119864

1= 04 119864

2= 02 and119898 = 075

32 Uniqueness of Limit Cycles It is known that for prey-predator systems existence and stability of a limit cycle arerelated to the existence and stability of a positive equilibriumIf the limit cycles do not exist in this case the equilibriumis globally asymptotically stable On the other hand if thepositive equilibrium exists and is unstable there must occurat least one limit cycle

Let us consider system (8) in the form

119889119909

119889119905= 119909119892 (119909) minus 119910ℎ (119909) 119909 (0) gt 0

119889119910

119889119905= 119910 (minus119889 minus 119902

21198642+ 119902 (119909)) 119910 (0) gt 0

(15)

where119892(119909) = 120572(1minus119909119896)minus2120572119898119896minus11990211198641+(1119909)(120572119898minus120572119898

2119896minus

11990211198641119898) ℎ(119909) = 120573119909(1 + 119886119909) and 119902(119909) = 119888120573119909(1 + 119886119909)Nowwe consider the following theorem (see [21]) regard-

ing uniqueness of limit cycles of the previous system

Theorem 4 Suppose for system (15) that

119889

119889119909(1199091198921015840(119909) + 119892 (119909) minus 119909119892 (119909) (ℎ

1015840(119909) ℎ (119909))

minus119889 minus 11990221198642+ 119902 (119909)

) le 0 (16)

ISRN Biomathematics 5

in 0 le 119909 lt 119909lowast and 119909lowast lt 119909 le 119896 Then the previous system has

exactly one limit cycle which is globally asymptotically stablewith respect to the set

(119909 119910) 119909 gt 0 119910 gt 0 119875lowast(119909lowast 119910lowast) (17)

Following Theorem 4 we may state that when 119898 lt 11987211198962120572

system (8) has unique globally stable limit cycle Thus wesee that when the system is unstable there exists a uniqueglobally stable limit cycle

4 Hopf Bifurcation

To discuss Hopf bifurcation of the system (8) we take the helpof the paper [22]

Let us now consider system (8) in the form

119889119909

119889119905= ℎ (119909) (119891 (119909) minus 119910)

119889119910

119889119905= 119888119910 (ℎ (119909) minus 119889

1015840)

(18)

where 119891(119909) = (120572119896120573)(119896 minus 119909)(1 + 119886119909) minus (2120572119898119896120573)(1 +

119886119909) minus (11990211198641120573)(1 + 119886119909) + (1119909120573)(120572119898 minus 120572119898

2119896 minus 119902

11198641119898)(1 +

119886119909) ℎ(119909) = 120573119909(1 + 119886119909) and 1198891015840 = (119889 + 11990221198642)119888

Let (120575 119910120575) be the interior equilibrium of the system (18)

where

120575 =119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910120575=1 + 119886120575

120573119896120575(120572 (120575 + 119898) (119896 minus 119898 minus 120575) minus 119902

11198641119896 (120575 + 119898))

(19)

In this section we examined the Hopf bifurcation occurringat (120575 119910

120575) taking 120575 as a bifurcation parameter

Variational matrix of the system (18) at (120575 119910120575) is

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0) (20)

where 119882(120575) = ℎ(120575)1198911015840(120575) 119884(120575) = minusℎ(120575) and 119885(120575) =

119888119891(120575)ℎ1015840(120575)

The characteristic equation of the variational matrix 119869 isgiven by 1205832 minus 120583119879 + 119863 = 0 where 119879 = tr 119869 = 119882(120575) and 119863 =

det 119869 = minus119884(120575)119885(120575)Now if 119879 lt 0 and 119863 gt 0 then (120575 119910

120575) is locally asymp-

totically stable On the other hand if 119879 gt 0 and 119863 gt 0 then(120575 119910120575) is unstable Also 119879(120575) = 0 and the Jacobian matrix

119869(120575) has a pair of imaginary eigenvalues 120583 = plusmn119894radicminus119884(120575)119885(120575)Let 120583 = 120573(120575) plusmn 119894120596(120575) be the roots of 1205832 minus 120583119879 + 119863 = 0 when 120575is near 120575 then

120573 (120575) =119860 (120575)

2 120596 (120575) =

radicminus4119884 (120575) 119885 (120575) minus 119882 (120575)2

2

1205731015840(120575) =

ℎ (120575) 1198911015840(120575)1015840

2=ℎ1015840(120575) 1198911015840(120575) + ℎ (120575) 119891

10158401015840(120575)

2

(21)

120575 is a root of 1198911015840(120575) = 0 (as ℎ(120575) = 1198891015840= 0) that is 120575 = 120575

satisfies the equation

21205721198861205753+ (2120572119886119898 + 119902

11198641119886119896 + 120572 minus 119886120572119896) 120575

2

+ (120572119898119896 minus 1205721198982minus 11990211198641119896119898) = 0

(22)

and also 11989110158401015840

(120575) = 0Therefore 1205731015840(120575)|

120575=120575= ℎ(120575)119891

10158401015840(120575)2 = 0 as ℎ(120575) = 0

The system (18) undergoes a Hopf bifurcation at (120575 119910120575)

when 120575 = 120575 By further analysis we determined that thebifurcation is either supercritical or subcritical by the firstLyapunov coefficient (see [23ndash25])

119886 (120575) =1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(23)

The computation of 119886(120575) is technical and the detailedcalculations are given in the appendix and only the results arestated in the following theorem

Theorem 5 The system (18) undergoes a Hopf bifurcationat (120575 119910

120575) the Hopf bifurcation is supercritical and backward

(subcritical and forward resp) if 119886(120575) lt 0 (119886(120575) gt 0) where119886(120575) is defined in (23)

5 Influence of the Parameter 119898 andHarvesting Efforts 119864

1and 119864

2

Interior equilibrium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast)

minus11990211198641119896 (119909lowast+ 119898))

(24)

Notice that the scaling from system (3) to system (8) is 119883 =

119909 minus 119898 Let the interior equilibrium of the system (3) be11987510158401015840(11990910158401015840 11991010158401015840) where

11990910158401015840=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

+ 119898

11991010158401015840=1 + 119886 (119909

10158401015840minus 119898)

120573119896 (11990910158401015840 minus 119898)(12057211990910158401015840(119896 minus 119909

10158401015840) minus 1199021119864111989611990910158401015840)

(25)

The model (3) is a one-prey-one-predator model with con-stant prey refuge Now we discuss the influence of the preyrefuge on the model dynamics

Now 11988911990910158401015840119889119898 = 1 gt 0

Therefore 11990910158401015840 is a strictly increasing function of119898

6 ISRN Biomathematics

Table 1

119898 11990910158401015840

11991010158401015840

01 103578 3084702 113578 3375390398969 133578 39500406 153578 45256075 168578 4951751 193578 5655723 393578 11005210 1093578 25775320 2093578 3620882331422lowast 2425002 36898024 2493578 36868525 2593578 36719730 3093578 34093435 3593578 28329740 4093578 19428847 4793578 739071475 4843578 0195172lowastAt119898 = 2331422 the value of 11991010158401015840 is maximum

That is increasing the amount of prey refuge can increaseprey population

To check the effect of 119898 on predator differentiating 11991010158401015840with respect to119898 we get11988911991010158401015840119889119898 = minus(2120572(1+119886119909

lowast)120573119909lowast119896)(119898+

119909lowastminus1198962+119902

111986411198962120572) where119909lowast = (119889+119902

21198642)(120573119888minus119886119889minus119886119902

21198642)

Case 1 If119898 gt 1198962minus119909lowastminus119902111986411198962120572 then11991010158401015840 is amonotonically

decreasing function of119898That is if the value of prey refuge 119898 gradually increases

above threshold value 1198962minus119909lowast minus119902111986411198962120572 then the predator

population gradually decreases

Case 2 If 119898 lt 1198962 minus 119909lowastminus 119902111986411198962120572 then 119910

10158401015840 is a mono-tonically increasing function of119898

That is if the value of prey refuge 119898 gradually decreasesbelow threshold value 1198962minus119909lowastminus119902

111986411198962120572 then the predator

population gradually increases

Case 3 If = 1198962 minus 119909lowastminus 119902111986411198962120572 then11991010158401015840 has a maximum

value

To construct Table 1 and Figure 3 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02

1198641= 04 and119864

2= 02 in appropriate units From the analysis

shown we see that increasing the amount of prey refuge canincrease prey population and that increasing the amount ofprey refuge can increase the density of predator species andthis happened due to predator species still having enoughfood for predation with 119898 being small but if the prey refugeis larger than a threshold that is as the prey refuge becomeslarge enough then the increasing amount of prey refuge candecrease predator species and this happened due to the lossof food for predator species

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

y998400998400

x998400998400

andy998400998400

m

x998400998400

Figure 3 Change of 11990910158401015840 and 11991010158401015840 with the change of the refuge

parameter119898

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

35

4

45

5ylowast

E1

Figure 4 Change of 119910lowast with the change of the harvesting effort 1198641

Influence of the Harvesting Efforts E1and E

2 Interior equilib-

rium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast) minus 11990211198641119896 (119909lowast+ 119898))

(26)

At first let119898 and 1198642be fixed

It is observed that 119909lowast is independent of the parameter 1198641

whereas the value of 119910lowast depends on 1198641 Therefore 119864

1has no

effect on the interior equilibrium level on 119909To check the effect of 119864

1on predator differentiating 119910lowast

with respect to 1198641 we get

119889119910lowast

1198891198641

= minus1199021(1 + 119886119909

lowast) (119898 + 119909

lowast)

120573119909lowastlt 0 (27)

This shows that 119910lowast is a strictly decreasing function of 1198641

ISRN Biomathematics 7

Table 2

1198641

119909lowast

119910lowast

1 093578 3627553 093578 3022485 093578 241746 093578 211487762878 093578 1622196 093578 10257411 093578 060218612 093578 029965125 093578 0148382129 093578 00273671

Table 3

1198642

119909lowast

119910lowast

1 104762 3799685 176471 41513910 333333 52381915 714286 77939417 108 977236194068 226799 12578196 246667 12499320 3000 115087205 406667 611798207 472308 0539829

To construct Table 2 and Figure 4 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02 119898 =

035 and 1198642= 02 in appropriate units From the analysis

shown we see that as the harvesting effort 1198641increases the

prey species remain unchanged but predator species decreaseand this happens due to loss of food for predator species andgoes to extinction when 119864

1is large

Now let119898 and 1198641be fixed then we have

119889119909lowast

1198891198642

=1205731198881199022

(120573119888 minus 119886119889 minus 11988611986421199022)2gt 0 (28)

Therefore 119909lowast is a strictly increasing function of 1198642

Also

119889119910lowast

1198891198642

= [1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572 minus 120572

minus2120572119886119909lowastminus 11990211198641119896119886 minus

119896120572119898

(119909lowast)2]119889119909lowast

1198891198642

(29)

0 2 4 6 8 10 12 14 16 18 20 220

10

20

30

40

50

60

ylowast

xlowast

E2

xlowast

andylowast

Figure 5 Variation of 119909lowast and 119910lowast with the change of the harvestingeffort 119864

2

Since 119889119909lowast1198891198642gt 0 therefore

119889119910lowast

1198891198642

gt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

gt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

119889119910lowast

1198891198642

lt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

lt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

(30)

Thus as the harvesting effort 1198642increases the predator

population increases when 1198642is smaller than the threshold

value But if the harvesting effort 1198642gradually increases

above the threshold value that is as the harvesting effort1198642becomes large enough then the increasing amount of the

harvesting effort 1198642can decrease predator population

To construct Table 3 and Figure 5 we take 120572 = 8 119886 =

05 120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2=

02 1198641= 05 and 119898 = 035 in appropriate units From

the analysis shown we see that as the harvesting effort 1198642

increases the prey and predator species increase But if theharvesting effort 119864

2increases larger than a threshold value

that is as the harvesting effort1198642becomes large enough then

increasing amount of the harvesting effort 1198642can increase

the prey species but will decrease predator species and go toextinction of the predator species when 119864

2is large

6 Numerical Simulation

As the problem is not a case study the real-world data arenot available for this model We therefore take here somehypothetical data with the sole purpose of illustrating theresults that we have established in the previous sections Letus consider the parameters of the system as 120572 = 8 119886 = 05

8 ISRN Biomathematics

0 02 04 06 08 1 12 14 16 18 24

5

6

7

8

9

10

y

x

m = 075 gt 0398969

(093578 495175)

Figure 6 Phase space trajectories corresponding to different initiallevels which shows that (093578 495175) is a global attractor

120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2= 02 119864

1= 04

and 1198642= 02 in appropriate units For these value of para-

meters we get the critical value of 119898 as 119898lowast = 0398969Thus it is easy to verify that for this set of parameters thesystem (8) is locally asymptotically stable around its interiorequilibrium 119875

lowast(119909lowast 119910lowast) for 119898 gt 119898

lowast and is unstable for 119898 lt

119898lowast Thus for 119898 = 119898

lowast= 0398969 the system (8) undergoes

a Hopf bifurcation Now for 119898 = 075 we have interiorequilibrium (093578 495175)which is asymptotically stable(see Figure 6) but 119898 = 02 and the interior equilibrium(093578 337539) is unstable (see Figure 7) Thus taking119898 as a control parameter it is possible to drive the prey-predator system to require equilibrium and to prevent thecycle behaviour of the system FromFigures 8 9 10 and 11 wesee that 119864

1and 119864

2may also be used as controls for the system

(8) Hopf bifurcation occurs when 1198641= 119864lowast

1= 762878 (here

1198642= 02 119898 = 035) and 119864

2= 119864lowast

2= 194068 (here 119864

1= 05

119898 = 035) For the previous values of parameters and 119898 =

0398969 we obtained one value of 120575 say 120575 = 093578 and119886(120575) = minus0332712 lt 0 Thus we may conclude that the Hopfbifurcation around the interior equilibrium is supercriticaland backward

7 Concluding Remarks

This paper deals with a prey-predator model with Hollingtype II functional response incorporating a constant preyrefuge and independent harvesting in either species Oscil-latory behavior and existence of limit cycles in harvestedprey-predator system are common in nature It is notedthat constant prey refuge plays an important role in thedynamics of the proposed model system It is also observedfrom the obtained results that constant prey refuge cancause an unstable equilibrium to become stable and evena simple Hopf bifurcation occurred when the parameter 119898passes through its critical value There exists a threshold

0 5 10 150

5

10

15

20

25

30m = 02 lt 0398969

x

y

Figure 7There is a stable limit cycle surrounding (093578 33754)with119898 = 02

0 05 1 15 2 25 3 35 405

1

15

2

25

3

35

4

45

5

55E1 = 96 gt 762878

x

y

Figure 8 Phase space trajectories corresponding to different initiallevels Here 119864

1= 96 119864

2= 02 and119898 = 035

value of 119898 such that for the prey refuge smaller than thisthreshold increasing the amount of prey refuge can increasethe predator population and if the prey refuge is largerthan the threshold increasing the amount of prey refugecan decrease the predator population We have proved thatexactly one stable limit cycle occurs when the positiveequilibrium is unstableWe also determined the critical valueof 120575 at which Hopf bifurcation occurs and observed that thebifurcation is supercritical and backward It was also foundthat it is possible to control the system in such a way that thesystem approaches a required state using the efforts 119864

1and

1198642as controlsOur analytical results and numerical simulation also indi-

cate that dynamic behavior of the model not only depends onthe prey refuge parameter 119898 but also depends on harvestingefforts 119864

1and 119864

2 Hence it is possible to control the system in

ISRN Biomathematics 9

0 05 1 15 2 25 3 35 41

15

2

25

3

35

4

45

5

55

6E1 = 6 lt 762878

x

y

Figure 9 There is a stable limit cycle surrounding (093578

211487) with 1198641= 6 119864

2= 02 and119898 = 035

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14E2 = 196 gt 194068

x

y

Figure 10 Phase space trajectories corresponding to different initiallevels with 119864

1= 05 119864

2= 196 and119898 = 035

such a way that the system approaches a required state usingthe harvesting efforts 119864

1and 119864

2or prey refuge119898 as controls

In our model we have considered the catch-rate functionbased on catch-per-unit-effort hypothesis that is ℎ = 119902119864119906 (119906and 119864 denote the prey or predator population and harvestingeffort resp) But this type of catch-rate function embodiessome defects in that (i) it assumes random search for fish (ii)it assumes equal likelihood of being captured for every fish(iii) there is unbounded linear increase of ℎ with respect to 119864for fixed 119906 and (iv) there is unbounded linear increase ℎwithrespect to 119906 for a fixed119864These unrealistic features can largelybe removed by adopting the alternative functional form ℎ =

119902119864119906(1198991119864 + 1198992119906) where 119899

1and 1198992are positive constants but

we leave it for our future research work The entire study ofthe paper is mainly based on the deterministic frameworkOn the other hand it will be more realistic if it is possible

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50

Time

Prey

Predator

PreyPredator

xy

E2 = 17 lt 194068

Figure 11 There exist Hopf-bifurcating small amplitude periodicsolutions with 119864

1= 05 119864

2= 17 and119898 = 035

to consider the model system in the stochastic environmentdue to some ecological fluctuations and other factors Thusa future research problem would be considered in stochasticenvironment

Appendix

Detailed Calculation of Formula (23)First we translate the equilibrium (120575 119910

120575) to the origin by

translation 119909 = 119909 minus 120575 119910 = 119910 minus 119910120575 (Still denote 119909 and 119910 by 119909

and 119910 resp) Thus the system (18) becomes

119889119909

119889119905= ℎ (119909 + 120575) (119891 (119909 + 120575) minus (119910 + 119910

120575))

119889119910

119889119905= 119888 (119910 + 119910

120575) (ℎ (119909 + 120575) minus 119889

1015840)

(A1)

We write the system (A1) as follows

119889119909

119889119905= ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

minus (ℎ (119909 + 120575) (119910 + 119910120575) minus ℎ (120575) 119910

120575)

119889119910

119889119905= minus119888119889

1015840119910 + (119888ℎ (119909 + 120575) (119910 + 119910

120575) minus 119888ℎ (120575) 119910

120575)

(A2)

where 119891(120575) = 119910120575and ℎ(120575) = 119889

1015840

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

Submit your manuscripts athttpwwwhindawicom

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Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Biomathematics 3

(3) All possible equilibria of the system (8) and the stabilitycriterion at those equilibria and uniqueness of limit cycles atinterior equilibrium are discussed in Section 3 In Section 4we have discussed Hopf bifurcation at interior equilibriumInfluence of refuge parameter 119898 and harvesting efforts 119864

1

and 1198642are discussed in Section 5 Numerical simulations are

given in Section 6 A brief concluding remark is given inSection 7

2 Boundedness of the System

Boundedness of amodel guarantees its validityThe followingtheorem establishes the uniform boundedness of the modelsystem (3)

Theorem 1 All solutions of the system (3) which start in 1198772

+

are uniformly bounded

Proof Let (119909(119905) 119910(119905)) be any solution of the system withpositive initial conditions 119909(0) gt 119898 and 119910(0) gt 0

Now we define the function119882 = 119909 + 119910119888Therefore time derivative gives

119889119882

119889119905=119889119909

119889119905+1

119888

119889119910

119889119905

= 120572119909 minus1205721199092

119896minus 11990211198641119909 minus

119889119910

119888minus11990221198642119910

119888

(4)

Now for each V gt 0 we have

119889119882

119889119905+ V119882 = 120572119909 minus

1205721199092

119896minus 11990211198641119909

+ V119909 +V119910

119888minus119889119910

119888minus11990221198642119910

119888

(5)

We have 119889119882119889119905 + V119882 le (1198964120572)(120572 + V minus 11990211198641)2minus (1119888)(119889 +

11990221198642minus V)

Let us choose V gt 119889 + 11990221198642 then the right-hand side is

positive As we assume that both 1198641and 119864

2are bounded the

right-hand side is bounded for all (119909 119910) isin 1198772+

Thus we choose a 119906 gt 0 such that 119889119882119889119905 + V119882 lt 119906Applying the theory of differential inequality [20] we

obtain

0 lt 119882(119909 119910) lt119906

V(1 minus 119890

minusV119905) +

119882(119909 (0) 119910 (0))

119890V119905

for 119905 997888rarr infin 0 lt 119882 lt119906

V

(6)

Thus all solutions of the system (3) that start in 1198772

+are

confined to the region 119861 where

119861 = (119909 119910) isin 1198772

+ 119882 =

119906

V+ 120576 for any 120576 gt 0 (7)

This completes the theorem

Now for simplicity let us introduce119883 = 119909 minus 119898 then thesystem (3) of equations changes to (still denote119883 = 119909)

119889119909

119889119905= 120572 (119909 + 119898) (1 minus

119909 + 119898

119896)

minus120573119909119910

1 + 119886119909minus 11990211198641(119909 + 119898)

119889119910

119889119905= minus119889119910 +

119888120573119909119910

1 + 119886119909minus 11990221198642119910

(8)

3 The Steady States and Their Stability

We now study the existence and nature of the steady statesParticularly we are interested in the interior equilibrium ofthe system To begin with we list all possible steady states ofthe system (8) as follows

(i) equilibrium in the absence of predator 1198751(1199091 0)

where 1199091= (119896120572 minus 119898120572 minus 119902

11198641119896)120572 exists provided

119896120572 gt 119898120572 + 11990211198641119896

(ii) the interior equilibrium 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast)

minus11990211198641119896 (119909lowast+ 119898))

(9)

If 120573119888 gt 119886119889 + 11988611990221198642holds and any of the following conditions

hold

(H1) 119860 gt 0 119861 gt 0 1198602 ge 4119861 and (119860 minus radic1198602 minus 4119861)2 lt 119898 lt

(119860 + radic1198602 minus 4119861)2

(H2) 119860 gt 0 119861 lt 0 and 0 lt 119898 lt (119860 + radic1198602 minus 4119861)2

(H3) 119860 lt 0 119861 lt 0 and 0 lt 119898 lt (119860 + radic1198602 minus 4119861)2

where119860 = 119896minus2119909lowastminus11990211198641119896120572 and119861 = (119909

lowast)2minus119896119909lowast+11990211198641119896119909lowast120572

then the interior equilibrium 119875lowast(119909lowast 119910lowast) exists

For the stability analysis of the equilibria let the varia-tional matrix of the system (8) at an arbitrary point (119909 119910) be119872(119909 119910)

At 1198751(1199091 0) the eigenvalues of the variational matrix

119872(1199091 0) are 119864

11199021minus 120572 and 119888120573119909

1(1 + 119886119909

1) minus 119889 minus 119902

21198642

Now if (1205721199021)(119896minus119898)119896minus(119889+119902

21198642)119896(119888120573minus119886119889minus119886119902

21198642) lt

1198641lt 120572119902

1and 120573119888 gt 119886119889 + 119886119902

21198642 then both eigenvalues are

negative and hence 1198751(1199091 0) is locally asymptotically stable

To determine the local stability character of the interiorequilibrium 119875

lowast(119909lowast 119910lowast) we compute the variational matrix

119872(119909lowast 119910lowast) at (119909lowast 119910lowast)

The characteristic equation of the variational matrix119872(119909lowast 119910lowast) is given by

1205742+ 120574 (

2119898120572

119896minus1198721) +119872

2= 0 (10)

4 ISRN Biomathematics

where

1198721= 120572 minus 119902

11198641minus

120573119910lowast

(1 + 119886119909)2minus2119909lowast120572

119896

1198722=

1198881205732119909lowast119910lowast

(1 + 119886119909lowast)3gt 0

(11)

Routh-Hurwitz criterion states that all roots of the character-istic equation (10) have negative real parts if (2119898120572119896 minus119872

1) gt

0Therefore interior equilibrium 119875

lowast(119909lowast 119910lowast)will be asymp-

totically stable if119898 gt 11987211198962120572 and unstable if119898 lt 119872

11198962120572

Theorem 2 If 119875lowast(119909lowast 119910lowast) exists with 119898 gt 11987211198962120572 then

119875lowast(119909lowast 119910lowast) is locally asymptotically stable

31 Global Stability for the Interior Equilibrium

Theorem 3 If 119877(119909) gt 0 then the system (8) will be globallyasymptotically stable around the interior equilibrium 119875

lowast(119909lowast

119910lowast)

Proof To show the global stability of system (8) we define aLyapunov function as follows

119881 (119909 119910) = 1198601(119909 minus 119909

lowastminus 119909lowast ln 119909

119909lowast)

+ 1198611(119910 minus 119910

lowastminus 119910lowast ln

119910

119910lowast)

(12)

where 1198601and 119861

1are positive constants to be determined in

the subsequent stepsThe time derivative along the trajectories of (8) is

119889119881

119889119905= 1198601

119909 minus 119909lowast

119909

119889119909

119889119905+ 1198611

119910 minus 119910lowast

119910

119889119910

119889119905

= 1198601(119909 minus 119909

lowast) [120572 (1 +

119898

119909)(1 minus

119909 + 119898

119896)

minus120573119910

1 + 119886119909minus 11990211198641minus11990211198641119898

119909]

+ 1198611(119910 minus 119910

lowast) (minus119889 minus 119902

21198642+

119888120573119909

1 + 119886119909)

= minus1198601

(119909 minus 119909lowast)2

119909lowast[120572119909lowast

119896+120572119898

119909(1 minus

119898

119896) minus

11990211198641119898

119909

minus119886120573119909lowast119910lowast

(1 + 119886119909) (1 + 119886119909lowast)]

+ 120573 [1198881198611minus 1198601(1 + 119886119909

lowast)]

(119909 minus 119909lowast) (119910 minus 119910

lowast)

(1 + 119886119909) (1 + 119886119909lowast)

= minus1198601

(119909 minus 119909lowast)2

119909lowast119877 (119909)

+ 120573 [1198881198611minus 1198601(1 + 119886119909

lowast)]

(119909 minus 119909lowast) (119910 minus 119910

lowast)

(1 + 119886119909) (1 + 119886119909lowast)

(13)

150 200 250 300 350 400 450 5000

002

004

006

008

01

012

014

016

018

02

R(x)

x

Figure 2 Graph of 119877(119909)

where 119877(119909) = [120572119909lowast119896 + (120572119898119909)(1 minus 119898119896) minus 119902

11198641119898119909 minus

119886120573119909lowast119910lowast(1 + 119886119909)(1 + 119886119909

lowast)]

Choosing 1198601= 1 and 119861

1= (1 + 119886119909

lowast)119888 we have

119889119881

119889119905= minus

(119909 minus 119909lowast)2

119909lowast119877 (119909) (14)

Thus if 119877(119909) gt 0 then 119889119881119889119905 lt 0 This completes the proof

In Figure 2 we show that 119877(119909) may be positive for somepositive value of 119909 by using the numerical value of theparameters 120572 = 8 119886 = 05 120573 = 4 119888 = 08 119889 = 2 119896 = 501199021= 06 119902

2= 02 119864

1= 04 119864

2= 02 and119898 = 075

32 Uniqueness of Limit Cycles It is known that for prey-predator systems existence and stability of a limit cycle arerelated to the existence and stability of a positive equilibriumIf the limit cycles do not exist in this case the equilibriumis globally asymptotically stable On the other hand if thepositive equilibrium exists and is unstable there must occurat least one limit cycle

Let us consider system (8) in the form

119889119909

119889119905= 119909119892 (119909) minus 119910ℎ (119909) 119909 (0) gt 0

119889119910

119889119905= 119910 (minus119889 minus 119902

21198642+ 119902 (119909)) 119910 (0) gt 0

(15)

where119892(119909) = 120572(1minus119909119896)minus2120572119898119896minus11990211198641+(1119909)(120572119898minus120572119898

2119896minus

11990211198641119898) ℎ(119909) = 120573119909(1 + 119886119909) and 119902(119909) = 119888120573119909(1 + 119886119909)Nowwe consider the following theorem (see [21]) regard-

ing uniqueness of limit cycles of the previous system

Theorem 4 Suppose for system (15) that

119889

119889119909(1199091198921015840(119909) + 119892 (119909) minus 119909119892 (119909) (ℎ

1015840(119909) ℎ (119909))

minus119889 minus 11990221198642+ 119902 (119909)

) le 0 (16)

ISRN Biomathematics 5

in 0 le 119909 lt 119909lowast and 119909lowast lt 119909 le 119896 Then the previous system has

exactly one limit cycle which is globally asymptotically stablewith respect to the set

(119909 119910) 119909 gt 0 119910 gt 0 119875lowast(119909lowast 119910lowast) (17)

Following Theorem 4 we may state that when 119898 lt 11987211198962120572

system (8) has unique globally stable limit cycle Thus wesee that when the system is unstable there exists a uniqueglobally stable limit cycle

4 Hopf Bifurcation

To discuss Hopf bifurcation of the system (8) we take the helpof the paper [22]

Let us now consider system (8) in the form

119889119909

119889119905= ℎ (119909) (119891 (119909) minus 119910)

119889119910

119889119905= 119888119910 (ℎ (119909) minus 119889

1015840)

(18)

where 119891(119909) = (120572119896120573)(119896 minus 119909)(1 + 119886119909) minus (2120572119898119896120573)(1 +

119886119909) minus (11990211198641120573)(1 + 119886119909) + (1119909120573)(120572119898 minus 120572119898

2119896 minus 119902

11198641119898)(1 +

119886119909) ℎ(119909) = 120573119909(1 + 119886119909) and 1198891015840 = (119889 + 11990221198642)119888

Let (120575 119910120575) be the interior equilibrium of the system (18)

where

120575 =119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910120575=1 + 119886120575

120573119896120575(120572 (120575 + 119898) (119896 minus 119898 minus 120575) minus 119902

11198641119896 (120575 + 119898))

(19)

In this section we examined the Hopf bifurcation occurringat (120575 119910

120575) taking 120575 as a bifurcation parameter

Variational matrix of the system (18) at (120575 119910120575) is

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0) (20)

where 119882(120575) = ℎ(120575)1198911015840(120575) 119884(120575) = minusℎ(120575) and 119885(120575) =

119888119891(120575)ℎ1015840(120575)

The characteristic equation of the variational matrix 119869 isgiven by 1205832 minus 120583119879 + 119863 = 0 where 119879 = tr 119869 = 119882(120575) and 119863 =

det 119869 = minus119884(120575)119885(120575)Now if 119879 lt 0 and 119863 gt 0 then (120575 119910

120575) is locally asymp-

totically stable On the other hand if 119879 gt 0 and 119863 gt 0 then(120575 119910120575) is unstable Also 119879(120575) = 0 and the Jacobian matrix

119869(120575) has a pair of imaginary eigenvalues 120583 = plusmn119894radicminus119884(120575)119885(120575)Let 120583 = 120573(120575) plusmn 119894120596(120575) be the roots of 1205832 minus 120583119879 + 119863 = 0 when 120575is near 120575 then

120573 (120575) =119860 (120575)

2 120596 (120575) =

radicminus4119884 (120575) 119885 (120575) minus 119882 (120575)2

2

1205731015840(120575) =

ℎ (120575) 1198911015840(120575)1015840

2=ℎ1015840(120575) 1198911015840(120575) + ℎ (120575) 119891

10158401015840(120575)

2

(21)

120575 is a root of 1198911015840(120575) = 0 (as ℎ(120575) = 1198891015840= 0) that is 120575 = 120575

satisfies the equation

21205721198861205753+ (2120572119886119898 + 119902

11198641119886119896 + 120572 minus 119886120572119896) 120575

2

+ (120572119898119896 minus 1205721198982minus 11990211198641119896119898) = 0

(22)

and also 11989110158401015840

(120575) = 0Therefore 1205731015840(120575)|

120575=120575= ℎ(120575)119891

10158401015840(120575)2 = 0 as ℎ(120575) = 0

The system (18) undergoes a Hopf bifurcation at (120575 119910120575)

when 120575 = 120575 By further analysis we determined that thebifurcation is either supercritical or subcritical by the firstLyapunov coefficient (see [23ndash25])

119886 (120575) =1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(23)

The computation of 119886(120575) is technical and the detailedcalculations are given in the appendix and only the results arestated in the following theorem

Theorem 5 The system (18) undergoes a Hopf bifurcationat (120575 119910

120575) the Hopf bifurcation is supercritical and backward

(subcritical and forward resp) if 119886(120575) lt 0 (119886(120575) gt 0) where119886(120575) is defined in (23)

5 Influence of the Parameter 119898 andHarvesting Efforts 119864

1and 119864

2

Interior equilibrium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast)

minus11990211198641119896 (119909lowast+ 119898))

(24)

Notice that the scaling from system (3) to system (8) is 119883 =

119909 minus 119898 Let the interior equilibrium of the system (3) be11987510158401015840(11990910158401015840 11991010158401015840) where

11990910158401015840=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

+ 119898

11991010158401015840=1 + 119886 (119909

10158401015840minus 119898)

120573119896 (11990910158401015840 minus 119898)(12057211990910158401015840(119896 minus 119909

10158401015840) minus 1199021119864111989611990910158401015840)

(25)

The model (3) is a one-prey-one-predator model with con-stant prey refuge Now we discuss the influence of the preyrefuge on the model dynamics

Now 11988911990910158401015840119889119898 = 1 gt 0

Therefore 11990910158401015840 is a strictly increasing function of119898

6 ISRN Biomathematics

Table 1

119898 11990910158401015840

11991010158401015840

01 103578 3084702 113578 3375390398969 133578 39500406 153578 45256075 168578 4951751 193578 5655723 393578 11005210 1093578 25775320 2093578 3620882331422lowast 2425002 36898024 2493578 36868525 2593578 36719730 3093578 34093435 3593578 28329740 4093578 19428847 4793578 739071475 4843578 0195172lowastAt119898 = 2331422 the value of 11991010158401015840 is maximum

That is increasing the amount of prey refuge can increaseprey population

To check the effect of 119898 on predator differentiating 11991010158401015840with respect to119898 we get11988911991010158401015840119889119898 = minus(2120572(1+119886119909

lowast)120573119909lowast119896)(119898+

119909lowastminus1198962+119902

111986411198962120572) where119909lowast = (119889+119902

21198642)(120573119888minus119886119889minus119886119902

21198642)

Case 1 If119898 gt 1198962minus119909lowastminus119902111986411198962120572 then11991010158401015840 is amonotonically

decreasing function of119898That is if the value of prey refuge 119898 gradually increases

above threshold value 1198962minus119909lowast minus119902111986411198962120572 then the predator

population gradually decreases

Case 2 If 119898 lt 1198962 minus 119909lowastminus 119902111986411198962120572 then 119910

10158401015840 is a mono-tonically increasing function of119898

That is if the value of prey refuge 119898 gradually decreasesbelow threshold value 1198962minus119909lowastminus119902

111986411198962120572 then the predator

population gradually increases

Case 3 If = 1198962 minus 119909lowastminus 119902111986411198962120572 then11991010158401015840 has a maximum

value

To construct Table 1 and Figure 3 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02

1198641= 04 and119864

2= 02 in appropriate units From the analysis

shown we see that increasing the amount of prey refuge canincrease prey population and that increasing the amount ofprey refuge can increase the density of predator species andthis happened due to predator species still having enoughfood for predation with 119898 being small but if the prey refugeis larger than a threshold that is as the prey refuge becomeslarge enough then the increasing amount of prey refuge candecrease predator species and this happened due to the lossof food for predator species

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

y998400998400

x998400998400

andy998400998400

m

x998400998400

Figure 3 Change of 11990910158401015840 and 11991010158401015840 with the change of the refuge

parameter119898

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

35

4

45

5ylowast

E1

Figure 4 Change of 119910lowast with the change of the harvesting effort 1198641

Influence of the Harvesting Efforts E1and E

2 Interior equilib-

rium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast) minus 11990211198641119896 (119909lowast+ 119898))

(26)

At first let119898 and 1198642be fixed

It is observed that 119909lowast is independent of the parameter 1198641

whereas the value of 119910lowast depends on 1198641 Therefore 119864

1has no

effect on the interior equilibrium level on 119909To check the effect of 119864

1on predator differentiating 119910lowast

with respect to 1198641 we get

119889119910lowast

1198891198641

= minus1199021(1 + 119886119909

lowast) (119898 + 119909

lowast)

120573119909lowastlt 0 (27)

This shows that 119910lowast is a strictly decreasing function of 1198641

ISRN Biomathematics 7

Table 2

1198641

119909lowast

119910lowast

1 093578 3627553 093578 3022485 093578 241746 093578 211487762878 093578 1622196 093578 10257411 093578 060218612 093578 029965125 093578 0148382129 093578 00273671

Table 3

1198642

119909lowast

119910lowast

1 104762 3799685 176471 41513910 333333 52381915 714286 77939417 108 977236194068 226799 12578196 246667 12499320 3000 115087205 406667 611798207 472308 0539829

To construct Table 2 and Figure 4 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02 119898 =

035 and 1198642= 02 in appropriate units From the analysis

shown we see that as the harvesting effort 1198641increases the

prey species remain unchanged but predator species decreaseand this happens due to loss of food for predator species andgoes to extinction when 119864

1is large

Now let119898 and 1198641be fixed then we have

119889119909lowast

1198891198642

=1205731198881199022

(120573119888 minus 119886119889 minus 11988611986421199022)2gt 0 (28)

Therefore 119909lowast is a strictly increasing function of 1198642

Also

119889119910lowast

1198891198642

= [1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572 minus 120572

minus2120572119886119909lowastminus 11990211198641119896119886 minus

119896120572119898

(119909lowast)2]119889119909lowast

1198891198642

(29)

0 2 4 6 8 10 12 14 16 18 20 220

10

20

30

40

50

60

ylowast

xlowast

E2

xlowast

andylowast

Figure 5 Variation of 119909lowast and 119910lowast with the change of the harvestingeffort 119864

2

Since 119889119909lowast1198891198642gt 0 therefore

119889119910lowast

1198891198642

gt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

gt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

119889119910lowast

1198891198642

lt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

lt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

(30)

Thus as the harvesting effort 1198642increases the predator

population increases when 1198642is smaller than the threshold

value But if the harvesting effort 1198642gradually increases

above the threshold value that is as the harvesting effort1198642becomes large enough then the increasing amount of the

harvesting effort 1198642can decrease predator population

To construct Table 3 and Figure 5 we take 120572 = 8 119886 =

05 120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2=

02 1198641= 05 and 119898 = 035 in appropriate units From

the analysis shown we see that as the harvesting effort 1198642

increases the prey and predator species increase But if theharvesting effort 119864

2increases larger than a threshold value

that is as the harvesting effort1198642becomes large enough then

increasing amount of the harvesting effort 1198642can increase

the prey species but will decrease predator species and go toextinction of the predator species when 119864

2is large

6 Numerical Simulation

As the problem is not a case study the real-world data arenot available for this model We therefore take here somehypothetical data with the sole purpose of illustrating theresults that we have established in the previous sections Letus consider the parameters of the system as 120572 = 8 119886 = 05

8 ISRN Biomathematics

0 02 04 06 08 1 12 14 16 18 24

5

6

7

8

9

10

y

x

m = 075 gt 0398969

(093578 495175)

Figure 6 Phase space trajectories corresponding to different initiallevels which shows that (093578 495175) is a global attractor

120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2= 02 119864

1= 04

and 1198642= 02 in appropriate units For these value of para-

meters we get the critical value of 119898 as 119898lowast = 0398969Thus it is easy to verify that for this set of parameters thesystem (8) is locally asymptotically stable around its interiorequilibrium 119875

lowast(119909lowast 119910lowast) for 119898 gt 119898

lowast and is unstable for 119898 lt

119898lowast Thus for 119898 = 119898

lowast= 0398969 the system (8) undergoes

a Hopf bifurcation Now for 119898 = 075 we have interiorequilibrium (093578 495175)which is asymptotically stable(see Figure 6) but 119898 = 02 and the interior equilibrium(093578 337539) is unstable (see Figure 7) Thus taking119898 as a control parameter it is possible to drive the prey-predator system to require equilibrium and to prevent thecycle behaviour of the system FromFigures 8 9 10 and 11 wesee that 119864

1and 119864

2may also be used as controls for the system

(8) Hopf bifurcation occurs when 1198641= 119864lowast

1= 762878 (here

1198642= 02 119898 = 035) and 119864

2= 119864lowast

2= 194068 (here 119864

1= 05

119898 = 035) For the previous values of parameters and 119898 =

0398969 we obtained one value of 120575 say 120575 = 093578 and119886(120575) = minus0332712 lt 0 Thus we may conclude that the Hopfbifurcation around the interior equilibrium is supercriticaland backward

7 Concluding Remarks

This paper deals with a prey-predator model with Hollingtype II functional response incorporating a constant preyrefuge and independent harvesting in either species Oscil-latory behavior and existence of limit cycles in harvestedprey-predator system are common in nature It is notedthat constant prey refuge plays an important role in thedynamics of the proposed model system It is also observedfrom the obtained results that constant prey refuge cancause an unstable equilibrium to become stable and evena simple Hopf bifurcation occurred when the parameter 119898passes through its critical value There exists a threshold

0 5 10 150

5

10

15

20

25

30m = 02 lt 0398969

x

y

Figure 7There is a stable limit cycle surrounding (093578 33754)with119898 = 02

0 05 1 15 2 25 3 35 405

1

15

2

25

3

35

4

45

5

55E1 = 96 gt 762878

x

y

Figure 8 Phase space trajectories corresponding to different initiallevels Here 119864

1= 96 119864

2= 02 and119898 = 035

value of 119898 such that for the prey refuge smaller than thisthreshold increasing the amount of prey refuge can increasethe predator population and if the prey refuge is largerthan the threshold increasing the amount of prey refugecan decrease the predator population We have proved thatexactly one stable limit cycle occurs when the positiveequilibrium is unstableWe also determined the critical valueof 120575 at which Hopf bifurcation occurs and observed that thebifurcation is supercritical and backward It was also foundthat it is possible to control the system in such a way that thesystem approaches a required state using the efforts 119864

1and

1198642as controlsOur analytical results and numerical simulation also indi-

cate that dynamic behavior of the model not only depends onthe prey refuge parameter 119898 but also depends on harvestingefforts 119864

1and 119864

2 Hence it is possible to control the system in

ISRN Biomathematics 9

0 05 1 15 2 25 3 35 41

15

2

25

3

35

4

45

5

55

6E1 = 6 lt 762878

x

y

Figure 9 There is a stable limit cycle surrounding (093578

211487) with 1198641= 6 119864

2= 02 and119898 = 035

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14E2 = 196 gt 194068

x

y

Figure 10 Phase space trajectories corresponding to different initiallevels with 119864

1= 05 119864

2= 196 and119898 = 035

such a way that the system approaches a required state usingthe harvesting efforts 119864

1and 119864

2or prey refuge119898 as controls

In our model we have considered the catch-rate functionbased on catch-per-unit-effort hypothesis that is ℎ = 119902119864119906 (119906and 119864 denote the prey or predator population and harvestingeffort resp) But this type of catch-rate function embodiessome defects in that (i) it assumes random search for fish (ii)it assumes equal likelihood of being captured for every fish(iii) there is unbounded linear increase of ℎ with respect to 119864for fixed 119906 and (iv) there is unbounded linear increase ℎwithrespect to 119906 for a fixed119864These unrealistic features can largelybe removed by adopting the alternative functional form ℎ =

119902119864119906(1198991119864 + 1198992119906) where 119899

1and 1198992are positive constants but

we leave it for our future research work The entire study ofthe paper is mainly based on the deterministic frameworkOn the other hand it will be more realistic if it is possible

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50

Time

Prey

Predator

PreyPredator

xy

E2 = 17 lt 194068

Figure 11 There exist Hopf-bifurcating small amplitude periodicsolutions with 119864

1= 05 119864

2= 17 and119898 = 035

to consider the model system in the stochastic environmentdue to some ecological fluctuations and other factors Thusa future research problem would be considered in stochasticenvironment

Appendix

Detailed Calculation of Formula (23)First we translate the equilibrium (120575 119910

120575) to the origin by

translation 119909 = 119909 minus 120575 119910 = 119910 minus 119910120575 (Still denote 119909 and 119910 by 119909

and 119910 resp) Thus the system (18) becomes

119889119909

119889119905= ℎ (119909 + 120575) (119891 (119909 + 120575) minus (119910 + 119910

120575))

119889119910

119889119905= 119888 (119910 + 119910

120575) (ℎ (119909 + 120575) minus 119889

1015840)

(A1)

We write the system (A1) as follows

119889119909

119889119905= ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

minus (ℎ (119909 + 120575) (119910 + 119910120575) minus ℎ (120575) 119910

120575)

119889119910

119889119905= minus119888119889

1015840119910 + (119888ℎ (119909 + 120575) (119910 + 119910

120575) minus 119888ℎ (120575) 119910

120575)

(A2)

where 119891(120575) = 119910120575and ℎ(120575) = 119889

1015840

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

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4 ISRN Biomathematics

where

1198721= 120572 minus 119902

11198641minus

120573119910lowast

(1 + 119886119909)2minus2119909lowast120572

119896

1198722=

1198881205732119909lowast119910lowast

(1 + 119886119909lowast)3gt 0

(11)

Routh-Hurwitz criterion states that all roots of the character-istic equation (10) have negative real parts if (2119898120572119896 minus119872

1) gt

0Therefore interior equilibrium 119875

lowast(119909lowast 119910lowast)will be asymp-

totically stable if119898 gt 11987211198962120572 and unstable if119898 lt 119872

11198962120572

Theorem 2 If 119875lowast(119909lowast 119910lowast) exists with 119898 gt 11987211198962120572 then

119875lowast(119909lowast 119910lowast) is locally asymptotically stable

31 Global Stability for the Interior Equilibrium

Theorem 3 If 119877(119909) gt 0 then the system (8) will be globallyasymptotically stable around the interior equilibrium 119875

lowast(119909lowast

119910lowast)

Proof To show the global stability of system (8) we define aLyapunov function as follows

119881 (119909 119910) = 1198601(119909 minus 119909

lowastminus 119909lowast ln 119909

119909lowast)

+ 1198611(119910 minus 119910

lowastminus 119910lowast ln

119910

119910lowast)

(12)

where 1198601and 119861

1are positive constants to be determined in

the subsequent stepsThe time derivative along the trajectories of (8) is

119889119881

119889119905= 1198601

119909 minus 119909lowast

119909

119889119909

119889119905+ 1198611

119910 minus 119910lowast

119910

119889119910

119889119905

= 1198601(119909 minus 119909

lowast) [120572 (1 +

119898

119909)(1 minus

119909 + 119898

119896)

minus120573119910

1 + 119886119909minus 11990211198641minus11990211198641119898

119909]

+ 1198611(119910 minus 119910

lowast) (minus119889 minus 119902

21198642+

119888120573119909

1 + 119886119909)

= minus1198601

(119909 minus 119909lowast)2

119909lowast[120572119909lowast

119896+120572119898

119909(1 minus

119898

119896) minus

11990211198641119898

119909

minus119886120573119909lowast119910lowast

(1 + 119886119909) (1 + 119886119909lowast)]

+ 120573 [1198881198611minus 1198601(1 + 119886119909

lowast)]

(119909 minus 119909lowast) (119910 minus 119910

lowast)

(1 + 119886119909) (1 + 119886119909lowast)

= minus1198601

(119909 minus 119909lowast)2

119909lowast119877 (119909)

+ 120573 [1198881198611minus 1198601(1 + 119886119909

lowast)]

(119909 minus 119909lowast) (119910 minus 119910

lowast)

(1 + 119886119909) (1 + 119886119909lowast)

(13)

150 200 250 300 350 400 450 5000

002

004

006

008

01

012

014

016

018

02

R(x)

x

Figure 2 Graph of 119877(119909)

where 119877(119909) = [120572119909lowast119896 + (120572119898119909)(1 minus 119898119896) minus 119902

11198641119898119909 minus

119886120573119909lowast119910lowast(1 + 119886119909)(1 + 119886119909

lowast)]

Choosing 1198601= 1 and 119861

1= (1 + 119886119909

lowast)119888 we have

119889119881

119889119905= minus

(119909 minus 119909lowast)2

119909lowast119877 (119909) (14)

Thus if 119877(119909) gt 0 then 119889119881119889119905 lt 0 This completes the proof

In Figure 2 we show that 119877(119909) may be positive for somepositive value of 119909 by using the numerical value of theparameters 120572 = 8 119886 = 05 120573 = 4 119888 = 08 119889 = 2 119896 = 501199021= 06 119902

2= 02 119864

1= 04 119864

2= 02 and119898 = 075

32 Uniqueness of Limit Cycles It is known that for prey-predator systems existence and stability of a limit cycle arerelated to the existence and stability of a positive equilibriumIf the limit cycles do not exist in this case the equilibriumis globally asymptotically stable On the other hand if thepositive equilibrium exists and is unstable there must occurat least one limit cycle

Let us consider system (8) in the form

119889119909

119889119905= 119909119892 (119909) minus 119910ℎ (119909) 119909 (0) gt 0

119889119910

119889119905= 119910 (minus119889 minus 119902

21198642+ 119902 (119909)) 119910 (0) gt 0

(15)

where119892(119909) = 120572(1minus119909119896)minus2120572119898119896minus11990211198641+(1119909)(120572119898minus120572119898

2119896minus

11990211198641119898) ℎ(119909) = 120573119909(1 + 119886119909) and 119902(119909) = 119888120573119909(1 + 119886119909)Nowwe consider the following theorem (see [21]) regard-

ing uniqueness of limit cycles of the previous system

Theorem 4 Suppose for system (15) that

119889

119889119909(1199091198921015840(119909) + 119892 (119909) minus 119909119892 (119909) (ℎ

1015840(119909) ℎ (119909))

minus119889 minus 11990221198642+ 119902 (119909)

) le 0 (16)

ISRN Biomathematics 5

in 0 le 119909 lt 119909lowast and 119909lowast lt 119909 le 119896 Then the previous system has

exactly one limit cycle which is globally asymptotically stablewith respect to the set

(119909 119910) 119909 gt 0 119910 gt 0 119875lowast(119909lowast 119910lowast) (17)

Following Theorem 4 we may state that when 119898 lt 11987211198962120572

system (8) has unique globally stable limit cycle Thus wesee that when the system is unstable there exists a uniqueglobally stable limit cycle

4 Hopf Bifurcation

To discuss Hopf bifurcation of the system (8) we take the helpof the paper [22]

Let us now consider system (8) in the form

119889119909

119889119905= ℎ (119909) (119891 (119909) minus 119910)

119889119910

119889119905= 119888119910 (ℎ (119909) minus 119889

1015840)

(18)

where 119891(119909) = (120572119896120573)(119896 minus 119909)(1 + 119886119909) minus (2120572119898119896120573)(1 +

119886119909) minus (11990211198641120573)(1 + 119886119909) + (1119909120573)(120572119898 minus 120572119898

2119896 minus 119902

11198641119898)(1 +

119886119909) ℎ(119909) = 120573119909(1 + 119886119909) and 1198891015840 = (119889 + 11990221198642)119888

Let (120575 119910120575) be the interior equilibrium of the system (18)

where

120575 =119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910120575=1 + 119886120575

120573119896120575(120572 (120575 + 119898) (119896 minus 119898 minus 120575) minus 119902

11198641119896 (120575 + 119898))

(19)

In this section we examined the Hopf bifurcation occurringat (120575 119910

120575) taking 120575 as a bifurcation parameter

Variational matrix of the system (18) at (120575 119910120575) is

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0) (20)

where 119882(120575) = ℎ(120575)1198911015840(120575) 119884(120575) = minusℎ(120575) and 119885(120575) =

119888119891(120575)ℎ1015840(120575)

The characteristic equation of the variational matrix 119869 isgiven by 1205832 minus 120583119879 + 119863 = 0 where 119879 = tr 119869 = 119882(120575) and 119863 =

det 119869 = minus119884(120575)119885(120575)Now if 119879 lt 0 and 119863 gt 0 then (120575 119910

120575) is locally asymp-

totically stable On the other hand if 119879 gt 0 and 119863 gt 0 then(120575 119910120575) is unstable Also 119879(120575) = 0 and the Jacobian matrix

119869(120575) has a pair of imaginary eigenvalues 120583 = plusmn119894radicminus119884(120575)119885(120575)Let 120583 = 120573(120575) plusmn 119894120596(120575) be the roots of 1205832 minus 120583119879 + 119863 = 0 when 120575is near 120575 then

120573 (120575) =119860 (120575)

2 120596 (120575) =

radicminus4119884 (120575) 119885 (120575) minus 119882 (120575)2

2

1205731015840(120575) =

ℎ (120575) 1198911015840(120575)1015840

2=ℎ1015840(120575) 1198911015840(120575) + ℎ (120575) 119891

10158401015840(120575)

2

(21)

120575 is a root of 1198911015840(120575) = 0 (as ℎ(120575) = 1198891015840= 0) that is 120575 = 120575

satisfies the equation

21205721198861205753+ (2120572119886119898 + 119902

11198641119886119896 + 120572 minus 119886120572119896) 120575

2

+ (120572119898119896 minus 1205721198982minus 11990211198641119896119898) = 0

(22)

and also 11989110158401015840

(120575) = 0Therefore 1205731015840(120575)|

120575=120575= ℎ(120575)119891

10158401015840(120575)2 = 0 as ℎ(120575) = 0

The system (18) undergoes a Hopf bifurcation at (120575 119910120575)

when 120575 = 120575 By further analysis we determined that thebifurcation is either supercritical or subcritical by the firstLyapunov coefficient (see [23ndash25])

119886 (120575) =1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(23)

The computation of 119886(120575) is technical and the detailedcalculations are given in the appendix and only the results arestated in the following theorem

Theorem 5 The system (18) undergoes a Hopf bifurcationat (120575 119910

120575) the Hopf bifurcation is supercritical and backward

(subcritical and forward resp) if 119886(120575) lt 0 (119886(120575) gt 0) where119886(120575) is defined in (23)

5 Influence of the Parameter 119898 andHarvesting Efforts 119864

1and 119864

2

Interior equilibrium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast)

minus11990211198641119896 (119909lowast+ 119898))

(24)

Notice that the scaling from system (3) to system (8) is 119883 =

119909 minus 119898 Let the interior equilibrium of the system (3) be11987510158401015840(11990910158401015840 11991010158401015840) where

11990910158401015840=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

+ 119898

11991010158401015840=1 + 119886 (119909

10158401015840minus 119898)

120573119896 (11990910158401015840 minus 119898)(12057211990910158401015840(119896 minus 119909

10158401015840) minus 1199021119864111989611990910158401015840)

(25)

The model (3) is a one-prey-one-predator model with con-stant prey refuge Now we discuss the influence of the preyrefuge on the model dynamics

Now 11988911990910158401015840119889119898 = 1 gt 0

Therefore 11990910158401015840 is a strictly increasing function of119898

6 ISRN Biomathematics

Table 1

119898 11990910158401015840

11991010158401015840

01 103578 3084702 113578 3375390398969 133578 39500406 153578 45256075 168578 4951751 193578 5655723 393578 11005210 1093578 25775320 2093578 3620882331422lowast 2425002 36898024 2493578 36868525 2593578 36719730 3093578 34093435 3593578 28329740 4093578 19428847 4793578 739071475 4843578 0195172lowastAt119898 = 2331422 the value of 11991010158401015840 is maximum

That is increasing the amount of prey refuge can increaseprey population

To check the effect of 119898 on predator differentiating 11991010158401015840with respect to119898 we get11988911991010158401015840119889119898 = minus(2120572(1+119886119909

lowast)120573119909lowast119896)(119898+

119909lowastminus1198962+119902

111986411198962120572) where119909lowast = (119889+119902

21198642)(120573119888minus119886119889minus119886119902

21198642)

Case 1 If119898 gt 1198962minus119909lowastminus119902111986411198962120572 then11991010158401015840 is amonotonically

decreasing function of119898That is if the value of prey refuge 119898 gradually increases

above threshold value 1198962minus119909lowast minus119902111986411198962120572 then the predator

population gradually decreases

Case 2 If 119898 lt 1198962 minus 119909lowastminus 119902111986411198962120572 then 119910

10158401015840 is a mono-tonically increasing function of119898

That is if the value of prey refuge 119898 gradually decreasesbelow threshold value 1198962minus119909lowastminus119902

111986411198962120572 then the predator

population gradually increases

Case 3 If = 1198962 minus 119909lowastminus 119902111986411198962120572 then11991010158401015840 has a maximum

value

To construct Table 1 and Figure 3 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02

1198641= 04 and119864

2= 02 in appropriate units From the analysis

shown we see that increasing the amount of prey refuge canincrease prey population and that increasing the amount ofprey refuge can increase the density of predator species andthis happened due to predator species still having enoughfood for predation with 119898 being small but if the prey refugeis larger than a threshold that is as the prey refuge becomeslarge enough then the increasing amount of prey refuge candecrease predator species and this happened due to the lossof food for predator species

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

y998400998400

x998400998400

andy998400998400

m

x998400998400

Figure 3 Change of 11990910158401015840 and 11991010158401015840 with the change of the refuge

parameter119898

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

35

4

45

5ylowast

E1

Figure 4 Change of 119910lowast with the change of the harvesting effort 1198641

Influence of the Harvesting Efforts E1and E

2 Interior equilib-

rium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast) minus 11990211198641119896 (119909lowast+ 119898))

(26)

At first let119898 and 1198642be fixed

It is observed that 119909lowast is independent of the parameter 1198641

whereas the value of 119910lowast depends on 1198641 Therefore 119864

1has no

effect on the interior equilibrium level on 119909To check the effect of 119864

1on predator differentiating 119910lowast

with respect to 1198641 we get

119889119910lowast

1198891198641

= minus1199021(1 + 119886119909

lowast) (119898 + 119909

lowast)

120573119909lowastlt 0 (27)

This shows that 119910lowast is a strictly decreasing function of 1198641

ISRN Biomathematics 7

Table 2

1198641

119909lowast

119910lowast

1 093578 3627553 093578 3022485 093578 241746 093578 211487762878 093578 1622196 093578 10257411 093578 060218612 093578 029965125 093578 0148382129 093578 00273671

Table 3

1198642

119909lowast

119910lowast

1 104762 3799685 176471 41513910 333333 52381915 714286 77939417 108 977236194068 226799 12578196 246667 12499320 3000 115087205 406667 611798207 472308 0539829

To construct Table 2 and Figure 4 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02 119898 =

035 and 1198642= 02 in appropriate units From the analysis

shown we see that as the harvesting effort 1198641increases the

prey species remain unchanged but predator species decreaseand this happens due to loss of food for predator species andgoes to extinction when 119864

1is large

Now let119898 and 1198641be fixed then we have

119889119909lowast

1198891198642

=1205731198881199022

(120573119888 minus 119886119889 minus 11988611986421199022)2gt 0 (28)

Therefore 119909lowast is a strictly increasing function of 1198642

Also

119889119910lowast

1198891198642

= [1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572 minus 120572

minus2120572119886119909lowastminus 11990211198641119896119886 minus

119896120572119898

(119909lowast)2]119889119909lowast

1198891198642

(29)

0 2 4 6 8 10 12 14 16 18 20 220

10

20

30

40

50

60

ylowast

xlowast

E2

xlowast

andylowast

Figure 5 Variation of 119909lowast and 119910lowast with the change of the harvestingeffort 119864

2

Since 119889119909lowast1198891198642gt 0 therefore

119889119910lowast

1198891198642

gt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

gt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

119889119910lowast

1198891198642

lt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

lt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

(30)

Thus as the harvesting effort 1198642increases the predator

population increases when 1198642is smaller than the threshold

value But if the harvesting effort 1198642gradually increases

above the threshold value that is as the harvesting effort1198642becomes large enough then the increasing amount of the

harvesting effort 1198642can decrease predator population

To construct Table 3 and Figure 5 we take 120572 = 8 119886 =

05 120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2=

02 1198641= 05 and 119898 = 035 in appropriate units From

the analysis shown we see that as the harvesting effort 1198642

increases the prey and predator species increase But if theharvesting effort 119864

2increases larger than a threshold value

that is as the harvesting effort1198642becomes large enough then

increasing amount of the harvesting effort 1198642can increase

the prey species but will decrease predator species and go toextinction of the predator species when 119864

2is large

6 Numerical Simulation

As the problem is not a case study the real-world data arenot available for this model We therefore take here somehypothetical data with the sole purpose of illustrating theresults that we have established in the previous sections Letus consider the parameters of the system as 120572 = 8 119886 = 05

8 ISRN Biomathematics

0 02 04 06 08 1 12 14 16 18 24

5

6

7

8

9

10

y

x

m = 075 gt 0398969

(093578 495175)

Figure 6 Phase space trajectories corresponding to different initiallevels which shows that (093578 495175) is a global attractor

120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2= 02 119864

1= 04

and 1198642= 02 in appropriate units For these value of para-

meters we get the critical value of 119898 as 119898lowast = 0398969Thus it is easy to verify that for this set of parameters thesystem (8) is locally asymptotically stable around its interiorequilibrium 119875

lowast(119909lowast 119910lowast) for 119898 gt 119898

lowast and is unstable for 119898 lt

119898lowast Thus for 119898 = 119898

lowast= 0398969 the system (8) undergoes

a Hopf bifurcation Now for 119898 = 075 we have interiorequilibrium (093578 495175)which is asymptotically stable(see Figure 6) but 119898 = 02 and the interior equilibrium(093578 337539) is unstable (see Figure 7) Thus taking119898 as a control parameter it is possible to drive the prey-predator system to require equilibrium and to prevent thecycle behaviour of the system FromFigures 8 9 10 and 11 wesee that 119864

1and 119864

2may also be used as controls for the system

(8) Hopf bifurcation occurs when 1198641= 119864lowast

1= 762878 (here

1198642= 02 119898 = 035) and 119864

2= 119864lowast

2= 194068 (here 119864

1= 05

119898 = 035) For the previous values of parameters and 119898 =

0398969 we obtained one value of 120575 say 120575 = 093578 and119886(120575) = minus0332712 lt 0 Thus we may conclude that the Hopfbifurcation around the interior equilibrium is supercriticaland backward

7 Concluding Remarks

This paper deals with a prey-predator model with Hollingtype II functional response incorporating a constant preyrefuge and independent harvesting in either species Oscil-latory behavior and existence of limit cycles in harvestedprey-predator system are common in nature It is notedthat constant prey refuge plays an important role in thedynamics of the proposed model system It is also observedfrom the obtained results that constant prey refuge cancause an unstable equilibrium to become stable and evena simple Hopf bifurcation occurred when the parameter 119898passes through its critical value There exists a threshold

0 5 10 150

5

10

15

20

25

30m = 02 lt 0398969

x

y

Figure 7There is a stable limit cycle surrounding (093578 33754)with119898 = 02

0 05 1 15 2 25 3 35 405

1

15

2

25

3

35

4

45

5

55E1 = 96 gt 762878

x

y

Figure 8 Phase space trajectories corresponding to different initiallevels Here 119864

1= 96 119864

2= 02 and119898 = 035

value of 119898 such that for the prey refuge smaller than thisthreshold increasing the amount of prey refuge can increasethe predator population and if the prey refuge is largerthan the threshold increasing the amount of prey refugecan decrease the predator population We have proved thatexactly one stable limit cycle occurs when the positiveequilibrium is unstableWe also determined the critical valueof 120575 at which Hopf bifurcation occurs and observed that thebifurcation is supercritical and backward It was also foundthat it is possible to control the system in such a way that thesystem approaches a required state using the efforts 119864

1and

1198642as controlsOur analytical results and numerical simulation also indi-

cate that dynamic behavior of the model not only depends onthe prey refuge parameter 119898 but also depends on harvestingefforts 119864

1and 119864

2 Hence it is possible to control the system in

ISRN Biomathematics 9

0 05 1 15 2 25 3 35 41

15

2

25

3

35

4

45

5

55

6E1 = 6 lt 762878

x

y

Figure 9 There is a stable limit cycle surrounding (093578

211487) with 1198641= 6 119864

2= 02 and119898 = 035

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14E2 = 196 gt 194068

x

y

Figure 10 Phase space trajectories corresponding to different initiallevels with 119864

1= 05 119864

2= 196 and119898 = 035

such a way that the system approaches a required state usingthe harvesting efforts 119864

1and 119864

2or prey refuge119898 as controls

In our model we have considered the catch-rate functionbased on catch-per-unit-effort hypothesis that is ℎ = 119902119864119906 (119906and 119864 denote the prey or predator population and harvestingeffort resp) But this type of catch-rate function embodiessome defects in that (i) it assumes random search for fish (ii)it assumes equal likelihood of being captured for every fish(iii) there is unbounded linear increase of ℎ with respect to 119864for fixed 119906 and (iv) there is unbounded linear increase ℎwithrespect to 119906 for a fixed119864These unrealistic features can largelybe removed by adopting the alternative functional form ℎ =

119902119864119906(1198991119864 + 1198992119906) where 119899

1and 1198992are positive constants but

we leave it for our future research work The entire study ofthe paper is mainly based on the deterministic frameworkOn the other hand it will be more realistic if it is possible

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50

Time

Prey

Predator

PreyPredator

xy

E2 = 17 lt 194068

Figure 11 There exist Hopf-bifurcating small amplitude periodicsolutions with 119864

1= 05 119864

2= 17 and119898 = 035

to consider the model system in the stochastic environmentdue to some ecological fluctuations and other factors Thusa future research problem would be considered in stochasticenvironment

Appendix

Detailed Calculation of Formula (23)First we translate the equilibrium (120575 119910

120575) to the origin by

translation 119909 = 119909 minus 120575 119910 = 119910 minus 119910120575 (Still denote 119909 and 119910 by 119909

and 119910 resp) Thus the system (18) becomes

119889119909

119889119905= ℎ (119909 + 120575) (119891 (119909 + 120575) minus (119910 + 119910

120575))

119889119910

119889119905= 119888 (119910 + 119910

120575) (ℎ (119909 + 120575) minus 119889

1015840)

(A1)

We write the system (A1) as follows

119889119909

119889119905= ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

minus (ℎ (119909 + 120575) (119910 + 119910120575) minus ℎ (120575) 119910

120575)

119889119910

119889119905= minus119888119889

1015840119910 + (119888ℎ (119909 + 120575) (119910 + 119910

120575) minus 119888ℎ (120575) 119910

120575)

(A2)

where 119891(120575) = 119910120575and ℎ(120575) = 119889

1015840

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Biomathematics 5

in 0 le 119909 lt 119909lowast and 119909lowast lt 119909 le 119896 Then the previous system has

exactly one limit cycle which is globally asymptotically stablewith respect to the set

(119909 119910) 119909 gt 0 119910 gt 0 119875lowast(119909lowast 119910lowast) (17)

Following Theorem 4 we may state that when 119898 lt 11987211198962120572

system (8) has unique globally stable limit cycle Thus wesee that when the system is unstable there exists a uniqueglobally stable limit cycle

4 Hopf Bifurcation

To discuss Hopf bifurcation of the system (8) we take the helpof the paper [22]

Let us now consider system (8) in the form

119889119909

119889119905= ℎ (119909) (119891 (119909) minus 119910)

119889119910

119889119905= 119888119910 (ℎ (119909) minus 119889

1015840)

(18)

where 119891(119909) = (120572119896120573)(119896 minus 119909)(1 + 119886119909) minus (2120572119898119896120573)(1 +

119886119909) minus (11990211198641120573)(1 + 119886119909) + (1119909120573)(120572119898 minus 120572119898

2119896 minus 119902

11198641119898)(1 +

119886119909) ℎ(119909) = 120573119909(1 + 119886119909) and 1198891015840 = (119889 + 11990221198642)119888

Let (120575 119910120575) be the interior equilibrium of the system (18)

where

120575 =119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910120575=1 + 119886120575

120573119896120575(120572 (120575 + 119898) (119896 minus 119898 minus 120575) minus 119902

11198641119896 (120575 + 119898))

(19)

In this section we examined the Hopf bifurcation occurringat (120575 119910

120575) taking 120575 as a bifurcation parameter

Variational matrix of the system (18) at (120575 119910120575) is

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0) (20)

where 119882(120575) = ℎ(120575)1198911015840(120575) 119884(120575) = minusℎ(120575) and 119885(120575) =

119888119891(120575)ℎ1015840(120575)

The characteristic equation of the variational matrix 119869 isgiven by 1205832 minus 120583119879 + 119863 = 0 where 119879 = tr 119869 = 119882(120575) and 119863 =

det 119869 = minus119884(120575)119885(120575)Now if 119879 lt 0 and 119863 gt 0 then (120575 119910

120575) is locally asymp-

totically stable On the other hand if 119879 gt 0 and 119863 gt 0 then(120575 119910120575) is unstable Also 119879(120575) = 0 and the Jacobian matrix

119869(120575) has a pair of imaginary eigenvalues 120583 = plusmn119894radicminus119884(120575)119885(120575)Let 120583 = 120573(120575) plusmn 119894120596(120575) be the roots of 1205832 minus 120583119879 + 119863 = 0 when 120575is near 120575 then

120573 (120575) =119860 (120575)

2 120596 (120575) =

radicminus4119884 (120575) 119885 (120575) minus 119882 (120575)2

2

1205731015840(120575) =

ℎ (120575) 1198911015840(120575)1015840

2=ℎ1015840(120575) 1198911015840(120575) + ℎ (120575) 119891

10158401015840(120575)

2

(21)

120575 is a root of 1198911015840(120575) = 0 (as ℎ(120575) = 1198891015840= 0) that is 120575 = 120575

satisfies the equation

21205721198861205753+ (2120572119886119898 + 119902

11198641119886119896 + 120572 minus 119886120572119896) 120575

2

+ (120572119898119896 minus 1205721198982minus 11990211198641119896119898) = 0

(22)

and also 11989110158401015840

(120575) = 0Therefore 1205731015840(120575)|

120575=120575= ℎ(120575)119891

10158401015840(120575)2 = 0 as ℎ(120575) = 0

The system (18) undergoes a Hopf bifurcation at (120575 119910120575)

when 120575 = 120575 By further analysis we determined that thebifurcation is either supercritical or subcritical by the firstLyapunov coefficient (see [23ndash25])

119886 (120575) =1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(23)

The computation of 119886(120575) is technical and the detailedcalculations are given in the appendix and only the results arestated in the following theorem

Theorem 5 The system (18) undergoes a Hopf bifurcationat (120575 119910

120575) the Hopf bifurcation is supercritical and backward

(subcritical and forward resp) if 119886(120575) lt 0 (119886(120575) gt 0) where119886(120575) is defined in (23)

5 Influence of the Parameter 119898 andHarvesting Efforts 119864

1and 119864

2

Interior equilibrium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast)

minus11990211198641119896 (119909lowast+ 119898))

(24)

Notice that the scaling from system (3) to system (8) is 119883 =

119909 minus 119898 Let the interior equilibrium of the system (3) be11987510158401015840(11990910158401015840 11991010158401015840) where

11990910158401015840=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

+ 119898

11991010158401015840=1 + 119886 (119909

10158401015840minus 119898)

120573119896 (11990910158401015840 minus 119898)(12057211990910158401015840(119896 minus 119909

10158401015840) minus 1199021119864111989611990910158401015840)

(25)

The model (3) is a one-prey-one-predator model with con-stant prey refuge Now we discuss the influence of the preyrefuge on the model dynamics

Now 11988911990910158401015840119889119898 = 1 gt 0

Therefore 11990910158401015840 is a strictly increasing function of119898

6 ISRN Biomathematics

Table 1

119898 11990910158401015840

11991010158401015840

01 103578 3084702 113578 3375390398969 133578 39500406 153578 45256075 168578 4951751 193578 5655723 393578 11005210 1093578 25775320 2093578 3620882331422lowast 2425002 36898024 2493578 36868525 2593578 36719730 3093578 34093435 3593578 28329740 4093578 19428847 4793578 739071475 4843578 0195172lowastAt119898 = 2331422 the value of 11991010158401015840 is maximum

That is increasing the amount of prey refuge can increaseprey population

To check the effect of 119898 on predator differentiating 11991010158401015840with respect to119898 we get11988911991010158401015840119889119898 = minus(2120572(1+119886119909

lowast)120573119909lowast119896)(119898+

119909lowastminus1198962+119902

111986411198962120572) where119909lowast = (119889+119902

21198642)(120573119888minus119886119889minus119886119902

21198642)

Case 1 If119898 gt 1198962minus119909lowastminus119902111986411198962120572 then11991010158401015840 is amonotonically

decreasing function of119898That is if the value of prey refuge 119898 gradually increases

above threshold value 1198962minus119909lowast minus119902111986411198962120572 then the predator

population gradually decreases

Case 2 If 119898 lt 1198962 minus 119909lowastminus 119902111986411198962120572 then 119910

10158401015840 is a mono-tonically increasing function of119898

That is if the value of prey refuge 119898 gradually decreasesbelow threshold value 1198962minus119909lowastminus119902

111986411198962120572 then the predator

population gradually increases

Case 3 If = 1198962 minus 119909lowastminus 119902111986411198962120572 then11991010158401015840 has a maximum

value

To construct Table 1 and Figure 3 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02

1198641= 04 and119864

2= 02 in appropriate units From the analysis

shown we see that increasing the amount of prey refuge canincrease prey population and that increasing the amount ofprey refuge can increase the density of predator species andthis happened due to predator species still having enoughfood for predation with 119898 being small but if the prey refugeis larger than a threshold that is as the prey refuge becomeslarge enough then the increasing amount of prey refuge candecrease predator species and this happened due to the lossof food for predator species

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

y998400998400

x998400998400

andy998400998400

m

x998400998400

Figure 3 Change of 11990910158401015840 and 11991010158401015840 with the change of the refuge

parameter119898

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

35

4

45

5ylowast

E1

Figure 4 Change of 119910lowast with the change of the harvesting effort 1198641

Influence of the Harvesting Efforts E1and E

2 Interior equilib-

rium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast) minus 11990211198641119896 (119909lowast+ 119898))

(26)

At first let119898 and 1198642be fixed

It is observed that 119909lowast is independent of the parameter 1198641

whereas the value of 119910lowast depends on 1198641 Therefore 119864

1has no

effect on the interior equilibrium level on 119909To check the effect of 119864

1on predator differentiating 119910lowast

with respect to 1198641 we get

119889119910lowast

1198891198641

= minus1199021(1 + 119886119909

lowast) (119898 + 119909

lowast)

120573119909lowastlt 0 (27)

This shows that 119910lowast is a strictly decreasing function of 1198641

ISRN Biomathematics 7

Table 2

1198641

119909lowast

119910lowast

1 093578 3627553 093578 3022485 093578 241746 093578 211487762878 093578 1622196 093578 10257411 093578 060218612 093578 029965125 093578 0148382129 093578 00273671

Table 3

1198642

119909lowast

119910lowast

1 104762 3799685 176471 41513910 333333 52381915 714286 77939417 108 977236194068 226799 12578196 246667 12499320 3000 115087205 406667 611798207 472308 0539829

To construct Table 2 and Figure 4 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02 119898 =

035 and 1198642= 02 in appropriate units From the analysis

shown we see that as the harvesting effort 1198641increases the

prey species remain unchanged but predator species decreaseand this happens due to loss of food for predator species andgoes to extinction when 119864

1is large

Now let119898 and 1198641be fixed then we have

119889119909lowast

1198891198642

=1205731198881199022

(120573119888 minus 119886119889 minus 11988611986421199022)2gt 0 (28)

Therefore 119909lowast is a strictly increasing function of 1198642

Also

119889119910lowast

1198891198642

= [1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572 minus 120572

minus2120572119886119909lowastminus 11990211198641119896119886 minus

119896120572119898

(119909lowast)2]119889119909lowast

1198891198642

(29)

0 2 4 6 8 10 12 14 16 18 20 220

10

20

30

40

50

60

ylowast

xlowast

E2

xlowast

andylowast

Figure 5 Variation of 119909lowast and 119910lowast with the change of the harvestingeffort 119864

2

Since 119889119909lowast1198891198642gt 0 therefore

119889119910lowast

1198891198642

gt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

gt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

119889119910lowast

1198891198642

lt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

lt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

(30)

Thus as the harvesting effort 1198642increases the predator

population increases when 1198642is smaller than the threshold

value But if the harvesting effort 1198642gradually increases

above the threshold value that is as the harvesting effort1198642becomes large enough then the increasing amount of the

harvesting effort 1198642can decrease predator population

To construct Table 3 and Figure 5 we take 120572 = 8 119886 =

05 120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2=

02 1198641= 05 and 119898 = 035 in appropriate units From

the analysis shown we see that as the harvesting effort 1198642

increases the prey and predator species increase But if theharvesting effort 119864

2increases larger than a threshold value

that is as the harvesting effort1198642becomes large enough then

increasing amount of the harvesting effort 1198642can increase

the prey species but will decrease predator species and go toextinction of the predator species when 119864

2is large

6 Numerical Simulation

As the problem is not a case study the real-world data arenot available for this model We therefore take here somehypothetical data with the sole purpose of illustrating theresults that we have established in the previous sections Letus consider the parameters of the system as 120572 = 8 119886 = 05

8 ISRN Biomathematics

0 02 04 06 08 1 12 14 16 18 24

5

6

7

8

9

10

y

x

m = 075 gt 0398969

(093578 495175)

Figure 6 Phase space trajectories corresponding to different initiallevels which shows that (093578 495175) is a global attractor

120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2= 02 119864

1= 04

and 1198642= 02 in appropriate units For these value of para-

meters we get the critical value of 119898 as 119898lowast = 0398969Thus it is easy to verify that for this set of parameters thesystem (8) is locally asymptotically stable around its interiorequilibrium 119875

lowast(119909lowast 119910lowast) for 119898 gt 119898

lowast and is unstable for 119898 lt

119898lowast Thus for 119898 = 119898

lowast= 0398969 the system (8) undergoes

a Hopf bifurcation Now for 119898 = 075 we have interiorequilibrium (093578 495175)which is asymptotically stable(see Figure 6) but 119898 = 02 and the interior equilibrium(093578 337539) is unstable (see Figure 7) Thus taking119898 as a control parameter it is possible to drive the prey-predator system to require equilibrium and to prevent thecycle behaviour of the system FromFigures 8 9 10 and 11 wesee that 119864

1and 119864

2may also be used as controls for the system

(8) Hopf bifurcation occurs when 1198641= 119864lowast

1= 762878 (here

1198642= 02 119898 = 035) and 119864

2= 119864lowast

2= 194068 (here 119864

1= 05

119898 = 035) For the previous values of parameters and 119898 =

0398969 we obtained one value of 120575 say 120575 = 093578 and119886(120575) = minus0332712 lt 0 Thus we may conclude that the Hopfbifurcation around the interior equilibrium is supercriticaland backward

7 Concluding Remarks

This paper deals with a prey-predator model with Hollingtype II functional response incorporating a constant preyrefuge and independent harvesting in either species Oscil-latory behavior and existence of limit cycles in harvestedprey-predator system are common in nature It is notedthat constant prey refuge plays an important role in thedynamics of the proposed model system It is also observedfrom the obtained results that constant prey refuge cancause an unstable equilibrium to become stable and evena simple Hopf bifurcation occurred when the parameter 119898passes through its critical value There exists a threshold

0 5 10 150

5

10

15

20

25

30m = 02 lt 0398969

x

y

Figure 7There is a stable limit cycle surrounding (093578 33754)with119898 = 02

0 05 1 15 2 25 3 35 405

1

15

2

25

3

35

4

45

5

55E1 = 96 gt 762878

x

y

Figure 8 Phase space trajectories corresponding to different initiallevels Here 119864

1= 96 119864

2= 02 and119898 = 035

value of 119898 such that for the prey refuge smaller than thisthreshold increasing the amount of prey refuge can increasethe predator population and if the prey refuge is largerthan the threshold increasing the amount of prey refugecan decrease the predator population We have proved thatexactly one stable limit cycle occurs when the positiveequilibrium is unstableWe also determined the critical valueof 120575 at which Hopf bifurcation occurs and observed that thebifurcation is supercritical and backward It was also foundthat it is possible to control the system in such a way that thesystem approaches a required state using the efforts 119864

1and

1198642as controlsOur analytical results and numerical simulation also indi-

cate that dynamic behavior of the model not only depends onthe prey refuge parameter 119898 but also depends on harvestingefforts 119864

1and 119864

2 Hence it is possible to control the system in

ISRN Biomathematics 9

0 05 1 15 2 25 3 35 41

15

2

25

3

35

4

45

5

55

6E1 = 6 lt 762878

x

y

Figure 9 There is a stable limit cycle surrounding (093578

211487) with 1198641= 6 119864

2= 02 and119898 = 035

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14E2 = 196 gt 194068

x

y

Figure 10 Phase space trajectories corresponding to different initiallevels with 119864

1= 05 119864

2= 196 and119898 = 035

such a way that the system approaches a required state usingthe harvesting efforts 119864

1and 119864

2or prey refuge119898 as controls

In our model we have considered the catch-rate functionbased on catch-per-unit-effort hypothesis that is ℎ = 119902119864119906 (119906and 119864 denote the prey or predator population and harvestingeffort resp) But this type of catch-rate function embodiessome defects in that (i) it assumes random search for fish (ii)it assumes equal likelihood of being captured for every fish(iii) there is unbounded linear increase of ℎ with respect to 119864for fixed 119906 and (iv) there is unbounded linear increase ℎwithrespect to 119906 for a fixed119864These unrealistic features can largelybe removed by adopting the alternative functional form ℎ =

119902119864119906(1198991119864 + 1198992119906) where 119899

1and 1198992are positive constants but

we leave it for our future research work The entire study ofthe paper is mainly based on the deterministic frameworkOn the other hand it will be more realistic if it is possible

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50

Time

Prey

Predator

PreyPredator

xy

E2 = 17 lt 194068

Figure 11 There exist Hopf-bifurcating small amplitude periodicsolutions with 119864

1= 05 119864

2= 17 and119898 = 035

to consider the model system in the stochastic environmentdue to some ecological fluctuations and other factors Thusa future research problem would be considered in stochasticenvironment

Appendix

Detailed Calculation of Formula (23)First we translate the equilibrium (120575 119910

120575) to the origin by

translation 119909 = 119909 minus 120575 119910 = 119910 minus 119910120575 (Still denote 119909 and 119910 by 119909

and 119910 resp) Thus the system (18) becomes

119889119909

119889119905= ℎ (119909 + 120575) (119891 (119909 + 120575) minus (119910 + 119910

120575))

119889119910

119889119905= 119888 (119910 + 119910

120575) (ℎ (119909 + 120575) minus 119889

1015840)

(A1)

We write the system (A1) as follows

119889119909

119889119905= ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

minus (ℎ (119909 + 120575) (119910 + 119910120575) minus ℎ (120575) 119910

120575)

119889119910

119889119905= minus119888119889

1015840119910 + (119888ℎ (119909 + 120575) (119910 + 119910

120575) minus 119888ℎ (120575) 119910

120575)

(A2)

where 119891(120575) = 119910120575and ℎ(120575) = 119889

1015840

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 ISRN Biomathematics

Table 1

119898 11990910158401015840

11991010158401015840

01 103578 3084702 113578 3375390398969 133578 39500406 153578 45256075 168578 4951751 193578 5655723 393578 11005210 1093578 25775320 2093578 3620882331422lowast 2425002 36898024 2493578 36868525 2593578 36719730 3093578 34093435 3593578 28329740 4093578 19428847 4793578 739071475 4843578 0195172lowastAt119898 = 2331422 the value of 11991010158401015840 is maximum

That is increasing the amount of prey refuge can increaseprey population

To check the effect of 119898 on predator differentiating 11991010158401015840with respect to119898 we get11988911991010158401015840119889119898 = minus(2120572(1+119886119909

lowast)120573119909lowast119896)(119898+

119909lowastminus1198962+119902

111986411198962120572) where119909lowast = (119889+119902

21198642)(120573119888minus119886119889minus119886119902

21198642)

Case 1 If119898 gt 1198962minus119909lowastminus119902111986411198962120572 then11991010158401015840 is amonotonically

decreasing function of119898That is if the value of prey refuge 119898 gradually increases

above threshold value 1198962minus119909lowast minus119902111986411198962120572 then the predator

population gradually decreases

Case 2 If 119898 lt 1198962 minus 119909lowastminus 119902111986411198962120572 then 119910

10158401015840 is a mono-tonically increasing function of119898

That is if the value of prey refuge 119898 gradually decreasesbelow threshold value 1198962minus119909lowastminus119902

111986411198962120572 then the predator

population gradually increases

Case 3 If = 1198962 minus 119909lowastminus 119902111986411198962120572 then11991010158401015840 has a maximum

value

To construct Table 1 and Figure 3 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02

1198641= 04 and119864

2= 02 in appropriate units From the analysis

shown we see that increasing the amount of prey refuge canincrease prey population and that increasing the amount ofprey refuge can increase the density of predator species andthis happened due to predator species still having enoughfood for predation with 119898 being small but if the prey refugeis larger than a threshold that is as the prey refuge becomeslarge enough then the increasing amount of prey refuge candecrease predator species and this happened due to the lossof food for predator species

0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

y998400998400

x998400998400

andy998400998400

m

x998400998400

Figure 3 Change of 11990910158401015840 and 11991010158401015840 with the change of the refuge

parameter119898

0 5 10 15 20 25 30 35 40 45 500

05

1

15

2

25

3

35

4

45

5ylowast

E1

Figure 4 Change of 119910lowast with the change of the harvesting effort 1198641

Influence of the Harvesting Efforts E1and E

2 Interior equilib-

rium of the system (8) is 119875lowast(119909lowast 119910lowast) where

119909lowast=

119889 + 11990221198642

120573119888 minus 119886119889 minus 11988611990221198642

119910lowast=1 + 119886119909

lowast

120573119896119909lowast(120572 (119909lowast+ 119898) (119896 minus 119898 minus 119909

lowast) minus 11990211198641119896 (119909lowast+ 119898))

(26)

At first let119898 and 1198642be fixed

It is observed that 119909lowast is independent of the parameter 1198641

whereas the value of 119910lowast depends on 1198641 Therefore 119864

1has no

effect on the interior equilibrium level on 119909To check the effect of 119864

1on predator differentiating 119910lowast

with respect to 1198641 we get

119889119910lowast

1198891198641

= minus1199021(1 + 119886119909

lowast) (119898 + 119909

lowast)

120573119909lowastlt 0 (27)

This shows that 119910lowast is a strictly decreasing function of 1198641

ISRN Biomathematics 7

Table 2

1198641

119909lowast

119910lowast

1 093578 3627553 093578 3022485 093578 241746 093578 211487762878 093578 1622196 093578 10257411 093578 060218612 093578 029965125 093578 0148382129 093578 00273671

Table 3

1198642

119909lowast

119910lowast

1 104762 3799685 176471 41513910 333333 52381915 714286 77939417 108 977236194068 226799 12578196 246667 12499320 3000 115087205 406667 611798207 472308 0539829

To construct Table 2 and Figure 4 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02 119898 =

035 and 1198642= 02 in appropriate units From the analysis

shown we see that as the harvesting effort 1198641increases the

prey species remain unchanged but predator species decreaseand this happens due to loss of food for predator species andgoes to extinction when 119864

1is large

Now let119898 and 1198641be fixed then we have

119889119909lowast

1198891198642

=1205731198881199022

(120573119888 minus 119886119889 minus 11988611986421199022)2gt 0 (28)

Therefore 119909lowast is a strictly increasing function of 1198642

Also

119889119910lowast

1198891198642

= [1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572 minus 120572

minus2120572119886119909lowastminus 11990211198641119896119886 minus

119896120572119898

(119909lowast)2]119889119909lowast

1198891198642

(29)

0 2 4 6 8 10 12 14 16 18 20 220

10

20

30

40

50

60

ylowast

xlowast

E2

xlowast

andylowast

Figure 5 Variation of 119909lowast and 119910lowast with the change of the harvestingeffort 119864

2

Since 119889119909lowast1198891198642gt 0 therefore

119889119910lowast

1198891198642

gt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

gt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

119889119910lowast

1198891198642

lt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

lt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

(30)

Thus as the harvesting effort 1198642increases the predator

population increases when 1198642is smaller than the threshold

value But if the harvesting effort 1198642gradually increases

above the threshold value that is as the harvesting effort1198642becomes large enough then the increasing amount of the

harvesting effort 1198642can decrease predator population

To construct Table 3 and Figure 5 we take 120572 = 8 119886 =

05 120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2=

02 1198641= 05 and 119898 = 035 in appropriate units From

the analysis shown we see that as the harvesting effort 1198642

increases the prey and predator species increase But if theharvesting effort 119864

2increases larger than a threshold value

that is as the harvesting effort1198642becomes large enough then

increasing amount of the harvesting effort 1198642can increase

the prey species but will decrease predator species and go toextinction of the predator species when 119864

2is large

6 Numerical Simulation

As the problem is not a case study the real-world data arenot available for this model We therefore take here somehypothetical data with the sole purpose of illustrating theresults that we have established in the previous sections Letus consider the parameters of the system as 120572 = 8 119886 = 05

8 ISRN Biomathematics

0 02 04 06 08 1 12 14 16 18 24

5

6

7

8

9

10

y

x

m = 075 gt 0398969

(093578 495175)

Figure 6 Phase space trajectories corresponding to different initiallevels which shows that (093578 495175) is a global attractor

120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2= 02 119864

1= 04

and 1198642= 02 in appropriate units For these value of para-

meters we get the critical value of 119898 as 119898lowast = 0398969Thus it is easy to verify that for this set of parameters thesystem (8) is locally asymptotically stable around its interiorequilibrium 119875

lowast(119909lowast 119910lowast) for 119898 gt 119898

lowast and is unstable for 119898 lt

119898lowast Thus for 119898 = 119898

lowast= 0398969 the system (8) undergoes

a Hopf bifurcation Now for 119898 = 075 we have interiorequilibrium (093578 495175)which is asymptotically stable(see Figure 6) but 119898 = 02 and the interior equilibrium(093578 337539) is unstable (see Figure 7) Thus taking119898 as a control parameter it is possible to drive the prey-predator system to require equilibrium and to prevent thecycle behaviour of the system FromFigures 8 9 10 and 11 wesee that 119864

1and 119864

2may also be used as controls for the system

(8) Hopf bifurcation occurs when 1198641= 119864lowast

1= 762878 (here

1198642= 02 119898 = 035) and 119864

2= 119864lowast

2= 194068 (here 119864

1= 05

119898 = 035) For the previous values of parameters and 119898 =

0398969 we obtained one value of 120575 say 120575 = 093578 and119886(120575) = minus0332712 lt 0 Thus we may conclude that the Hopfbifurcation around the interior equilibrium is supercriticaland backward

7 Concluding Remarks

This paper deals with a prey-predator model with Hollingtype II functional response incorporating a constant preyrefuge and independent harvesting in either species Oscil-latory behavior and existence of limit cycles in harvestedprey-predator system are common in nature It is notedthat constant prey refuge plays an important role in thedynamics of the proposed model system It is also observedfrom the obtained results that constant prey refuge cancause an unstable equilibrium to become stable and evena simple Hopf bifurcation occurred when the parameter 119898passes through its critical value There exists a threshold

0 5 10 150

5

10

15

20

25

30m = 02 lt 0398969

x

y

Figure 7There is a stable limit cycle surrounding (093578 33754)with119898 = 02

0 05 1 15 2 25 3 35 405

1

15

2

25

3

35

4

45

5

55E1 = 96 gt 762878

x

y

Figure 8 Phase space trajectories corresponding to different initiallevels Here 119864

1= 96 119864

2= 02 and119898 = 035

value of 119898 such that for the prey refuge smaller than thisthreshold increasing the amount of prey refuge can increasethe predator population and if the prey refuge is largerthan the threshold increasing the amount of prey refugecan decrease the predator population We have proved thatexactly one stable limit cycle occurs when the positiveequilibrium is unstableWe also determined the critical valueof 120575 at which Hopf bifurcation occurs and observed that thebifurcation is supercritical and backward It was also foundthat it is possible to control the system in such a way that thesystem approaches a required state using the efforts 119864

1and

1198642as controlsOur analytical results and numerical simulation also indi-

cate that dynamic behavior of the model not only depends onthe prey refuge parameter 119898 but also depends on harvestingefforts 119864

1and 119864

2 Hence it is possible to control the system in

ISRN Biomathematics 9

0 05 1 15 2 25 3 35 41

15

2

25

3

35

4

45

5

55

6E1 = 6 lt 762878

x

y

Figure 9 There is a stable limit cycle surrounding (093578

211487) with 1198641= 6 119864

2= 02 and119898 = 035

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14E2 = 196 gt 194068

x

y

Figure 10 Phase space trajectories corresponding to different initiallevels with 119864

1= 05 119864

2= 196 and119898 = 035

such a way that the system approaches a required state usingthe harvesting efforts 119864

1and 119864

2or prey refuge119898 as controls

In our model we have considered the catch-rate functionbased on catch-per-unit-effort hypothesis that is ℎ = 119902119864119906 (119906and 119864 denote the prey or predator population and harvestingeffort resp) But this type of catch-rate function embodiessome defects in that (i) it assumes random search for fish (ii)it assumes equal likelihood of being captured for every fish(iii) there is unbounded linear increase of ℎ with respect to 119864for fixed 119906 and (iv) there is unbounded linear increase ℎwithrespect to 119906 for a fixed119864These unrealistic features can largelybe removed by adopting the alternative functional form ℎ =

119902119864119906(1198991119864 + 1198992119906) where 119899

1and 1198992are positive constants but

we leave it for our future research work The entire study ofthe paper is mainly based on the deterministic frameworkOn the other hand it will be more realistic if it is possible

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50

Time

Prey

Predator

PreyPredator

xy

E2 = 17 lt 194068

Figure 11 There exist Hopf-bifurcating small amplitude periodicsolutions with 119864

1= 05 119864

2= 17 and119898 = 035

to consider the model system in the stochastic environmentdue to some ecological fluctuations and other factors Thusa future research problem would be considered in stochasticenvironment

Appendix

Detailed Calculation of Formula (23)First we translate the equilibrium (120575 119910

120575) to the origin by

translation 119909 = 119909 minus 120575 119910 = 119910 minus 119910120575 (Still denote 119909 and 119910 by 119909

and 119910 resp) Thus the system (18) becomes

119889119909

119889119905= ℎ (119909 + 120575) (119891 (119909 + 120575) minus (119910 + 119910

120575))

119889119910

119889119905= 119888 (119910 + 119910

120575) (ℎ (119909 + 120575) minus 119889

1015840)

(A1)

We write the system (A1) as follows

119889119909

119889119905= ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

minus (ℎ (119909 + 120575) (119910 + 119910120575) minus ℎ (120575) 119910

120575)

119889119910

119889119905= minus119888119889

1015840119910 + (119888ℎ (119909 + 120575) (119910 + 119910

120575) minus 119888ℎ (120575) 119910

120575)

(A2)

where 119891(120575) = 119910120575and ℎ(120575) = 119889

1015840

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Biomathematics 7

Table 2

1198641

119909lowast

119910lowast

1 093578 3627553 093578 3022485 093578 241746 093578 211487762878 093578 1622196 093578 10257411 093578 060218612 093578 029965125 093578 0148382129 093578 00273671

Table 3

1198642

119909lowast

119910lowast

1 104762 3799685 176471 41513910 333333 52381915 714286 77939417 108 977236194068 226799 12578196 246667 12499320 3000 115087205 406667 611798207 472308 0539829

To construct Table 2 and Figure 4 we take 120572 = 8 119886 = 05120573 = 4 119888 = 08 119889 = 2 119896 = 50 119902

1= 06 119902

2= 02 119898 =

035 and 1198642= 02 in appropriate units From the analysis

shown we see that as the harvesting effort 1198641increases the

prey species remain unchanged but predator species decreaseand this happens due to loss of food for predator species andgoes to extinction when 119864

1is large

Now let119898 and 1198641be fixed then we have

119889119909lowast

1198891198642

=1205731198881199022

(120573119888 minus 119886119889 minus 11988611986421199022)2gt 0 (28)

Therefore 119909lowast is a strictly increasing function of 1198642

Also

119889119910lowast

1198891198642

= [1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572 minus 120572

minus2120572119886119909lowastminus 11990211198641119896119886 minus

119896120572119898

(119909lowast)2]119889119909lowast

1198891198642

(29)

0 2 4 6 8 10 12 14 16 18 20 220

10

20

30

40

50

60

ylowast

xlowast

E2

xlowast

andylowast

Figure 5 Variation of 119909lowast and 119910lowast with the change of the harvestingeffort 119864

2

Since 119889119909lowast1198891198642gt 0 therefore

119889119910lowast

1198891198642

gt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

gt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

119889119910lowast

1198891198642

lt 0 if 1205721198982

(119909lowast)2+11990211198641119896119898

(119909lowast)2

+ 2119886119898120572 + 119886119896120572

lt 120572 + 2119886120572119909lowast+ 11990211198641119896119886 +

119896120572119898

(119909lowast)2

(30)

Thus as the harvesting effort 1198642increases the predator

population increases when 1198642is smaller than the threshold

value But if the harvesting effort 1198642gradually increases

above the threshold value that is as the harvesting effort1198642becomes large enough then the increasing amount of the

harvesting effort 1198642can decrease predator population

To construct Table 3 and Figure 5 we take 120572 = 8 119886 =

05 120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2=

02 1198641= 05 and 119898 = 035 in appropriate units From

the analysis shown we see that as the harvesting effort 1198642

increases the prey and predator species increase But if theharvesting effort 119864

2increases larger than a threshold value

that is as the harvesting effort1198642becomes large enough then

increasing amount of the harvesting effort 1198642can increase

the prey species but will decrease predator species and go toextinction of the predator species when 119864

2is large

6 Numerical Simulation

As the problem is not a case study the real-world data arenot available for this model We therefore take here somehypothetical data with the sole purpose of illustrating theresults that we have established in the previous sections Letus consider the parameters of the system as 120572 = 8 119886 = 05

8 ISRN Biomathematics

0 02 04 06 08 1 12 14 16 18 24

5

6

7

8

9

10

y

x

m = 075 gt 0398969

(093578 495175)

Figure 6 Phase space trajectories corresponding to different initiallevels which shows that (093578 495175) is a global attractor

120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2= 02 119864

1= 04

and 1198642= 02 in appropriate units For these value of para-

meters we get the critical value of 119898 as 119898lowast = 0398969Thus it is easy to verify that for this set of parameters thesystem (8) is locally asymptotically stable around its interiorequilibrium 119875

lowast(119909lowast 119910lowast) for 119898 gt 119898

lowast and is unstable for 119898 lt

119898lowast Thus for 119898 = 119898

lowast= 0398969 the system (8) undergoes

a Hopf bifurcation Now for 119898 = 075 we have interiorequilibrium (093578 495175)which is asymptotically stable(see Figure 6) but 119898 = 02 and the interior equilibrium(093578 337539) is unstable (see Figure 7) Thus taking119898 as a control parameter it is possible to drive the prey-predator system to require equilibrium and to prevent thecycle behaviour of the system FromFigures 8 9 10 and 11 wesee that 119864

1and 119864

2may also be used as controls for the system

(8) Hopf bifurcation occurs when 1198641= 119864lowast

1= 762878 (here

1198642= 02 119898 = 035) and 119864

2= 119864lowast

2= 194068 (here 119864

1= 05

119898 = 035) For the previous values of parameters and 119898 =

0398969 we obtained one value of 120575 say 120575 = 093578 and119886(120575) = minus0332712 lt 0 Thus we may conclude that the Hopfbifurcation around the interior equilibrium is supercriticaland backward

7 Concluding Remarks

This paper deals with a prey-predator model with Hollingtype II functional response incorporating a constant preyrefuge and independent harvesting in either species Oscil-latory behavior and existence of limit cycles in harvestedprey-predator system are common in nature It is notedthat constant prey refuge plays an important role in thedynamics of the proposed model system It is also observedfrom the obtained results that constant prey refuge cancause an unstable equilibrium to become stable and evena simple Hopf bifurcation occurred when the parameter 119898passes through its critical value There exists a threshold

0 5 10 150

5

10

15

20

25

30m = 02 lt 0398969

x

y

Figure 7There is a stable limit cycle surrounding (093578 33754)with119898 = 02

0 05 1 15 2 25 3 35 405

1

15

2

25

3

35

4

45

5

55E1 = 96 gt 762878

x

y

Figure 8 Phase space trajectories corresponding to different initiallevels Here 119864

1= 96 119864

2= 02 and119898 = 035

value of 119898 such that for the prey refuge smaller than thisthreshold increasing the amount of prey refuge can increasethe predator population and if the prey refuge is largerthan the threshold increasing the amount of prey refugecan decrease the predator population We have proved thatexactly one stable limit cycle occurs when the positiveequilibrium is unstableWe also determined the critical valueof 120575 at which Hopf bifurcation occurs and observed that thebifurcation is supercritical and backward It was also foundthat it is possible to control the system in such a way that thesystem approaches a required state using the efforts 119864

1and

1198642as controlsOur analytical results and numerical simulation also indi-

cate that dynamic behavior of the model not only depends onthe prey refuge parameter 119898 but also depends on harvestingefforts 119864

1and 119864

2 Hence it is possible to control the system in

ISRN Biomathematics 9

0 05 1 15 2 25 3 35 41

15

2

25

3

35

4

45

5

55

6E1 = 6 lt 762878

x

y

Figure 9 There is a stable limit cycle surrounding (093578

211487) with 1198641= 6 119864

2= 02 and119898 = 035

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14E2 = 196 gt 194068

x

y

Figure 10 Phase space trajectories corresponding to different initiallevels with 119864

1= 05 119864

2= 196 and119898 = 035

such a way that the system approaches a required state usingthe harvesting efforts 119864

1and 119864

2or prey refuge119898 as controls

In our model we have considered the catch-rate functionbased on catch-per-unit-effort hypothesis that is ℎ = 119902119864119906 (119906and 119864 denote the prey or predator population and harvestingeffort resp) But this type of catch-rate function embodiessome defects in that (i) it assumes random search for fish (ii)it assumes equal likelihood of being captured for every fish(iii) there is unbounded linear increase of ℎ with respect to 119864for fixed 119906 and (iv) there is unbounded linear increase ℎwithrespect to 119906 for a fixed119864These unrealistic features can largelybe removed by adopting the alternative functional form ℎ =

119902119864119906(1198991119864 + 1198992119906) where 119899

1and 1198992are positive constants but

we leave it for our future research work The entire study ofthe paper is mainly based on the deterministic frameworkOn the other hand it will be more realistic if it is possible

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50

Time

Prey

Predator

PreyPredator

xy

E2 = 17 lt 194068

Figure 11 There exist Hopf-bifurcating small amplitude periodicsolutions with 119864

1= 05 119864

2= 17 and119898 = 035

to consider the model system in the stochastic environmentdue to some ecological fluctuations and other factors Thusa future research problem would be considered in stochasticenvironment

Appendix

Detailed Calculation of Formula (23)First we translate the equilibrium (120575 119910

120575) to the origin by

translation 119909 = 119909 minus 120575 119910 = 119910 minus 119910120575 (Still denote 119909 and 119910 by 119909

and 119910 resp) Thus the system (18) becomes

119889119909

119889119905= ℎ (119909 + 120575) (119891 (119909 + 120575) minus (119910 + 119910

120575))

119889119910

119889119905= 119888 (119910 + 119910

120575) (ℎ (119909 + 120575) minus 119889

1015840)

(A1)

We write the system (A1) as follows

119889119909

119889119905= ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

minus (ℎ (119909 + 120575) (119910 + 119910120575) minus ℎ (120575) 119910

120575)

119889119910

119889119905= minus119888119889

1015840119910 + (119888ℎ (119909 + 120575) (119910 + 119910

120575) minus 119888ℎ (120575) 119910

120575)

(A2)

where 119891(120575) = 119910120575and ℎ(120575) = 119889

1015840

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 ISRN Biomathematics

0 02 04 06 08 1 12 14 16 18 24

5

6

7

8

9

10

y

x

m = 075 gt 0398969

(093578 495175)

Figure 6 Phase space trajectories corresponding to different initiallevels which shows that (093578 495175) is a global attractor

120573 = 4 119888 = 08 119889 = 2 119896 = 50 1199021= 06 119902

2= 02 119864

1= 04

and 1198642= 02 in appropriate units For these value of para-

meters we get the critical value of 119898 as 119898lowast = 0398969Thus it is easy to verify that for this set of parameters thesystem (8) is locally asymptotically stable around its interiorequilibrium 119875

lowast(119909lowast 119910lowast) for 119898 gt 119898

lowast and is unstable for 119898 lt

119898lowast Thus for 119898 = 119898

lowast= 0398969 the system (8) undergoes

a Hopf bifurcation Now for 119898 = 075 we have interiorequilibrium (093578 495175)which is asymptotically stable(see Figure 6) but 119898 = 02 and the interior equilibrium(093578 337539) is unstable (see Figure 7) Thus taking119898 as a control parameter it is possible to drive the prey-predator system to require equilibrium and to prevent thecycle behaviour of the system FromFigures 8 9 10 and 11 wesee that 119864

1and 119864

2may also be used as controls for the system

(8) Hopf bifurcation occurs when 1198641= 119864lowast

1= 762878 (here

1198642= 02 119898 = 035) and 119864

2= 119864lowast

2= 194068 (here 119864

1= 05

119898 = 035) For the previous values of parameters and 119898 =

0398969 we obtained one value of 120575 say 120575 = 093578 and119886(120575) = minus0332712 lt 0 Thus we may conclude that the Hopfbifurcation around the interior equilibrium is supercriticaland backward

7 Concluding Remarks

This paper deals with a prey-predator model with Hollingtype II functional response incorporating a constant preyrefuge and independent harvesting in either species Oscil-latory behavior and existence of limit cycles in harvestedprey-predator system are common in nature It is notedthat constant prey refuge plays an important role in thedynamics of the proposed model system It is also observedfrom the obtained results that constant prey refuge cancause an unstable equilibrium to become stable and evena simple Hopf bifurcation occurred when the parameter 119898passes through its critical value There exists a threshold

0 5 10 150

5

10

15

20

25

30m = 02 lt 0398969

x

y

Figure 7There is a stable limit cycle surrounding (093578 33754)with119898 = 02

0 05 1 15 2 25 3 35 405

1

15

2

25

3

35

4

45

5

55E1 = 96 gt 762878

x

y

Figure 8 Phase space trajectories corresponding to different initiallevels Here 119864

1= 96 119864

2= 02 and119898 = 035

value of 119898 such that for the prey refuge smaller than thisthreshold increasing the amount of prey refuge can increasethe predator population and if the prey refuge is largerthan the threshold increasing the amount of prey refugecan decrease the predator population We have proved thatexactly one stable limit cycle occurs when the positiveequilibrium is unstableWe also determined the critical valueof 120575 at which Hopf bifurcation occurs and observed that thebifurcation is supercritical and backward It was also foundthat it is possible to control the system in such a way that thesystem approaches a required state using the efforts 119864

1and

1198642as controlsOur analytical results and numerical simulation also indi-

cate that dynamic behavior of the model not only depends onthe prey refuge parameter 119898 but also depends on harvestingefforts 119864

1and 119864

2 Hence it is possible to control the system in

ISRN Biomathematics 9

0 05 1 15 2 25 3 35 41

15

2

25

3

35

4

45

5

55

6E1 = 6 lt 762878

x

y

Figure 9 There is a stable limit cycle surrounding (093578

211487) with 1198641= 6 119864

2= 02 and119898 = 035

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14E2 = 196 gt 194068

x

y

Figure 10 Phase space trajectories corresponding to different initiallevels with 119864

1= 05 119864

2= 196 and119898 = 035

such a way that the system approaches a required state usingthe harvesting efforts 119864

1and 119864

2or prey refuge119898 as controls

In our model we have considered the catch-rate functionbased on catch-per-unit-effort hypothesis that is ℎ = 119902119864119906 (119906and 119864 denote the prey or predator population and harvestingeffort resp) But this type of catch-rate function embodiessome defects in that (i) it assumes random search for fish (ii)it assumes equal likelihood of being captured for every fish(iii) there is unbounded linear increase of ℎ with respect to 119864for fixed 119906 and (iv) there is unbounded linear increase ℎwithrespect to 119906 for a fixed119864These unrealistic features can largelybe removed by adopting the alternative functional form ℎ =

119902119864119906(1198991119864 + 1198992119906) where 119899

1and 1198992are positive constants but

we leave it for our future research work The entire study ofthe paper is mainly based on the deterministic frameworkOn the other hand it will be more realistic if it is possible

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50

Time

Prey

Predator

PreyPredator

xy

E2 = 17 lt 194068

Figure 11 There exist Hopf-bifurcating small amplitude periodicsolutions with 119864

1= 05 119864

2= 17 and119898 = 035

to consider the model system in the stochastic environmentdue to some ecological fluctuations and other factors Thusa future research problem would be considered in stochasticenvironment

Appendix

Detailed Calculation of Formula (23)First we translate the equilibrium (120575 119910

120575) to the origin by

translation 119909 = 119909 minus 120575 119910 = 119910 minus 119910120575 (Still denote 119909 and 119910 by 119909

and 119910 resp) Thus the system (18) becomes

119889119909

119889119905= ℎ (119909 + 120575) (119891 (119909 + 120575) minus (119910 + 119910

120575))

119889119910

119889119905= 119888 (119910 + 119910

120575) (ℎ (119909 + 120575) minus 119889

1015840)

(A1)

We write the system (A1) as follows

119889119909

119889119905= ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

minus (ℎ (119909 + 120575) (119910 + 119910120575) minus ℎ (120575) 119910

120575)

119889119910

119889119905= minus119888119889

1015840119910 + (119888ℎ (119909 + 120575) (119910 + 119910

120575) minus 119888ℎ (120575) 119910

120575)

(A2)

where 119891(120575) = 119910120575and ℎ(120575) = 119889

1015840

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Biomathematics 9

0 05 1 15 2 25 3 35 41

15

2

25

3

35

4

45

5

55

6E1 = 6 lt 762878

x

y

Figure 9 There is a stable limit cycle surrounding (093578

211487) with 1198641= 6 119864

2= 02 and119898 = 035

0 5 10 15 20 25 30 35 40 45 500

2

4

6

8

10

12

14E2 = 196 gt 194068

x

y

Figure 10 Phase space trajectories corresponding to different initiallevels with 119864

1= 05 119864

2= 196 and119898 = 035

such a way that the system approaches a required state usingthe harvesting efforts 119864

1and 119864

2or prey refuge119898 as controls

In our model we have considered the catch-rate functionbased on catch-per-unit-effort hypothesis that is ℎ = 119902119864119906 (119906and 119864 denote the prey or predator population and harvestingeffort resp) But this type of catch-rate function embodiessome defects in that (i) it assumes random search for fish (ii)it assumes equal likelihood of being captured for every fish(iii) there is unbounded linear increase of ℎ with respect to 119864for fixed 119906 and (iv) there is unbounded linear increase ℎwithrespect to 119906 for a fixed119864These unrealistic features can largelybe removed by adopting the alternative functional form ℎ =

119902119864119906(1198991119864 + 1198992119906) where 119899

1and 1198992are positive constants but

we leave it for our future research work The entire study ofthe paper is mainly based on the deterministic frameworkOn the other hand it will be more realistic if it is possible

0 50 100 150 200 250 3000

5

10

15

20

25

30

35

40

45

50

Time

Prey

Predator

PreyPredator

xy

E2 = 17 lt 194068

Figure 11 There exist Hopf-bifurcating small amplitude periodicsolutions with 119864

1= 05 119864

2= 17 and119898 = 035

to consider the model system in the stochastic environmentdue to some ecological fluctuations and other factors Thusa future research problem would be considered in stochasticenvironment

Appendix

Detailed Calculation of Formula (23)First we translate the equilibrium (120575 119910

120575) to the origin by

translation 119909 = 119909 minus 120575 119910 = 119910 minus 119910120575 (Still denote 119909 and 119910 by 119909

and 119910 resp) Thus the system (18) becomes

119889119909

119889119905= ℎ (119909 + 120575) (119891 (119909 + 120575) minus (119910 + 119910

120575))

119889119910

119889119905= 119888 (119910 + 119910

120575) (ℎ (119909 + 120575) minus 119889

1015840)

(A1)

We write the system (A1) as follows

119889119909

119889119905= ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

minus (ℎ (119909 + 120575) (119910 + 119910120575) minus ℎ (120575) 119910

120575)

119889119910

119889119905= minus119888119889

1015840119910 + (119888ℎ (119909 + 120575) (119910 + 119910

120575) minus 119888ℎ (120575) 119910

120575)

(A2)

where 119891(120575) = 119910120575and ℎ(120575) = 119889

1015840

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 ISRN Biomathematics

Now compute the Taylor expansion of related functions

(119910 + 119910120575) ℎ (119909 + 120575) minus ℎ (120575) 119910

120575

= 11988610119909 + 11988601119910 + 119886201199092+ 11988611119909119910 + 119886

301199093

+ 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

ℎ (119909 + 120575) 119891 (119909 + 120575) minus ℎ (120575) 119891 (120575)

= 11988710119909 + 119887201199092+ 119887301199093+ 119874 (|119909|

4)

(A3)

where

11988610= ℎ1015840(120575) 119910120575 11988620=1

2ℎ10158401015840(120575) 119910120575 11988630=1

6ℎ101584010158401015840(120575) 119910120575

11988601= ℎ (120575) 119886

11= ℎ1015840(120575) 119886

21=1

2ℎ10158401015840(120575)

11988710= (119891ℎ)

1015840

(120575) 11988720=1

2(119891ℎ)10158401015840

(120575) 11988730=1

6(119891ℎ)101584010158401015840

(120575)

(A4)

Then the system (A1) becomes

(

119889119909

119889119905

119889119910

119889119905

) = 119869(119909

119910) + (

1198651(119909 119910 120575)

1198652(119909 119910 120575)

) (A5)

where

119869 = (119882(120575) 119884 (120575)

119885 (120575) 0)

1198651(119909 119910 120575) = (119887

20minus 11988620) 1199092minus 11988611119909119910 + (119887

30minus 11988630) 1199093

minus 119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816)

1198652(119909 119910 120575) = 119888 (119886

201199092+ 11988611119909119910 + 119886

301199093

+119886211199092119910 + 119874 (|119909|

4 |119909|3 1003816100381610038161003816119910

1003816100381610038161003816))

(A6)

Now we define a matrix

119875 = (1 0

119873 119872) (A7)

where 119873 = minus119882(120575)2119884(120575) and 119872 =

minusradicminus4119884(120575)119885(120575) minus 1198822(120575)2119884(120575) then

119875minus1= (

1 0

minus119873

119872

1

119872

) (A8)

By using the linear transformation

(119909

119910) = 119875(

119906

V) (A9)

we have

(119909

119910) = (

119906

119873119906 +119872V) 119875

minus1119869119875(

119906

V) = 119869 (120575) (

119906

V) (A10)

where

119869 (120575) = (120573 (120575) minus120596 (120575)

120596 (120575) 120573 (120575)) (A11)

Then system (A5) becomes

(

119889119906

119889119905

119889V

119889119905

) = 119869 (120575) (119906

V) + (

1198651(119906 V 120575)

1198652(119906 V 120575)

) (A12)

where 120573(120575) and 120596(120575) are defined in (21) and

1198651(119906 V 120575) = 119865

1(119906119873119906 +119872V 120575)

= 119860201199062+ 11986011119906V + 119860

301199063

+ 119860211199062V + 119874 (|119906|

4 |119906|3|V|)

1198652(119906 V 120575) = minus

119873

1198721198651(119906119873119906 +119872V 120575)

+1

1198721198652(119906119873119906 +119872V 120575)

= 119861201199062+ 11986111119906V + 119861

301199063+ 119861211199062V

+ 119874 (|119906|4 |119906|3|V|)

(A13)

where

11986020= (11988720minus 11988620) minus 11988611119873 119860

11= minus11988611119872

11986030= (11988730minus 11988630) minus 11988621119873 119860

21= minus11988621119872

11986120=

119888

119872(11988620+ 11988611119873) minus

119873

119872(11988720minus 11988620minus 11988611119873)

11986111= 11988811988611+ 11988611119873

11986130=

119888

119872(11988630+ 11988621119873) minus

119873

119872(11988730minus 11988630minus 11988621119873)

11986121= 11988811988621+ 11988621119873

(A14)

Rewrite the system (A12) in a polar coordinate form as

119903 = 120573 (120575) 119903 + 119886 (120575) 1199033+ sdot sdot sdot

120579 = 120596 (120575) + 119888 (120575) 1199032+ sdot sdot sdot

(A15)

Then the Taylor expansion of (A15) at 120575 = 120575 yields

119903 = 1205731015840(120575) (120575 minus 120575) 119903 + 119886 (120575) 119903

3

+ 119874(11990310038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199033 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199035)

120579 = 120596 (120575) + 1205961015840(120575) (120575 minus 120575) + 119888 (120575) 119903

2

+ 119874(10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816

2

1199032 10038161003816100381610038161003816120575 minus 120575

10038161003816100381610038161003816 1199034)

(A16)

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

ISRN Biomathematics 11

In order to determine the stability of the periodic solutionwe need to calculate the sign of the coefficient 119886(120575) which isgiven by

119886 (120575) =1

16[1198651

119906119906119906+ 1198651

119906VV + 1198652

119906119906V + 1198652

VVV]

+1

16120596 (120575)[1198651

119906V (1198651

119906119906+ 1198651

VV) minus 1198652

119906V (1198652

119906119906+ 1198652

VV)

minus1198651

1199061199061198652

119906119906+ 1198651

VV1198652

VV]

(A17)

where all partial derivatives are evaluated at the bifurcationpoint that is (119906 V 120575) = (0 0 120575)

Since V is linear in both1198651(119906 V 120575) and1198652(119906 V 120575) we havethat

1198651

119906VV = 1198652

VVV = 1198651

VV = 1198652

VV = 0 (A18)

when 120575 = 120575 119873120575

= 119873|120575=120575

= 0 119872120575

= 119872|120575=120575

=

radic119888119891(120575)ℎ1015840(120575)ℎ(120575) also 120596(120575)119872120575= 119888119891(120575)ℎ

1015840(120575)

At (0 0 120575) we have by simple calculation that

1198651

119906119906119906+ 1198652

119906119906V = 611986030+ 211986121= 6 (119887

30minus 11988630) + 2119888119886

21

1198651

119906V1198651

119906119906= 21198601111986020= minus2 (119887

20minus 11988620) 11988611119872120575

1198652

119906V1198652

119906119906= 21198612011986111=21198882

119872120575

1198862011988611

1198651

1199061199061198652

119906119906= 41198602011986120=

4119888

119872120575

11988620(11988720minus 11988620)

(A19)

Now

119886 (120575) =1

16[1198651

119906119906119906+ 1198652

119906119906V]

+1

16120596 (120575)[1198651

119906V1198651

119906119906minus 1198652

119906V1198652

119906119906minus 1198651

1199061199061198652

119906119906]

=1

16[6 (11988730minus 11988630) + 2119888119886

21]

+1

16120596 (120575)[minus2 (119887

20minus 11988620) 11988611119872120575minus21198882

119872120575

1198861111988620

minus4119888

119872120575

11988620(11988720minus 11988620)]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575) + 119888ℎ

10158401015840(120575)]

+1

16119888119891 (120575) ℎ1015840 (120575)

times [

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)119888119891 (120575) ℎ

1015840(120575)2

ℎ (120575)

minus 1198882119891 (120575) ℎ

1015840(120575) ℎ10158401015840(120575)

minus119888119891 (120575) ℎ10158401015840(120575) (119891ℎ)

10158401015840

(120575) minus ℎ10158401015840(120575) 119891 (120575)]

]

=1

16[(119891ℎ)

101584010158401015840

(120575) minus ℎ101584010158401015840(120575) 119891 (120575)]

+1

16

[[

[

ℎ10158401015840(120575) 119891 (120575) minus (119891119892)

10158401015840

(120575)

times

ℎ1015840(120575)2

+ ℎ (120575) ℎ10158401015840(120575)

ℎ (120575) ℎ1015840 (120575)

]]

]

=1

16ℎ1015840 (120575)[119891101584010158401015840(120575) ℎ (120575) ℎ

1015840(120575) + 2ℎ

1015840(120575)2

11989110158401015840(120575)

minus11989110158401015840(120575) ℎ10158401015840(120575) ℎ (120575) ]

(A20)

since 1198911015840(120575) = 0

References

[1] S Zhang L Dong and L Chen ldquoThe study of predator-preysystem with defensive ability of prey and impulsive perturba-tions on the predatorrdquo Chaos Solitons and Fractals vol 23 no2 pp 631ndash643 2005

[2] C S Holling ldquoThe functional response of predator to prey den-sity and its role mimicry and population regulationrdquo Memoirsof the Entomological Society of Canada vol 45 pp 3ndash60 1965

[3] C W Clark Bioeconomic Modelling and Fisheries ManagementWiley New York NY USA 1985

[4] C W ClarkMathematical Bioeconomics The Optimal Manage-ment of Renewable Resource John Wiley and Sons New YorkNY USA 2nd edition 1990

[5] D Xiao W Li and M Han ldquoDynamics in a ratio-dependentpredator-prey model with predator harvestingrdquo Journal ofMathematical Analysis and Applications vol 324 no 1 pp 14ndash29 2006

[6] B Leard C Lewis and J Rebaza ldquoDynamics of ratio-dependentpredator prey models with nonconstant harvestingrdquo Discreteand Continuous Dynamical Systems Series S vol 1 pp 303ndash3152008

[7] J Xia Z Liu R Yuan and S Ruan ldquoThe effects of harvestingand time delay on predator-prey systems with Holling type IIfunctional responserdquo SIAM Journal on Applied Mathematicsvol 70 no 4 pp 1178ndash1200 2009

[8] U K Pahari and T K Kar ldquoConservation of a resource basedfishery through optimal taxationrdquo Nonlinear Dynamics vol 72pp 591ndash603 2013

[9] R J Taylor Predation Chapman and Hall New York NY USA1984

[10] J N McNair ldquoThe effects of refuges on predator-prey interac-tions a reconsiderationrdquoTheoretical Population Biology vol 29no 1 pp 38ndash63 1986

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 ISRN Biomathematics

[11] J N McNair ldquoStability effects of prey refuges with entry-exitdynamicsrdquo Journal of Theoretical Biology vol 125 no 4 pp449ndash464 1987

[12] T K Kar ldquoModelling and analysis of a harvested prey-predatorsystem incorporating a prey refugerdquo Journal of Computationaland Applied Mathematics vol 185 no 1 pp 19ndash33 2006

[13] H Wang W Morrison A Singh and H Weiss ldquoModelinginverted biomass pyramids and refuges in ecosystemsrdquo Ecolog-ical Modelling vol 220 no 11 pp 1376ndash1382 2009

[14] L Ji and C Wu ldquoQualitative analysis of a predator-prey modelwith constant-rate prey harvesting incorporating a constantprey refugerdquo Nonlinear Analysis Real World Applications vol11 no 4 pp 2285ndash2295 2010

[15] Y Huang F Chen and L Zhong ldquoStability analysis of a prey-predator model with holling type III response function incor-porating a prey refugerdquo Applied Mathematics and Computationvol 182 no 1 pp 672ndash683 2006

[16] J Wang and L Pan ldquoQualitative analysis of a harvestedpredator-prey system with Holling-type III functional responseincorporating a prey refugerdquo Advances in Difference Equationsvol 96 pp 1ndash14 2012

[17] T K Kar A Ghorai and S Jana ldquoDynamics consequences ofprey refuges in a two predator one prey systemrdquo Journal ofBiological Systems vol 21 no 2 Article ID 1350013 28 pages2013

[18] E Gonzalez-Olivares and R Ramos-Jiliberto ldquoDynamic conse-quences of prey refuges in a simple model system more preyfewer predators and enhanced stabilityrdquo Ecological Modellingvol 166 no 1-2 pp 135ndash146 2003

[19] L Chen F Chen and L Chen ldquoQualitative analysis of apredator-prey model with Holling type II functional responseincorporating a constant prey refugerdquo Nonlinear Analysis RealWorld Applications vol 11 no 1 pp 246ndash252 2010

[20] G Birkoff and G C Rota Ordinary Differential EquationsGinn Cambridge UK 1982

[21] Y Kuang and H I Freedman ldquoUniqueness of limit cycles inGause-type models of predator-prey systemsrdquo MathematicalBiosciences vol 88 no 1 pp 67ndash84 1988

[22] J Wang J Shi and J Wei ldquoPredator-prey system with strongAllee effect in preyrdquo Journal of Mathematical Biology vol 62no 3 pp 291ndash331 2011

[23] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems Chaos Texts in AppliedMathematics Springer New YorkNY USA 2nd edition 2003

[24] BDHassardNDKazarinoff andYWanTheory andApplica-tions of Hopf Bifurcation vol 41 of LondonMathematical SocietyLecture Note Series Cambridge University Press CambridgeUK 1981

[25] Y A Kuznetsov Elements of Applied BifurcationTheory vol 112ofAppliedMathematical Sciences SpringerNewYorkNYUSA2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of