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
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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
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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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
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Differential EquationsInternational Journal of
Volume 2014
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
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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