Research ArticleBifurcation in a Discrete Competition System

Li Xu Lianjun Zou Zhongxiang Chang Shanshan Lou Xiangwei Peng and Guang Zhang

School of Science Tianjin University of Commerce Tianjin 300134 China

Correspondence should be addressed to Li Xu beifang xl163com

Received 9 February 2014 Accepted 9 April 2014 Published 28 April 2014

Academic Editor Xiaochen Sun

Copyright copy 2014 Li Xu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Anewdifference system is induced fromadifferential competition systemby different discretemethodsWe give theoretical analysisfor local bifurcation of the fixed points and derive the conditions under which the local bifurcations such as flip occur at thefixed points Furthermore one- and two-dimensional diffusion systems are given when diffusion terms are added We provide theTuring instability conditions by linearizationmethod and inner product technique for the diffusion systemwith periodic boundaryconditions A series of numerical simulations are performed that not only verify the theoretical analysis but also display someinteresting dynamics

1 Introduction

Interactions of different species may take many forms such ascompetition predation parasitism and mutualism One ofthe most important interactions is the competition relation-ship The dynamic relationship between the two competitionspecies is one of the dominant subjects in mathematicalecology due to its universal existence and importance Lotka-Volterra competition systems are ecological models thatdescribe the interaction among various competing speciesand have been extensively investigated in recent years (see[1ndash3] and the references therein) In the earlier literaturethe two-competing species competition models are oftenformulated in the form of ordinary differential systems asfollows

1199061015840

(119905) = 119906 (119905) (1199031minus 11988611119906 (119905) minus 119886

12V (119905))

V1015840 (119905) = V (119905) (1199032minus 11988621V (119905) minus 119886

22V (119905))

(1)

for 119905 isin [0 +infin) 119886119894119895ge 0 119894 119895 = 1 2 where 119906(119905) and V(119905) are the

quantities of the two species at time 119905 1199031gt 0 and 119903

2gt 0 are

growth rates of the respective species 11988611

and 11988622

representthe strength of the intraspecific competition and 119886

12and 11988621

represent the strength of the interspecific competitionThe discrete timemodels governed by difference equation

are more realistic than the continuous ones when the pop-ulations have nonoverlapping generations or the populationstatistics are compiled from given time intervals and not

continuously Moreover since the discrete time models canalso provide efficient computational models of continuousmodels for numerical simulations it is reasonable to studydiscrete time models governed by difference equations

Applying forward Euler scheme to the first equation ofsystem (1) and obtaining a discrete analog of the secondequation by considering a variation with piecewise constantarguments for certain terms on the right side (exponentialdiscrete form) [4] we obtain the following equation

119906119905+1

= 119906119905(1199031minus 11988611119906119905minus 11988612V119905)

V119905+1

= V119905exp (119903

2minus 11988621119906119905minus 11988622V119905)

(2)

For the sake of simplicity let

119906119905+1

= 119906119905(1199031+ 1 minus 119886

11119906119905minus 11988612V119905)

V119905+1

= V119905exp (119903

2minus 11988621119906119905minus 11988622V119905)

(3)

By setting119880119905= 11988611119906119905and119881119905= 11988622V119905and 1199031= 1199032 11988611= 11988622

and 11988612= 11988621 we have the following form

119906119905+1

= 119906119905(119903 + 1 minus 119906

119905minus 119886V119905) = 119891 (119906

119905 V119905)

V119905+1

= V119905exp (119903 minus 119886119906

119905minus V119905) = 119892 (119906

119905 V119905)

(4)

Although numerical variations of system (1) have beenextensively studied (see eg thework in [5ndash8]) somediscrete

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 193143 7 pageshttpdxdoiorg1011552014193143

2 Discrete Dynamics in Nature and Society

analogs may be found in [9ndash15] regarding attractivity persis-tence global stabilities of equilibrium and other dynamicsUp to now to the best of our knowledge the discrete system(4) has not been investigated

Since the pioneering theoretical works of Skellam [15] andTuring [16] many works have focused on the effect of spatialfactors which play a crucial role in the stability of populations[17ndash19] Many important epidemiological and ecologicalphenomena are strongly influenced by spatial heterogeneitiesbecause of the localized nature of transmission or other formsof interaction Thus spatial models are more suitable fordescribing the process of population development Impact ofspatial component on system has been widely investigated(eg see [20ndash22]) It may be a case in reality that the motionof individuals is random and isotropic that is without anypreferred direction the individuals are also absolute ones inmicroscopic sense and each isolated individual exchangesmaterials by diffusion with its neighbors [19 23] Thusit is reasonable to consider a 1D or 2D spatially discretereaction diffusion system to explain the population systemCorresponding to the above analysis we can obtain thefollowing one-dimensional diffusion systems

119906119905+1

119894= 119906119905

119894(119903 + 1 minus 119906

119905

119894minus 119886V119905119894) + 1198891nabla2119906119905

119894

V119905+1119894

= V119905119894exp (119903 minus 119886119906119905

119894minus V119905119894) + 1198892nabla2V119905119894

(5)

for 119894 isin 1 2 119898 = [1119898] 119905 isin 119885+ and nabla2119906119905119894= 119906119905

119894+1minus 2119906119905

119894+

119906119905

119894minus1 nabla2V119905119894= V119905119894+1

minus 2V119905119894+ V119905119894minus1

and two-dimensional diffusionsystems

119906119905+1

119894119895= 119906119905

119894119895(119903 + 1 minus 119906

119905

119894119895minus 119886V119905119894119895) + 1198891nabla2119906119905

119894119895

V119905+1119894119895

= V119905119894119895exp (119903 minus 119886119906119905

119894119895minus V119905119894119895) + 1198892nabla2V119905119894119895

(6)

for 119894 119895 isin 1 2 119898 = [1119898] 119905 isin 119877+ = [0infin) and nabla2119906119905119894119895=

119906119905

119894+1119895+ 119906119905

119894119895+1+ 119906119905

119894minus1119895+ 119906119905

119894119895minus1minus 4119906119905

119894119895 nabla2V119905119894119895= V119905119894+1119895

+ V119905119894119895+1

+

V119905119894minus1119895

+ V119905119894119895minus1

minus 4V119905119894119895

In this paper we will study the dynamical behaviors ofmodels (4) (5) and (6) By using the theory of differenceequation the theory of bifurcation and the center manifoldtheoremwewill establish the series of criteria on the existenceand local stability of equilibria flip bifurcation for the system(4) For the one- or two-dimensional diffusion systemswith periodic boundary conditions the Turing instability(or Turing bifurcation) theory analysis will be given Tur-ing instability conditions can then be deduced combininglinearization method and inner product technique Further-more bymeans of the numerical simulationsmethod we willindicate the correctness and rationality of our results

The paper is organized as follows In Section 2 westudy the existence and stability of equilibria points and theconditions of existence for flip bifurcation are verified forsystem (4) Turing instability conditions will be illustratedby linearization method and inner product technique for thesystem (5) and (6) with periodic boundary conditions inSection 3 A series of numerical simulations are performedthat not only verify the theoretical analysis but also dis-play some interesting dynamics For the system (4) the

bifurcation diagrams are given The impact of the systemparameters and diffusion coefficients on patterns can alsobe observed visually for the given diffusion systems Finallysome conclusions are given

2 Analysis of Equilibria and Flip Bifurcation

Clearly the system (4) has four possible steady states that is1198640= (0 0) exclusion points 119864

1= (119903 0) 119864

2= (0 119903) and

nontrivial coexistence point 1198643= (119906lowast Vlowast) where

119906lowast= Vlowast =

119903

119886 + 1

(7)

The linearized form of (4) is then

119906119905+1

= 119891119906119906119905+ 119891VV119905

V119905+1

= 119892119906119906119905+ 119892VV119905

(8)

which has the Jacobian matrix

119869119864119894= [

119891119906119891V

119892119906119892V]

119875119894

= [

1 minus 119906 minus119886119906

minus119886V 1 minus V]119875119894

119894 = 0 1 2 3 (9)

The characteristic equation of the Jacobian matrix 119869 canbe written as

1205822+ 119901120582 + 119902 = 0 (10)

where 119901 = minus(119891119906+ 119892V) and 119902 = 119891119906119892V minus 119891V119892119906

In order to discuss the stability of the fixed points of (4)we also need the following definitions [20]

(1) if |1205821| lt 1 and |120582

2| lt 1 then steady state 119864 is called a

sink and 119864 is locally asymptotically stable(2) if |120582

1| gt 1 and |120582

2| gt 1 then 119864 is called a source and

119864 is unstable(3) if |120582

1| gt 1 and |120582

2| lt 1 (or |120582

1| lt 1 and |120582

2| gt 1)

then 119864 is called a saddle(4) if either |120582

1| = 1 and |120582

2| = 1 or |120582

2| = 1 and |120582

1| = 1

then 119864 is called nonhyperbolic

Case 1 (the fixed point 1198640= (0 0)) The linearization of (4)

about 1198640has the Jacobian matrix

1198691198640= [

1 0

0 1] (11)

which has two eigenvalues

1205821= 1205822= 1 (12)

The fact means that the system is resonance at the fixed point1198750

Case 2 (the fixed point 1198641= (0 119903)) At the fixed point the

Jacobian matrix has the form

1198691198641= [

1 0

minus119886119903 1 minus 119903] (13)

Discrete Dynamics in Nature and Society 3

and the corresponding eigenvalues of (13) are

1205821= 1 120582

2= 1 minus 119903 (14)

119903 is a bifurcation parameter And 119903 = 2 implies 1205822= minus 1 and

the fixed point 1198641is nonhyperbolic

Case 3 (the third fixed point 1198642= (119903 0)) The linearization of

(4) about 1198642has the Jacobian matrix

1198691198641= [

1 minus 119903 minus119886119903

0 1] (15)

and the eigenvalues of (15) are

1205821= 1 minus 119903 120582

2= 1 (16)

119903 is a bifurcation parameter And 119903 = 2 implies 1205821= minus 1 and

the fixed point 1198642is nonhyperbolic

Case 4 (the fixed point 1198643= (119903(119886 + 1) 119903(119886 + 1))) The

linearization of (4) about 1198643has the Jacobian matrix

1198691198643=[

[

[

1 minus

119903

119886 + 1

minus

119886119903

119886 + 1

minus

119886119903

119886 + 1

1 minus

119903

119886 + 1

]

]

]

(17)

and the eigenvalues of (17) are

1205821= 1 minus 119903 120582

2= 1 +

(119886 minus 1) 119903

119886 + 1

(18)

then we have the following results

(1) |1205821| lt 1 |120582

2| lt 1 if and only if 0 lt 119903 lt 2 0 lt 119886 lt 1

(2) 1205821= minus1 |120582

2| = 1 if and only if 119903 = 2 119886 = 1

(3) 1205822= 1 |120582

1| = 1 if and only if 119886 = 2 119903 = 2

(4) 1205822= minus1 |120582

1| = 1 if and only if 119886 = (119903minus2)(119903+2) 119903 = 2

(5) |1205821| lt 1 |120582

2| gt 1 if and only if 0 lt 119903 lt 2 119886 gt 1

(6) |1205821| gt 1 |120582

2| lt 1 if and only if 2 lt 119903 lt (119886+ 1)(119886 minus 1)

(7) |1205821| gt 1 |120582

2| gt 1 if and only if 119903 gt 2 119886 gt 1

The following theorem is the case that the fixed point 1198643is a

flip bifurcation point

Theorem 1 The positive fixed point 1198643undergoes a flip

bifurcation at the threshold 119903119865= 2

Proof Let 120577119899= 119906119899minus 119906lowast 120578119899= V119899minus Vlowast and 120583

119899= 119903 minus 2 and

parameter 120583119899is a new and dependent variable the system (4)

becomes

(

120577119899+1

120578119899+1

120583119899+1

) =(

(120577119899+

120583119899+ 2

119886 + 1

) (1 minus 120577119899minus 119886120578119899) minus

120583119899+ 2

119886 + 1

(120578119899+

120583119899+ 2

119886 + 1

) exp (minus119886120577119899minus 120578119899) minus

120583119899+ 2

119886 + 1

120583119899

)

(19)

Let

119879 =[

[

1 minus1 0

1 1 0

0 0 1

]

]

(20)

then

119879minus1=

[

[

[

[

[

[

[

1

2

1

2

0

minus

1

2

1

2

0

0 0 1

]

]

]

]

]

]

]

(21)

By the following transformation

(

120577119899

120578119899

120583119899

) = 119879(

119909119899

119910119899

120575119899

) (22)

the system (19) can be changed into

(

119909119899+1

119910119899+1

120575119899+1

) = (

minus1 0 0

0

3119886 minus 1

119886 + 1

0

0 0 1

)(

119909119899

119910119899

120575119899

) + (

119891 (119909119899 119910119899 120575119899)

119892 (119909119899 119910119899 120575119899)

0

)

(23)

where

119891 (119909119899 119910119899 120575119899) = minus (119886 + 1) 119909

2

119899minus 2 (119886 + 1) 119909

119899119910119899minus 119909119899120575119899

+ (119886 minus 1 +

(119886 minus 1)2

119886 + 1

)1199102

119899

+ 119900 ((1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816119910119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

119892 (119909119899 119910119899 120575119899) = (119886 + 1) 119909

2

119899minus 2 (119886 + 1) 119909

119899119910119899

minus

2 (119886 minus 1)

119886 + 1

119910119899120575119899+

(119886 minus 1)2

119886 + 1

1199102

119899

+ 119900 ((1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816119910119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

(24)

Then we can consider

119910119899= ℎ (119909

119899 120575119899) = 11988611199092

119899+ 1198862119909119899120575119899+ 11988631205752

119899+ 119900 ((

1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

(25)

which must satisfy

ℎ (minus119909119899+ 119891 (119909

119899 119910119899 120575119899) 120575119899+1)

=

3119886 minus 1

119886 + 1

ℎ (119909119899 120575119899) + (119886 + 1) 119909

2

119899

minus 2 (119886 + 1) 119909119899ℎ (119909119899 120575119899)

minus

2 (119886 minus 1)

119886 + 1

ℎ (119909119899 120575119899) 120575119899

+

(119886 minus 1)2

119886 + 1

ℎ2(119909119899 120575119899)

+ 119900 ((1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816119910119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

(26)

4 Discrete Dynamics in Nature and Society

By calculating we can get that

1198861=

(119886 + 1)2

2 (1 minus 119886)

1198862= 0 119886

3= 0 (27)

and the system (19) is restricted to the center manifold whichis given by

119891 119909119899+1

= minus 119909119899+

1198863+ 41198862+ 3119886

119886 minus 1

1199092

119899+ (119886 + 1) 119909

119899120575119899

+

1

4

(119886 + 1)31199094

119899+ 119900 (

1003816100381610038161003816119909119899

1003816100381610038161003816

4

)

(28)

Since

(

120597119891

120597120575

1205972119891

1205971199092+ 2

1205972119891

120597119909120597120575

)

100381610038161003816100381610038161003816100381610038161003816(00)

= 2 (119886 + 1) = 0

(

1

2

(

1205972119891

1205971199092)

2

+

1

3

1205973119891

1205971199093)

1003816100381610038161003816100381610038161003816100381610038161003816(00)

= 2(

1198863+ 41198862+ 3119886

119886 minus 1

)

2

gt 0

(29)

system (4) undergoes a flip bifurcation at 1198643 The proof is

completed

3 Turing Bifurcation

In this section we discuss the Turing bifurcation Turingrsquostheory shows that diffusion could destabilize an otherwisestable equilibrium of the reaction-diffusion system and leadto nonuniform spatial patterns This kind of instability isusually calledTuring instability or diffusion-driven instability[16]

31 One-Dimensional Case We consider the following diffu-sion system

119906119905+1

119894= 119906119905

119894(119903 + 1 minus 119906

119905

119894minus 119886V119905119894) + 1198891nabla2119906119905

119894

V119905+1119894

= V119905119894exp (119903 minus 119886119906119905

119894minus V119905119894) + 1198892nabla2V119905119894

(30)

with the periodic boundary conditions

119906119905

0= 119906119905

119898 119906

119905

1= 119906119905

119898+1

V1199050= V119905119898 V119905

1= V119905119898+1

(31)

for 119894 isin 1 2 119898 = [1119898] and 119905 isin 119885+ where119898 is a positiveinteger

nabla2119906119905

119894= 119906119905

119894+1minus 2119906119905

119894+ 119906119905

119894minus1

nabla2V119905119894= V119905119894+1

minus 2V119905119894+ V119905119894minus1

(32)

In order to study Turing instability of (30) and (31) wefirstly consider eigenvalues of the following equation

nabla2119883119894+ 120582119883119894= 0 (33)

with the periodic boundary conditions

1198830= 119883119898 119883

1= 119883119898+1

(34)

By calculating the eigenvalue problem (33)-(34) has theeigenvalues

120582119904= 4sin2 (119904 minus 1) 120587

119898

for 119904 isin [1119898] (35)

We linearise at the steady state to get

119908119905+1

119894= 119869119908119905

119894+ 119863nabla2119908119905

119894 119863 = (

11988910

0 1198892

) (36)

with the periodic boundary conditions

119908119905

0= 119908119905

119898 119908

119905

1= 119908119905

119898+1 (37)

where

119908119905

119894= (

119906119905

119894minus 119906lowast

V119905119894minus Vlowast

) = (

119909119905

119894

119910119905

119894

) (38)

Then respectively taking the inner product of (36) withthe corresponding eigenfunction 119883119894

119904of the eigenvalue 120582

119904 we

see that119898

sum

119894=1

119883119894

119904119909119905+1

119894= 119891119906

119898

sum

119894=1

119883119894

119904119909119905

119894+ 119891V

119898

sum

119894=1

119883119894

119904119910119905

119894+ 1198891

119898

sum

119894=1

119883119894

119904nabla2119909119905

119894

119898

sum

119894=1

119883119894

119904119910119905+1

119894= 119892119906

119898

sum

119894=1

119883119894

119904119909119905

119894+ 119892V

119898

sum

119894=1

119883119894

119904119910119905

119894+ 1198892

119898

sum

119894=1

119883119894

119904nabla2119910119905

119894

(39)

Let 119880119905 = sum119898119894=1119883119894

119904119909119905

119894and 119881119905 = sum119898

119894=1119883119894

119904119910119905

119894and use the periodic

boundary conditions (34) and (37) then we have

119880119905+1

= 119891119906119880119905+ 119891V119881

119905minus 1198891120582119904119880119905

119881119905+1

= 119892119906119880119905+ 119892V119881

119905minus 1198892120582119904119881119905

(40)

or

119880119905+1

= (119891119906minus 1198891120582119904) 119880119905+ 119891V119881

119905

119881119905+1

= 119892119906119880119905+ (119892V minus 1198892120582119904) 119881

119905

(41)

Thus the following fact can be obtained

Proposition 2 If (119906119905119894 V119905119894) is a solution of the problem of (30)

and (31) then

(119880119905=

119898

sum

119894=1

119883119894

119904119909119905

119894 119881119905=

119898

sum

119894=1

119883119894

119904119910119905

119894) (42)

is a solution of (41) where 120582119904is some eigenvalue of (33)-(34)

and119883119894119904is the corresponding eigenfunction For some eigenvalue

120582119904of (33)-(34) if (119880119905 119881119905) is a solution of the system (41) then

(119906119905

119894= 119880119905119883119894

119904 V119905119894= 119881119905119883119894

119904) (43)

is a solution of (30)with the periodic boundary conditions (31)

Discrete Dynamics in Nature and Society 5

Proposition 3 If there exist positive numbers 1198891 1198892and the

eigenvalue 120582119904of the problem (33)-(34) such that one of the

conditions

ℎ (120582119904) lt (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (120582119904) lt minus (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(44)

or

ℎ (120582119904) gt 1 (45)

holds then the problem (30) and (31) at the fixed point (119906lowast Vlowast)is unstable where

ℎ (120582119904) = 119889111988921205822

119904minus (1198891119892Vlowast + 1198892119891119906lowast) 120582119904 + (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

(46)

For the system (5) we have the following results aboutinstability of the positive equilibrium of system

Theorem 4 0 lt 119903 lt 2 0 lt 119886 lt 1 and Proposition 3 mean orshow that the problem (30) and (31) is diffusion-driven unstableor Turing unstable

32 Two-Dimensional Case In this subsection we will payour attention to the Turing instability analysis for the follow-ing two-dimensional system

119906119905+1

119894119895= 119906119905

119894119895(119903 + 1 minus 119906

119905

119894119895minus 119886V119905119894119895) + 1198891nabla2119906119905

119894119895

V119905+1119894119895

= V119905119894119895exp (119903 minus 119886119906119905

119894119895minus V119905119894119895) + 1198892nabla2V119905119894119895

(47)

with the periodic boundary conditions

119906119905

1198940= 119906119905

119894119898 119906

119905

1198941= 119906119905

119894119898+1

119906119905

0119895= 119906119905

119898119895 119906

119905

1119895= 119906119905

119898+1119895

V1199051198940= V119905119894119898 V119905

1198941= V119905119894119898+1

V1199050119895= V119905119898119895 V119905

1119895= V119905119898+1119895

(48)

for 119894 119895 isin 1 2 119898 = [1119898] and 119905 isin 119885+ where 119898 is a

positive integer

nabla2119906119905

119894119895= 119906119905

119894+1119895+ 119906119905

119894119895+1+ 119906119905

119894minus1119895+ 119906119905

119894119895minus1minus 4119906119905

119894119895

nabla2V119905119894119895= V119905119894+1119895

+ V119905119894119895+1

+ V119905119894minus1119895

+ V119905119894119895minus1

minus 4V119905119894119895

(49)

The following theoremwill show that the system (47) alsoundergoes Turing instability Since the analysis is very similarto the one-dimensional case the proof is omitted

Theorem 5 If there exist positive numbers 1198891 1198892and the

eigenvalue 1198962119897119904of the corresponding characteristic equation such

that one of the conditions

ℎ (1198962

119897119904) lt (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (1198962

119897119904) lt minus (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(50)

4

r

u

35

35

3

3

25

25

2

15

15

1

21

05

05

0

Figure 1 Bifurcation diagram for 119903 minus 119906119905with 119886 = 0511

or

ℎ (1198962

119897119904) gt 1 (51)

holds and 0 lt 119903 lt 2 0 lt 119886 lt 1 then the problem (47)-(48) atthe fixed point (119906lowast Vlowast) is diffusion-driven unstable or Turingunstable where

ℎ (1198962

119897119904) = 119889111988921198964

119897119904minus (1198891119892Vlowast + 1198892119891119906lowast) 119896

2

119897119904+ (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

1198962

119897119904= 120582119897119904= 4(sin2 ((119897 minus 1) 120587

119898

) + sin2 ((119904 minus 1) 120587119898

))

119891119900119903 119897 119904 isin [1119898]

(52)

4 Numerical Simulation

As is known to all the bifurcation diagram provides a generalview of the evolution process of the dynamical behaviorsby plotting a state variable with the abscissa being oneparameter As a parameter varies the dynamics of the systemwe concerned change through a local or global bifurcationwhich leads to the change of stability at the same time

Now 119903 is considered as a parameter with the range 05ndash35 for the system (4) Since the bifurcation diagrams of 119903minus119906

119905

are similar to the bifurcation diagrams of 119903 minus V119905 we will only

show the former which can be seen from Figure 1Next we performed a series of simulations for the

reaction-diffusion systems and in each the initial condi-tion was always a small amplitude random perturbation1 around the steady state As a numerical example weconsider the bifurcation of the two-dimensional system (47)-(48) It is well known that Turing instability (bifurcation) isdiffusion-driven instability thus the diffusion rate is vital tothe pattern formation To investigate the effect of diffusioncoefficients on patterns by keeping all the other parametersof the system fixed (119886 = 055 119903 = 071 and 119889

1= 022)

we change a diffusion coefficient in the Turing instability

6 Discrete Dynamics in Nature and Society

20

20

40

40

60

60

80

80

100

100

120

120

(a)

20 40 60 80 100 120

20

40

60

80

100

120

(b)

20 40 60 80 100 120

20

40

60

80

100

120

(c)

20 40 60 80 100 120

20

40

60

80

100

120

(d)

20 40 60 80 100 120

20

40

60

80

100

120

(e)

20 40 60 80 100 120

20

40

60

80

100

120

(f)

Figure 2 Pattern selection with the increase of 1198892in the Turing instability region when 119886 = 055 119903 = 071 and 119889

1= 022 (a) 119889

2= 0195

(b) 1198892= 020 (c) 119889

2= 0205 (d) 119889

2= 021 (e) 119889

2= 0215 (f) 119889

2= 0217

region (parameter space which satisfies Turing instability)Figures 2(a)ndash2(f) exhibit in detail the different distributionof patterns with varying values of 119889 If we let 119889

2= 019

a stable pattern of square shapes namely stationary wave

is observed in Figure 2(a) With the increase of 1198892 some

tips appear which attempt to form self-centered spiral waves(Figure 2(b))Then some tips vanish and some tend to evolveto regular spiral waves shown in Figure 2(c) Then with the

Discrete Dynamics in Nature and Society 7

parameter evolution proceeding the size of spiral waves risesbut the density of them decreases (see Figure 2(d)) If 119889

2is

further increased we observe that in parts of patterns spiralwaves begin to break up Hardly can spiral waves be seendisorder and chaotic structure are depicted in Figures 2(e)and 2(f)

5 Discussion and Conclusion

In this paper we have applied different discrete schemes toconvert the continuous Lotka-Volterra competition modelto a new discrete model and studied the dynamical charac-teristic of the discrete model Our theoretical analysis andnumerical simulations have demonstrated that the discretecompetition model undergoes flip bifurcation Furthermorewhen the effects of spatial factors are considered we discussthe Turing instability conditions combining linearizationmethod and inner product technique The impact of the dif-fusion coefficients on patterns can also be observed visuallyand some interesting situations can be observed Indeed thenew discrete model can result in a rich set of patterns and weexpect that it is more effective in practice

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Dr Y D Jiao for the valuable suggestionsand thank the referees for their helpful comments This workwas financially supported by TianjinUniversity of Commercewith the Grant no X0803 the National Natural ScienceFoundation of China under Grant nos 11001072 11371277and Cultivation Program for Excellent Youth Teacher inUniversity Tianjin (507-125RCPY0314)

References

[1] Y Takeuchi Global Dynamical Properties of Lotka-VolterraSystems World Scientific Publishing Singapore 1996

[2] PWaltman CompetitionModels in Populationbiology vol 45 ofCBMS-NSF Regional Conference Series in Applied MathematicsSIAM Philadelphia Pa USA 1985

[3] M E FisherAnalysis of difference equationmodels in populationdynamics [PhD thesis] University of Western Australia 1982

[4] P Liu and X Cui ldquoA discrete model of competitionrdquoMathemat-ics and Computers in Simulation vol 49 no 1-2 pp 1ndash12 1999

[5] F B Hildebrand Finite-Difference Equations and SimulationsPrentice Hall Englewood Cliffs NJ USA 1968

[6] J K Hale and A S Somolinos ldquoCompetition for fluctuatingnutrientrdquo Journal ofMathematical Biology vol 18 no 3 pp 255ndash280 1983

[7] K P Hadeler and I Gerstmann ldquoThe discrete RosenzweigmodelrdquoMathematical Biosciences vol 98 no 1 pp 49ndash72 1990

[8] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics vol 191 Academic Press New York NYUSA 1993

[9] S N Elaydi An Introduction to Difference Equations SpringerBerlin Germany 1996

[10] J Maynard Smith Models in Ecology Cambridge UniversityPress 1974

[11] P Z Liu and S N Elaydi ldquoDiscrete competitive and coopera-tive models of Lotka-Volterra Typerdquo Journal of ComputationalAnalysis and Applications vol 3 pp 53ndash73 2001

[12] Y S Wang and H Wu ldquoDynamics of competitive Lotka-Volte-rra systems that can be projected to a linerdquo Computers ampMathematics with Applications vol 47 pp 1263ndash1271 2004

[13] X Xiong and Z Zhang ldquoPeriodic solutions of a discrete two-species competitive model with stage structurerdquo Mathematicaland Computer Modelling vol 48 no 3-4 pp 333ndash343 2008

[14] C Niu and X Chen ldquoAlmost periodic sequence solutions ofa discrete Lotka-Volterra competitive system with feedbackcontrolrdquoNonlinear Analysis RealWorldApplications vol 10 no5 pp 3152ndash3161 2009

[15] J G Skellam ldquoRandom dispersal in theoretical populationsrdquoBiometrika vol 38 pp 196ndash218 1951

[16] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society B vol 237 pp 37ndash721953

[17] J Bascompte and R V Sole ldquoSpatially induced bifurcations insingle-species population dynamicsrdquo Journal of Animal Ecologyvol 63 pp 256ndash264 1994

[18] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[19] Y T Han B Han L Zhang L Xu M F Li and G ZhangldquoTuring instability and labyrinthine patterns for a symmetricdiscrete comptitive Lotka-Volterra systemrdquo WSEAS Transac-tions on Mathematics vol 10 pp 181ndash189 2011

[20] G Q Sun Z Jin Q X Liu and L Li ldquoDynamical complexityof a spatial predator-prey model with migrationrdquo EcologicalModelling vol 219 pp 248ndash255 2008

[21] G Q Sun Z Jin Q X Liu and L Li ldquoSpatial pattern of anepidemic model with cross-diffusionrdquo Chinese Physics B vol 17pp 3936ndash3941 2008

[22] Y-X Wang and W-T Li ldquoEffect of cross-diffusion on thestationary problem of a diffusive competition model with aprotection zonerdquo Nonlinear Analysis Real World Applicationsvol 14 no 1 pp 224ndash245 2013

[23] M Li B Han L Xu and G Zhang ldquoSpiral patterns nearTuring instability in a discrete reaction diffusion systemrdquoChaosSolitons amp Fractals vol 49 pp 1ndash6 2013

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

2 Discrete Dynamics in Nature and Society

analogs may be found in [9ndash15] regarding attractivity persis-tence global stabilities of equilibrium and other dynamicsUp to now to the best of our knowledge the discrete system(4) has not been investigated

Since the pioneering theoretical works of Skellam [15] andTuring [16] many works have focused on the effect of spatialfactors which play a crucial role in the stability of populations[17ndash19] Many important epidemiological and ecologicalphenomena are strongly influenced by spatial heterogeneitiesbecause of the localized nature of transmission or other formsof interaction Thus spatial models are more suitable fordescribing the process of population development Impact ofspatial component on system has been widely investigated(eg see [20ndash22]) It may be a case in reality that the motionof individuals is random and isotropic that is without anypreferred direction the individuals are also absolute ones inmicroscopic sense and each isolated individual exchangesmaterials by diffusion with its neighbors [19 23] Thusit is reasonable to consider a 1D or 2D spatially discretereaction diffusion system to explain the population systemCorresponding to the above analysis we can obtain thefollowing one-dimensional diffusion systems

119906119905+1

119894= 119906119905

119894(119903 + 1 minus 119906

119905

119894minus 119886V119905119894) + 1198891nabla2119906119905

119894

V119905+1119894

= V119905119894exp (119903 minus 119886119906119905

119894minus V119905119894) + 1198892nabla2V119905119894

(5)

for 119894 isin 1 2 119898 = [1119898] 119905 isin 119885+ and nabla2119906119905119894= 119906119905

119894+1minus 2119906119905

119894+

119906119905

119894minus1 nabla2V119905119894= V119905119894+1

minus 2V119905119894+ V119905119894minus1

and two-dimensional diffusionsystems

119906119905+1

119894119895= 119906119905

119894119895(119903 + 1 minus 119906

119905

119894119895minus 119886V119905119894119895) + 1198891nabla2119906119905

119894119895

V119905+1119894119895

= V119905119894119895exp (119903 minus 119886119906119905

119894119895minus V119905119894119895) + 1198892nabla2V119905119894119895

(6)

for 119894 119895 isin 1 2 119898 = [1119898] 119905 isin 119877+ = [0infin) and nabla2119906119905119894119895=

119906119905

119894+1119895+ 119906119905

119894119895+1+ 119906119905

119894minus1119895+ 119906119905

119894119895minus1minus 4119906119905

119894119895 nabla2V119905119894119895= V119905119894+1119895

+ V119905119894119895+1

+

V119905119894minus1119895

+ V119905119894119895minus1

minus 4V119905119894119895

In this paper we will study the dynamical behaviors ofmodels (4) (5) and (6) By using the theory of differenceequation the theory of bifurcation and the center manifoldtheoremwewill establish the series of criteria on the existenceand local stability of equilibria flip bifurcation for the system(4) For the one- or two-dimensional diffusion systemswith periodic boundary conditions the Turing instability(or Turing bifurcation) theory analysis will be given Tur-ing instability conditions can then be deduced combininglinearization method and inner product technique Further-more bymeans of the numerical simulationsmethod we willindicate the correctness and rationality of our results

The paper is organized as follows In Section 2 westudy the existence and stability of equilibria points and theconditions of existence for flip bifurcation are verified forsystem (4) Turing instability conditions will be illustratedby linearization method and inner product technique for thesystem (5) and (6) with periodic boundary conditions inSection 3 A series of numerical simulations are performedthat not only verify the theoretical analysis but also dis-play some interesting dynamics For the system (4) the

bifurcation diagrams are given The impact of the systemparameters and diffusion coefficients on patterns can alsobe observed visually for the given diffusion systems Finallysome conclusions are given

2 Analysis of Equilibria and Flip Bifurcation

Clearly the system (4) has four possible steady states that is1198640= (0 0) exclusion points 119864

1= (119903 0) 119864

2= (0 119903) and

nontrivial coexistence point 1198643= (119906lowast Vlowast) where

119906lowast= Vlowast =

119903

119886 + 1

(7)

The linearized form of (4) is then

119906119905+1

= 119891119906119906119905+ 119891VV119905

V119905+1

= 119892119906119906119905+ 119892VV119905

(8)

which has the Jacobian matrix

119869119864119894= [

119891119906119891V

119892119906119892V]

119875119894

= [

1 minus 119906 minus119886119906

minus119886V 1 minus V]119875119894

119894 = 0 1 2 3 (9)

The characteristic equation of the Jacobian matrix 119869 canbe written as

1205822+ 119901120582 + 119902 = 0 (10)

where 119901 = minus(119891119906+ 119892V) and 119902 = 119891119906119892V minus 119891V119892119906

In order to discuss the stability of the fixed points of (4)we also need the following definitions [20]

(1) if |1205821| lt 1 and |120582

2| lt 1 then steady state 119864 is called a

sink and 119864 is locally asymptotically stable(2) if |120582

1| gt 1 and |120582

2| gt 1 then 119864 is called a source and

119864 is unstable(3) if |120582

1| gt 1 and |120582

2| lt 1 (or |120582

1| lt 1 and |120582

2| gt 1)

then 119864 is called a saddle(4) if either |120582

1| = 1 and |120582

2| = 1 or |120582

2| = 1 and |120582

1| = 1

then 119864 is called nonhyperbolic

Case 1 (the fixed point 1198640= (0 0)) The linearization of (4)

about 1198640has the Jacobian matrix

1198691198640= [

1 0

0 1] (11)

which has two eigenvalues

1205821= 1205822= 1 (12)

The fact means that the system is resonance at the fixed point1198750

Case 2 (the fixed point 1198641= (0 119903)) At the fixed point the

Jacobian matrix has the form

1198691198641= [

1 0

minus119886119903 1 minus 119903] (13)

Discrete Dynamics in Nature and Society 3

and the corresponding eigenvalues of (13) are

1205821= 1 120582

2= 1 minus 119903 (14)

119903 is a bifurcation parameter And 119903 = 2 implies 1205822= minus 1 and

the fixed point 1198641is nonhyperbolic

Case 3 (the third fixed point 1198642= (119903 0)) The linearization of

(4) about 1198642has the Jacobian matrix

1198691198641= [

1 minus 119903 minus119886119903

0 1] (15)

and the eigenvalues of (15) are

1205821= 1 minus 119903 120582

2= 1 (16)

119903 is a bifurcation parameter And 119903 = 2 implies 1205821= minus 1 and

the fixed point 1198642is nonhyperbolic

Case 4 (the fixed point 1198643= (119903(119886 + 1) 119903(119886 + 1))) The

linearization of (4) about 1198643has the Jacobian matrix

1198691198643=[

[

[

1 minus

119903

119886 + 1

minus

119886119903

119886 + 1

minus

119886119903

119886 + 1

1 minus

119903

119886 + 1

]

]

]

(17)

and the eigenvalues of (17) are

1205821= 1 minus 119903 120582

2= 1 +

(119886 minus 1) 119903

119886 + 1

(18)

then we have the following results

(1) |1205821| lt 1 |120582

2| lt 1 if and only if 0 lt 119903 lt 2 0 lt 119886 lt 1

(2) 1205821= minus1 |120582

2| = 1 if and only if 119903 = 2 119886 = 1

(3) 1205822= 1 |120582

1| = 1 if and only if 119886 = 2 119903 = 2

(4) 1205822= minus1 |120582

1| = 1 if and only if 119886 = (119903minus2)(119903+2) 119903 = 2

(5) |1205821| lt 1 |120582

2| gt 1 if and only if 0 lt 119903 lt 2 119886 gt 1

(6) |1205821| gt 1 |120582

2| lt 1 if and only if 2 lt 119903 lt (119886+ 1)(119886 minus 1)

(7) |1205821| gt 1 |120582

2| gt 1 if and only if 119903 gt 2 119886 gt 1

The following theorem is the case that the fixed point 1198643is a

flip bifurcation point

Theorem 1 The positive fixed point 1198643undergoes a flip

bifurcation at the threshold 119903119865= 2

Proof Let 120577119899= 119906119899minus 119906lowast 120578119899= V119899minus Vlowast and 120583

119899= 119903 minus 2 and

parameter 120583119899is a new and dependent variable the system (4)

becomes

(

120577119899+1

120578119899+1

120583119899+1

) =(

(120577119899+

120583119899+ 2

119886 + 1

) (1 minus 120577119899minus 119886120578119899) minus

120583119899+ 2

119886 + 1

(120578119899+

120583119899+ 2

119886 + 1

) exp (minus119886120577119899minus 120578119899) minus

120583119899+ 2

119886 + 1

120583119899

)

(19)

Let

119879 =[

[

1 minus1 0

1 1 0

0 0 1

]

]

(20)

then

119879minus1=

[

[

[

[

[

[

[

1

2

1

2

0

minus

1

2

1

2

0

0 0 1

]

]

]

]

]

]

]

(21)

By the following transformation

(

120577119899

120578119899

120583119899

) = 119879(

119909119899

119910119899

120575119899

) (22)

the system (19) can be changed into

(

119909119899+1

119910119899+1

120575119899+1

) = (

minus1 0 0

0

3119886 minus 1

119886 + 1

0

0 0 1

)(

119909119899

119910119899

120575119899

) + (

119891 (119909119899 119910119899 120575119899)

119892 (119909119899 119910119899 120575119899)

0

)

(23)

where

119891 (119909119899 119910119899 120575119899) = minus (119886 + 1) 119909

2

119899minus 2 (119886 + 1) 119909

119899119910119899minus 119909119899120575119899

+ (119886 minus 1 +

(119886 minus 1)2

119886 + 1

)1199102

119899

+ 119900 ((1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816119910119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

119892 (119909119899 119910119899 120575119899) = (119886 + 1) 119909

2

119899minus 2 (119886 + 1) 119909

119899119910119899

minus

2 (119886 minus 1)

119886 + 1

119910119899120575119899+

(119886 minus 1)2

119886 + 1

1199102

119899

+ 119900 ((1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816119910119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

(24)

Then we can consider

119910119899= ℎ (119909

119899 120575119899) = 11988611199092

119899+ 1198862119909119899120575119899+ 11988631205752

119899+ 119900 ((

1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

(25)

which must satisfy

ℎ (minus119909119899+ 119891 (119909

119899 119910119899 120575119899) 120575119899+1)

=

3119886 minus 1

119886 + 1

ℎ (119909119899 120575119899) + (119886 + 1) 119909

2

119899

minus 2 (119886 + 1) 119909119899ℎ (119909119899 120575119899)

minus

2 (119886 minus 1)

119886 + 1

ℎ (119909119899 120575119899) 120575119899

+

(119886 minus 1)2

119886 + 1

ℎ2(119909119899 120575119899)

+ 119900 ((1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816119910119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

(26)

4 Discrete Dynamics in Nature and Society

By calculating we can get that

1198861=

(119886 + 1)2

2 (1 minus 119886)

1198862= 0 119886

3= 0 (27)

and the system (19) is restricted to the center manifold whichis given by

119891 119909119899+1

= minus 119909119899+

1198863+ 41198862+ 3119886

119886 minus 1

1199092

119899+ (119886 + 1) 119909

119899120575119899

+

1

4

(119886 + 1)31199094

119899+ 119900 (

1003816100381610038161003816119909119899

1003816100381610038161003816

4

)

(28)

Since

(

120597119891

120597120575

1205972119891

1205971199092+ 2

1205972119891

120597119909120597120575

)

100381610038161003816100381610038161003816100381610038161003816(00)

= 2 (119886 + 1) = 0

(

1

2

(

1205972119891

1205971199092)

2

+

1

3

1205973119891

1205971199093)

1003816100381610038161003816100381610038161003816100381610038161003816(00)

= 2(

1198863+ 41198862+ 3119886

119886 minus 1

)

2

gt 0

(29)

system (4) undergoes a flip bifurcation at 1198643 The proof is

completed

3 Turing Bifurcation

In this section we discuss the Turing bifurcation Turingrsquostheory shows that diffusion could destabilize an otherwisestable equilibrium of the reaction-diffusion system and leadto nonuniform spatial patterns This kind of instability isusually calledTuring instability or diffusion-driven instability[16]

31 One-Dimensional Case We consider the following diffu-sion system

119906119905+1

119894= 119906119905

119894(119903 + 1 minus 119906

119905

119894minus 119886V119905119894) + 1198891nabla2119906119905

119894

V119905+1119894

= V119905119894exp (119903 minus 119886119906119905

119894minus V119905119894) + 1198892nabla2V119905119894

(30)

with the periodic boundary conditions

119906119905

0= 119906119905

119898 119906

119905

1= 119906119905

119898+1

V1199050= V119905119898 V119905

1= V119905119898+1

(31)

for 119894 isin 1 2 119898 = [1119898] and 119905 isin 119885+ where119898 is a positiveinteger

nabla2119906119905

119894= 119906119905

119894+1minus 2119906119905

119894+ 119906119905

119894minus1

nabla2V119905119894= V119905119894+1

minus 2V119905119894+ V119905119894minus1

(32)

In order to study Turing instability of (30) and (31) wefirstly consider eigenvalues of the following equation

nabla2119883119894+ 120582119883119894= 0 (33)

with the periodic boundary conditions

1198830= 119883119898 119883

1= 119883119898+1

(34)

By calculating the eigenvalue problem (33)-(34) has theeigenvalues

120582119904= 4sin2 (119904 minus 1) 120587

119898

for 119904 isin [1119898] (35)

We linearise at the steady state to get

119908119905+1

119894= 119869119908119905

119894+ 119863nabla2119908119905

119894 119863 = (

11988910

0 1198892

) (36)

with the periodic boundary conditions

119908119905

0= 119908119905

119898 119908

119905

1= 119908119905

119898+1 (37)

where

119908119905

119894= (

119906119905

119894minus 119906lowast

V119905119894minus Vlowast

) = (

119909119905

119894

119910119905

119894

) (38)

Then respectively taking the inner product of (36) withthe corresponding eigenfunction 119883119894

119904of the eigenvalue 120582

119904 we

see that119898

sum

119894=1

119883119894

119904119909119905+1

119894= 119891119906

119898

sum

119894=1

119883119894

119904119909119905

119894+ 119891V

119898

sum

119894=1

119883119894

119904119910119905

119894+ 1198891

119898

sum

119894=1

119883119894

119904nabla2119909119905

119894

119898

sum

119894=1

119883119894

119904119910119905+1

119894= 119892119906

119898

sum

119894=1

119883119894

119904119909119905

119894+ 119892V

119898

sum

119894=1

119883119894

119904119910119905

119894+ 1198892

119898

sum

119894=1

119883119894

119904nabla2119910119905

119894

(39)

Let 119880119905 = sum119898119894=1119883119894

119904119909119905

119894and 119881119905 = sum119898

119894=1119883119894

119904119910119905

119894and use the periodic

boundary conditions (34) and (37) then we have

119880119905+1

= 119891119906119880119905+ 119891V119881

119905minus 1198891120582119904119880119905

119881119905+1

= 119892119906119880119905+ 119892V119881

119905minus 1198892120582119904119881119905

(40)

or

119880119905+1

= (119891119906minus 1198891120582119904) 119880119905+ 119891V119881

119905

119881119905+1

= 119892119906119880119905+ (119892V minus 1198892120582119904) 119881

119905

(41)

Thus the following fact can be obtained

Proposition 2 If (119906119905119894 V119905119894) is a solution of the problem of (30)

and (31) then

(119880119905=

119898

sum

119894=1

119883119894

119904119909119905

119894 119881119905=

119898

sum

119894=1

119883119894

119904119910119905

119894) (42)

is a solution of (41) where 120582119904is some eigenvalue of (33)-(34)

and119883119894119904is the corresponding eigenfunction For some eigenvalue

120582119904of (33)-(34) if (119880119905 119881119905) is a solution of the system (41) then

(119906119905

119894= 119880119905119883119894

119904 V119905119894= 119881119905119883119894

119904) (43)

is a solution of (30)with the periodic boundary conditions (31)

Discrete Dynamics in Nature and Society 5

Proposition 3 If there exist positive numbers 1198891 1198892and the

eigenvalue 120582119904of the problem (33)-(34) such that one of the

conditions

ℎ (120582119904) lt (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (120582119904) lt minus (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(44)

or

ℎ (120582119904) gt 1 (45)

holds then the problem (30) and (31) at the fixed point (119906lowast Vlowast)is unstable where

ℎ (120582119904) = 119889111988921205822

119904minus (1198891119892Vlowast + 1198892119891119906lowast) 120582119904 + (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

(46)

For the system (5) we have the following results aboutinstability of the positive equilibrium of system

Theorem 4 0 lt 119903 lt 2 0 lt 119886 lt 1 and Proposition 3 mean orshow that the problem (30) and (31) is diffusion-driven unstableor Turing unstable

32 Two-Dimensional Case In this subsection we will payour attention to the Turing instability analysis for the follow-ing two-dimensional system

119906119905+1

119894119895= 119906119905

119894119895(119903 + 1 minus 119906

119905

119894119895minus 119886V119905119894119895) + 1198891nabla2119906119905

119894119895

V119905+1119894119895

= V119905119894119895exp (119903 minus 119886119906119905

119894119895minus V119905119894119895) + 1198892nabla2V119905119894119895

(47)

with the periodic boundary conditions

119906119905

1198940= 119906119905

119894119898 119906

119905

1198941= 119906119905

119894119898+1

119906119905

0119895= 119906119905

119898119895 119906

119905

1119895= 119906119905

119898+1119895

V1199051198940= V119905119894119898 V119905

1198941= V119905119894119898+1

V1199050119895= V119905119898119895 V119905

1119895= V119905119898+1119895

(48)

for 119894 119895 isin 1 2 119898 = [1119898] and 119905 isin 119885+ where 119898 is a

positive integer

nabla2119906119905

119894119895= 119906119905

119894+1119895+ 119906119905

119894119895+1+ 119906119905

119894minus1119895+ 119906119905

119894119895minus1minus 4119906119905

119894119895

nabla2V119905119894119895= V119905119894+1119895

+ V119905119894119895+1

+ V119905119894minus1119895

+ V119905119894119895minus1

minus 4V119905119894119895

(49)

The following theoremwill show that the system (47) alsoundergoes Turing instability Since the analysis is very similarto the one-dimensional case the proof is omitted

Theorem 5 If there exist positive numbers 1198891 1198892and the

eigenvalue 1198962119897119904of the corresponding characteristic equation such

that one of the conditions

ℎ (1198962

119897119904) lt (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (1198962

119897119904) lt minus (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(50)

4

r

u

35

35

3

3

25

25

2

15

15

1

21

05

05

0

Figure 1 Bifurcation diagram for 119903 minus 119906119905with 119886 = 0511

or

ℎ (1198962

119897119904) gt 1 (51)

holds and 0 lt 119903 lt 2 0 lt 119886 lt 1 then the problem (47)-(48) atthe fixed point (119906lowast Vlowast) is diffusion-driven unstable or Turingunstable where

ℎ (1198962

119897119904) = 119889111988921198964

119897119904minus (1198891119892Vlowast + 1198892119891119906lowast) 119896

2

119897119904+ (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

1198962

119897119904= 120582119897119904= 4(sin2 ((119897 minus 1) 120587

119898

) + sin2 ((119904 minus 1) 120587119898

))

119891119900119903 119897 119904 isin [1119898]

(52)

4 Numerical Simulation

As is known to all the bifurcation diagram provides a generalview of the evolution process of the dynamical behaviorsby plotting a state variable with the abscissa being oneparameter As a parameter varies the dynamics of the systemwe concerned change through a local or global bifurcationwhich leads to the change of stability at the same time

Now 119903 is considered as a parameter with the range 05ndash35 for the system (4) Since the bifurcation diagrams of 119903minus119906

119905

are similar to the bifurcation diagrams of 119903 minus V119905 we will only

show the former which can be seen from Figure 1Next we performed a series of simulations for the

reaction-diffusion systems and in each the initial condi-tion was always a small amplitude random perturbation1 around the steady state As a numerical example weconsider the bifurcation of the two-dimensional system (47)-(48) It is well known that Turing instability (bifurcation) isdiffusion-driven instability thus the diffusion rate is vital tothe pattern formation To investigate the effect of diffusioncoefficients on patterns by keeping all the other parametersof the system fixed (119886 = 055 119903 = 071 and 119889

1= 022)

we change a diffusion coefficient in the Turing instability

6 Discrete Dynamics in Nature and Society

20

20

40

40

60

60

80

80

100

100

120

120

(a)

20 40 60 80 100 120

20

40

60

80

100

120

(b)

20 40 60 80 100 120

20

40

60

80

100

120

(c)

20 40 60 80 100 120

20

40

60

80

100

120

(d)

20 40 60 80 100 120

20

40

60

80

100

120

(e)

20 40 60 80 100 120

20

40

60

80

100

120

(f)

Figure 2 Pattern selection with the increase of 1198892in the Turing instability region when 119886 = 055 119903 = 071 and 119889

1= 022 (a) 119889

2= 0195

(b) 1198892= 020 (c) 119889

2= 0205 (d) 119889

2= 021 (e) 119889

2= 0215 (f) 119889

2= 0217

region (parameter space which satisfies Turing instability)Figures 2(a)ndash2(f) exhibit in detail the different distributionof patterns with varying values of 119889 If we let 119889

2= 019

a stable pattern of square shapes namely stationary wave

is observed in Figure 2(a) With the increase of 1198892 some

tips appear which attempt to form self-centered spiral waves(Figure 2(b))Then some tips vanish and some tend to evolveto regular spiral waves shown in Figure 2(c) Then with the

Discrete Dynamics in Nature and Society 7

parameter evolution proceeding the size of spiral waves risesbut the density of them decreases (see Figure 2(d)) If 119889

2is

further increased we observe that in parts of patterns spiralwaves begin to break up Hardly can spiral waves be seendisorder and chaotic structure are depicted in Figures 2(e)and 2(f)

5 Discussion and Conclusion

In this paper we have applied different discrete schemes toconvert the continuous Lotka-Volterra competition modelto a new discrete model and studied the dynamical charac-teristic of the discrete model Our theoretical analysis andnumerical simulations have demonstrated that the discretecompetition model undergoes flip bifurcation Furthermorewhen the effects of spatial factors are considered we discussthe Turing instability conditions combining linearizationmethod and inner product technique The impact of the dif-fusion coefficients on patterns can also be observed visuallyand some interesting situations can be observed Indeed thenew discrete model can result in a rich set of patterns and weexpect that it is more effective in practice

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Dr Y D Jiao for the valuable suggestionsand thank the referees for their helpful comments This workwas financially supported by TianjinUniversity of Commercewith the Grant no X0803 the National Natural ScienceFoundation of China under Grant nos 11001072 11371277and Cultivation Program for Excellent Youth Teacher inUniversity Tianjin (507-125RCPY0314)

References

[1] Y Takeuchi Global Dynamical Properties of Lotka-VolterraSystems World Scientific Publishing Singapore 1996

[2] PWaltman CompetitionModels in Populationbiology vol 45 ofCBMS-NSF Regional Conference Series in Applied MathematicsSIAM Philadelphia Pa USA 1985

[3] M E FisherAnalysis of difference equationmodels in populationdynamics [PhD thesis] University of Western Australia 1982

[4] P Liu and X Cui ldquoA discrete model of competitionrdquoMathemat-ics and Computers in Simulation vol 49 no 1-2 pp 1ndash12 1999

[5] F B Hildebrand Finite-Difference Equations and SimulationsPrentice Hall Englewood Cliffs NJ USA 1968

[6] J K Hale and A S Somolinos ldquoCompetition for fluctuatingnutrientrdquo Journal ofMathematical Biology vol 18 no 3 pp 255ndash280 1983

[7] K P Hadeler and I Gerstmann ldquoThe discrete RosenzweigmodelrdquoMathematical Biosciences vol 98 no 1 pp 49ndash72 1990

[8] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics vol 191 Academic Press New York NYUSA 1993

[9] S N Elaydi An Introduction to Difference Equations SpringerBerlin Germany 1996

[10] J Maynard Smith Models in Ecology Cambridge UniversityPress 1974

[11] P Z Liu and S N Elaydi ldquoDiscrete competitive and coopera-tive models of Lotka-Volterra Typerdquo Journal of ComputationalAnalysis and Applications vol 3 pp 53ndash73 2001

[12] Y S Wang and H Wu ldquoDynamics of competitive Lotka-Volte-rra systems that can be projected to a linerdquo Computers ampMathematics with Applications vol 47 pp 1263ndash1271 2004

[13] X Xiong and Z Zhang ldquoPeriodic solutions of a discrete two-species competitive model with stage structurerdquo Mathematicaland Computer Modelling vol 48 no 3-4 pp 333ndash343 2008

[14] C Niu and X Chen ldquoAlmost periodic sequence solutions ofa discrete Lotka-Volterra competitive system with feedbackcontrolrdquoNonlinear Analysis RealWorldApplications vol 10 no5 pp 3152ndash3161 2009

[15] J G Skellam ldquoRandom dispersal in theoretical populationsrdquoBiometrika vol 38 pp 196ndash218 1951

[16] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society B vol 237 pp 37ndash721953

[17] J Bascompte and R V Sole ldquoSpatially induced bifurcations insingle-species population dynamicsrdquo Journal of Animal Ecologyvol 63 pp 256ndash264 1994

[18] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[19] Y T Han B Han L Zhang L Xu M F Li and G ZhangldquoTuring instability and labyrinthine patterns for a symmetricdiscrete comptitive Lotka-Volterra systemrdquo WSEAS Transac-tions on Mathematics vol 10 pp 181ndash189 2011

[20] G Q Sun Z Jin Q X Liu and L Li ldquoDynamical complexityof a spatial predator-prey model with migrationrdquo EcologicalModelling vol 219 pp 248ndash255 2008

[21] G Q Sun Z Jin Q X Liu and L Li ldquoSpatial pattern of anepidemic model with cross-diffusionrdquo Chinese Physics B vol 17pp 3936ndash3941 2008

[22] Y-X Wang and W-T Li ldquoEffect of cross-diffusion on thestationary problem of a diffusive competition model with aprotection zonerdquo Nonlinear Analysis Real World Applicationsvol 14 no 1 pp 224ndash245 2013

[23] M Li B Han L Xu and G Zhang ldquoSpiral patterns nearTuring instability in a discrete reaction diffusion systemrdquoChaosSolitons amp Fractals vol 49 pp 1ndash6 2013

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

Discrete Dynamics in Nature and Society 3

and the corresponding eigenvalues of (13) are

1205821= 1 120582

2= 1 minus 119903 (14)

119903 is a bifurcation parameter And 119903 = 2 implies 1205822= minus 1 and

the fixed point 1198641is nonhyperbolic

Case 3 (the third fixed point 1198642= (119903 0)) The linearization of

(4) about 1198642has the Jacobian matrix

1198691198641= [

1 minus 119903 minus119886119903

0 1] (15)

and the eigenvalues of (15) are

1205821= 1 minus 119903 120582

2= 1 (16)

119903 is a bifurcation parameter And 119903 = 2 implies 1205821= minus 1 and

the fixed point 1198642is nonhyperbolic

Case 4 (the fixed point 1198643= (119903(119886 + 1) 119903(119886 + 1))) The

linearization of (4) about 1198643has the Jacobian matrix

1198691198643=[

[

[

1 minus

119903

119886 + 1

minus

119886119903

119886 + 1

minus

119886119903

119886 + 1

1 minus

119903

119886 + 1

]

]

]

(17)

and the eigenvalues of (17) are

1205821= 1 minus 119903 120582

2= 1 +

(119886 minus 1) 119903

119886 + 1

(18)

then we have the following results

(1) |1205821| lt 1 |120582

2| lt 1 if and only if 0 lt 119903 lt 2 0 lt 119886 lt 1

(2) 1205821= minus1 |120582

2| = 1 if and only if 119903 = 2 119886 = 1

(3) 1205822= 1 |120582

1| = 1 if and only if 119886 = 2 119903 = 2

(4) 1205822= minus1 |120582

1| = 1 if and only if 119886 = (119903minus2)(119903+2) 119903 = 2

(5) |1205821| lt 1 |120582

2| gt 1 if and only if 0 lt 119903 lt 2 119886 gt 1

(6) |1205821| gt 1 |120582

2| lt 1 if and only if 2 lt 119903 lt (119886+ 1)(119886 minus 1)

(7) |1205821| gt 1 |120582

2| gt 1 if and only if 119903 gt 2 119886 gt 1

The following theorem is the case that the fixed point 1198643is a

flip bifurcation point

Theorem 1 The positive fixed point 1198643undergoes a flip

bifurcation at the threshold 119903119865= 2

Proof Let 120577119899= 119906119899minus 119906lowast 120578119899= V119899minus Vlowast and 120583

119899= 119903 minus 2 and

parameter 120583119899is a new and dependent variable the system (4)

becomes

(

120577119899+1

120578119899+1

120583119899+1

) =(

(120577119899+

120583119899+ 2

119886 + 1

) (1 minus 120577119899minus 119886120578119899) minus

120583119899+ 2

119886 + 1

(120578119899+

120583119899+ 2

119886 + 1

) exp (minus119886120577119899minus 120578119899) minus

120583119899+ 2

119886 + 1

120583119899

)

(19)

Let

119879 =[

[

1 minus1 0

1 1 0

0 0 1

]

]

(20)

then

119879minus1=

[

[

[

[

[

[

[

1

2

1

2

0

minus

1

2

1

2

0

0 0 1

]

]

]

]

]

]

]

(21)

By the following transformation

(

120577119899

120578119899

120583119899

) = 119879(

119909119899

119910119899

120575119899

) (22)

the system (19) can be changed into

(

119909119899+1

119910119899+1

120575119899+1

) = (

minus1 0 0

0

3119886 minus 1

119886 + 1

0

0 0 1

)(

119909119899

119910119899

120575119899

) + (

119891 (119909119899 119910119899 120575119899)

119892 (119909119899 119910119899 120575119899)

0

)

(23)

where

119891 (119909119899 119910119899 120575119899) = minus (119886 + 1) 119909

2

119899minus 2 (119886 + 1) 119909

119899119910119899minus 119909119899120575119899

+ (119886 minus 1 +

(119886 minus 1)2

119886 + 1

)1199102

119899

+ 119900 ((1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816119910119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

119892 (119909119899 119910119899 120575119899) = (119886 + 1) 119909

2

119899minus 2 (119886 + 1) 119909

119899119910119899

minus

2 (119886 minus 1)

119886 + 1

119910119899120575119899+

(119886 minus 1)2

119886 + 1

1199102

119899

+ 119900 ((1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816119910119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

(24)

Then we can consider

119910119899= ℎ (119909

119899 120575119899) = 11988611199092

119899+ 1198862119909119899120575119899+ 11988631205752

119899+ 119900 ((

1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

(25)

which must satisfy

ℎ (minus119909119899+ 119891 (119909

119899 119910119899 120575119899) 120575119899+1)

=

3119886 minus 1

119886 + 1

ℎ (119909119899 120575119899) + (119886 + 1) 119909

2

119899

minus 2 (119886 + 1) 119909119899ℎ (119909119899 120575119899)

minus

2 (119886 minus 1)

119886 + 1

ℎ (119909119899 120575119899) 120575119899

+

(119886 minus 1)2

119886 + 1

ℎ2(119909119899 120575119899)

+ 119900 ((1003816100381610038161003816119909119899

1003816100381610038161003816+1003816100381610038161003816119910119899

1003816100381610038161003816+1003816100381610038161003816120575119899

1003816100381610038161003816)3

)

(26)

4 Discrete Dynamics in Nature and Society

By calculating we can get that

1198861=

(119886 + 1)2

2 (1 minus 119886)

1198862= 0 119886

3= 0 (27)

and the system (19) is restricted to the center manifold whichis given by

119891 119909119899+1

= minus 119909119899+

1198863+ 41198862+ 3119886

119886 minus 1

1199092

119899+ (119886 + 1) 119909

119899120575119899

+

1

4

(119886 + 1)31199094

119899+ 119900 (

1003816100381610038161003816119909119899

1003816100381610038161003816

4

)

(28)

Since

(

120597119891

120597120575

1205972119891

1205971199092+ 2

1205972119891

120597119909120597120575

)

100381610038161003816100381610038161003816100381610038161003816(00)

= 2 (119886 + 1) = 0

(

1

2

(

1205972119891

1205971199092)

2

+

1

3

1205973119891

1205971199093)

1003816100381610038161003816100381610038161003816100381610038161003816(00)

= 2(

1198863+ 41198862+ 3119886

119886 minus 1

)

2

gt 0

(29)

system (4) undergoes a flip bifurcation at 1198643 The proof is

completed

3 Turing Bifurcation

In this section we discuss the Turing bifurcation Turingrsquostheory shows that diffusion could destabilize an otherwisestable equilibrium of the reaction-diffusion system and leadto nonuniform spatial patterns This kind of instability isusually calledTuring instability or diffusion-driven instability[16]

31 One-Dimensional Case We consider the following diffu-sion system

119906119905+1

119894= 119906119905

119894(119903 + 1 minus 119906

119905

119894minus 119886V119905119894) + 1198891nabla2119906119905

119894

V119905+1119894

= V119905119894exp (119903 minus 119886119906119905

119894minus V119905119894) + 1198892nabla2V119905119894

(30)

with the periodic boundary conditions

119906119905

0= 119906119905

119898 119906

119905

1= 119906119905

119898+1

V1199050= V119905119898 V119905

1= V119905119898+1

(31)

for 119894 isin 1 2 119898 = [1119898] and 119905 isin 119885+ where119898 is a positiveinteger

nabla2119906119905

119894= 119906119905

119894+1minus 2119906119905

119894+ 119906119905

119894minus1

nabla2V119905119894= V119905119894+1

minus 2V119905119894+ V119905119894minus1

(32)

In order to study Turing instability of (30) and (31) wefirstly consider eigenvalues of the following equation

nabla2119883119894+ 120582119883119894= 0 (33)

with the periodic boundary conditions

1198830= 119883119898 119883

1= 119883119898+1

(34)

By calculating the eigenvalue problem (33)-(34) has theeigenvalues

120582119904= 4sin2 (119904 minus 1) 120587

119898

for 119904 isin [1119898] (35)

We linearise at the steady state to get

119908119905+1

119894= 119869119908119905

119894+ 119863nabla2119908119905

119894 119863 = (

11988910

0 1198892

) (36)

with the periodic boundary conditions

119908119905

0= 119908119905

119898 119908

119905

1= 119908119905

119898+1 (37)

where

119908119905

119894= (

119906119905

119894minus 119906lowast

V119905119894minus Vlowast

) = (

119909119905

119894

119910119905

119894

) (38)

Then respectively taking the inner product of (36) withthe corresponding eigenfunction 119883119894

119904of the eigenvalue 120582

119904 we

see that119898

sum

119894=1

119883119894

119904119909119905+1

119894= 119891119906

119898

sum

119894=1

119883119894

119904119909119905

119894+ 119891V

119898

sum

119894=1

119883119894

119904119910119905

119894+ 1198891

119898

sum

119894=1

119883119894

119904nabla2119909119905

119894

119898

sum

119894=1

119883119894

119904119910119905+1

119894= 119892119906

119898

sum

119894=1

119883119894

119904119909119905

119894+ 119892V

119898

sum

119894=1

119883119894

119904119910119905

119894+ 1198892

119898

sum

119894=1

119883119894

119904nabla2119910119905

119894

(39)

Let 119880119905 = sum119898119894=1119883119894

119904119909119905

119894and 119881119905 = sum119898

119894=1119883119894

119904119910119905

119894and use the periodic

boundary conditions (34) and (37) then we have

119880119905+1

= 119891119906119880119905+ 119891V119881

119905minus 1198891120582119904119880119905

119881119905+1

= 119892119906119880119905+ 119892V119881

119905minus 1198892120582119904119881119905

(40)

or

119880119905+1

= (119891119906minus 1198891120582119904) 119880119905+ 119891V119881

119905

119881119905+1

= 119892119906119880119905+ (119892V minus 1198892120582119904) 119881

119905

(41)

Thus the following fact can be obtained

Proposition 2 If (119906119905119894 V119905119894) is a solution of the problem of (30)

and (31) then

(119880119905=

119898

sum

119894=1

119883119894

119904119909119905

119894 119881119905=

119898

sum

119894=1

119883119894

119904119910119905

119894) (42)

is a solution of (41) where 120582119904is some eigenvalue of (33)-(34)

and119883119894119904is the corresponding eigenfunction For some eigenvalue

120582119904of (33)-(34) if (119880119905 119881119905) is a solution of the system (41) then

(119906119905

119894= 119880119905119883119894

119904 V119905119894= 119881119905119883119894

119904) (43)

is a solution of (30)with the periodic boundary conditions (31)

Discrete Dynamics in Nature and Society 5

Proposition 3 If there exist positive numbers 1198891 1198892and the

eigenvalue 120582119904of the problem (33)-(34) such that one of the

conditions

ℎ (120582119904) lt (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (120582119904) lt minus (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(44)

or

ℎ (120582119904) gt 1 (45)

holds then the problem (30) and (31) at the fixed point (119906lowast Vlowast)is unstable where

ℎ (120582119904) = 119889111988921205822

119904minus (1198891119892Vlowast + 1198892119891119906lowast) 120582119904 + (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

(46)

For the system (5) we have the following results aboutinstability of the positive equilibrium of system

Theorem 4 0 lt 119903 lt 2 0 lt 119886 lt 1 and Proposition 3 mean orshow that the problem (30) and (31) is diffusion-driven unstableor Turing unstable

32 Two-Dimensional Case In this subsection we will payour attention to the Turing instability analysis for the follow-ing two-dimensional system

119906119905+1

119894119895= 119906119905

119894119895(119903 + 1 minus 119906

119905

119894119895minus 119886V119905119894119895) + 1198891nabla2119906119905

119894119895

V119905+1119894119895

= V119905119894119895exp (119903 minus 119886119906119905

119894119895minus V119905119894119895) + 1198892nabla2V119905119894119895

(47)

with the periodic boundary conditions

119906119905

1198940= 119906119905

119894119898 119906

119905

1198941= 119906119905

119894119898+1

119906119905

0119895= 119906119905

119898119895 119906

119905

1119895= 119906119905

119898+1119895

V1199051198940= V119905119894119898 V119905

1198941= V119905119894119898+1

V1199050119895= V119905119898119895 V119905

1119895= V119905119898+1119895

(48)

for 119894 119895 isin 1 2 119898 = [1119898] and 119905 isin 119885+ where 119898 is a

positive integer

nabla2119906119905

119894119895= 119906119905

119894+1119895+ 119906119905

119894119895+1+ 119906119905

119894minus1119895+ 119906119905

119894119895minus1minus 4119906119905

119894119895

nabla2V119905119894119895= V119905119894+1119895

+ V119905119894119895+1

+ V119905119894minus1119895

+ V119905119894119895minus1

minus 4V119905119894119895

(49)

The following theoremwill show that the system (47) alsoundergoes Turing instability Since the analysis is very similarto the one-dimensional case the proof is omitted

Theorem 5 If there exist positive numbers 1198891 1198892and the

eigenvalue 1198962119897119904of the corresponding characteristic equation such

that one of the conditions

ℎ (1198962

119897119904) lt (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (1198962

119897119904) lt minus (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(50)

4

r

u

35

35

3

3

25

25

2

15

15

1

21

05

05

0

Figure 1 Bifurcation diagram for 119903 minus 119906119905with 119886 = 0511

or

ℎ (1198962

119897119904) gt 1 (51)

holds and 0 lt 119903 lt 2 0 lt 119886 lt 1 then the problem (47)-(48) atthe fixed point (119906lowast Vlowast) is diffusion-driven unstable or Turingunstable where

ℎ (1198962

119897119904) = 119889111988921198964

119897119904minus (1198891119892Vlowast + 1198892119891119906lowast) 119896

2

119897119904+ (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

1198962

119897119904= 120582119897119904= 4(sin2 ((119897 minus 1) 120587

119898

) + sin2 ((119904 minus 1) 120587119898

))

119891119900119903 119897 119904 isin [1119898]

(52)

4 Numerical Simulation

As is known to all the bifurcation diagram provides a generalview of the evolution process of the dynamical behaviorsby plotting a state variable with the abscissa being oneparameter As a parameter varies the dynamics of the systemwe concerned change through a local or global bifurcationwhich leads to the change of stability at the same time

Now 119903 is considered as a parameter with the range 05ndash35 for the system (4) Since the bifurcation diagrams of 119903minus119906

119905

are similar to the bifurcation diagrams of 119903 minus V119905 we will only

show the former which can be seen from Figure 1Next we performed a series of simulations for the

reaction-diffusion systems and in each the initial condi-tion was always a small amplitude random perturbation1 around the steady state As a numerical example weconsider the bifurcation of the two-dimensional system (47)-(48) It is well known that Turing instability (bifurcation) isdiffusion-driven instability thus the diffusion rate is vital tothe pattern formation To investigate the effect of diffusioncoefficients on patterns by keeping all the other parametersof the system fixed (119886 = 055 119903 = 071 and 119889

1= 022)

we change a diffusion coefficient in the Turing instability

6 Discrete Dynamics in Nature and Society

20

20

40

40

60

60

80

80

100

100

120

120

(a)

20 40 60 80 100 120

20

40

60

80

100

120

(b)

20 40 60 80 100 120

20

40

60

80

100

120

(c)

20 40 60 80 100 120

20

40

60

80

100

120

(d)

20 40 60 80 100 120

20

40

60

80

100

120

(e)

20 40 60 80 100 120

20

40

60

80

100

120

(f)

Figure 2 Pattern selection with the increase of 1198892in the Turing instability region when 119886 = 055 119903 = 071 and 119889

1= 022 (a) 119889

2= 0195

(b) 1198892= 020 (c) 119889

2= 0205 (d) 119889

2= 021 (e) 119889

2= 0215 (f) 119889

2= 0217

region (parameter space which satisfies Turing instability)Figures 2(a)ndash2(f) exhibit in detail the different distributionof patterns with varying values of 119889 If we let 119889

2= 019

a stable pattern of square shapes namely stationary wave

is observed in Figure 2(a) With the increase of 1198892 some

tips appear which attempt to form self-centered spiral waves(Figure 2(b))Then some tips vanish and some tend to evolveto regular spiral waves shown in Figure 2(c) Then with the

Discrete Dynamics in Nature and Society 7

parameter evolution proceeding the size of spiral waves risesbut the density of them decreases (see Figure 2(d)) If 119889

2is

further increased we observe that in parts of patterns spiralwaves begin to break up Hardly can spiral waves be seendisorder and chaotic structure are depicted in Figures 2(e)and 2(f)

5 Discussion and Conclusion

In this paper we have applied different discrete schemes toconvert the continuous Lotka-Volterra competition modelto a new discrete model and studied the dynamical charac-teristic of the discrete model Our theoretical analysis andnumerical simulations have demonstrated that the discretecompetition model undergoes flip bifurcation Furthermorewhen the effects of spatial factors are considered we discussthe Turing instability conditions combining linearizationmethod and inner product technique The impact of the dif-fusion coefficients on patterns can also be observed visuallyand some interesting situations can be observed Indeed thenew discrete model can result in a rich set of patterns and weexpect that it is more effective in practice

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Dr Y D Jiao for the valuable suggestionsand thank the referees for their helpful comments This workwas financially supported by TianjinUniversity of Commercewith the Grant no X0803 the National Natural ScienceFoundation of China under Grant nos 11001072 11371277and Cultivation Program for Excellent Youth Teacher inUniversity Tianjin (507-125RCPY0314)

References

[1] Y Takeuchi Global Dynamical Properties of Lotka-VolterraSystems World Scientific Publishing Singapore 1996

[2] PWaltman CompetitionModels in Populationbiology vol 45 ofCBMS-NSF Regional Conference Series in Applied MathematicsSIAM Philadelphia Pa USA 1985

[3] M E FisherAnalysis of difference equationmodels in populationdynamics [PhD thesis] University of Western Australia 1982

[4] P Liu and X Cui ldquoA discrete model of competitionrdquoMathemat-ics and Computers in Simulation vol 49 no 1-2 pp 1ndash12 1999

[5] F B Hildebrand Finite-Difference Equations and SimulationsPrentice Hall Englewood Cliffs NJ USA 1968

[6] J K Hale and A S Somolinos ldquoCompetition for fluctuatingnutrientrdquo Journal ofMathematical Biology vol 18 no 3 pp 255ndash280 1983

[7] K P Hadeler and I Gerstmann ldquoThe discrete RosenzweigmodelrdquoMathematical Biosciences vol 98 no 1 pp 49ndash72 1990

[8] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics vol 191 Academic Press New York NYUSA 1993

[9] S N Elaydi An Introduction to Difference Equations SpringerBerlin Germany 1996

[10] J Maynard Smith Models in Ecology Cambridge UniversityPress 1974

[11] P Z Liu and S N Elaydi ldquoDiscrete competitive and coopera-tive models of Lotka-Volterra Typerdquo Journal of ComputationalAnalysis and Applications vol 3 pp 53ndash73 2001

[12] Y S Wang and H Wu ldquoDynamics of competitive Lotka-Volte-rra systems that can be projected to a linerdquo Computers ampMathematics with Applications vol 47 pp 1263ndash1271 2004

[13] X Xiong and Z Zhang ldquoPeriodic solutions of a discrete two-species competitive model with stage structurerdquo Mathematicaland Computer Modelling vol 48 no 3-4 pp 333ndash343 2008

[14] C Niu and X Chen ldquoAlmost periodic sequence solutions ofa discrete Lotka-Volterra competitive system with feedbackcontrolrdquoNonlinear Analysis RealWorldApplications vol 10 no5 pp 3152ndash3161 2009

[15] J G Skellam ldquoRandom dispersal in theoretical populationsrdquoBiometrika vol 38 pp 196ndash218 1951

[16] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society B vol 237 pp 37ndash721953

[17] J Bascompte and R V Sole ldquoSpatially induced bifurcations insingle-species population dynamicsrdquo Journal of Animal Ecologyvol 63 pp 256ndash264 1994

[18] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[19] Y T Han B Han L Zhang L Xu M F Li and G ZhangldquoTuring instability and labyrinthine patterns for a symmetricdiscrete comptitive Lotka-Volterra systemrdquo WSEAS Transac-tions on Mathematics vol 10 pp 181ndash189 2011

[20] G Q Sun Z Jin Q X Liu and L Li ldquoDynamical complexityof a spatial predator-prey model with migrationrdquo EcologicalModelling vol 219 pp 248ndash255 2008

[21] G Q Sun Z Jin Q X Liu and L Li ldquoSpatial pattern of anepidemic model with cross-diffusionrdquo Chinese Physics B vol 17pp 3936ndash3941 2008

[22] Y-X Wang and W-T Li ldquoEffect of cross-diffusion on thestationary problem of a diffusive competition model with aprotection zonerdquo Nonlinear Analysis Real World Applicationsvol 14 no 1 pp 224ndash245 2013

[23] M Li B Han L Xu and G Zhang ldquoSpiral patterns nearTuring instability in a discrete reaction diffusion systemrdquoChaosSolitons amp Fractals vol 49 pp 1ndash6 2013

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

4 Discrete Dynamics in Nature and Society

By calculating we can get that

1198861=

(119886 + 1)2

2 (1 minus 119886)

1198862= 0 119886

3= 0 (27)

and the system (19) is restricted to the center manifold whichis given by

119891 119909119899+1

= minus 119909119899+

1198863+ 41198862+ 3119886

119886 minus 1

1199092

119899+ (119886 + 1) 119909

119899120575119899

+

1

4

(119886 + 1)31199094

119899+ 119900 (

1003816100381610038161003816119909119899

1003816100381610038161003816

4

)

(28)

Since

(

120597119891

120597120575

1205972119891

1205971199092+ 2

1205972119891

120597119909120597120575

)

100381610038161003816100381610038161003816100381610038161003816(00)

= 2 (119886 + 1) = 0

(

1

2

(

1205972119891

1205971199092)

2

+

1

3

1205973119891

1205971199093)

1003816100381610038161003816100381610038161003816100381610038161003816(00)

= 2(

1198863+ 41198862+ 3119886

119886 minus 1

)

2

gt 0

(29)

system (4) undergoes a flip bifurcation at 1198643 The proof is

completed

3 Turing Bifurcation

In this section we discuss the Turing bifurcation Turingrsquostheory shows that diffusion could destabilize an otherwisestable equilibrium of the reaction-diffusion system and leadto nonuniform spatial patterns This kind of instability isusually calledTuring instability or diffusion-driven instability[16]

31 One-Dimensional Case We consider the following diffu-sion system

119906119905+1

119894= 119906119905

119894(119903 + 1 minus 119906

119905

119894minus 119886V119905119894) + 1198891nabla2119906119905

119894

V119905+1119894

= V119905119894exp (119903 minus 119886119906119905

119894minus V119905119894) + 1198892nabla2V119905119894

(30)

with the periodic boundary conditions

119906119905

0= 119906119905

119898 119906

119905

1= 119906119905

119898+1

V1199050= V119905119898 V119905

1= V119905119898+1

(31)

for 119894 isin 1 2 119898 = [1119898] and 119905 isin 119885+ where119898 is a positiveinteger

nabla2119906119905

119894= 119906119905

119894+1minus 2119906119905

119894+ 119906119905

119894minus1

nabla2V119905119894= V119905119894+1

minus 2V119905119894+ V119905119894minus1

(32)

In order to study Turing instability of (30) and (31) wefirstly consider eigenvalues of the following equation

nabla2119883119894+ 120582119883119894= 0 (33)

with the periodic boundary conditions

1198830= 119883119898 119883

1= 119883119898+1

(34)

By calculating the eigenvalue problem (33)-(34) has theeigenvalues

120582119904= 4sin2 (119904 minus 1) 120587

119898

for 119904 isin [1119898] (35)

We linearise at the steady state to get

119908119905+1

119894= 119869119908119905

119894+ 119863nabla2119908119905

119894 119863 = (

11988910

0 1198892

) (36)

with the periodic boundary conditions

119908119905

0= 119908119905

119898 119908

119905

1= 119908119905

119898+1 (37)

where

119908119905

119894= (

119906119905

119894minus 119906lowast

V119905119894minus Vlowast

) = (

119909119905

119894

119910119905

119894

) (38)

Then respectively taking the inner product of (36) withthe corresponding eigenfunction 119883119894

119904of the eigenvalue 120582

119904 we

see that119898

sum

119894=1

119883119894

119904119909119905+1

119894= 119891119906

119898

sum

119894=1

119883119894

119904119909119905

119894+ 119891V

119898

sum

119894=1

119883119894

119904119910119905

119894+ 1198891

119898

sum

119894=1

119883119894

119904nabla2119909119905

119894

119898

sum

119894=1

119883119894

119904119910119905+1

119894= 119892119906

119898

sum

119894=1

119883119894

119904119909119905

119894+ 119892V

119898

sum

119894=1

119883119894

119904119910119905

119894+ 1198892

119898

sum

119894=1

119883119894

119904nabla2119910119905

119894

(39)

Let 119880119905 = sum119898119894=1119883119894

119904119909119905

119894and 119881119905 = sum119898

119894=1119883119894

119904119910119905

119894and use the periodic

boundary conditions (34) and (37) then we have

119880119905+1

= 119891119906119880119905+ 119891V119881

119905minus 1198891120582119904119880119905

119881119905+1

= 119892119906119880119905+ 119892V119881

119905minus 1198892120582119904119881119905

(40)

or

119880119905+1

= (119891119906minus 1198891120582119904) 119880119905+ 119891V119881

119905

119881119905+1

= 119892119906119880119905+ (119892V minus 1198892120582119904) 119881

119905

(41)

Thus the following fact can be obtained

Proposition 2 If (119906119905119894 V119905119894) is a solution of the problem of (30)

and (31) then

(119880119905=

119898

sum

119894=1

119883119894

119904119909119905

119894 119881119905=

119898

sum

119894=1

119883119894

119904119910119905

119894) (42)

is a solution of (41) where 120582119904is some eigenvalue of (33)-(34)

and119883119894119904is the corresponding eigenfunction For some eigenvalue

120582119904of (33)-(34) if (119880119905 119881119905) is a solution of the system (41) then

(119906119905

119894= 119880119905119883119894

119904 V119905119894= 119881119905119883119894

119904) (43)

is a solution of (30)with the periodic boundary conditions (31)

Discrete Dynamics in Nature and Society 5

Proposition 3 If there exist positive numbers 1198891 1198892and the

eigenvalue 120582119904of the problem (33)-(34) such that one of the

conditions

ℎ (120582119904) lt (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (120582119904) lt minus (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(44)

or

ℎ (120582119904) gt 1 (45)

holds then the problem (30) and (31) at the fixed point (119906lowast Vlowast)is unstable where

ℎ (120582119904) = 119889111988921205822

119904minus (1198891119892Vlowast + 1198892119891119906lowast) 120582119904 + (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

(46)

For the system (5) we have the following results aboutinstability of the positive equilibrium of system

Theorem 4 0 lt 119903 lt 2 0 lt 119886 lt 1 and Proposition 3 mean orshow that the problem (30) and (31) is diffusion-driven unstableor Turing unstable

32 Two-Dimensional Case In this subsection we will payour attention to the Turing instability analysis for the follow-ing two-dimensional system

119906119905+1

119894119895= 119906119905

119894119895(119903 + 1 minus 119906

119905

119894119895minus 119886V119905119894119895) + 1198891nabla2119906119905

119894119895

V119905+1119894119895

= V119905119894119895exp (119903 minus 119886119906119905

119894119895minus V119905119894119895) + 1198892nabla2V119905119894119895

(47)

with the periodic boundary conditions

119906119905

1198940= 119906119905

119894119898 119906

119905

1198941= 119906119905

119894119898+1

119906119905

0119895= 119906119905

119898119895 119906

119905

1119895= 119906119905

119898+1119895

V1199051198940= V119905119894119898 V119905

1198941= V119905119894119898+1

V1199050119895= V119905119898119895 V119905

1119895= V119905119898+1119895

(48)

for 119894 119895 isin 1 2 119898 = [1119898] and 119905 isin 119885+ where 119898 is a

positive integer

nabla2119906119905

119894119895= 119906119905

119894+1119895+ 119906119905

119894119895+1+ 119906119905

119894minus1119895+ 119906119905

119894119895minus1minus 4119906119905

119894119895

nabla2V119905119894119895= V119905119894+1119895

+ V119905119894119895+1

+ V119905119894minus1119895

+ V119905119894119895minus1

minus 4V119905119894119895

(49)

The following theoremwill show that the system (47) alsoundergoes Turing instability Since the analysis is very similarto the one-dimensional case the proof is omitted

Theorem 5 If there exist positive numbers 1198891 1198892and the

eigenvalue 1198962119897119904of the corresponding characteristic equation such

that one of the conditions

ℎ (1198962

119897119904) lt (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (1198962

119897119904) lt minus (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(50)

4

r

u

35

35

3

3

25

25

2

15

15

1

21

05

05

0

Figure 1 Bifurcation diagram for 119903 minus 119906119905with 119886 = 0511

or

ℎ (1198962

119897119904) gt 1 (51)

holds and 0 lt 119903 lt 2 0 lt 119886 lt 1 then the problem (47)-(48) atthe fixed point (119906lowast Vlowast) is diffusion-driven unstable or Turingunstable where

ℎ (1198962

119897119904) = 119889111988921198964

119897119904minus (1198891119892Vlowast + 1198892119891119906lowast) 119896

2

119897119904+ (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

1198962

119897119904= 120582119897119904= 4(sin2 ((119897 minus 1) 120587

119898

) + sin2 ((119904 minus 1) 120587119898

))

119891119900119903 119897 119904 isin [1119898]

(52)

4 Numerical Simulation

As is known to all the bifurcation diagram provides a generalview of the evolution process of the dynamical behaviorsby plotting a state variable with the abscissa being oneparameter As a parameter varies the dynamics of the systemwe concerned change through a local or global bifurcationwhich leads to the change of stability at the same time

Now 119903 is considered as a parameter with the range 05ndash35 for the system (4) Since the bifurcation diagrams of 119903minus119906

119905

are similar to the bifurcation diagrams of 119903 minus V119905 we will only

show the former which can be seen from Figure 1Next we performed a series of simulations for the

reaction-diffusion systems and in each the initial condi-tion was always a small amplitude random perturbation1 around the steady state As a numerical example weconsider the bifurcation of the two-dimensional system (47)-(48) It is well known that Turing instability (bifurcation) isdiffusion-driven instability thus the diffusion rate is vital tothe pattern formation To investigate the effect of diffusioncoefficients on patterns by keeping all the other parametersof the system fixed (119886 = 055 119903 = 071 and 119889

1= 022)

we change a diffusion coefficient in the Turing instability

6 Discrete Dynamics in Nature and Society

20

20

40

40

60

60

80

80

100

100

120

120

(a)

20 40 60 80 100 120

20

40

60

80

100

120

(b)

20 40 60 80 100 120

20

40

60

80

100

120

(c)

20 40 60 80 100 120

20

40

60

80

100

120

(d)

20 40 60 80 100 120

20

40

60

80

100

120

(e)

20 40 60 80 100 120

20

40

60

80

100

120

(f)

Figure 2 Pattern selection with the increase of 1198892in the Turing instability region when 119886 = 055 119903 = 071 and 119889

1= 022 (a) 119889

2= 0195

(b) 1198892= 020 (c) 119889

2= 0205 (d) 119889

2= 021 (e) 119889

2= 0215 (f) 119889

2= 0217

region (parameter space which satisfies Turing instability)Figures 2(a)ndash2(f) exhibit in detail the different distributionof patterns with varying values of 119889 If we let 119889

2= 019

a stable pattern of square shapes namely stationary wave

is observed in Figure 2(a) With the increase of 1198892 some

tips appear which attempt to form self-centered spiral waves(Figure 2(b))Then some tips vanish and some tend to evolveto regular spiral waves shown in Figure 2(c) Then with the

Discrete Dynamics in Nature and Society 7

parameter evolution proceeding the size of spiral waves risesbut the density of them decreases (see Figure 2(d)) If 119889

2is

further increased we observe that in parts of patterns spiralwaves begin to break up Hardly can spiral waves be seendisorder and chaotic structure are depicted in Figures 2(e)and 2(f)

5 Discussion and Conclusion

In this paper we have applied different discrete schemes toconvert the continuous Lotka-Volterra competition modelto a new discrete model and studied the dynamical charac-teristic of the discrete model Our theoretical analysis andnumerical simulations have demonstrated that the discretecompetition model undergoes flip bifurcation Furthermorewhen the effects of spatial factors are considered we discussthe Turing instability conditions combining linearizationmethod and inner product technique The impact of the dif-fusion coefficients on patterns can also be observed visuallyand some interesting situations can be observed Indeed thenew discrete model can result in a rich set of patterns and weexpect that it is more effective in practice

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Dr Y D Jiao for the valuable suggestionsand thank the referees for their helpful comments This workwas financially supported by TianjinUniversity of Commercewith the Grant no X0803 the National Natural ScienceFoundation of China under Grant nos 11001072 11371277and Cultivation Program for Excellent Youth Teacher inUniversity Tianjin (507-125RCPY0314)

References

[1] Y Takeuchi Global Dynamical Properties of Lotka-VolterraSystems World Scientific Publishing Singapore 1996

[2] PWaltman CompetitionModels in Populationbiology vol 45 ofCBMS-NSF Regional Conference Series in Applied MathematicsSIAM Philadelphia Pa USA 1985

[3] M E FisherAnalysis of difference equationmodels in populationdynamics [PhD thesis] University of Western Australia 1982

[4] P Liu and X Cui ldquoA discrete model of competitionrdquoMathemat-ics and Computers in Simulation vol 49 no 1-2 pp 1ndash12 1999

[5] F B Hildebrand Finite-Difference Equations and SimulationsPrentice Hall Englewood Cliffs NJ USA 1968

[6] J K Hale and A S Somolinos ldquoCompetition for fluctuatingnutrientrdquo Journal ofMathematical Biology vol 18 no 3 pp 255ndash280 1983

[7] K P Hadeler and I Gerstmann ldquoThe discrete RosenzweigmodelrdquoMathematical Biosciences vol 98 no 1 pp 49ndash72 1990

[8] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics vol 191 Academic Press New York NYUSA 1993

[9] S N Elaydi An Introduction to Difference Equations SpringerBerlin Germany 1996

[10] J Maynard Smith Models in Ecology Cambridge UniversityPress 1974

[11] P Z Liu and S N Elaydi ldquoDiscrete competitive and coopera-tive models of Lotka-Volterra Typerdquo Journal of ComputationalAnalysis and Applications vol 3 pp 53ndash73 2001

[12] Y S Wang and H Wu ldquoDynamics of competitive Lotka-Volte-rra systems that can be projected to a linerdquo Computers ampMathematics with Applications vol 47 pp 1263ndash1271 2004

[13] X Xiong and Z Zhang ldquoPeriodic solutions of a discrete two-species competitive model with stage structurerdquo Mathematicaland Computer Modelling vol 48 no 3-4 pp 333ndash343 2008

[14] C Niu and X Chen ldquoAlmost periodic sequence solutions ofa discrete Lotka-Volterra competitive system with feedbackcontrolrdquoNonlinear Analysis RealWorldApplications vol 10 no5 pp 3152ndash3161 2009

[15] J G Skellam ldquoRandom dispersal in theoretical populationsrdquoBiometrika vol 38 pp 196ndash218 1951

[16] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society B vol 237 pp 37ndash721953

[17] J Bascompte and R V Sole ldquoSpatially induced bifurcations insingle-species population dynamicsrdquo Journal of Animal Ecologyvol 63 pp 256ndash264 1994

[18] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[19] Y T Han B Han L Zhang L Xu M F Li and G ZhangldquoTuring instability and labyrinthine patterns for a symmetricdiscrete comptitive Lotka-Volterra systemrdquo WSEAS Transac-tions on Mathematics vol 10 pp 181ndash189 2011

[20] G Q Sun Z Jin Q X Liu and L Li ldquoDynamical complexityof a spatial predator-prey model with migrationrdquo EcologicalModelling vol 219 pp 248ndash255 2008

[21] G Q Sun Z Jin Q X Liu and L Li ldquoSpatial pattern of anepidemic model with cross-diffusionrdquo Chinese Physics B vol 17pp 3936ndash3941 2008

[22] Y-X Wang and W-T Li ldquoEffect of cross-diffusion on thestationary problem of a diffusive competition model with aprotection zonerdquo Nonlinear Analysis Real World Applicationsvol 14 no 1 pp 224ndash245 2013

[23] M Li B Han L Xu and G Zhang ldquoSpiral patterns nearTuring instability in a discrete reaction diffusion systemrdquoChaosSolitons amp Fractals vol 49 pp 1ndash6 2013

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

Discrete Dynamics in Nature and Society 5

Proposition 3 If there exist positive numbers 1198891 1198892and the

eigenvalue 120582119904of the problem (33)-(34) such that one of the

conditions

ℎ (120582119904) lt (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (120582119904) lt minus (120582

119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(44)

or

ℎ (120582119904) gt 1 (45)

holds then the problem (30) and (31) at the fixed point (119906lowast Vlowast)is unstable where

ℎ (120582119904) = 119889111988921205822

119904minus (1198891119892Vlowast + 1198892119891119906lowast) 120582119904 + (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

(46)

For the system (5) we have the following results aboutinstability of the positive equilibrium of system

Theorem 4 0 lt 119903 lt 2 0 lt 119886 lt 1 and Proposition 3 mean orshow that the problem (30) and (31) is diffusion-driven unstableor Turing unstable

32 Two-Dimensional Case In this subsection we will payour attention to the Turing instability analysis for the follow-ing two-dimensional system

119906119905+1

119894119895= 119906119905

119894119895(119903 + 1 minus 119906

119905

119894119895minus 119886V119905119894119895) + 1198891nabla2119906119905

119894119895

V119905+1119894119895

= V119905119894119895exp (119903 minus 119886119906119905

119894119895minus V119905119894119895) + 1198892nabla2V119905119894119895

(47)

with the periodic boundary conditions

119906119905

1198940= 119906119905

119894119898 119906

119905

1198941= 119906119905

119894119898+1

119906119905

0119895= 119906119905

119898119895 119906

119905

1119895= 119906119905

119898+1119895

V1199051198940= V119905119894119898 V119905

1198941= V119905119894119898+1

V1199050119895= V119905119898119895 V119905

1119895= V119905119898+1119895

(48)

for 119894 119895 isin 1 2 119898 = [1119898] and 119905 isin 119885+ where 119898 is a

positive integer

nabla2119906119905

119894119895= 119906119905

119894+1119895+ 119906119905

119894119895+1+ 119906119905

119894minus1119895+ 119906119905

119894119895minus1minus 4119906119905

119894119895

nabla2V119905119894119895= V119905119894+1119895

+ V119905119894119895+1

+ V119905119894minus1119895

+ V119905119894119895minus1

minus 4V119905119894119895

(49)

The following theoremwill show that the system (47) alsoundergoes Turing instability Since the analysis is very similarto the one-dimensional case the proof is omitted

Theorem 5 If there exist positive numbers 1198891 1198892and the

eigenvalue 1198962119897119904of the corresponding characteristic equation such

that one of the conditions

ℎ (1198962

119897119904) lt (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

ℎ (1198962

119897119904) lt minus (119896

2

119897119904(1198891+ 1198892) minus (119891

119906lowast + 119892Vlowast)) minus 1

(50)

4

r

u

35

35

3

3

25

25

2

15

15

1

21

05

05

0

Figure 1 Bifurcation diagram for 119903 minus 119906119905with 119886 = 0511

or

ℎ (1198962

119897119904) gt 1 (51)

holds and 0 lt 119903 lt 2 0 lt 119886 lt 1 then the problem (47)-(48) atthe fixed point (119906lowast Vlowast) is diffusion-driven unstable or Turingunstable where

ℎ (1198962

119897119904) = 119889111988921198964

119897119904minus (1198891119892Vlowast + 1198892119891119906lowast) 119896

2

119897119904+ (119891119906lowast119892Vlowast minus 119891Vlowast119892119906lowast)

1198962

119897119904= 120582119897119904= 4(sin2 ((119897 minus 1) 120587

119898

) + sin2 ((119904 minus 1) 120587119898

))

119891119900119903 119897 119904 isin [1119898]

(52)

4 Numerical Simulation

As is known to all the bifurcation diagram provides a generalview of the evolution process of the dynamical behaviorsby plotting a state variable with the abscissa being oneparameter As a parameter varies the dynamics of the systemwe concerned change through a local or global bifurcationwhich leads to the change of stability at the same time

Now 119903 is considered as a parameter with the range 05ndash35 for the system (4) Since the bifurcation diagrams of 119903minus119906

119905

are similar to the bifurcation diagrams of 119903 minus V119905 we will only

show the former which can be seen from Figure 1Next we performed a series of simulations for the

reaction-diffusion systems and in each the initial condi-tion was always a small amplitude random perturbation1 around the steady state As a numerical example weconsider the bifurcation of the two-dimensional system (47)-(48) It is well known that Turing instability (bifurcation) isdiffusion-driven instability thus the diffusion rate is vital tothe pattern formation To investigate the effect of diffusioncoefficients on patterns by keeping all the other parametersof the system fixed (119886 = 055 119903 = 071 and 119889

1= 022)

we change a diffusion coefficient in the Turing instability

6 Discrete Dynamics in Nature and Society

20

20

40

40

60

60

80

80

100

100

120

120

(a)

20 40 60 80 100 120

20

40

60

80

100

120

(b)

20 40 60 80 100 120

20

40

60

80

100

120

(c)

20 40 60 80 100 120

20

40

60

80

100

120

(d)

20 40 60 80 100 120

20

40

60

80

100

120

(e)

20 40 60 80 100 120

20

40

60

80

100

120

(f)

Figure 2 Pattern selection with the increase of 1198892in the Turing instability region when 119886 = 055 119903 = 071 and 119889

1= 022 (a) 119889

2= 0195

(b) 1198892= 020 (c) 119889

2= 0205 (d) 119889

2= 021 (e) 119889

2= 0215 (f) 119889

2= 0217

region (parameter space which satisfies Turing instability)Figures 2(a)ndash2(f) exhibit in detail the different distributionof patterns with varying values of 119889 If we let 119889

2= 019

a stable pattern of square shapes namely stationary wave

is observed in Figure 2(a) With the increase of 1198892 some

tips appear which attempt to form self-centered spiral waves(Figure 2(b))Then some tips vanish and some tend to evolveto regular spiral waves shown in Figure 2(c) Then with the

Discrete Dynamics in Nature and Society 7

parameter evolution proceeding the size of spiral waves risesbut the density of them decreases (see Figure 2(d)) If 119889

2is

further increased we observe that in parts of patterns spiralwaves begin to break up Hardly can spiral waves be seendisorder and chaotic structure are depicted in Figures 2(e)and 2(f)

5 Discussion and Conclusion

In this paper we have applied different discrete schemes toconvert the continuous Lotka-Volterra competition modelto a new discrete model and studied the dynamical charac-teristic of the discrete model Our theoretical analysis andnumerical simulations have demonstrated that the discretecompetition model undergoes flip bifurcation Furthermorewhen the effects of spatial factors are considered we discussthe Turing instability conditions combining linearizationmethod and inner product technique The impact of the dif-fusion coefficients on patterns can also be observed visuallyand some interesting situations can be observed Indeed thenew discrete model can result in a rich set of patterns and weexpect that it is more effective in practice

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Dr Y D Jiao for the valuable suggestionsand thank the referees for their helpful comments This workwas financially supported by TianjinUniversity of Commercewith the Grant no X0803 the National Natural ScienceFoundation of China under Grant nos 11001072 11371277and Cultivation Program for Excellent Youth Teacher inUniversity Tianjin (507-125RCPY0314)

References

[1] Y Takeuchi Global Dynamical Properties of Lotka-VolterraSystems World Scientific Publishing Singapore 1996

[2] PWaltman CompetitionModels in Populationbiology vol 45 ofCBMS-NSF Regional Conference Series in Applied MathematicsSIAM Philadelphia Pa USA 1985

[3] M E FisherAnalysis of difference equationmodels in populationdynamics [PhD thesis] University of Western Australia 1982

[4] P Liu and X Cui ldquoA discrete model of competitionrdquoMathemat-ics and Computers in Simulation vol 49 no 1-2 pp 1ndash12 1999

[5] F B Hildebrand Finite-Difference Equations and SimulationsPrentice Hall Englewood Cliffs NJ USA 1968

[6] J K Hale and A S Somolinos ldquoCompetition for fluctuatingnutrientrdquo Journal ofMathematical Biology vol 18 no 3 pp 255ndash280 1983

[7] K P Hadeler and I Gerstmann ldquoThe discrete RosenzweigmodelrdquoMathematical Biosciences vol 98 no 1 pp 49ndash72 1990

[8] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics vol 191 Academic Press New York NYUSA 1993

[9] S N Elaydi An Introduction to Difference Equations SpringerBerlin Germany 1996

[10] J Maynard Smith Models in Ecology Cambridge UniversityPress 1974

[11] P Z Liu and S N Elaydi ldquoDiscrete competitive and coopera-tive models of Lotka-Volterra Typerdquo Journal of ComputationalAnalysis and Applications vol 3 pp 53ndash73 2001

[12] Y S Wang and H Wu ldquoDynamics of competitive Lotka-Volte-rra systems that can be projected to a linerdquo Computers ampMathematics with Applications vol 47 pp 1263ndash1271 2004

[13] X Xiong and Z Zhang ldquoPeriodic solutions of a discrete two-species competitive model with stage structurerdquo Mathematicaland Computer Modelling vol 48 no 3-4 pp 333ndash343 2008

[14] C Niu and X Chen ldquoAlmost periodic sequence solutions ofa discrete Lotka-Volterra competitive system with feedbackcontrolrdquoNonlinear Analysis RealWorldApplications vol 10 no5 pp 3152ndash3161 2009

[15] J G Skellam ldquoRandom dispersal in theoretical populationsrdquoBiometrika vol 38 pp 196ndash218 1951

[16] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society B vol 237 pp 37ndash721953

[17] J Bascompte and R V Sole ldquoSpatially induced bifurcations insingle-species population dynamicsrdquo Journal of Animal Ecologyvol 63 pp 256ndash264 1994

[18] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[19] Y T Han B Han L Zhang L Xu M F Li and G ZhangldquoTuring instability and labyrinthine patterns for a symmetricdiscrete comptitive Lotka-Volterra systemrdquo WSEAS Transac-tions on Mathematics vol 10 pp 181ndash189 2011

[20] G Q Sun Z Jin Q X Liu and L Li ldquoDynamical complexityof a spatial predator-prey model with migrationrdquo EcologicalModelling vol 219 pp 248ndash255 2008

[21] G Q Sun Z Jin Q X Liu and L Li ldquoSpatial pattern of anepidemic model with cross-diffusionrdquo Chinese Physics B vol 17pp 3936ndash3941 2008

[22] Y-X Wang and W-T Li ldquoEffect of cross-diffusion on thestationary problem of a diffusive competition model with aprotection zonerdquo Nonlinear Analysis Real World Applicationsvol 14 no 1 pp 224ndash245 2013

[23] M Li B Han L Xu and G Zhang ldquoSpiral patterns nearTuring instability in a discrete reaction diffusion systemrdquoChaosSolitons amp Fractals vol 49 pp 1ndash6 2013

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

6 Discrete Dynamics in Nature and Society

20

20

40

40

60

60

80

80

100

100

120

120

(a)

20 40 60 80 100 120

20

40

60

80

100

120

(b)

20 40 60 80 100 120

20

40

60

80

100

120

(c)

20 40 60 80 100 120

20

40

60

80

100

120

(d)

20 40 60 80 100 120

20

40

60

80

100

120

(e)

20 40 60 80 100 120

20

40

60

80

100

120

(f)

Figure 2 Pattern selection with the increase of 1198892in the Turing instability region when 119886 = 055 119903 = 071 and 119889

1= 022 (a) 119889

2= 0195

(b) 1198892= 020 (c) 119889

2= 0205 (d) 119889

2= 021 (e) 119889

2= 0215 (f) 119889

2= 0217

region (parameter space which satisfies Turing instability)Figures 2(a)ndash2(f) exhibit in detail the different distributionof patterns with varying values of 119889 If we let 119889

2= 019

a stable pattern of square shapes namely stationary wave

is observed in Figure 2(a) With the increase of 1198892 some

tips appear which attempt to form self-centered spiral waves(Figure 2(b))Then some tips vanish and some tend to evolveto regular spiral waves shown in Figure 2(c) Then with the

Discrete Dynamics in Nature and Society 7

parameter evolution proceeding the size of spiral waves risesbut the density of them decreases (see Figure 2(d)) If 119889

2is

further increased we observe that in parts of patterns spiralwaves begin to break up Hardly can spiral waves be seendisorder and chaotic structure are depicted in Figures 2(e)and 2(f)

5 Discussion and Conclusion

In this paper we have applied different discrete schemes toconvert the continuous Lotka-Volterra competition modelto a new discrete model and studied the dynamical charac-teristic of the discrete model Our theoretical analysis andnumerical simulations have demonstrated that the discretecompetition model undergoes flip bifurcation Furthermorewhen the effects of spatial factors are considered we discussthe Turing instability conditions combining linearizationmethod and inner product technique The impact of the dif-fusion coefficients on patterns can also be observed visuallyand some interesting situations can be observed Indeed thenew discrete model can result in a rich set of patterns and weexpect that it is more effective in practice

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Dr Y D Jiao for the valuable suggestionsand thank the referees for their helpful comments This workwas financially supported by TianjinUniversity of Commercewith the Grant no X0803 the National Natural ScienceFoundation of China under Grant nos 11001072 11371277and Cultivation Program for Excellent Youth Teacher inUniversity Tianjin (507-125RCPY0314)

References

[1] Y Takeuchi Global Dynamical Properties of Lotka-VolterraSystems World Scientific Publishing Singapore 1996

[2] PWaltman CompetitionModels in Populationbiology vol 45 ofCBMS-NSF Regional Conference Series in Applied MathematicsSIAM Philadelphia Pa USA 1985

[3] M E FisherAnalysis of difference equationmodels in populationdynamics [PhD thesis] University of Western Australia 1982

[4] P Liu and X Cui ldquoA discrete model of competitionrdquoMathemat-ics and Computers in Simulation vol 49 no 1-2 pp 1ndash12 1999

[5] F B Hildebrand Finite-Difference Equations and SimulationsPrentice Hall Englewood Cliffs NJ USA 1968

[6] J K Hale and A S Somolinos ldquoCompetition for fluctuatingnutrientrdquo Journal ofMathematical Biology vol 18 no 3 pp 255ndash280 1983

[7] K P Hadeler and I Gerstmann ldquoThe discrete RosenzweigmodelrdquoMathematical Biosciences vol 98 no 1 pp 49ndash72 1990

[8] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics vol 191 Academic Press New York NYUSA 1993

[9] S N Elaydi An Introduction to Difference Equations SpringerBerlin Germany 1996

[10] J Maynard Smith Models in Ecology Cambridge UniversityPress 1974

[11] P Z Liu and S N Elaydi ldquoDiscrete competitive and coopera-tive models of Lotka-Volterra Typerdquo Journal of ComputationalAnalysis and Applications vol 3 pp 53ndash73 2001

[12] Y S Wang and H Wu ldquoDynamics of competitive Lotka-Volte-rra systems that can be projected to a linerdquo Computers ampMathematics with Applications vol 47 pp 1263ndash1271 2004

[13] X Xiong and Z Zhang ldquoPeriodic solutions of a discrete two-species competitive model with stage structurerdquo Mathematicaland Computer Modelling vol 48 no 3-4 pp 333ndash343 2008

[14] C Niu and X Chen ldquoAlmost periodic sequence solutions ofa discrete Lotka-Volterra competitive system with feedbackcontrolrdquoNonlinear Analysis RealWorldApplications vol 10 no5 pp 3152ndash3161 2009

[15] J G Skellam ldquoRandom dispersal in theoretical populationsrdquoBiometrika vol 38 pp 196ndash218 1951

[16] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society B vol 237 pp 37ndash721953

[17] J Bascompte and R V Sole ldquoSpatially induced bifurcations insingle-species population dynamicsrdquo Journal of Animal Ecologyvol 63 pp 256ndash264 1994

[18] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[19] Y T Han B Han L Zhang L Xu M F Li and G ZhangldquoTuring instability and labyrinthine patterns for a symmetricdiscrete comptitive Lotka-Volterra systemrdquo WSEAS Transac-tions on Mathematics vol 10 pp 181ndash189 2011

[20] G Q Sun Z Jin Q X Liu and L Li ldquoDynamical complexityof a spatial predator-prey model with migrationrdquo EcologicalModelling vol 219 pp 248ndash255 2008

[21] G Q Sun Z Jin Q X Liu and L Li ldquoSpatial pattern of anepidemic model with cross-diffusionrdquo Chinese Physics B vol 17pp 3936ndash3941 2008

[22] Y-X Wang and W-T Li ldquoEffect of cross-diffusion on thestationary problem of a diffusive competition model with aprotection zonerdquo Nonlinear Analysis Real World Applicationsvol 14 no 1 pp 224ndash245 2013

[23] M Li B Han L Xu and G Zhang ldquoSpiral patterns nearTuring instability in a discrete reaction diffusion systemrdquoChaosSolitons amp Fractals vol 49 pp 1ndash6 2013

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

Discrete Dynamics in Nature and Society 7

parameter evolution proceeding the size of spiral waves risesbut the density of them decreases (see Figure 2(d)) If 119889

2is

further increased we observe that in parts of patterns spiralwaves begin to break up Hardly can spiral waves be seendisorder and chaotic structure are depicted in Figures 2(e)and 2(f)

5 Discussion and Conclusion

In this paper we have applied different discrete schemes toconvert the continuous Lotka-Volterra competition modelto a new discrete model and studied the dynamical charac-teristic of the discrete model Our theoretical analysis andnumerical simulations have demonstrated that the discretecompetition model undergoes flip bifurcation Furthermorewhen the effects of spatial factors are considered we discussthe Turing instability conditions combining linearizationmethod and inner product technique The impact of the dif-fusion coefficients on patterns can also be observed visuallyand some interesting situations can be observed Indeed thenew discrete model can result in a rich set of patterns and weexpect that it is more effective in practice

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors thank Dr Y D Jiao for the valuable suggestionsand thank the referees for their helpful comments This workwas financially supported by TianjinUniversity of Commercewith the Grant no X0803 the National Natural ScienceFoundation of China under Grant nos 11001072 11371277and Cultivation Program for Excellent Youth Teacher inUniversity Tianjin (507-125RCPY0314)

References

[1] Y Takeuchi Global Dynamical Properties of Lotka-VolterraSystems World Scientific Publishing Singapore 1996

[2] PWaltman CompetitionModels in Populationbiology vol 45 ofCBMS-NSF Regional Conference Series in Applied MathematicsSIAM Philadelphia Pa USA 1985

[3] M E FisherAnalysis of difference equationmodels in populationdynamics [PhD thesis] University of Western Australia 1982

[4] P Liu and X Cui ldquoA discrete model of competitionrdquoMathemat-ics and Computers in Simulation vol 49 no 1-2 pp 1ndash12 1999

[5] F B Hildebrand Finite-Difference Equations and SimulationsPrentice Hall Englewood Cliffs NJ USA 1968

[6] J K Hale and A S Somolinos ldquoCompetition for fluctuatingnutrientrdquo Journal ofMathematical Biology vol 18 no 3 pp 255ndash280 1983

[7] K P Hadeler and I Gerstmann ldquoThe discrete RosenzweigmodelrdquoMathematical Biosciences vol 98 no 1 pp 49ndash72 1990

[8] Y Kuang Delay Differential Equations with Applications inPopulation Dynamics vol 191 Academic Press New York NYUSA 1993

[9] S N Elaydi An Introduction to Difference Equations SpringerBerlin Germany 1996

[10] J Maynard Smith Models in Ecology Cambridge UniversityPress 1974

[11] P Z Liu and S N Elaydi ldquoDiscrete competitive and coopera-tive models of Lotka-Volterra Typerdquo Journal of ComputationalAnalysis and Applications vol 3 pp 53ndash73 2001

[12] Y S Wang and H Wu ldquoDynamics of competitive Lotka-Volte-rra systems that can be projected to a linerdquo Computers ampMathematics with Applications vol 47 pp 1263ndash1271 2004

[13] X Xiong and Z Zhang ldquoPeriodic solutions of a discrete two-species competitive model with stage structurerdquo Mathematicaland Computer Modelling vol 48 no 3-4 pp 333ndash343 2008

[14] C Niu and X Chen ldquoAlmost periodic sequence solutions ofa discrete Lotka-Volterra competitive system with feedbackcontrolrdquoNonlinear Analysis RealWorldApplications vol 10 no5 pp 3152ndash3161 2009

[15] J G Skellam ldquoRandom dispersal in theoretical populationsrdquoBiometrika vol 38 pp 196ndash218 1951

[16] A M Turing ldquoThe chemical basis of morphogenesisrdquo Philo-sophical Transactions of the Royal Society B vol 237 pp 37ndash721953

[17] J Bascompte and R V Sole ldquoSpatially induced bifurcations insingle-species population dynamicsrdquo Journal of Animal Ecologyvol 63 pp 256ndash264 1994

[18] L A D Rodrigues D C Mistro and S Petrovskii ldquoPattern for-mation long-term transients and the Turing-Hopf bifurcationin a space- and time-discrete predator-prey systemrdquo Bulletin ofMathematical Biology vol 73 no 8 pp 1812ndash1840 2011

[19] Y T Han B Han L Zhang L Xu M F Li and G ZhangldquoTuring instability and labyrinthine patterns for a symmetricdiscrete comptitive Lotka-Volterra systemrdquo WSEAS Transac-tions on Mathematics vol 10 pp 181ndash189 2011

[20] G Q Sun Z Jin Q X Liu and L Li ldquoDynamical complexityof a spatial predator-prey model with migrationrdquo EcologicalModelling vol 219 pp 248ndash255 2008

[21] G Q Sun Z Jin Q X Liu and L Li ldquoSpatial pattern of anepidemic model with cross-diffusionrdquo Chinese Physics B vol 17pp 3936ndash3941 2008

[22] Y-X Wang and W-T Li ldquoEffect of cross-diffusion on thestationary problem of a diffusive competition model with aprotection zonerdquo Nonlinear Analysis Real World Applicationsvol 14 no 1 pp 224ndash245 2013

[23] M Li B Han L Xu and G Zhang ldquoSpiral patterns nearTuring instability in a discrete reaction diffusion systemrdquoChaosSolitons amp Fractals vol 49 pp 1ndash6 2013

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