Research ArticleBifurcation Analysis of Gene Propagation Model Governed byReaction-Diffusion Equations

Guichen Lu

School of Mathematics and Statistics Chongqing University of Technology Chongqing 400054 China

Correspondence should be addressed to Guichen Lu bromn006gmailcom

Received 3 April 2016 Revised 15 June 2016 Accepted 22 June 2016

Academic Editor Andrew Pickering

Copyright copy 2016 Guichen Lu 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

We present a theoretical analysis of the attractor bifurcation for gene propagation model governed by reaction-diffusion equationsWe investigate the dynamical transition problems of the model under the homogeneous boundary conditions By using thedynamical transition theory we give a complete characterization of the bifurcated objects in terms of the biological parametersof the problem

1 Introduction

As the field of gene technology develops the gene propaga-tion problems continue to be relevant Some recent advancesand problems include the following the genetic engineeringfor improving crop pest and disease resistance the bacteriahave developed a tolerance to widely prescribed antibioticsthe human genome project will enable us to deduce moreinformation on human bodies and to deduce historicalpatterns of migration by archaeologists Lots of papers devel-oped equations to describe the changes in the frequency ofalleles in a population that has several possible alleles at thelocus in question Fisher [1] proposed a reaction-diffusionequation with quadratic source term that models the spreadof a recessive advantageous gene through a population thatpreviously had only one allele at the locus in question Fisherrsquosequation is

120597119906

120597119905= 120579

1205972119906

1205971199092+ 119903119906 (1 minus 119906) (1)

where 119906 is the frequency of the new mutant gene 120579 is thediffusion coefficient and 119903 is the intensity of selection in favorof the mutant gene

In [2ndash5] the authors have claimed that a cubic sourceterm was more appropriate than a quadratic source termAlthough the cubic source term is implicit as one possibilityin the general genetic dispersion equations derived by others

its significance has not been highlighted and the differencebetween cubic and quadratic source terms has not beenexamined Based on the Fitzhugh-Nagumo equation andHuxley equation by using the methods of a continuum limitof a discrete generation model direct continuum modellingand Fickrsquos laws for random motion Bradshaw-Hajek andBroadbridge [6ndash8] have derived a reaction-diffusion equationdescribing the spread of a new mutant gene that is

120597119906

120597119905= 120579

1205972119906

1205971199092+ 1199031199062(1 minus 119906) (2)

where 119906 is the frequency of the new mutant gene 120579 is thediffusion coefficient and 119903 is the intensity of selection in favorof the mutant gene

In [9ndash11] the authors have discussed the two possiblealleles while some others recently investigate another casein which there are more than two possible alleles at thelocus in question For three possible alleles Littler [12] hasmostly used stochastic models while Bradshaw-Hajek andBroadbridge [6ndash8] have developed the reaction-diffusion-convection models

In this paper we will follow the work of Bradshaw-Hajeket al [7] and investigate the gene propagation model ofthree possible alleles at the locus By introducing the spatialtwo-dimensional domains we will give a detailed analysisof the dynamical properties for the model and consider theattractor bifurcation to show a complete characterization of

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2016 Article ID 9840297 9 pageshttpdxdoiorg10115520169840297

2 Discrete Dynamics in Nature and Society

the attractors and their basins of attraction in terms of thephysical parameters of the problemwhich is developed byMaand Wang [13 14]

The paper is organized as follows In Section 2 we brieflysummarize the two-dimensional spatial gene propagationmodel and give some mathematical settings Section 3 statesprinciple of exchange of stability for system Section 4 is themain results of the phase transition theorems based on theattractor bifurcation theory An example with the computersimulation of the pattern formation is given in the concludingremark section to illustrate our main results

2 Modelling Analysis

In order to describe the spread of a new mutant gene basedon Skellamrsquos method Bradshaw-Hajek and Broadbridge[6] have developed a one-dimensional population geneticsmodel governed by reaction-diffusion equation describingthe changes in allelic frequencies For a population having onenewmutant allele119860

1and twooriginal alleles119860

2 1198603 there are

six possible genotypes

11986011198601 11986011198602 11986011198603 11986021198602 11986021198603 11986031198603 (3)

Let 120588119894119895(1199091 1199092 119905) (119894 119895 = 1 2 3) denote the frequency of indi-

viduals of the genotype119860119894119860119895on the spatial two-dimensional

domain We follow the ideas of [6] and write the genotypeequations as

12059712058811

120597119905= Δ12058811minus 12058312058811+ 120574111199062

1120588

12059712058812

120597119905= Δ12058812minus 12058312058812+ 21205741211990611199062120588

12059712058813

120597119905= Δ12058813minus 12058312058813+ 2120574131199061(1 minus 119906

1minus 1199062) 120588

12059712058822

120597119905= Δ12058822minus 12058312058822+ 2120574121199062

2120588

12059712058823

120597119905= Δ12058823minus 12058312058823+ 2120574231199062(1 minus 119906

1minus 1199062) 120588

12059712058833

120597119905= Δ12058833minus 12058312058833+ 212057433(1 minus 119906

1minus 1199062)2

120588

(4)

where Δ = 12059721205971199091

2+ 12059721205971199092

2 119906119894(1199091 1199092 119905) is the frequency

of allele 119860119894 which can be expressed as

1199061=212058811+ 12058812+ 12058813

2120588

1199062=12058812+ 212058822+ 12058823

2120588

1199063= 1 minus 119906

1minus 1199062

(5)

120588(119909 119905) is the total population density 120583 is the common deathrate and 120574

119894119895is the reproductive success rate of individuals

with genotype 119860119894119860119895for 119894 119895 = 1 2 3 respectively

From (5) we can simplify these above six equations intothe following two coupled equations describing the change infrequency of two of alleles

1205971199061

120597119905= Δ1199061+2

120588

120597120588

120597119909

1205971199061

120597119909+ Φ (119906

1 1199062)

1205971199062

120597119905= Δ1199062+2

120588

120597120588

120597119909

1205971199062

120597119909+ Ψ (119906

1 1199062)

(6)

where

Φ(1199061 1199062) = (120574

13minus 12057433) 1199061+ (12057411minus 312057413+ 212057433) 1199062

1

+ (12057412minus 12057413minus 212057423+ 212057433) 11990611199062

+ (minus12057422+ 212057423minus 12057433) 11990611199062

2

+ (minus212057412+ 212057413+ 212057423minus 212057433) 1199062

11199062

+ (minus12057411+ 212057413minus 12057433) 1199063

1

Ψ (1199061 1199062) = (120574

23minus 12057433) 1199062+ (12057422minus 312057423+ 212057433) 1199062

2

+ (12057412minus 212057413minus 12057423+ 212057433) 11990611199062

+ (minus12057411+ 212057413minus 12057433) 1199062

11199062

+ (minus212057412+ 212057413+ 212057423minus 212057433) 11990611199062

2

+ (minus12057422+ 212057423minus 12057433) 1199063

2

(7)

Assume that the total population density is constantacross the range (so that 120597120588120597119909 = 0) system (6) becomes

1205971199061

120597119905= Δ1199061+ Φ (119906

1 1199062)

1205971199062

120597119905= Δ1199062+ Ψ (119906

1 1199062)

(8)

One of the attractions of (8) tomathematicians is to studythe diffusion induced instability introduced by Turing in his1952 seminal paper [15] For showing the diffusion effect onstability we will consider a modified equation of (8)

1205971199061

120597119905= Δ1199061+ (12057413minus 12057433) 1199061+ 11988611199062

1+ 119886211990611199062+ 119886311990611199062

2

+ 11988641199062

11199062+ 11988651199063

1

1205971199062

120597119905= 119889Δ119906

2+ (12057423minus 12057433) 1199062+ 11988711199062

2+ 119887211990611199062+ 119886411990611199062

2

+ 11988651199062

11199062+ 11988631199063

2

(9)

where 1198861= 12057411minus 312057413+ 212057433 1198862= 12057412minus 12057413minus 212057423+ 212057433

1198863= minus12057422+ 212057423minus 12057433 1198864= minus2120574

12+ 212057413+ 212057423minus 212057433

1198865= minus12057411+ 212057413minus 12057433 1198871= 12057422minus 312057423+ 212057433 1198872= 12057412minus 212057413minus

12057423+ 212057433 and 1198862

3+ 1198862

4+ 1198862

5= 0 119889 is the diffusion coefficient

which measures the dispersal rate of allele 1198602 On the

other hand diffusive terms can be considered as describing

Discrete Dynamics in Nature and Society 3

the ability of the allele 119860119894to occupy different zones in 2-

dimensional space either through the action of small-scalemechanism or by some native transport device

System (9) has seven constant solutions (1199061 1199062) =

(0 0) (0 1) (1 0) (0 120572) (120573 0) (119886lowast 119887lowast) (119886 119887) where 120572 120573 119886lowast

119887lowast 119886 and 119887 are complicated expressions of the reproductive

success rate 120574119894119895and 119886 119887 are given by

119886 =Δ1

Δ0

119887 =Δ2

Δ0

(10)

where

Δ1= minus1205741312057423+ 1205741312057422minus 1205741212057423+ 1205741212057433minus 1205742212057433

+ 12057423

2

Δ2= minus1205741212057413+ 1205741212057433minus 1205741312057423minus 1205741112057433+ 1205741112057423

+ 12057413

2

Δ0= 12057412

2+ 12057413

2+ 12057423

2minus 21205741212057413minus 21205741212057423minus 1205741112057433

+ 21205741212057433minus 1205741112057422minus 21205741312057423minus 1205742212057433

+ 21205741312057422+ 21205741112057423

(11)

Considering the biological context we assume that the steadystate solution (119886 119887) is positive and 0 lt 119906

1 1199062lt 1

In the present paper we focus on the bifurcation from theconstant solution (119886 119887) Let

1199061= 119886 + 119906

1015840

1

1199062= 119887 + 119906

1015840

2

(12)

Omitting the primes then system (9) becomes

1205971199061

120597119905= Δ1199061+ 119886 (120574

11minus 12057413) 1199061+ 119886 (120574

12minus 12057413) 1199062

+ 1198661(1199061 1199062)

1205971199062

120597119905= 119889Δ119906

2+ 119887 (120574

12minus 12057423) 1199061+ 119887 (120574

22minus 12057423) 1199062

+ 1198662(1199061 1199062)

(13)

where

1198661(1199061 1199062) = (119886

4119887 + 3119886

5119886 + 1198861) 1199061

2

+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

+ 11988651199061

3+ 119906111988631199062

2+ 11988641199061

21199062

1198662(1199061 1199062) = 119887119886

51199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062

+ (1198871+ 31198863119887 + 1198864119886) 1199062

2+ 11988631199062

3

+ 119886411990611199062

2+ 119886511990621199061

2

(14)

We assume that system (13) is satisfied on an openbounded domainΩ sub 119877

2 There are two types of biologicallysound boundary conditions the Dirichlet boundary condi-tion

1199061

1003816100381610038161003816120597Ω = 0

1199062

1003816100381610038161003816120597Ω = 0(15)

which means that the frequency is extinct in the boundary ofrange and the Neumann boundary condition

1205971199061

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

1205971199062

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(16)

which means that the frequency is invariant in the boundaryof range in biological significant

Define the function spaces

119867 = 1198712(Ω 1198772)

1198671=

1198672(Ω 1198772) cap 119867

1

0(Ω 1198772) for boundary conditions (15)

119906 isin 1198672(Ω 1198772)

10038161003816100381610038161003816100381610038161003816

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0 for boundary conditions (16)

(17)

It is clear that119867 and1198671are two Hilbert space and119867

1997893rarr 119867

is dense and compact inclusionLater we choose the bifurcation parameter 120582 to be the

diffusion coefficient 119889 that is 120582 = 119889 Let 119871120582 1198671rarr 119867 be

defined by

119871120582fl minus119860

120582+ 119861 (18)

whereminus119860120582119906 = (Δ119906

1 120582Δ1199062)119879

119861119906 = (11988611

11988612

11988621

11988622

)(1199061

1199062

)

(19)

where 119906 = (1199061 1199062)119879isin 1198671 11988611= 119886(12057411minus 12057413) 11988612= 119886(12057412minus

12057413) 11988621= 119887(12057412minus 12057423) and 119886

22= 119887(12057422minus 12057423)

4 Discrete Dynamics in Nature and Society

Furthermore let 119866(1199061 1199062) be given by

119866 (1199061 1199062) = (119866

2

1(1199061 1199062) + 1198663

1(1199061 1199062) 1198662

2(1199061 1199062)

+ 1198663

2(1199061 1199062))119879

(20)

where

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(1198661

3(1199061 1199062)

1198662

3(1199061 1199062)) = (

11988651199061

3+ 119906111988631199062

2+ 11988641199061

21199062

11988631199062

3+ 119886411990611199062

2+ 11988651199061

21199062

)

(21)

Then (13) can be written in the following operator form

119889119906

119889119905= 119871120582119906 + 119866 (119906) (22)

3 Principle of Exchange of Stability

From the theoretical ecology it is interesting to study thebifurcation of system (9) at steady state (119886 119887) Bifurcationmeans that a change in the stability or in the types ofsteady state which occurs as a parameter is varied in adissipative dynamic system that is the state changes duringthe biology conditions The classical bifurcation types areHopf bifurcation and Turing bifurcation Ma and Wang[13 14] have developed new methods to study bifurca-tions and transitions which are called attractor bifurcationsThis theory yields complete information about bifurcationstransitions stability and persistence including informationabout transient states in terms of the physical parametersof the system Therefore in this section we consider theattractor bifurcation of system (9) at (119886 119887) and from thetransformation we only need to discuss system (13) at (0 0)

Firstly we consider the linear system of (13)

1205971199061

120597119905= Δ1199061+ 119886111199061+ 119886121199062

1205971199062

120597119905= 120582Δ119906

2+ 119886211199061+ 119886221199062

(23)

and its eigenvalue problem

119871120582120593 = 120573 (120582) 120593 120593 isin 119867

1 (24)

minusΔ119890119896= 120588119896119890119896 (25)

with the Dirichlet boundary condition (15) or the Neumannboundary condition (16)

Let119872119896be defined by

119872119896= (

11988611minus 120588119896

11988612

11988621

11988622minus 120582120588119896

) (26)

It is easy to see that any eigenvector 120601119896and eigenvalue 120573

119896

of (24) can be expressed as

120601119896= (

1205851198961119890119896

1205851198962119890119896

)

119872120582

119896(1205851198961

1205851198962

) = 120573119896(1205851198961

1205851198962

)

(27)

where 119890119896is as in (25) and 120573

119896is also the eigenvalue of 119861120582

119896 By

(24) 120573119896can be written as

1205731119896= minus

1

2[(120588119896minus 11988611) + (120582120588

119896minus 11988622)]

+1

2

radic[(120588119896minus 11988611) minus (120582120588

119896minus 11988622)]2

+ 41198861211988621

1205732119896= minus

1

2[(120588119896minus 11988611) + (120582120588

119896minus 11988622)]

minus1

2

radic[(120588119896minus 11988611) minus (120582120588

119896minus 11988622)]2

+ 41198861211988621

(28)

It is clear that 1205732119896(120582) lt 120573

1119896(120582) = 0 if and only if

(120588119896minus 11988611) + (120582120588

119896minus 11988622) gt 0

(120588119896minus 11988611) (120582120588119896minus 11988622) minus 1198861211988621= 0

(29)

We define a parameter

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(30)

where 120588119896= 120588119870such that 120594

119870(120582) attains its minimum values

120594119870(120582) = min

120588119896

120582120588119896+1198861111988622minus 1198861211988621

120588119896

minus (12058211988611+ 11988622)

= 120582120588119870+1198861111988622minus 1198861211988621

120588119870

minus (12058211988611+ 11988622)

(31)

Theorem 1 Let 1205820be the number given in (30) such that (29)

is satisfied and let119870 ge 1 be the integer such that the minimumis achieved at 120588

119870 Then 120573

1119870(120582) is the first real eigenvalue of 119871

120582

near 120582 = 1205820satisfying that

1205731119870(120582)

lt 0 120582 isin Λ+

119870

= 0 120582 = 1205820

gt 0 120582 isin Λminus

119870

Re1205732119896(1205820) lt 0 forall119896 isin N

Re1205731119896(1205820) lt 0 forall119896 = 119870

(32)

Here Λ+119870= 120582 | 120594

119870(120582) gt 0 and Λminus

119870= 120582 | 120594

119870(120582) lt 0

In the absence of diffusion system (23) becomes thespatial homogeneous system

1198891199061

119889119905= 119886111199061+ 119886121199062

1198891199062

119889119905= 119886211199061+ 119886221199062

(33)

Discrete Dynamics in Nature and Society 5

System (33) is local asymptotic stability if

11988611+ 11988622lt 0

1198861111988622minus 1198861211988621gt 0

(34)

and the Hopf bifurcation occurs when

11988611+ 11988622= 0

1198861111988622minus 1198861211988621lt 0

(35)

From Theorem 1 we can infer that if condition (34) and120582 isin Λ

minus

119870hold then the homogeneous attracting equilibrium

loses stability due to the interaction of diffusion processes andsystem (23) undergoes a Turing bifurcation

4 Phase Transition on Homogeneous State

Hereafter we always assume that the eigenvalue 1205731119870(120582) in

(24) is simple Based onTheorem 1 as 120582 isin Λminus

119870the transition

of (22) occurs at 120582 = 1205820 which is from real eigenvalues

The following is the main theorem in this paper whichprovides not only a precise criterion for the transition typesof (22) but also globally dynamical behaviors

Theorem 2 Let 120588119870be defined in Theorem 1 and 119890

119870is the

corresponding eigenvector to 120588119870of (25) satisfying

intΩ

1198903

119870119889119909 = 0 (36)

For system (22) we have the following assertions(1) Equation (22) has a mixed transition from (0 120582

0) more

precisely there exists a neighborhood119880 isin 119883 of 119906 = 0 such that119880 is separated into two disjoint open sets 119880120582

1and 119880

120582

2by the

stable manifold Γ120582of 119906 = 0 satisfying the following

(a) 119880 = 119880120582

1+119880120582

2+ Γ120582 (b) The transition in 119880120582

1is jump (c)

The transition in 1198801205822is continuous

(2) Equation (22) bifurcates in 119880120582

2to a unique singular

point V120582 on120582 isin Λ+

119870which is attractor such that for any120601 isin 119880

120582

2

lim119905rarrinfin

10038171003817100381710038171003817119906 (119905 120601) minus V120582

10038171003817100381710038171003817119883= 0 (37)

(3) Equation (22) bifurcates on 120582 isin Λminus

119870to a unique saddle

point with morse index 1(4) The bifurcated singular point V120582 can be expressed by

V120582 = minus1205731119870(120582)

119887120593119870+ 119900 (

10038171003817100381710038171205731119870 (120582)1003817100381710038171003817)

(38)

Here

119887 =1

1198861211988621+ (11988611minus 120588119870)2[minus119887 (119887

2+ 1198864119887 + 2119886

5119886)

sdot (1198861198863(119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (1198862+ 21198863119887 + 2119886

4119886) 119886 (119886

2+ 21198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870) + (119886

4119887 + 3119886

5119886 + 1198861)

sdot 1198862(1198862+ 21198863119887 + 1198864119886)2

) + (119886 (1198861+ 1198864119887 + 2119886

5119886)

minus 120588119870) ((1198871+ 31198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (21198865119886 + 1198872+ 21198864119887)

sdot 119886 (1198862+ 21198863119887 + 1198864119886) (119886 (119886

1+ 1198864119887 + 2119886

5119886) minus 120588

119870)

+ 11988711988651198862(1198862+ 21198863119887 + 1198864119886)2

)]

(39)

Proof First we need to get the reduced equation of (22) near120582 = 120582

0

Let 119906 = 119909 sdot 120601119870+ 119910 where 119910 = sum

119899

119894=1119910119894120601119894and 120601

119870is the

eigenvector of (24) corresponding to 1205731119870(120582) at 120582 = 120582

0 Then

the reduced equation of (22) reads

119889119909

119889t= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870+ 119910) 120601

lowast

119870⟩ (40)

Here 120601lowast119870is the conjugate eigenvector of 120601

119870

By Implicit FunctionTheorem we can obtain that

119910 = Φ (119909 120582) = 119900 (119909) (41)

Substituting (41) into (40) we get the bifurcation equation of(22) as follows

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870) 120601lowast

119870⟩

+ 119900 (2)

(42)

By (27) 120601119870is written as

120601119870= (1205851119890119870 1205852119890119870)119879

(43)

with (1205851 1205852) satisfying

(11988611minus 120588119870

11988612

11988621

11988622minus 120582120588K

)(1205851

1205852

) = 1205731119870(1205820) (

1205851

1205852

) (44)

from which we get

(1205851 1205852) = (minus119886

12 11988611minus 120588119870) (45)

Likewise 120601lowast119870is

120601lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(46)

6 Discrete Dynamics in Nature and Society

with (120585lowast1 120585lowast

2) satisfying

(11988611minus 120588119870

11988621

11988612

11988622minus 120582120588119870

)(1205851

1205852

) = 1205731119870(1205820) (

120585lowast

1

120585lowast

2

) (47)

which yields

(120585lowast

1 120585lowast

2) = (minus119886

21 11988611minus 120588119870) (48)

By (22) the nonlinear operator 119866 is

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(49)

Then in view of (45) and (48) by direct computation wederive that

⟨119866 (1199091205851119890119870 1199091205852119890119870) 120601lowast

119870⟩ = [(119886

4119887 + 1198861+ 31198865119886) 1205851

2120585lowast

1

+ 120585lowast

211988651198871205851

2+ (1198872+ 21198865119886 + 2119886

4119887) 12058521205851120585lowast

2

+ (1198871+ 1198864119886 + 3119886

3119887) 1205852

2120585lowast

2

+ (21198863119887 + 1198862+ 21198864119886) 12058521205851120585lowast

1+ 120585lowast

111988631198861205852

2]

sdot 1199092intΩ

1198903

119870119889119909 + 119900 (119909

2)

(50)

By (45) and (48) we have

⟨120601 120601lowast⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909 (51)

Hence by (50) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 119887119909

2+ 119900 (119909

2) (52)

where 119887 is the parameter as in (39) Based onTheorem A2 in[16] this theorem follows from (52) The proof is complete

Remark 3 If the domain Ω = (0 119871) times 119863 with 119863 isin 119877

being a bounded open set then condition (36) holds true forboundary conditions (15) and (16)

If the domainΩ = (0 119871)times119863with119863 isin 119877 being a boundedopen set then

intΩ

1198903

119870119889119909 = 0 (53)

holds true for Neumann condition (16) and Dirichlet condi-tion (15) when the number119898 in 120588

1198960= 119898212058721198712+ 1205881015840

1198960is even

Hereafter we consider the Neumann condition (16) andlet the domain Ω = (0 119871

1) times (0 119871

2) with 119871

1= 1198712

The following eigenvalue problem (24) with the equation

minusΔ119890119896= 120588119896119890119896

120597119890119896

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(54)

has eigenfunctions 119890119896and eigenvalues 120588

119896as follows

119890119896= cos11989611205871199091

1198711

cos119896212058711990921198712

119896 = (1198961 1198962) 1198961 1198962= 0 1 2

(55)

120588119896= 1205872(1198962

1

11987121

+1198962

2

11987122

) (56)

Here 1205880= 0 and 119890

0= 1

Denote 119870 = (1198701 1198702) the pair of integers satisfying (30)

and simply denote 1198701= (1198701 0) and 119870

2= (0 119870

2) then we

have the following

Theorem 4 Let 120578 be the parameter defined by

120578 =[(916) (120585 120585

lowast) + (14)119867 (120585 120585

lowast)]

1198861211988621+ (11988611minus 120588119870)2

(57)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

2120585112058522+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix and 120585 120585lowast are

defined in the proof

Assume that the eigenvalue 1205731119870(120582) satisfying (28) is

simple then for system (22) we have the following assertions

(1) This system has a transition from (0 1205820) that is 119906 = 0

is asymptotically stable for 120582 isin Λminus

119870and unstable for

120582 isin Λ+

119870which transits to a stable equilibrium at 120582 =

1205820

(2) If 120578 gt 0 then (22) has a jump transition from (0 1205820)

and bifurcates on 120582 isin Λ+

119870to exactly two saddle points

V1205821and V1205822with the Morse index one

(3) If 120578 lt 0 then (22) has a continuous transition from(0 1205820) which is an attractor bifurcation

(4) The bifurcated singular points V1205821and V1205822in the above

cases can be expressed in the following form

V12058212

= plusmn

10038161003816100381610038161003816100381610038161003816

1205731119870(120582)

120578

10038161003816100381610038161003816100381610038161003816

12

119890119870+ 119900 (

10038171003817100381710038171205731119870100381710038171003817100381712

) (58)

Proof Assertion (1) follows from (32) To prove assertions (2)and (3) we need to get the reduced equation of (22) to thecenter manifold near 120582 = 120582

0

Let 119906 = 119909 sdot 120593119870+ Φ where 120593

119870is the eigenvector of (25)

corresponding to 1205731119870(120582) at 120582 = 120582

0and Φ(119909) the center

manifold function of (22) Then the reduced equation of (22)takes the following form

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120593119870 120593lowast119870⟩⟨119866 (119909 sdot 120593

119870+ Φ) 120593

lowast

119870⟩ (59)

Here 120593lowast119870is the conjugate eigenvector of 120593

119870

By (27) 120593119870is written as

120593119870= (1205851119890119870 1205852119890119870)119879

(60)

with (1205851 1205852) satisfying (45)

Discrete Dynamics in Nature and Society 7

Likewise 120593lowast119870is

120593lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(61)

with (120585lowast1 120585lowast

2) satisfying (46)

It is known that the center manifold function

Φ (119909) = (Φ1(119909) Φ

2(119909)) = 119900 (119909

2) (62)

By using the formula for center manifold functions in [1314 16] Φ = (Φ

1 Φ2) satisfies

1198711205820Φ = minus119909

21198662(1205851119890119896 1205852119890119870) + 119900 (119909

2)

= minus11990921198902

119870(1205781

1205782

) + 119900 (1199092)

(63)

where

1205781= 11988611988631205852

2+ (1198862+ 21198864119886 + 2119886

3119887) 12058511205852

+ (1198864119887 + 3119886

5119886 + 1198861) 1205851

2

1205782= (1198871+ 31198863119887 + 1198864119886) 1205852

2+ (21198865119886 + 2119886

4119887 + 1198872) 12058511205852

+ 11988711988651205851

2

(64)

From (55) we see that

1198902

119870=1

4(1 + 119890

21198701) (1 + 119890

21198702)

=1

4(1198900+ 11989021198701

+ 11989021198702

+ 1198902119870)

(65)

Let

(Φ1

Φ2

) = (Φ0

1

Φ0

2

)1198900+ (

Φ21198701

1

Φ21198701

2

)11989021198701

+ (Φ21198702

1

Φ21198702

2

)11989021198702

+ (Φ2119870

1

Φ2119870

2

)1198902119870

(66)

It is clear that

1198711205820(Φ119896

1

Φ119896

2

)119890119896= 119872119896(Φ119896

1

Φ119896

2

)119890119896 (67)

where 119872119896is the matrix given by (26) Then it follows from

(65) and (66) that

(Φ119896

1

Φ119896

2

) = minus1

4119872minus1

119896(1205781

1205782

) + 119900 (1199092) (68)

for 119896 = 1198701198701 1198702 where

119872minus1

119896=

1

det (119872119896)[minus1205820120588119896+ 11988622

minus11988612

minus11988621

minus120588119896+ 11988611

]

det (119872119896) = 1205820120588119896

2minus 12058811989611988622minus 119886111205820120588119896+ 1198861111988622

minus 1198861211988621

(69)

Direct calculation shows that

(Φ0

1

Φ0

2

) =1199092

4 (1198861111988622minus 1198861211988621)(minus119886221205782minus 119886121205781

minus119886211205782minus 119886111205781

)

(Φ21198701

1

Φ21198701

2

)

=1199092

4det (11987221198701

)((120582012058821198701

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198701

minus 11988611) 1205781

)

(Φ21198702

1

Φ21198702

2

)

=1199092

4det (11987221198702

)((120582012058821198702

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198702

minus 11988611) 1205781

)

(Φ2119870

1

Φ2119870

2

) =1199092

4det (1198722119870)((12058201205882119870

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (1205882119870

minus 11988611) 1205781

)

(70)

Inserting (66) into (59) by (60) and (61) we get

⟨119866 (119909120593119870+ Φ) 120593

lowast

119870⟩ =

911987111198712

641199093 (120585 120585

lowast)

+11987111198712

161199093119867(120585 120585

lowast) + 119900 (119909

3)

(71)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

212058511205852

2+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix

In view of (70) it follows that

⟨120601119870 120601lowast

119870⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909

=11987111198712

4[1198861211988621+ (11988611minus 120588119870)2

]

(72)

Hence by (73) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 120578119909

3+ 119900 (119909

3) (73)

Based onTheoremA1 in [16] this theorem follows from (73)The proof is complete

5 Concluding Remarks and Example

In this paper we have studied the Turing bifurcation intro-duced by Alan Turing and attractor bifurcation developedby Ma and Wang [13] of gene propagation population modelgoverned by reaction-diffusion equation

Theorems 2 and 4 tell us that the critical value 1205820of

diffusion constant given by

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(74)

8 Discrete Dynamics in Nature and Society

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

05228

Time step = 300

(a) 120582 = 13

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

Time step = 1200

(b) 120582 = 1586209390

Figure 1 Turing patterns in two dimensions of system (9) with 120574119894119895

defined by (75)

plays a crucial role in determining the attractor bifurcationand Turing bifurcation when the diffusion parameter 120582

crosses the critical value 1205820 and the uniform stationary state

(119886 119887) loses its stability which yields the Turing or attractorbifurcation

We give some examples to illustrate our main theories

Example 5 In system (9) we let

12057411=2773376069253

499999999994

12057422=3789926287167

499999999994

12057433=810843022017

45454545454

12057423=4789523227559

249999999997

12057412=6952222073553

499999999994

12057413=436265576135

249999999997

(75)

and then the steady state (119886 119887) of system (9) is

119886 =732070686076807741207077

1417067211664436233221844

119887 =380119934758603124485687

708533605832218116610922

(76)

where 0 lt 119886 and 119887 lt 1

If we set Ω = (0 40120587) times (0 60120587) then 120588119870= 1314400

where119870 = (1 1) and then

1205820=814910923617150223684573395768233809936382690140395479526347826117214934665200

262055733351368170238516966160634367981740693955623134829331749178881678669asymp 3109685536 (77)

By Theorem 1 we obtain that the transition conditions aresatisfied

FromTheorem 4 120578 asymp minus4527181405 lt 0 and system (22)has a continuous transition from (0 120582

0) which is an attractor

bifurcationSince120588

119870+11988611lt 0 we infer that if120582 lt 120582

0 then120573

1119870(120582) gt 0

and the Turing instability occurs By numerical simulationwe have shown that the gene population model is able tosustain Turing patterns (Figure 1)

Appendix

The Expressions of 119867(120585120585lowast)

The Expressions of119867(120585 120585lowast) are as follows

119867(120585 120585lowast) = 4119886

3120585lowast

1119887Φ21198701

11205852+ 4119887120585lowast

2Φ21198701

111988651205852

+ 4120585lowast

211988651198871205851Φ21198701

2+ 4120585lowast

211988641198871205851Φ21198702

1

+ 4120585lowast

21198864119887Φ21198702

11205852+ 4120585lowast

211988641198871205851Φ21198702

2

+ 2120585lowast

211988651198871205852Φ2119870

2+ 2120585lowast

11198864119887Φ2119870

11205852

+ 4120585lowast

111988631198861205852Φ21198702

2+ 4120585lowast

111988631198871205852Φ21198701

2

+ 4120585lowast

211988651198871205851Φ21198702

2+ 6120585lowast

111988651198871205851Φ2119870

1

+ 2120585lowast

21198864119886Φ2119870

11205852+ 4120585lowast

111988641198871205851Φ21198701

1

+ 12120585lowast

111988651198871205851Φ21198701

1+ 2120585lowast

111988631198871205852Φ2119870

2

+ 2120585lowast

111988641198871205851Φ2119870

1+ 24120585lowast

111988651198871205851Φ0

1

+ 2120585lowast

111988631198871205851Φ2119870

2+ 121198863119887120585lowast

21205852Φ21198701

2

+ 241198863119887Φ0

2120585lowast

21205852+ 2120585lowast

211988721205851Φ21198702

2

+ 4120585lowast

21198864119887Φ21198701

11205852+ 4120585lowast

111988641198871205851Φ21198701

2

+ 12120585lowast

111988651198871205851Φ21198702

1+ 4120585lowast

11198864119887Φ21198702

11205852

Discrete Dynamics in Nature and Society 9

+ 8120585lowast

111988641198871205851Φ0

1+ 8120585lowast

211988651198871205851Φ0

2

+ 2120585lowast

211988641198871205851Φ2119870

1+ 8120585lowast

211988641198871205851Φ0

2

+ 4120585lowast

111988641198871205851Φ21198702

2+ 41198863Φ21198702

1120585lowast

11198871205852

+ 121198863119887120585lowast

21205852Φ21198702

2+ 4120585lowast

211988641198871205851Φ21198701

2

+ 4120585lowast

111988631198871205851Φ21198701

2+ 4120585lowast

11198864119887Φ21198701

11205852

+ 4Φ21198702

1119887120585lowast

211988651205852+ 4120585lowast

211988651198871205852Φ21198702

2

+ 4120585lowast

111988641198871205851Φ21198702

1+ 6120585lowast

211988631198871205852Φ2119870

2

+ 4120585lowast

111988631198871205851Φ21198702

2+ 4120585lowast

211988651198871205852Φ21198701

2

+ 2Φ2119870

1119887120585lowast

211988651205852+ 4120585lowast

21198872Φ0

11205852

+ 8120585lowast

111988611205851Φ0

1+ 2120585lowast

111988621205851Φ21198702

2

+ 2120585lowast

11198862Φ21198702

11205852+ 21198872Φ21198702

1120585lowast

21205852

+ 4120585lowast

211988721205851Φ0

2+ 2120585lowast

111988621205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198702

2+ 4120585lowast

11198862Φ0

11205852

+ 4120585lowast

111988611205851Φ21198701

1+ 2120585lowast

11198862Φ21198701

11205852

+ Φ2119870

11198872120585lowast

21205852+ 4120585lowast

111988611205851Φ21198702

1

+ 120585lowast

211988721205851Φ2119870

2+ 2120585lowast

211988711205852Φ2119870

2

+ 4120585lowast

111988621205851Φ0

2+ 120585lowast

11198862Φ2119870

11205852

+ 21198872120585lowast

2Φ21198701

11205852+ 2120585lowast

111988611205851Φ2119870

1

+ 8Φ0

2120585lowast

211988711205852+ 2120585lowast

211988721205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198701

2+ 8120585lowast

111988631198871205852Φ0

2

+ 2120585lowast

111988641198871205851Φ2119870

2+ 2120585lowast

211988651198871205851Φ2119870

2

+ 2120585lowast

211988641198871205851Φ2119870

2+ 2Φ2119870

11198863120585lowast

11198871205852

+ 120585lowast

111988621205851Φ2119870

2+ 8120585lowast

11198863119887Φ0

11205852

+ 8120585lowast

111988631198871205851Φ0

2+ 8120585lowast

111988641198871205851Φ0

2

+ 8120585lowast

211988641198871205851Φ0

1+ 8120585lowast

11198864119887Φ0

11205852

+ 8120585lowast

21198865119887Φ0

11205852+ 8120585lowast

21198864119887Φ0

11205852

+ 8120585lowast

211988651198871205852Φ0

2+ 4120585lowast

211988641198871205851Φ21198701

1

(A1)

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was partially supported by National Natu-ral Science Foundation of China (Grants no 11401062and no 11371386) Research Fund for the National Nat-ural Science Foundation of Chongqing CSTC (Grant nocstc2014jcyjA0080) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(Grant no KJ1400937) and the Scientific Research Founda-tion of CQUT (Grant no 2012ZD37)

References

[1] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 no 4 pp 355ndash369 1937

[2] A D Bazykin ldquoHypothetical mechanism of speciatonrdquo Evolu-tion vol 23 no 4 pp 685ndash687 1969

[3] J Pialek andNH Barton ldquoThe spread of an advantageous alleleacross a barrier the effects of randomdrift and selection againstheterozygotesrdquo Genetics vol 145 no 2 pp 493ndash504 1997

[4] T Nagylaki ldquoConditions for the existence of clinesrdquo Geneticsvol 80 no 3 pp 595ndash615 1975

[5] T Nagylaki and J F Crow ldquoContinuous selective modelsrdquoTheoretical Population Biology vol 5 no 2 pp 257ndash283 1974

[6] B H Bradshaw-Hajek and P Broadbridge ldquoA robust cubicreaction-diffusion system for gene propagationrdquo Mathematicaland Computer Modelling vol 39 no 9-10 pp 1151ndash1163 2004

[7] B H Bradshaw-Hajek P Broadbridge and G H WilliamsldquoEvolving gene frequencies in a population with three possiblealleles at a locusrdquo Mathematical and Computer Modelling vol47 no 1-2 pp 210ndash217 2008

[8] P Broadbridge B H Bradshaw G R Fulford and G K AldisldquoHuxley and Fisher equations for gene propagation an exactsolutionrdquoThe Anziam Journal vol 44 no 1 pp 11ndash20 2002

[9] A Kolmogorov I Petrovsky and N Piscounov ldquoEtudes delrsquoequation aved croissance de la quantite de matiere et sonapplication a un probleme biologiquerdquo Moscow UniversityMathematics Bulletin vol 1 pp 1ndash25 1937

[10] M Slatkin ldquoGene flow and selection in a clinerdquoGenetics vol 75no 4 pp 733ndash756 1973

[11] J M SmithMathematical Ideas in Biology Cambridge Univer-sity Press Cambridge UK 1968

[12] R A Littler ldquoLoss of variability at one locus in a finitepopulationrdquo Mathematical Biosciences vol 25 no 1-2 pp 151ndash163 1975

[13] T Ma and S Wang BifurcationTheory and Applications vol 53of Nonlinear Science Series A World Scientific Beijing China2005

[14] T Ma and S Wang Stability and Bifurcation of the NonlinearEvolution Equations Science Press Singapore 2007 (Chinese)

[15] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society of London B vol 237 no 641pp 37ndash52 1952

[16] T Ma and S Wang ldquoDynamic phase transition theory in PVTsystemsrdquo Indiana University Mathematics Journal vol 57 no 6pp 2861ndash2889 2008

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

the attractors and their basins of attraction in terms of thephysical parameters of the problemwhich is developed byMaand Wang [13 14]

The paper is organized as follows In Section 2 we brieflysummarize the two-dimensional spatial gene propagationmodel and give some mathematical settings Section 3 statesprinciple of exchange of stability for system Section 4 is themain results of the phase transition theorems based on theattractor bifurcation theory An example with the computersimulation of the pattern formation is given in the concludingremark section to illustrate our main results

2 Modelling Analysis

In order to describe the spread of a new mutant gene basedon Skellamrsquos method Bradshaw-Hajek and Broadbridge[6] have developed a one-dimensional population geneticsmodel governed by reaction-diffusion equation describingthe changes in allelic frequencies For a population having onenewmutant allele119860

1and twooriginal alleles119860

2 1198603 there are

six possible genotypes

11986011198601 11986011198602 11986011198603 11986021198602 11986021198603 11986031198603 (3)

Let 120588119894119895(1199091 1199092 119905) (119894 119895 = 1 2 3) denote the frequency of indi-

viduals of the genotype119860119894119860119895on the spatial two-dimensional

domain We follow the ideas of [6] and write the genotypeequations as

12059712058811

120597119905= Δ12058811minus 12058312058811+ 120574111199062

1120588

12059712058812

120597119905= Δ12058812minus 12058312058812+ 21205741211990611199062120588

12059712058813

120597119905= Δ12058813minus 12058312058813+ 2120574131199061(1 minus 119906

1minus 1199062) 120588

12059712058822

120597119905= Δ12058822minus 12058312058822+ 2120574121199062

2120588

12059712058823

120597119905= Δ12058823minus 12058312058823+ 2120574231199062(1 minus 119906

1minus 1199062) 120588

12059712058833

120597119905= Δ12058833minus 12058312058833+ 212057433(1 minus 119906

1minus 1199062)2

120588

(4)

where Δ = 12059721205971199091

2+ 12059721205971199092

2 119906119894(1199091 1199092 119905) is the frequency

of allele 119860119894 which can be expressed as

1199061=212058811+ 12058812+ 12058813

2120588

1199062=12058812+ 212058822+ 12058823

2120588

1199063= 1 minus 119906

1minus 1199062

(5)

120588(119909 119905) is the total population density 120583 is the common deathrate and 120574

119894119895is the reproductive success rate of individuals

with genotype 119860119894119860119895for 119894 119895 = 1 2 3 respectively

From (5) we can simplify these above six equations intothe following two coupled equations describing the change infrequency of two of alleles

1205971199061

120597119905= Δ1199061+2

120588

120597120588

120597119909

1205971199061

120597119909+ Φ (119906

1 1199062)

1205971199062

120597119905= Δ1199062+2

120588

120597120588

120597119909

1205971199062

120597119909+ Ψ (119906

1 1199062)

(6)

where

Φ(1199061 1199062) = (120574

13minus 12057433) 1199061+ (12057411minus 312057413+ 212057433) 1199062

1

+ (12057412minus 12057413minus 212057423+ 212057433) 11990611199062

+ (minus12057422+ 212057423minus 12057433) 11990611199062

2

+ (minus212057412+ 212057413+ 212057423minus 212057433) 1199062

11199062

+ (minus12057411+ 212057413minus 12057433) 1199063

1

Ψ (1199061 1199062) = (120574

23minus 12057433) 1199062+ (12057422minus 312057423+ 212057433) 1199062

2

+ (12057412minus 212057413minus 12057423+ 212057433) 11990611199062

+ (minus12057411+ 212057413minus 12057433) 1199062

11199062

+ (minus212057412+ 212057413+ 212057423minus 212057433) 11990611199062

2

+ (minus12057422+ 212057423minus 12057433) 1199063

2

(7)

Assume that the total population density is constantacross the range (so that 120597120588120597119909 = 0) system (6) becomes

1205971199061

120597119905= Δ1199061+ Φ (119906

1 1199062)

1205971199062

120597119905= Δ1199062+ Ψ (119906

1 1199062)

(8)

One of the attractions of (8) tomathematicians is to studythe diffusion induced instability introduced by Turing in his1952 seminal paper [15] For showing the diffusion effect onstability we will consider a modified equation of (8)

1205971199061

120597119905= Δ1199061+ (12057413minus 12057433) 1199061+ 11988611199062

1+ 119886211990611199062+ 119886311990611199062

2

+ 11988641199062

11199062+ 11988651199063

1

1205971199062

120597119905= 119889Δ119906

2+ (12057423minus 12057433) 1199062+ 11988711199062

2+ 119887211990611199062+ 119886411990611199062

2

+ 11988651199062

11199062+ 11988631199063

2

(9)

where 1198861= 12057411minus 312057413+ 212057433 1198862= 12057412minus 12057413minus 212057423+ 212057433

1198863= minus12057422+ 212057423minus 12057433 1198864= minus2120574

12+ 212057413+ 212057423minus 212057433

1198865= minus12057411+ 212057413minus 12057433 1198871= 12057422minus 312057423+ 212057433 1198872= 12057412minus 212057413minus

12057423+ 212057433 and 1198862

3+ 1198862

4+ 1198862

5= 0 119889 is the diffusion coefficient

which measures the dispersal rate of allele 1198602 On the

other hand diffusive terms can be considered as describing

Discrete Dynamics in Nature and Society 3

the ability of the allele 119860119894to occupy different zones in 2-

dimensional space either through the action of small-scalemechanism or by some native transport device

System (9) has seven constant solutions (1199061 1199062) =

(0 0) (0 1) (1 0) (0 120572) (120573 0) (119886lowast 119887lowast) (119886 119887) where 120572 120573 119886lowast

119887lowast 119886 and 119887 are complicated expressions of the reproductive

success rate 120574119894119895and 119886 119887 are given by

119886 =Δ1

Δ0

119887 =Δ2

Δ0

(10)

where

Δ1= minus1205741312057423+ 1205741312057422minus 1205741212057423+ 1205741212057433minus 1205742212057433

+ 12057423

2

Δ2= minus1205741212057413+ 1205741212057433minus 1205741312057423minus 1205741112057433+ 1205741112057423

+ 12057413

2

Δ0= 12057412

2+ 12057413

2+ 12057423

2minus 21205741212057413minus 21205741212057423minus 1205741112057433

+ 21205741212057433minus 1205741112057422minus 21205741312057423minus 1205742212057433

+ 21205741312057422+ 21205741112057423

(11)

Considering the biological context we assume that the steadystate solution (119886 119887) is positive and 0 lt 119906

1 1199062lt 1

In the present paper we focus on the bifurcation from theconstant solution (119886 119887) Let

1199061= 119886 + 119906

1015840

1

1199062= 119887 + 119906

1015840

2

(12)

Omitting the primes then system (9) becomes

1205971199061

120597119905= Δ1199061+ 119886 (120574

11minus 12057413) 1199061+ 119886 (120574

12minus 12057413) 1199062

+ 1198661(1199061 1199062)

1205971199062

120597119905= 119889Δ119906

2+ 119887 (120574

12minus 12057423) 1199061+ 119887 (120574

22minus 12057423) 1199062

+ 1198662(1199061 1199062)

(13)

where

1198661(1199061 1199062) = (119886

4119887 + 3119886

5119886 + 1198861) 1199061

2

+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

+ 11988651199061

3+ 119906111988631199062

2+ 11988641199061

21199062

1198662(1199061 1199062) = 119887119886

51199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062

+ (1198871+ 31198863119887 + 1198864119886) 1199062

2+ 11988631199062

3

+ 119886411990611199062

2+ 119886511990621199061

2

(14)

We assume that system (13) is satisfied on an openbounded domainΩ sub 119877

2 There are two types of biologicallysound boundary conditions the Dirichlet boundary condi-tion

1199061

1003816100381610038161003816120597Ω = 0

1199062

1003816100381610038161003816120597Ω = 0(15)

which means that the frequency is extinct in the boundary ofrange and the Neumann boundary condition

1205971199061

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

1205971199062

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(16)

which means that the frequency is invariant in the boundaryof range in biological significant

Define the function spaces

119867 = 1198712(Ω 1198772)

1198671=

1198672(Ω 1198772) cap 119867

1

0(Ω 1198772) for boundary conditions (15)

119906 isin 1198672(Ω 1198772)

10038161003816100381610038161003816100381610038161003816

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0 for boundary conditions (16)

(17)

It is clear that119867 and1198671are two Hilbert space and119867

1997893rarr 119867

is dense and compact inclusionLater we choose the bifurcation parameter 120582 to be the

diffusion coefficient 119889 that is 120582 = 119889 Let 119871120582 1198671rarr 119867 be

defined by

119871120582fl minus119860

120582+ 119861 (18)

whereminus119860120582119906 = (Δ119906

1 120582Δ1199062)119879

119861119906 = (11988611

11988612

11988621

11988622

)(1199061

1199062

)

(19)

where 119906 = (1199061 1199062)119879isin 1198671 11988611= 119886(12057411minus 12057413) 11988612= 119886(12057412minus

12057413) 11988621= 119887(12057412minus 12057423) and 119886

22= 119887(12057422minus 12057423)

4 Discrete Dynamics in Nature and Society

Furthermore let 119866(1199061 1199062) be given by

119866 (1199061 1199062) = (119866

2

1(1199061 1199062) + 1198663

1(1199061 1199062) 1198662

2(1199061 1199062)

+ 1198663

2(1199061 1199062))119879

(20)

where

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(1198661

3(1199061 1199062)

1198662

3(1199061 1199062)) = (

11988651199061

3+ 119906111988631199062

2+ 11988641199061

21199062

11988631199062

3+ 119886411990611199062

2+ 11988651199061

21199062

)

(21)

Then (13) can be written in the following operator form

119889119906

119889119905= 119871120582119906 + 119866 (119906) (22)

3 Principle of Exchange of Stability

From the theoretical ecology it is interesting to study thebifurcation of system (9) at steady state (119886 119887) Bifurcationmeans that a change in the stability or in the types ofsteady state which occurs as a parameter is varied in adissipative dynamic system that is the state changes duringthe biology conditions The classical bifurcation types areHopf bifurcation and Turing bifurcation Ma and Wang[13 14] have developed new methods to study bifurca-tions and transitions which are called attractor bifurcationsThis theory yields complete information about bifurcationstransitions stability and persistence including informationabout transient states in terms of the physical parametersof the system Therefore in this section we consider theattractor bifurcation of system (9) at (119886 119887) and from thetransformation we only need to discuss system (13) at (0 0)

Firstly we consider the linear system of (13)

1205971199061

120597119905= Δ1199061+ 119886111199061+ 119886121199062

1205971199062

120597119905= 120582Δ119906

2+ 119886211199061+ 119886221199062

(23)

and its eigenvalue problem

119871120582120593 = 120573 (120582) 120593 120593 isin 119867

1 (24)

minusΔ119890119896= 120588119896119890119896 (25)

with the Dirichlet boundary condition (15) or the Neumannboundary condition (16)

Let119872119896be defined by

119872119896= (

11988611minus 120588119896

11988612

11988621

11988622minus 120582120588119896

) (26)

It is easy to see that any eigenvector 120601119896and eigenvalue 120573

119896

of (24) can be expressed as

120601119896= (

1205851198961119890119896

1205851198962119890119896

)

119872120582

119896(1205851198961

1205851198962

) = 120573119896(1205851198961

1205851198962

)

(27)

where 119890119896is as in (25) and 120573

119896is also the eigenvalue of 119861120582

119896 By

(24) 120573119896can be written as

1205731119896= minus

1

2[(120588119896minus 11988611) + (120582120588

119896minus 11988622)]

+1

2

radic[(120588119896minus 11988611) minus (120582120588

119896minus 11988622)]2

+ 41198861211988621

1205732119896= minus

1

2[(120588119896minus 11988611) + (120582120588

119896minus 11988622)]

minus1

2

radic[(120588119896minus 11988611) minus (120582120588

119896minus 11988622)]2

+ 41198861211988621

(28)

It is clear that 1205732119896(120582) lt 120573

1119896(120582) = 0 if and only if

(120588119896minus 11988611) + (120582120588

119896minus 11988622) gt 0

(120588119896minus 11988611) (120582120588119896minus 11988622) minus 1198861211988621= 0

(29)

We define a parameter

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(30)

where 120588119896= 120588119870such that 120594

119870(120582) attains its minimum values

120594119870(120582) = min

120588119896

120582120588119896+1198861111988622minus 1198861211988621

120588119896

minus (12058211988611+ 11988622)

= 120582120588119870+1198861111988622minus 1198861211988621

120588119870

minus (12058211988611+ 11988622)

(31)

Theorem 1 Let 1205820be the number given in (30) such that (29)

is satisfied and let119870 ge 1 be the integer such that the minimumis achieved at 120588

119870 Then 120573

1119870(120582) is the first real eigenvalue of 119871

120582

near 120582 = 1205820satisfying that

1205731119870(120582)

lt 0 120582 isin Λ+

119870

= 0 120582 = 1205820

gt 0 120582 isin Λminus

119870

Re1205732119896(1205820) lt 0 forall119896 isin N

Re1205731119896(1205820) lt 0 forall119896 = 119870

(32)

Here Λ+119870= 120582 | 120594

119870(120582) gt 0 and Λminus

119870= 120582 | 120594

119870(120582) lt 0

In the absence of diffusion system (23) becomes thespatial homogeneous system

1198891199061

119889119905= 119886111199061+ 119886121199062

1198891199062

119889119905= 119886211199061+ 119886221199062

(33)

Discrete Dynamics in Nature and Society 5

System (33) is local asymptotic stability if

11988611+ 11988622lt 0

1198861111988622minus 1198861211988621gt 0

(34)

and the Hopf bifurcation occurs when

11988611+ 11988622= 0

1198861111988622minus 1198861211988621lt 0

(35)

From Theorem 1 we can infer that if condition (34) and120582 isin Λ

minus

119870hold then the homogeneous attracting equilibrium

loses stability due to the interaction of diffusion processes andsystem (23) undergoes a Turing bifurcation

4 Phase Transition on Homogeneous State

Hereafter we always assume that the eigenvalue 1205731119870(120582) in

(24) is simple Based onTheorem 1 as 120582 isin Λminus

119870the transition

of (22) occurs at 120582 = 1205820 which is from real eigenvalues

The following is the main theorem in this paper whichprovides not only a precise criterion for the transition typesof (22) but also globally dynamical behaviors

Theorem 2 Let 120588119870be defined in Theorem 1 and 119890

119870is the

corresponding eigenvector to 120588119870of (25) satisfying

intΩ

1198903

119870119889119909 = 0 (36)

For system (22) we have the following assertions(1) Equation (22) has a mixed transition from (0 120582

0) more

precisely there exists a neighborhood119880 isin 119883 of 119906 = 0 such that119880 is separated into two disjoint open sets 119880120582

1and 119880

120582

2by the

stable manifold Γ120582of 119906 = 0 satisfying the following

(a) 119880 = 119880120582

1+119880120582

2+ Γ120582 (b) The transition in 119880120582

1is jump (c)

The transition in 1198801205822is continuous

(2) Equation (22) bifurcates in 119880120582

2to a unique singular

point V120582 on120582 isin Λ+

119870which is attractor such that for any120601 isin 119880

120582

2

lim119905rarrinfin

10038171003817100381710038171003817119906 (119905 120601) minus V120582

10038171003817100381710038171003817119883= 0 (37)

(3) Equation (22) bifurcates on 120582 isin Λminus

119870to a unique saddle

point with morse index 1(4) The bifurcated singular point V120582 can be expressed by

V120582 = minus1205731119870(120582)

119887120593119870+ 119900 (

10038171003817100381710038171205731119870 (120582)1003817100381710038171003817)

(38)

Here

119887 =1

1198861211988621+ (11988611minus 120588119870)2[minus119887 (119887

2+ 1198864119887 + 2119886

5119886)

sdot (1198861198863(119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (1198862+ 21198863119887 + 2119886

4119886) 119886 (119886

2+ 21198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870) + (119886

4119887 + 3119886

5119886 + 1198861)

sdot 1198862(1198862+ 21198863119887 + 1198864119886)2

) + (119886 (1198861+ 1198864119887 + 2119886

5119886)

minus 120588119870) ((1198871+ 31198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (21198865119886 + 1198872+ 21198864119887)

sdot 119886 (1198862+ 21198863119887 + 1198864119886) (119886 (119886

1+ 1198864119887 + 2119886

5119886) minus 120588

119870)

+ 11988711988651198862(1198862+ 21198863119887 + 1198864119886)2

)]

(39)

Proof First we need to get the reduced equation of (22) near120582 = 120582

0

Let 119906 = 119909 sdot 120601119870+ 119910 where 119910 = sum

119899

119894=1119910119894120601119894and 120601

119870is the

eigenvector of (24) corresponding to 1205731119870(120582) at 120582 = 120582

0 Then

the reduced equation of (22) reads

119889119909

119889t= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870+ 119910) 120601

lowast

119870⟩ (40)

Here 120601lowast119870is the conjugate eigenvector of 120601

119870

By Implicit FunctionTheorem we can obtain that

119910 = Φ (119909 120582) = 119900 (119909) (41)

Substituting (41) into (40) we get the bifurcation equation of(22) as follows

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870) 120601lowast

119870⟩

+ 119900 (2)

(42)

By (27) 120601119870is written as

120601119870= (1205851119890119870 1205852119890119870)119879

(43)

with (1205851 1205852) satisfying

(11988611minus 120588119870

11988612

11988621

11988622minus 120582120588K

)(1205851

1205852

) = 1205731119870(1205820) (

1205851

1205852

) (44)

from which we get

(1205851 1205852) = (minus119886

12 11988611minus 120588119870) (45)

Likewise 120601lowast119870is

120601lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(46)

6 Discrete Dynamics in Nature and Society

with (120585lowast1 120585lowast

2) satisfying

(11988611minus 120588119870

11988621

11988612

11988622minus 120582120588119870

)(1205851

1205852

) = 1205731119870(1205820) (

120585lowast

1

120585lowast

2

) (47)

which yields

(120585lowast

1 120585lowast

2) = (minus119886

21 11988611minus 120588119870) (48)

By (22) the nonlinear operator 119866 is

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(49)

Then in view of (45) and (48) by direct computation wederive that

⟨119866 (1199091205851119890119870 1199091205852119890119870) 120601lowast

119870⟩ = [(119886

4119887 + 1198861+ 31198865119886) 1205851

2120585lowast

1

+ 120585lowast

211988651198871205851

2+ (1198872+ 21198865119886 + 2119886

4119887) 12058521205851120585lowast

2

+ (1198871+ 1198864119886 + 3119886

3119887) 1205852

2120585lowast

2

+ (21198863119887 + 1198862+ 21198864119886) 12058521205851120585lowast

1+ 120585lowast

111988631198861205852

2]

sdot 1199092intΩ

1198903

119870119889119909 + 119900 (119909

2)

(50)

By (45) and (48) we have

⟨120601 120601lowast⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909 (51)

Hence by (50) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 119887119909

2+ 119900 (119909

2) (52)

where 119887 is the parameter as in (39) Based onTheorem A2 in[16] this theorem follows from (52) The proof is complete

Remark 3 If the domain Ω = (0 119871) times 119863 with 119863 isin 119877

being a bounded open set then condition (36) holds true forboundary conditions (15) and (16)

If the domainΩ = (0 119871)times119863with119863 isin 119877 being a boundedopen set then

intΩ

1198903

119870119889119909 = 0 (53)

holds true for Neumann condition (16) and Dirichlet condi-tion (15) when the number119898 in 120588

1198960= 119898212058721198712+ 1205881015840

1198960is even

Hereafter we consider the Neumann condition (16) andlet the domain Ω = (0 119871

1) times (0 119871

2) with 119871

1= 1198712

The following eigenvalue problem (24) with the equation

minusΔ119890119896= 120588119896119890119896

120597119890119896

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(54)

has eigenfunctions 119890119896and eigenvalues 120588

119896as follows

119890119896= cos11989611205871199091

1198711

cos119896212058711990921198712

119896 = (1198961 1198962) 1198961 1198962= 0 1 2

(55)

120588119896= 1205872(1198962

1

11987121

+1198962

2

11987122

) (56)

Here 1205880= 0 and 119890

0= 1

Denote 119870 = (1198701 1198702) the pair of integers satisfying (30)

and simply denote 1198701= (1198701 0) and 119870

2= (0 119870

2) then we

have the following

Theorem 4 Let 120578 be the parameter defined by

120578 =[(916) (120585 120585

lowast) + (14)119867 (120585 120585

lowast)]

1198861211988621+ (11988611minus 120588119870)2

(57)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

2120585112058522+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix and 120585 120585lowast are

defined in the proof

Assume that the eigenvalue 1205731119870(120582) satisfying (28) is

simple then for system (22) we have the following assertions

(1) This system has a transition from (0 1205820) that is 119906 = 0

is asymptotically stable for 120582 isin Λminus

119870and unstable for

120582 isin Λ+

119870which transits to a stable equilibrium at 120582 =

1205820

(2) If 120578 gt 0 then (22) has a jump transition from (0 1205820)

and bifurcates on 120582 isin Λ+

119870to exactly two saddle points

V1205821and V1205822with the Morse index one

(3) If 120578 lt 0 then (22) has a continuous transition from(0 1205820) which is an attractor bifurcation

(4) The bifurcated singular points V1205821and V1205822in the above

cases can be expressed in the following form

V12058212

= plusmn

10038161003816100381610038161003816100381610038161003816

1205731119870(120582)

120578

10038161003816100381610038161003816100381610038161003816

12

119890119870+ 119900 (

10038171003817100381710038171205731119870100381710038171003817100381712

) (58)

Proof Assertion (1) follows from (32) To prove assertions (2)and (3) we need to get the reduced equation of (22) to thecenter manifold near 120582 = 120582

0

Let 119906 = 119909 sdot 120593119870+ Φ where 120593

119870is the eigenvector of (25)

corresponding to 1205731119870(120582) at 120582 = 120582

0and Φ(119909) the center

manifold function of (22) Then the reduced equation of (22)takes the following form

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120593119870 120593lowast119870⟩⟨119866 (119909 sdot 120593

119870+ Φ) 120593

lowast

119870⟩ (59)

Here 120593lowast119870is the conjugate eigenvector of 120593

119870

By (27) 120593119870is written as

120593119870= (1205851119890119870 1205852119890119870)119879

(60)

with (1205851 1205852) satisfying (45)

Discrete Dynamics in Nature and Society 7

Likewise 120593lowast119870is

120593lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(61)

with (120585lowast1 120585lowast

2) satisfying (46)

It is known that the center manifold function

Φ (119909) = (Φ1(119909) Φ

2(119909)) = 119900 (119909

2) (62)

By using the formula for center manifold functions in [1314 16] Φ = (Φ

1 Φ2) satisfies

1198711205820Φ = minus119909

21198662(1205851119890119896 1205852119890119870) + 119900 (119909

2)

= minus11990921198902

119870(1205781

1205782

) + 119900 (1199092)

(63)

where

1205781= 11988611988631205852

2+ (1198862+ 21198864119886 + 2119886

3119887) 12058511205852

+ (1198864119887 + 3119886

5119886 + 1198861) 1205851

2

1205782= (1198871+ 31198863119887 + 1198864119886) 1205852

2+ (21198865119886 + 2119886

4119887 + 1198872) 12058511205852

+ 11988711988651205851

2

(64)

From (55) we see that

1198902

119870=1

4(1 + 119890

21198701) (1 + 119890

21198702)

=1

4(1198900+ 11989021198701

+ 11989021198702

+ 1198902119870)

(65)

Let

(Φ1

Φ2

) = (Φ0

1

Φ0

2

)1198900+ (

Φ21198701

1

Φ21198701

2

)11989021198701

+ (Φ21198702

1

Φ21198702

2

)11989021198702

+ (Φ2119870

1

Φ2119870

2

)1198902119870

(66)

It is clear that

1198711205820(Φ119896

1

Φ119896

2

)119890119896= 119872119896(Φ119896

1

Φ119896

2

)119890119896 (67)

where 119872119896is the matrix given by (26) Then it follows from

(65) and (66) that

(Φ119896

1

Φ119896

2

) = minus1

4119872minus1

119896(1205781

1205782

) + 119900 (1199092) (68)

for 119896 = 1198701198701 1198702 where

119872minus1

119896=

1

det (119872119896)[minus1205820120588119896+ 11988622

minus11988612

minus11988621

minus120588119896+ 11988611

]

det (119872119896) = 1205820120588119896

2minus 12058811989611988622minus 119886111205820120588119896+ 1198861111988622

minus 1198861211988621

(69)

Direct calculation shows that

(Φ0

1

Φ0

2

) =1199092

4 (1198861111988622minus 1198861211988621)(minus119886221205782minus 119886121205781

minus119886211205782minus 119886111205781

)

(Φ21198701

1

Φ21198701

2

)

=1199092

4det (11987221198701

)((120582012058821198701

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198701

minus 11988611) 1205781

)

(Φ21198702

1

Φ21198702

2

)

=1199092

4det (11987221198702

)((120582012058821198702

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198702

minus 11988611) 1205781

)

(Φ2119870

1

Φ2119870

2

) =1199092

4det (1198722119870)((12058201205882119870

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (1205882119870

minus 11988611) 1205781

)

(70)

Inserting (66) into (59) by (60) and (61) we get

⟨119866 (119909120593119870+ Φ) 120593

lowast

119870⟩ =

911987111198712

641199093 (120585 120585

lowast)

+11987111198712

161199093119867(120585 120585

lowast) + 119900 (119909

3)

(71)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

212058511205852

2+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix

In view of (70) it follows that

⟨120601119870 120601lowast

119870⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909

=11987111198712

4[1198861211988621+ (11988611minus 120588119870)2

]

(72)

Hence by (73) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 120578119909

3+ 119900 (119909

3) (73)

Based onTheoremA1 in [16] this theorem follows from (73)The proof is complete

5 Concluding Remarks and Example

In this paper we have studied the Turing bifurcation intro-duced by Alan Turing and attractor bifurcation developedby Ma and Wang [13] of gene propagation population modelgoverned by reaction-diffusion equation

Theorems 2 and 4 tell us that the critical value 1205820of

diffusion constant given by

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(74)

8 Discrete Dynamics in Nature and Society

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

05228

Time step = 300

(a) 120582 = 13

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

Time step = 1200

(b) 120582 = 1586209390

Figure 1 Turing patterns in two dimensions of system (9) with 120574119894119895

defined by (75)

plays a crucial role in determining the attractor bifurcationand Turing bifurcation when the diffusion parameter 120582

crosses the critical value 1205820 and the uniform stationary state

(119886 119887) loses its stability which yields the Turing or attractorbifurcation

We give some examples to illustrate our main theories

Example 5 In system (9) we let

12057411=2773376069253

499999999994

12057422=3789926287167

499999999994

12057433=810843022017

45454545454

12057423=4789523227559

249999999997

12057412=6952222073553

499999999994

12057413=436265576135

249999999997

(75)

and then the steady state (119886 119887) of system (9) is

119886 =732070686076807741207077

1417067211664436233221844

119887 =380119934758603124485687

708533605832218116610922

(76)

where 0 lt 119886 and 119887 lt 1

If we set Ω = (0 40120587) times (0 60120587) then 120588119870= 1314400

where119870 = (1 1) and then

1205820=814910923617150223684573395768233809936382690140395479526347826117214934665200

262055733351368170238516966160634367981740693955623134829331749178881678669asymp 3109685536 (77)

By Theorem 1 we obtain that the transition conditions aresatisfied

FromTheorem 4 120578 asymp minus4527181405 lt 0 and system (22)has a continuous transition from (0 120582

0) which is an attractor

bifurcationSince120588

119870+11988611lt 0 we infer that if120582 lt 120582

0 then120573

1119870(120582) gt 0

and the Turing instability occurs By numerical simulationwe have shown that the gene population model is able tosustain Turing patterns (Figure 1)

Appendix

The Expressions of 119867(120585120585lowast)

The Expressions of119867(120585 120585lowast) are as follows

119867(120585 120585lowast) = 4119886

3120585lowast

1119887Φ21198701

11205852+ 4119887120585lowast

2Φ21198701

111988651205852

+ 4120585lowast

211988651198871205851Φ21198701

2+ 4120585lowast

211988641198871205851Φ21198702

1

+ 4120585lowast

21198864119887Φ21198702

11205852+ 4120585lowast

211988641198871205851Φ21198702

2

+ 2120585lowast

211988651198871205852Φ2119870

2+ 2120585lowast

11198864119887Φ2119870

11205852

+ 4120585lowast

111988631198861205852Φ21198702

2+ 4120585lowast

111988631198871205852Φ21198701

2

+ 4120585lowast

211988651198871205851Φ21198702

2+ 6120585lowast

111988651198871205851Φ2119870

1

+ 2120585lowast

21198864119886Φ2119870

11205852+ 4120585lowast

111988641198871205851Φ21198701

1

+ 12120585lowast

111988651198871205851Φ21198701

1+ 2120585lowast

111988631198871205852Φ2119870

2

+ 2120585lowast

111988641198871205851Φ2119870

1+ 24120585lowast

111988651198871205851Φ0

1

+ 2120585lowast

111988631198871205851Φ2119870

2+ 121198863119887120585lowast

21205852Φ21198701

2

+ 241198863119887Φ0

2120585lowast

21205852+ 2120585lowast

211988721205851Φ21198702

2

+ 4120585lowast

21198864119887Φ21198701

11205852+ 4120585lowast

111988641198871205851Φ21198701

2

+ 12120585lowast

111988651198871205851Φ21198702

1+ 4120585lowast

11198864119887Φ21198702

11205852

Discrete Dynamics in Nature and Society 9

+ 8120585lowast

111988641198871205851Φ0

1+ 8120585lowast

211988651198871205851Φ0

2

+ 2120585lowast

211988641198871205851Φ2119870

1+ 8120585lowast

211988641198871205851Φ0

2

+ 4120585lowast

111988641198871205851Φ21198702

2+ 41198863Φ21198702

1120585lowast

11198871205852

+ 121198863119887120585lowast

21205852Φ21198702

2+ 4120585lowast

211988641198871205851Φ21198701

2

+ 4120585lowast

111988631198871205851Φ21198701

2+ 4120585lowast

11198864119887Φ21198701

11205852

+ 4Φ21198702

1119887120585lowast

211988651205852+ 4120585lowast

211988651198871205852Φ21198702

2

+ 4120585lowast

111988641198871205851Φ21198702

1+ 6120585lowast

211988631198871205852Φ2119870

2

+ 4120585lowast

111988631198871205851Φ21198702

2+ 4120585lowast

211988651198871205852Φ21198701

2

+ 2Φ2119870

1119887120585lowast

211988651205852+ 4120585lowast

21198872Φ0

11205852

+ 8120585lowast

111988611205851Φ0

1+ 2120585lowast

111988621205851Φ21198702

2

+ 2120585lowast

11198862Φ21198702

11205852+ 21198872Φ21198702

1120585lowast

21205852

+ 4120585lowast

211988721205851Φ0

2+ 2120585lowast

111988621205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198702

2+ 4120585lowast

11198862Φ0

11205852

+ 4120585lowast

111988611205851Φ21198701

1+ 2120585lowast

11198862Φ21198701

11205852

+ Φ2119870

11198872120585lowast

21205852+ 4120585lowast

111988611205851Φ21198702

1

+ 120585lowast

211988721205851Φ2119870

2+ 2120585lowast

211988711205852Φ2119870

2

+ 4120585lowast

111988621205851Φ0

2+ 120585lowast

11198862Φ2119870

11205852

+ 21198872120585lowast

2Φ21198701

11205852+ 2120585lowast

111988611205851Φ2119870

1

+ 8Φ0

2120585lowast

211988711205852+ 2120585lowast

211988721205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198701

2+ 8120585lowast

111988631198871205852Φ0

2

+ 2120585lowast

111988641198871205851Φ2119870

2+ 2120585lowast

211988651198871205851Φ2119870

2

+ 2120585lowast

211988641198871205851Φ2119870

2+ 2Φ2119870

11198863120585lowast

11198871205852

+ 120585lowast

111988621205851Φ2119870

2+ 8120585lowast

11198863119887Φ0

11205852

+ 8120585lowast

111988631198871205851Φ0

2+ 8120585lowast

111988641198871205851Φ0

2

+ 8120585lowast

211988641198871205851Φ0

1+ 8120585lowast

11198864119887Φ0

11205852

+ 8120585lowast

21198865119887Φ0

11205852+ 8120585lowast

21198864119887Φ0

11205852

+ 8120585lowast

211988651198871205852Φ0

2+ 4120585lowast

211988641198871205851Φ21198701

1

(A1)

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was partially supported by National Natu-ral Science Foundation of China (Grants no 11401062and no 11371386) Research Fund for the National Nat-ural Science Foundation of Chongqing CSTC (Grant nocstc2014jcyjA0080) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(Grant no KJ1400937) and the Scientific Research Founda-tion of CQUT (Grant no 2012ZD37)

References

[1] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 no 4 pp 355ndash369 1937

[2] A D Bazykin ldquoHypothetical mechanism of speciatonrdquo Evolu-tion vol 23 no 4 pp 685ndash687 1969

[3] J Pialek andNH Barton ldquoThe spread of an advantageous alleleacross a barrier the effects of randomdrift and selection againstheterozygotesrdquo Genetics vol 145 no 2 pp 493ndash504 1997

[4] T Nagylaki ldquoConditions for the existence of clinesrdquo Geneticsvol 80 no 3 pp 595ndash615 1975

[5] T Nagylaki and J F Crow ldquoContinuous selective modelsrdquoTheoretical Population Biology vol 5 no 2 pp 257ndash283 1974

[6] B H Bradshaw-Hajek and P Broadbridge ldquoA robust cubicreaction-diffusion system for gene propagationrdquo Mathematicaland Computer Modelling vol 39 no 9-10 pp 1151ndash1163 2004

[7] B H Bradshaw-Hajek P Broadbridge and G H WilliamsldquoEvolving gene frequencies in a population with three possiblealleles at a locusrdquo Mathematical and Computer Modelling vol47 no 1-2 pp 210ndash217 2008

[8] P Broadbridge B H Bradshaw G R Fulford and G K AldisldquoHuxley and Fisher equations for gene propagation an exactsolutionrdquoThe Anziam Journal vol 44 no 1 pp 11ndash20 2002

[9] A Kolmogorov I Petrovsky and N Piscounov ldquoEtudes delrsquoequation aved croissance de la quantite de matiere et sonapplication a un probleme biologiquerdquo Moscow UniversityMathematics Bulletin vol 1 pp 1ndash25 1937

[10] M Slatkin ldquoGene flow and selection in a clinerdquoGenetics vol 75no 4 pp 733ndash756 1973

[11] J M SmithMathematical Ideas in Biology Cambridge Univer-sity Press Cambridge UK 1968

[12] R A Littler ldquoLoss of variability at one locus in a finitepopulationrdquo Mathematical Biosciences vol 25 no 1-2 pp 151ndash163 1975

[13] T Ma and S Wang BifurcationTheory and Applications vol 53of Nonlinear Science Series A World Scientific Beijing China2005

[14] T Ma and S Wang Stability and Bifurcation of the NonlinearEvolution Equations Science Press Singapore 2007 (Chinese)

[15] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society of London B vol 237 no 641pp 37ndash52 1952

[16] T Ma and S Wang ldquoDynamic phase transition theory in PVTsystemsrdquo Indiana University Mathematics Journal vol 57 no 6pp 2861ndash2889 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

the ability of the allele 119860119894to occupy different zones in 2-

dimensional space either through the action of small-scalemechanism or by some native transport device

System (9) has seven constant solutions (1199061 1199062) =

(0 0) (0 1) (1 0) (0 120572) (120573 0) (119886lowast 119887lowast) (119886 119887) where 120572 120573 119886lowast

119887lowast 119886 and 119887 are complicated expressions of the reproductive

success rate 120574119894119895and 119886 119887 are given by

119886 =Δ1

Δ0

119887 =Δ2

Δ0

(10)

where

Δ1= minus1205741312057423+ 1205741312057422minus 1205741212057423+ 1205741212057433minus 1205742212057433

+ 12057423

2

Δ2= minus1205741212057413+ 1205741212057433minus 1205741312057423minus 1205741112057433+ 1205741112057423

+ 12057413

2

Δ0= 12057412

2+ 12057413

2+ 12057423

2minus 21205741212057413minus 21205741212057423minus 1205741112057433

+ 21205741212057433minus 1205741112057422minus 21205741312057423minus 1205742212057433

+ 21205741312057422+ 21205741112057423

(11)

Considering the biological context we assume that the steadystate solution (119886 119887) is positive and 0 lt 119906

1 1199062lt 1

In the present paper we focus on the bifurcation from theconstant solution (119886 119887) Let

1199061= 119886 + 119906

1015840

1

1199062= 119887 + 119906

1015840

2

(12)

Omitting the primes then system (9) becomes

1205971199061

120597119905= Δ1199061+ 119886 (120574

11minus 12057413) 1199061+ 119886 (120574

12minus 12057413) 1199062

+ 1198661(1199061 1199062)

1205971199062

120597119905= 119889Δ119906

2+ 119887 (120574

12minus 12057423) 1199061+ 119887 (120574

22minus 12057423) 1199062

+ 1198662(1199061 1199062)

(13)

where

1198661(1199061 1199062) = (119886

4119887 + 3119886

5119886 + 1198861) 1199061

2

+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

+ 11988651199061

3+ 119906111988631199062

2+ 11988641199061

21199062

1198662(1199061 1199062) = 119887119886

51199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062

+ (1198871+ 31198863119887 + 1198864119886) 1199062

2+ 11988631199062

3

+ 119886411990611199062

2+ 119886511990621199061

2

(14)

We assume that system (13) is satisfied on an openbounded domainΩ sub 119877

2 There are two types of biologicallysound boundary conditions the Dirichlet boundary condi-tion

1199061

1003816100381610038161003816120597Ω = 0

1199062

1003816100381610038161003816120597Ω = 0(15)

which means that the frequency is extinct in the boundary ofrange and the Neumann boundary condition

1205971199061

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

1205971199062

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(16)

which means that the frequency is invariant in the boundaryof range in biological significant

Define the function spaces

119867 = 1198712(Ω 1198772)

1198671=

1198672(Ω 1198772) cap 119867

1

0(Ω 1198772) for boundary conditions (15)

119906 isin 1198672(Ω 1198772)

10038161003816100381610038161003816100381610038161003816

120597119906

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0 for boundary conditions (16)

(17)

It is clear that119867 and1198671are two Hilbert space and119867

1997893rarr 119867

is dense and compact inclusionLater we choose the bifurcation parameter 120582 to be the

diffusion coefficient 119889 that is 120582 = 119889 Let 119871120582 1198671rarr 119867 be

defined by

119871120582fl minus119860

120582+ 119861 (18)

whereminus119860120582119906 = (Δ119906

1 120582Δ1199062)119879

119861119906 = (11988611

11988612

11988621

11988622

)(1199061

1199062

)

(19)

where 119906 = (1199061 1199062)119879isin 1198671 11988611= 119886(12057411minus 12057413) 11988612= 119886(12057412minus

12057413) 11988621= 119887(12057412minus 12057423) and 119886

22= 119887(12057422minus 12057423)

4 Discrete Dynamics in Nature and Society

Furthermore let 119866(1199061 1199062) be given by

119866 (1199061 1199062) = (119866

2

1(1199061 1199062) + 1198663

1(1199061 1199062) 1198662

2(1199061 1199062)

+ 1198663

2(1199061 1199062))119879

(20)

where

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(1198661

3(1199061 1199062)

1198662

3(1199061 1199062)) = (

11988651199061

3+ 119906111988631199062

2+ 11988641199061

21199062

11988631199062

3+ 119886411990611199062

2+ 11988651199061

21199062

)

(21)

Then (13) can be written in the following operator form

119889119906

119889119905= 119871120582119906 + 119866 (119906) (22)

3 Principle of Exchange of Stability

From the theoretical ecology it is interesting to study thebifurcation of system (9) at steady state (119886 119887) Bifurcationmeans that a change in the stability or in the types ofsteady state which occurs as a parameter is varied in adissipative dynamic system that is the state changes duringthe biology conditions The classical bifurcation types areHopf bifurcation and Turing bifurcation Ma and Wang[13 14] have developed new methods to study bifurca-tions and transitions which are called attractor bifurcationsThis theory yields complete information about bifurcationstransitions stability and persistence including informationabout transient states in terms of the physical parametersof the system Therefore in this section we consider theattractor bifurcation of system (9) at (119886 119887) and from thetransformation we only need to discuss system (13) at (0 0)

Firstly we consider the linear system of (13)

1205971199061

120597119905= Δ1199061+ 119886111199061+ 119886121199062

1205971199062

120597119905= 120582Δ119906

2+ 119886211199061+ 119886221199062

(23)

and its eigenvalue problem

119871120582120593 = 120573 (120582) 120593 120593 isin 119867

1 (24)

minusΔ119890119896= 120588119896119890119896 (25)

with the Dirichlet boundary condition (15) or the Neumannboundary condition (16)

Let119872119896be defined by

119872119896= (

11988611minus 120588119896

11988612

11988621

11988622minus 120582120588119896

) (26)

It is easy to see that any eigenvector 120601119896and eigenvalue 120573

119896

of (24) can be expressed as

120601119896= (

1205851198961119890119896

1205851198962119890119896

)

119872120582

119896(1205851198961

1205851198962

) = 120573119896(1205851198961

1205851198962

)

(27)

where 119890119896is as in (25) and 120573

119896is also the eigenvalue of 119861120582

119896 By

(24) 120573119896can be written as

1205731119896= minus

1

2[(120588119896minus 11988611) + (120582120588

119896minus 11988622)]

+1

2

radic[(120588119896minus 11988611) minus (120582120588

119896minus 11988622)]2

+ 41198861211988621

1205732119896= minus

1

2[(120588119896minus 11988611) + (120582120588

119896minus 11988622)]

minus1

2

radic[(120588119896minus 11988611) minus (120582120588

119896minus 11988622)]2

+ 41198861211988621

(28)

It is clear that 1205732119896(120582) lt 120573

1119896(120582) = 0 if and only if

(120588119896minus 11988611) + (120582120588

119896minus 11988622) gt 0

(120588119896minus 11988611) (120582120588119896minus 11988622) minus 1198861211988621= 0

(29)

We define a parameter

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(30)

where 120588119896= 120588119870such that 120594

119870(120582) attains its minimum values

120594119870(120582) = min

120588119896

120582120588119896+1198861111988622minus 1198861211988621

120588119896

minus (12058211988611+ 11988622)

= 120582120588119870+1198861111988622minus 1198861211988621

120588119870

minus (12058211988611+ 11988622)

(31)

Theorem 1 Let 1205820be the number given in (30) such that (29)

is satisfied and let119870 ge 1 be the integer such that the minimumis achieved at 120588

119870 Then 120573

1119870(120582) is the first real eigenvalue of 119871

120582

near 120582 = 1205820satisfying that

1205731119870(120582)

lt 0 120582 isin Λ+

119870

= 0 120582 = 1205820

gt 0 120582 isin Λminus

119870

Re1205732119896(1205820) lt 0 forall119896 isin N

Re1205731119896(1205820) lt 0 forall119896 = 119870

(32)

Here Λ+119870= 120582 | 120594

119870(120582) gt 0 and Λminus

119870= 120582 | 120594

119870(120582) lt 0

In the absence of diffusion system (23) becomes thespatial homogeneous system

1198891199061

119889119905= 119886111199061+ 119886121199062

1198891199062

119889119905= 119886211199061+ 119886221199062

(33)

Discrete Dynamics in Nature and Society 5

System (33) is local asymptotic stability if

11988611+ 11988622lt 0

1198861111988622minus 1198861211988621gt 0

(34)

and the Hopf bifurcation occurs when

11988611+ 11988622= 0

1198861111988622minus 1198861211988621lt 0

(35)

From Theorem 1 we can infer that if condition (34) and120582 isin Λ

minus

119870hold then the homogeneous attracting equilibrium

loses stability due to the interaction of diffusion processes andsystem (23) undergoes a Turing bifurcation

4 Phase Transition on Homogeneous State

Hereafter we always assume that the eigenvalue 1205731119870(120582) in

(24) is simple Based onTheorem 1 as 120582 isin Λminus

119870the transition

of (22) occurs at 120582 = 1205820 which is from real eigenvalues

The following is the main theorem in this paper whichprovides not only a precise criterion for the transition typesof (22) but also globally dynamical behaviors

Theorem 2 Let 120588119870be defined in Theorem 1 and 119890

119870is the

corresponding eigenvector to 120588119870of (25) satisfying

intΩ

1198903

119870119889119909 = 0 (36)

For system (22) we have the following assertions(1) Equation (22) has a mixed transition from (0 120582

0) more

precisely there exists a neighborhood119880 isin 119883 of 119906 = 0 such that119880 is separated into two disjoint open sets 119880120582

1and 119880

120582

2by the

stable manifold Γ120582of 119906 = 0 satisfying the following

(a) 119880 = 119880120582

1+119880120582

2+ Γ120582 (b) The transition in 119880120582

1is jump (c)

The transition in 1198801205822is continuous

(2) Equation (22) bifurcates in 119880120582

2to a unique singular

point V120582 on120582 isin Λ+

119870which is attractor such that for any120601 isin 119880

120582

2

lim119905rarrinfin

10038171003817100381710038171003817119906 (119905 120601) minus V120582

10038171003817100381710038171003817119883= 0 (37)

(3) Equation (22) bifurcates on 120582 isin Λminus

119870to a unique saddle

point with morse index 1(4) The bifurcated singular point V120582 can be expressed by

V120582 = minus1205731119870(120582)

119887120593119870+ 119900 (

10038171003817100381710038171205731119870 (120582)1003817100381710038171003817)

(38)

Here

119887 =1

1198861211988621+ (11988611minus 120588119870)2[minus119887 (119887

2+ 1198864119887 + 2119886

5119886)

sdot (1198861198863(119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (1198862+ 21198863119887 + 2119886

4119886) 119886 (119886

2+ 21198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870) + (119886

4119887 + 3119886

5119886 + 1198861)

sdot 1198862(1198862+ 21198863119887 + 1198864119886)2

) + (119886 (1198861+ 1198864119887 + 2119886

5119886)

minus 120588119870) ((1198871+ 31198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (21198865119886 + 1198872+ 21198864119887)

sdot 119886 (1198862+ 21198863119887 + 1198864119886) (119886 (119886

1+ 1198864119887 + 2119886

5119886) minus 120588

119870)

+ 11988711988651198862(1198862+ 21198863119887 + 1198864119886)2

)]

(39)

Proof First we need to get the reduced equation of (22) near120582 = 120582

0

Let 119906 = 119909 sdot 120601119870+ 119910 where 119910 = sum

119899

119894=1119910119894120601119894and 120601

119870is the

eigenvector of (24) corresponding to 1205731119870(120582) at 120582 = 120582

0 Then

the reduced equation of (22) reads

119889119909

119889t= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870+ 119910) 120601

lowast

119870⟩ (40)

Here 120601lowast119870is the conjugate eigenvector of 120601

119870

By Implicit FunctionTheorem we can obtain that

119910 = Φ (119909 120582) = 119900 (119909) (41)

Substituting (41) into (40) we get the bifurcation equation of(22) as follows

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870) 120601lowast

119870⟩

+ 119900 (2)

(42)

By (27) 120601119870is written as

120601119870= (1205851119890119870 1205852119890119870)119879

(43)

with (1205851 1205852) satisfying

(11988611minus 120588119870

11988612

11988621

11988622minus 120582120588K

)(1205851

1205852

) = 1205731119870(1205820) (

1205851

1205852

) (44)

from which we get

(1205851 1205852) = (minus119886

12 11988611minus 120588119870) (45)

Likewise 120601lowast119870is

120601lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(46)

6 Discrete Dynamics in Nature and Society

with (120585lowast1 120585lowast

2) satisfying

(11988611minus 120588119870

11988621

11988612

11988622minus 120582120588119870

)(1205851

1205852

) = 1205731119870(1205820) (

120585lowast

1

120585lowast

2

) (47)

which yields

(120585lowast

1 120585lowast

2) = (minus119886

21 11988611minus 120588119870) (48)

By (22) the nonlinear operator 119866 is

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(49)

Then in view of (45) and (48) by direct computation wederive that

⟨119866 (1199091205851119890119870 1199091205852119890119870) 120601lowast

119870⟩ = [(119886

4119887 + 1198861+ 31198865119886) 1205851

2120585lowast

1

+ 120585lowast

211988651198871205851

2+ (1198872+ 21198865119886 + 2119886

4119887) 12058521205851120585lowast

2

+ (1198871+ 1198864119886 + 3119886

3119887) 1205852

2120585lowast

2

+ (21198863119887 + 1198862+ 21198864119886) 12058521205851120585lowast

1+ 120585lowast

111988631198861205852

2]

sdot 1199092intΩ

1198903

119870119889119909 + 119900 (119909

2)

(50)

By (45) and (48) we have

⟨120601 120601lowast⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909 (51)

Hence by (50) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 119887119909

2+ 119900 (119909

2) (52)

where 119887 is the parameter as in (39) Based onTheorem A2 in[16] this theorem follows from (52) The proof is complete

Remark 3 If the domain Ω = (0 119871) times 119863 with 119863 isin 119877

being a bounded open set then condition (36) holds true forboundary conditions (15) and (16)

If the domainΩ = (0 119871)times119863with119863 isin 119877 being a boundedopen set then

intΩ

1198903

119870119889119909 = 0 (53)

holds true for Neumann condition (16) and Dirichlet condi-tion (15) when the number119898 in 120588

1198960= 119898212058721198712+ 1205881015840

1198960is even

Hereafter we consider the Neumann condition (16) andlet the domain Ω = (0 119871

1) times (0 119871

2) with 119871

1= 1198712

The following eigenvalue problem (24) with the equation

minusΔ119890119896= 120588119896119890119896

120597119890119896

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(54)

has eigenfunctions 119890119896and eigenvalues 120588

119896as follows

119890119896= cos11989611205871199091

1198711

cos119896212058711990921198712

119896 = (1198961 1198962) 1198961 1198962= 0 1 2

(55)

120588119896= 1205872(1198962

1

11987121

+1198962

2

11987122

) (56)

Here 1205880= 0 and 119890

0= 1

Denote 119870 = (1198701 1198702) the pair of integers satisfying (30)

and simply denote 1198701= (1198701 0) and 119870

2= (0 119870

2) then we

have the following

Theorem 4 Let 120578 be the parameter defined by

120578 =[(916) (120585 120585

lowast) + (14)119867 (120585 120585

lowast)]

1198861211988621+ (11988611minus 120588119870)2

(57)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

2120585112058522+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix and 120585 120585lowast are

defined in the proof

Assume that the eigenvalue 1205731119870(120582) satisfying (28) is

simple then for system (22) we have the following assertions

(1) This system has a transition from (0 1205820) that is 119906 = 0

is asymptotically stable for 120582 isin Λminus

119870and unstable for

120582 isin Λ+

119870which transits to a stable equilibrium at 120582 =

1205820

(2) If 120578 gt 0 then (22) has a jump transition from (0 1205820)

and bifurcates on 120582 isin Λ+

119870to exactly two saddle points

V1205821and V1205822with the Morse index one

(3) If 120578 lt 0 then (22) has a continuous transition from(0 1205820) which is an attractor bifurcation

(4) The bifurcated singular points V1205821and V1205822in the above

cases can be expressed in the following form

V12058212

= plusmn

10038161003816100381610038161003816100381610038161003816

1205731119870(120582)

120578

10038161003816100381610038161003816100381610038161003816

12

119890119870+ 119900 (

10038171003817100381710038171205731119870100381710038171003817100381712

) (58)

Proof Assertion (1) follows from (32) To prove assertions (2)and (3) we need to get the reduced equation of (22) to thecenter manifold near 120582 = 120582

0

Let 119906 = 119909 sdot 120593119870+ Φ where 120593

119870is the eigenvector of (25)

corresponding to 1205731119870(120582) at 120582 = 120582

0and Φ(119909) the center

manifold function of (22) Then the reduced equation of (22)takes the following form

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120593119870 120593lowast119870⟩⟨119866 (119909 sdot 120593

119870+ Φ) 120593

lowast

119870⟩ (59)

Here 120593lowast119870is the conjugate eigenvector of 120593

119870

By (27) 120593119870is written as

120593119870= (1205851119890119870 1205852119890119870)119879

(60)

with (1205851 1205852) satisfying (45)

Discrete Dynamics in Nature and Society 7

Likewise 120593lowast119870is

120593lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(61)

with (120585lowast1 120585lowast

2) satisfying (46)

It is known that the center manifold function

Φ (119909) = (Φ1(119909) Φ

2(119909)) = 119900 (119909

2) (62)

By using the formula for center manifold functions in [1314 16] Φ = (Φ

1 Φ2) satisfies

1198711205820Φ = minus119909

21198662(1205851119890119896 1205852119890119870) + 119900 (119909

2)

= minus11990921198902

119870(1205781

1205782

) + 119900 (1199092)

(63)

where

1205781= 11988611988631205852

2+ (1198862+ 21198864119886 + 2119886

3119887) 12058511205852

+ (1198864119887 + 3119886

5119886 + 1198861) 1205851

2

1205782= (1198871+ 31198863119887 + 1198864119886) 1205852

2+ (21198865119886 + 2119886

4119887 + 1198872) 12058511205852

+ 11988711988651205851

2

(64)

From (55) we see that

1198902

119870=1

4(1 + 119890

21198701) (1 + 119890

21198702)

=1

4(1198900+ 11989021198701

+ 11989021198702

+ 1198902119870)

(65)

Let

(Φ1

Φ2

) = (Φ0

1

Φ0

2

)1198900+ (

Φ21198701

1

Φ21198701

2

)11989021198701

+ (Φ21198702

1

Φ21198702

2

)11989021198702

+ (Φ2119870

1

Φ2119870

2

)1198902119870

(66)

It is clear that

1198711205820(Φ119896

1

Φ119896

2

)119890119896= 119872119896(Φ119896

1

Φ119896

2

)119890119896 (67)

where 119872119896is the matrix given by (26) Then it follows from

(65) and (66) that

(Φ119896

1

Φ119896

2

) = minus1

4119872minus1

119896(1205781

1205782

) + 119900 (1199092) (68)

for 119896 = 1198701198701 1198702 where

119872minus1

119896=

1

det (119872119896)[minus1205820120588119896+ 11988622

minus11988612

minus11988621

minus120588119896+ 11988611

]

det (119872119896) = 1205820120588119896

2minus 12058811989611988622minus 119886111205820120588119896+ 1198861111988622

minus 1198861211988621

(69)

Direct calculation shows that

(Φ0

1

Φ0

2

) =1199092

4 (1198861111988622minus 1198861211988621)(minus119886221205782minus 119886121205781

minus119886211205782minus 119886111205781

)

(Φ21198701

1

Φ21198701

2

)

=1199092

4det (11987221198701

)((120582012058821198701

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198701

minus 11988611) 1205781

)

(Φ21198702

1

Φ21198702

2

)

=1199092

4det (11987221198702

)((120582012058821198702

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198702

minus 11988611) 1205781

)

(Φ2119870

1

Φ2119870

2

) =1199092

4det (1198722119870)((12058201205882119870

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (1205882119870

minus 11988611) 1205781

)

(70)

Inserting (66) into (59) by (60) and (61) we get

⟨119866 (119909120593119870+ Φ) 120593

lowast

119870⟩ =

911987111198712

641199093 (120585 120585

lowast)

+11987111198712

161199093119867(120585 120585

lowast) + 119900 (119909

3)

(71)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

212058511205852

2+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix

In view of (70) it follows that

⟨120601119870 120601lowast

119870⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909

=11987111198712

4[1198861211988621+ (11988611minus 120588119870)2

]

(72)

Hence by (73) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 120578119909

3+ 119900 (119909

3) (73)

Based onTheoremA1 in [16] this theorem follows from (73)The proof is complete

5 Concluding Remarks and Example

In this paper we have studied the Turing bifurcation intro-duced by Alan Turing and attractor bifurcation developedby Ma and Wang [13] of gene propagation population modelgoverned by reaction-diffusion equation

Theorems 2 and 4 tell us that the critical value 1205820of

diffusion constant given by

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(74)

8 Discrete Dynamics in Nature and Society

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

05228

Time step = 300

(a) 120582 = 13

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

Time step = 1200

(b) 120582 = 1586209390

Figure 1 Turing patterns in two dimensions of system (9) with 120574119894119895

defined by (75)

plays a crucial role in determining the attractor bifurcationand Turing bifurcation when the diffusion parameter 120582

crosses the critical value 1205820 and the uniform stationary state

(119886 119887) loses its stability which yields the Turing or attractorbifurcation

We give some examples to illustrate our main theories

Example 5 In system (9) we let

12057411=2773376069253

499999999994

12057422=3789926287167

499999999994

12057433=810843022017

45454545454

12057423=4789523227559

249999999997

12057412=6952222073553

499999999994

12057413=436265576135

249999999997

(75)

and then the steady state (119886 119887) of system (9) is

119886 =732070686076807741207077

1417067211664436233221844

119887 =380119934758603124485687

708533605832218116610922

(76)

where 0 lt 119886 and 119887 lt 1

If we set Ω = (0 40120587) times (0 60120587) then 120588119870= 1314400

where119870 = (1 1) and then

1205820=814910923617150223684573395768233809936382690140395479526347826117214934665200

262055733351368170238516966160634367981740693955623134829331749178881678669asymp 3109685536 (77)

By Theorem 1 we obtain that the transition conditions aresatisfied

FromTheorem 4 120578 asymp minus4527181405 lt 0 and system (22)has a continuous transition from (0 120582

0) which is an attractor

bifurcationSince120588

119870+11988611lt 0 we infer that if120582 lt 120582

0 then120573

1119870(120582) gt 0

and the Turing instability occurs By numerical simulationwe have shown that the gene population model is able tosustain Turing patterns (Figure 1)

Appendix

The Expressions of 119867(120585120585lowast)

The Expressions of119867(120585 120585lowast) are as follows

119867(120585 120585lowast) = 4119886

3120585lowast

1119887Φ21198701

11205852+ 4119887120585lowast

2Φ21198701

111988651205852

+ 4120585lowast

211988651198871205851Φ21198701

2+ 4120585lowast

211988641198871205851Φ21198702

1

+ 4120585lowast

21198864119887Φ21198702

11205852+ 4120585lowast

211988641198871205851Φ21198702

2

+ 2120585lowast

211988651198871205852Φ2119870

2+ 2120585lowast

11198864119887Φ2119870

11205852

+ 4120585lowast

111988631198861205852Φ21198702

2+ 4120585lowast

111988631198871205852Φ21198701

2

+ 4120585lowast

211988651198871205851Φ21198702

2+ 6120585lowast

111988651198871205851Φ2119870

1

+ 2120585lowast

21198864119886Φ2119870

11205852+ 4120585lowast

111988641198871205851Φ21198701

1

+ 12120585lowast

111988651198871205851Φ21198701

1+ 2120585lowast

111988631198871205852Φ2119870

2

+ 2120585lowast

111988641198871205851Φ2119870

1+ 24120585lowast

111988651198871205851Φ0

1

+ 2120585lowast

111988631198871205851Φ2119870

2+ 121198863119887120585lowast

21205852Φ21198701

2

+ 241198863119887Φ0

2120585lowast

21205852+ 2120585lowast

211988721205851Φ21198702

2

+ 4120585lowast

21198864119887Φ21198701

11205852+ 4120585lowast

111988641198871205851Φ21198701

2

+ 12120585lowast

111988651198871205851Φ21198702

1+ 4120585lowast

11198864119887Φ21198702

11205852

Discrete Dynamics in Nature and Society 9

+ 8120585lowast

111988641198871205851Φ0

1+ 8120585lowast

211988651198871205851Φ0

2

+ 2120585lowast

211988641198871205851Φ2119870

1+ 8120585lowast

211988641198871205851Φ0

2

+ 4120585lowast

111988641198871205851Φ21198702

2+ 41198863Φ21198702

1120585lowast

11198871205852

+ 121198863119887120585lowast

21205852Φ21198702

2+ 4120585lowast

211988641198871205851Φ21198701

2

+ 4120585lowast

111988631198871205851Φ21198701

2+ 4120585lowast

11198864119887Φ21198701

11205852

+ 4Φ21198702

1119887120585lowast

211988651205852+ 4120585lowast

211988651198871205852Φ21198702

2

+ 4120585lowast

111988641198871205851Φ21198702

1+ 6120585lowast

211988631198871205852Φ2119870

2

+ 4120585lowast

111988631198871205851Φ21198702

2+ 4120585lowast

211988651198871205852Φ21198701

2

+ 2Φ2119870

1119887120585lowast

211988651205852+ 4120585lowast

21198872Φ0

11205852

+ 8120585lowast

111988611205851Φ0

1+ 2120585lowast

111988621205851Φ21198702

2

+ 2120585lowast

11198862Φ21198702

11205852+ 21198872Φ21198702

1120585lowast

21205852

+ 4120585lowast

211988721205851Φ0

2+ 2120585lowast

111988621205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198702

2+ 4120585lowast

11198862Φ0

11205852

+ 4120585lowast

111988611205851Φ21198701

1+ 2120585lowast

11198862Φ21198701

11205852

+ Φ2119870

11198872120585lowast

21205852+ 4120585lowast

111988611205851Φ21198702

1

+ 120585lowast

211988721205851Φ2119870

2+ 2120585lowast

211988711205852Φ2119870

2

+ 4120585lowast

111988621205851Φ0

2+ 120585lowast

11198862Φ2119870

11205852

+ 21198872120585lowast

2Φ21198701

11205852+ 2120585lowast

111988611205851Φ2119870

1

+ 8Φ0

2120585lowast

211988711205852+ 2120585lowast

211988721205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198701

2+ 8120585lowast

111988631198871205852Φ0

2

+ 2120585lowast

111988641198871205851Φ2119870

2+ 2120585lowast

211988651198871205851Φ2119870

2

+ 2120585lowast

211988641198871205851Φ2119870

2+ 2Φ2119870

11198863120585lowast

11198871205852

+ 120585lowast

111988621205851Φ2119870

2+ 8120585lowast

11198863119887Φ0

11205852

+ 8120585lowast

111988631198871205851Φ0

2+ 8120585lowast

111988641198871205851Φ0

2

+ 8120585lowast

211988641198871205851Φ0

1+ 8120585lowast

11198864119887Φ0

11205852

+ 8120585lowast

21198865119887Φ0

11205852+ 8120585lowast

21198864119887Φ0

11205852

+ 8120585lowast

211988651198871205852Φ0

2+ 4120585lowast

211988641198871205851Φ21198701

1

(A1)

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was partially supported by National Natu-ral Science Foundation of China (Grants no 11401062and no 11371386) Research Fund for the National Nat-ural Science Foundation of Chongqing CSTC (Grant nocstc2014jcyjA0080) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(Grant no KJ1400937) and the Scientific Research Founda-tion of CQUT (Grant no 2012ZD37)

References

[1] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 no 4 pp 355ndash369 1937

[2] A D Bazykin ldquoHypothetical mechanism of speciatonrdquo Evolu-tion vol 23 no 4 pp 685ndash687 1969

[3] J Pialek andNH Barton ldquoThe spread of an advantageous alleleacross a barrier the effects of randomdrift and selection againstheterozygotesrdquo Genetics vol 145 no 2 pp 493ndash504 1997

[4] T Nagylaki ldquoConditions for the existence of clinesrdquo Geneticsvol 80 no 3 pp 595ndash615 1975

[5] T Nagylaki and J F Crow ldquoContinuous selective modelsrdquoTheoretical Population Biology vol 5 no 2 pp 257ndash283 1974

[6] B H Bradshaw-Hajek and P Broadbridge ldquoA robust cubicreaction-diffusion system for gene propagationrdquo Mathematicaland Computer Modelling vol 39 no 9-10 pp 1151ndash1163 2004

[7] B H Bradshaw-Hajek P Broadbridge and G H WilliamsldquoEvolving gene frequencies in a population with three possiblealleles at a locusrdquo Mathematical and Computer Modelling vol47 no 1-2 pp 210ndash217 2008

[8] P Broadbridge B H Bradshaw G R Fulford and G K AldisldquoHuxley and Fisher equations for gene propagation an exactsolutionrdquoThe Anziam Journal vol 44 no 1 pp 11ndash20 2002

[9] A Kolmogorov I Petrovsky and N Piscounov ldquoEtudes delrsquoequation aved croissance de la quantite de matiere et sonapplication a un probleme biologiquerdquo Moscow UniversityMathematics Bulletin vol 1 pp 1ndash25 1937

[10] M Slatkin ldquoGene flow and selection in a clinerdquoGenetics vol 75no 4 pp 733ndash756 1973

[11] J M SmithMathematical Ideas in Biology Cambridge Univer-sity Press Cambridge UK 1968

[12] R A Littler ldquoLoss of variability at one locus in a finitepopulationrdquo Mathematical Biosciences vol 25 no 1-2 pp 151ndash163 1975

[13] T Ma and S Wang BifurcationTheory and Applications vol 53of Nonlinear Science Series A World Scientific Beijing China2005

[14] T Ma and S Wang Stability and Bifurcation of the NonlinearEvolution Equations Science Press Singapore 2007 (Chinese)

[15] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society of London B vol 237 no 641pp 37ndash52 1952

[16] T Ma and S Wang ldquoDynamic phase transition theory in PVTsystemsrdquo Indiana University Mathematics Journal vol 57 no 6pp 2861ndash2889 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Furthermore let 119866(1199061 1199062) be given by

119866 (1199061 1199062) = (119866

2

1(1199061 1199062) + 1198663

1(1199061 1199062) 1198662

2(1199061 1199062)

+ 1198663

2(1199061 1199062))119879

(20)

where

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(1198661

3(1199061 1199062)

1198662

3(1199061 1199062)) = (

11988651199061

3+ 119906111988631199062

2+ 11988641199061

21199062

11988631199062

3+ 119886411990611199062

2+ 11988651199061

21199062

)

(21)

Then (13) can be written in the following operator form

119889119906

119889119905= 119871120582119906 + 119866 (119906) (22)

3 Principle of Exchange of Stability

From the theoretical ecology it is interesting to study thebifurcation of system (9) at steady state (119886 119887) Bifurcationmeans that a change in the stability or in the types ofsteady state which occurs as a parameter is varied in adissipative dynamic system that is the state changes duringthe biology conditions The classical bifurcation types areHopf bifurcation and Turing bifurcation Ma and Wang[13 14] have developed new methods to study bifurca-tions and transitions which are called attractor bifurcationsThis theory yields complete information about bifurcationstransitions stability and persistence including informationabout transient states in terms of the physical parametersof the system Therefore in this section we consider theattractor bifurcation of system (9) at (119886 119887) and from thetransformation we only need to discuss system (13) at (0 0)

Firstly we consider the linear system of (13)

1205971199061

120597119905= Δ1199061+ 119886111199061+ 119886121199062

1205971199062

120597119905= 120582Δ119906

2+ 119886211199061+ 119886221199062

(23)

and its eigenvalue problem

119871120582120593 = 120573 (120582) 120593 120593 isin 119867

1 (24)

minusΔ119890119896= 120588119896119890119896 (25)

with the Dirichlet boundary condition (15) or the Neumannboundary condition (16)

Let119872119896be defined by

119872119896= (

11988611minus 120588119896

11988612

11988621

11988622minus 120582120588119896

) (26)

It is easy to see that any eigenvector 120601119896and eigenvalue 120573

119896

of (24) can be expressed as

120601119896= (

1205851198961119890119896

1205851198962119890119896

)

119872120582

119896(1205851198961

1205851198962

) = 120573119896(1205851198961

1205851198962

)

(27)

where 119890119896is as in (25) and 120573

119896is also the eigenvalue of 119861120582

119896 By

(24) 120573119896can be written as

1205731119896= minus

1

2[(120588119896minus 11988611) + (120582120588

119896minus 11988622)]

+1

2

radic[(120588119896minus 11988611) minus (120582120588

119896minus 11988622)]2

+ 41198861211988621

1205732119896= minus

1

2[(120588119896minus 11988611) + (120582120588

119896minus 11988622)]

minus1

2

radic[(120588119896minus 11988611) minus (120582120588

119896minus 11988622)]2

+ 41198861211988621

(28)

It is clear that 1205732119896(120582) lt 120573

1119896(120582) = 0 if and only if

(120588119896minus 11988611) + (120582120588

119896minus 11988622) gt 0

(120588119896minus 11988611) (120582120588119896minus 11988622) minus 1198861211988621= 0

(29)

We define a parameter

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(30)

where 120588119896= 120588119870such that 120594

119870(120582) attains its minimum values

120594119870(120582) = min

120588119896

120582120588119896+1198861111988622minus 1198861211988621

120588119896

minus (12058211988611+ 11988622)

= 120582120588119870+1198861111988622minus 1198861211988621

120588119870

minus (12058211988611+ 11988622)

(31)

Theorem 1 Let 1205820be the number given in (30) such that (29)

is satisfied and let119870 ge 1 be the integer such that the minimumis achieved at 120588

119870 Then 120573

1119870(120582) is the first real eigenvalue of 119871

120582

near 120582 = 1205820satisfying that

1205731119870(120582)

lt 0 120582 isin Λ+

119870

= 0 120582 = 1205820

gt 0 120582 isin Λminus

119870

Re1205732119896(1205820) lt 0 forall119896 isin N

Re1205731119896(1205820) lt 0 forall119896 = 119870

(32)

Here Λ+119870= 120582 | 120594

119870(120582) gt 0 and Λminus

119870= 120582 | 120594

119870(120582) lt 0

In the absence of diffusion system (23) becomes thespatial homogeneous system

1198891199061

119889119905= 119886111199061+ 119886121199062

1198891199062

119889119905= 119886211199061+ 119886221199062

(33)

Discrete Dynamics in Nature and Society 5

System (33) is local asymptotic stability if

11988611+ 11988622lt 0

1198861111988622minus 1198861211988621gt 0

(34)

and the Hopf bifurcation occurs when

11988611+ 11988622= 0

1198861111988622minus 1198861211988621lt 0

(35)

From Theorem 1 we can infer that if condition (34) and120582 isin Λ

minus

119870hold then the homogeneous attracting equilibrium

loses stability due to the interaction of diffusion processes andsystem (23) undergoes a Turing bifurcation

4 Phase Transition on Homogeneous State

Hereafter we always assume that the eigenvalue 1205731119870(120582) in

(24) is simple Based onTheorem 1 as 120582 isin Λminus

119870the transition

of (22) occurs at 120582 = 1205820 which is from real eigenvalues

The following is the main theorem in this paper whichprovides not only a precise criterion for the transition typesof (22) but also globally dynamical behaviors

Theorem 2 Let 120588119870be defined in Theorem 1 and 119890

119870is the

corresponding eigenvector to 120588119870of (25) satisfying

intΩ

1198903

119870119889119909 = 0 (36)

For system (22) we have the following assertions(1) Equation (22) has a mixed transition from (0 120582

0) more

precisely there exists a neighborhood119880 isin 119883 of 119906 = 0 such that119880 is separated into two disjoint open sets 119880120582

1and 119880

120582

2by the

stable manifold Γ120582of 119906 = 0 satisfying the following

(a) 119880 = 119880120582

1+119880120582

2+ Γ120582 (b) The transition in 119880120582

1is jump (c)

The transition in 1198801205822is continuous

(2) Equation (22) bifurcates in 119880120582

2to a unique singular

point V120582 on120582 isin Λ+

119870which is attractor such that for any120601 isin 119880

120582

2

lim119905rarrinfin

10038171003817100381710038171003817119906 (119905 120601) minus V120582

10038171003817100381710038171003817119883= 0 (37)

(3) Equation (22) bifurcates on 120582 isin Λminus

119870to a unique saddle

point with morse index 1(4) The bifurcated singular point V120582 can be expressed by

V120582 = minus1205731119870(120582)

119887120593119870+ 119900 (

10038171003817100381710038171205731119870 (120582)1003817100381710038171003817)

(38)

Here

119887 =1

1198861211988621+ (11988611minus 120588119870)2[minus119887 (119887

2+ 1198864119887 + 2119886

5119886)

sdot (1198861198863(119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (1198862+ 21198863119887 + 2119886

4119886) 119886 (119886

2+ 21198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870) + (119886

4119887 + 3119886

5119886 + 1198861)

sdot 1198862(1198862+ 21198863119887 + 1198864119886)2

) + (119886 (1198861+ 1198864119887 + 2119886

5119886)

minus 120588119870) ((1198871+ 31198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (21198865119886 + 1198872+ 21198864119887)

sdot 119886 (1198862+ 21198863119887 + 1198864119886) (119886 (119886

1+ 1198864119887 + 2119886

5119886) minus 120588

119870)

+ 11988711988651198862(1198862+ 21198863119887 + 1198864119886)2

)]

(39)

Proof First we need to get the reduced equation of (22) near120582 = 120582

0

Let 119906 = 119909 sdot 120601119870+ 119910 where 119910 = sum

119899

119894=1119910119894120601119894and 120601

119870is the

eigenvector of (24) corresponding to 1205731119870(120582) at 120582 = 120582

0 Then

the reduced equation of (22) reads

119889119909

119889t= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870+ 119910) 120601

lowast

119870⟩ (40)

Here 120601lowast119870is the conjugate eigenvector of 120601

119870

By Implicit FunctionTheorem we can obtain that

119910 = Φ (119909 120582) = 119900 (119909) (41)

Substituting (41) into (40) we get the bifurcation equation of(22) as follows

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870) 120601lowast

119870⟩

+ 119900 (2)

(42)

By (27) 120601119870is written as

120601119870= (1205851119890119870 1205852119890119870)119879

(43)

with (1205851 1205852) satisfying

(11988611minus 120588119870

11988612

11988621

11988622minus 120582120588K

)(1205851

1205852

) = 1205731119870(1205820) (

1205851

1205852

) (44)

from which we get

(1205851 1205852) = (minus119886

12 11988611minus 120588119870) (45)

Likewise 120601lowast119870is

120601lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(46)

6 Discrete Dynamics in Nature and Society

with (120585lowast1 120585lowast

2) satisfying

(11988611minus 120588119870

11988621

11988612

11988622minus 120582120588119870

)(1205851

1205852

) = 1205731119870(1205820) (

120585lowast

1

120585lowast

2

) (47)

which yields

(120585lowast

1 120585lowast

2) = (minus119886

21 11988611minus 120588119870) (48)

By (22) the nonlinear operator 119866 is

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(49)

Then in view of (45) and (48) by direct computation wederive that

⟨119866 (1199091205851119890119870 1199091205852119890119870) 120601lowast

119870⟩ = [(119886

4119887 + 1198861+ 31198865119886) 1205851

2120585lowast

1

+ 120585lowast

211988651198871205851

2+ (1198872+ 21198865119886 + 2119886

4119887) 12058521205851120585lowast

2

+ (1198871+ 1198864119886 + 3119886

3119887) 1205852

2120585lowast

2

+ (21198863119887 + 1198862+ 21198864119886) 12058521205851120585lowast

1+ 120585lowast

111988631198861205852

2]

sdot 1199092intΩ

1198903

119870119889119909 + 119900 (119909

2)

(50)

By (45) and (48) we have

⟨120601 120601lowast⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909 (51)

Hence by (50) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 119887119909

2+ 119900 (119909

2) (52)

where 119887 is the parameter as in (39) Based onTheorem A2 in[16] this theorem follows from (52) The proof is complete

Remark 3 If the domain Ω = (0 119871) times 119863 with 119863 isin 119877

being a bounded open set then condition (36) holds true forboundary conditions (15) and (16)

If the domainΩ = (0 119871)times119863with119863 isin 119877 being a boundedopen set then

intΩ

1198903

119870119889119909 = 0 (53)

holds true for Neumann condition (16) and Dirichlet condi-tion (15) when the number119898 in 120588

1198960= 119898212058721198712+ 1205881015840

1198960is even

Hereafter we consider the Neumann condition (16) andlet the domain Ω = (0 119871

1) times (0 119871

2) with 119871

1= 1198712

The following eigenvalue problem (24) with the equation

minusΔ119890119896= 120588119896119890119896

120597119890119896

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(54)

has eigenfunctions 119890119896and eigenvalues 120588

119896as follows

119890119896= cos11989611205871199091

1198711

cos119896212058711990921198712

119896 = (1198961 1198962) 1198961 1198962= 0 1 2

(55)

120588119896= 1205872(1198962

1

11987121

+1198962

2

11987122

) (56)

Here 1205880= 0 and 119890

0= 1

Denote 119870 = (1198701 1198702) the pair of integers satisfying (30)

and simply denote 1198701= (1198701 0) and 119870

2= (0 119870

2) then we

have the following

Theorem 4 Let 120578 be the parameter defined by

120578 =[(916) (120585 120585

lowast) + (14)119867 (120585 120585

lowast)]

1198861211988621+ (11988611minus 120588119870)2

(57)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

2120585112058522+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix and 120585 120585lowast are

defined in the proof

Assume that the eigenvalue 1205731119870(120582) satisfying (28) is

simple then for system (22) we have the following assertions

(1) This system has a transition from (0 1205820) that is 119906 = 0

is asymptotically stable for 120582 isin Λminus

119870and unstable for

120582 isin Λ+

119870which transits to a stable equilibrium at 120582 =

1205820

(2) If 120578 gt 0 then (22) has a jump transition from (0 1205820)

and bifurcates on 120582 isin Λ+

119870to exactly two saddle points

V1205821and V1205822with the Morse index one

(3) If 120578 lt 0 then (22) has a continuous transition from(0 1205820) which is an attractor bifurcation

(4) The bifurcated singular points V1205821and V1205822in the above

cases can be expressed in the following form

V12058212

= plusmn

10038161003816100381610038161003816100381610038161003816

1205731119870(120582)

120578

10038161003816100381610038161003816100381610038161003816

12

119890119870+ 119900 (

10038171003817100381710038171205731119870100381710038171003817100381712

) (58)

Proof Assertion (1) follows from (32) To prove assertions (2)and (3) we need to get the reduced equation of (22) to thecenter manifold near 120582 = 120582

0

Let 119906 = 119909 sdot 120593119870+ Φ where 120593

119870is the eigenvector of (25)

corresponding to 1205731119870(120582) at 120582 = 120582

0and Φ(119909) the center

manifold function of (22) Then the reduced equation of (22)takes the following form

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120593119870 120593lowast119870⟩⟨119866 (119909 sdot 120593

119870+ Φ) 120593

lowast

119870⟩ (59)

Here 120593lowast119870is the conjugate eigenvector of 120593

119870

By (27) 120593119870is written as

120593119870= (1205851119890119870 1205852119890119870)119879

(60)

with (1205851 1205852) satisfying (45)

Discrete Dynamics in Nature and Society 7

Likewise 120593lowast119870is

120593lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(61)

with (120585lowast1 120585lowast

2) satisfying (46)

It is known that the center manifold function

Φ (119909) = (Φ1(119909) Φ

2(119909)) = 119900 (119909

2) (62)

By using the formula for center manifold functions in [1314 16] Φ = (Φ

1 Φ2) satisfies

1198711205820Φ = minus119909

21198662(1205851119890119896 1205852119890119870) + 119900 (119909

2)

= minus11990921198902

119870(1205781

1205782

) + 119900 (1199092)

(63)

where

1205781= 11988611988631205852

2+ (1198862+ 21198864119886 + 2119886

3119887) 12058511205852

+ (1198864119887 + 3119886

5119886 + 1198861) 1205851

2

1205782= (1198871+ 31198863119887 + 1198864119886) 1205852

2+ (21198865119886 + 2119886

4119887 + 1198872) 12058511205852

+ 11988711988651205851

2

(64)

From (55) we see that

1198902

119870=1

4(1 + 119890

21198701) (1 + 119890

21198702)

=1

4(1198900+ 11989021198701

+ 11989021198702

+ 1198902119870)

(65)

Let

(Φ1

Φ2

) = (Φ0

1

Φ0

2

)1198900+ (

Φ21198701

1

Φ21198701

2

)11989021198701

+ (Φ21198702

1

Φ21198702

2

)11989021198702

+ (Φ2119870

1

Φ2119870

2

)1198902119870

(66)

It is clear that

1198711205820(Φ119896

1

Φ119896

2

)119890119896= 119872119896(Φ119896

1

Φ119896

2

)119890119896 (67)

where 119872119896is the matrix given by (26) Then it follows from

(65) and (66) that

(Φ119896

1

Φ119896

2

) = minus1

4119872minus1

119896(1205781

1205782

) + 119900 (1199092) (68)

for 119896 = 1198701198701 1198702 where

119872minus1

119896=

1

det (119872119896)[minus1205820120588119896+ 11988622

minus11988612

minus11988621

minus120588119896+ 11988611

]

det (119872119896) = 1205820120588119896

2minus 12058811989611988622minus 119886111205820120588119896+ 1198861111988622

minus 1198861211988621

(69)

Direct calculation shows that

(Φ0

1

Φ0

2

) =1199092

4 (1198861111988622minus 1198861211988621)(minus119886221205782minus 119886121205781

minus119886211205782minus 119886111205781

)

(Φ21198701

1

Φ21198701

2

)

=1199092

4det (11987221198701

)((120582012058821198701

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198701

minus 11988611) 1205781

)

(Φ21198702

1

Φ21198702

2

)

=1199092

4det (11987221198702

)((120582012058821198702

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198702

minus 11988611) 1205781

)

(Φ2119870

1

Φ2119870

2

) =1199092

4det (1198722119870)((12058201205882119870

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (1205882119870

minus 11988611) 1205781

)

(70)

Inserting (66) into (59) by (60) and (61) we get

⟨119866 (119909120593119870+ Φ) 120593

lowast

119870⟩ =

911987111198712

641199093 (120585 120585

lowast)

+11987111198712

161199093119867(120585 120585

lowast) + 119900 (119909

3)

(71)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

212058511205852

2+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix

In view of (70) it follows that

⟨120601119870 120601lowast

119870⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909

=11987111198712

4[1198861211988621+ (11988611minus 120588119870)2

]

(72)

Hence by (73) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 120578119909

3+ 119900 (119909

3) (73)

Based onTheoremA1 in [16] this theorem follows from (73)The proof is complete

5 Concluding Remarks and Example

In this paper we have studied the Turing bifurcation intro-duced by Alan Turing and attractor bifurcation developedby Ma and Wang [13] of gene propagation population modelgoverned by reaction-diffusion equation

Theorems 2 and 4 tell us that the critical value 1205820of

diffusion constant given by

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(74)

8 Discrete Dynamics in Nature and Society

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

05228

Time step = 300

(a) 120582 = 13

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

Time step = 1200

(b) 120582 = 1586209390

Figure 1 Turing patterns in two dimensions of system (9) with 120574119894119895

defined by (75)

plays a crucial role in determining the attractor bifurcationand Turing bifurcation when the diffusion parameter 120582

crosses the critical value 1205820 and the uniform stationary state

(119886 119887) loses its stability which yields the Turing or attractorbifurcation

We give some examples to illustrate our main theories

Example 5 In system (9) we let

12057411=2773376069253

499999999994

12057422=3789926287167

499999999994

12057433=810843022017

45454545454

12057423=4789523227559

249999999997

12057412=6952222073553

499999999994

12057413=436265576135

249999999997

(75)

and then the steady state (119886 119887) of system (9) is

119886 =732070686076807741207077

1417067211664436233221844

119887 =380119934758603124485687

708533605832218116610922

(76)

where 0 lt 119886 and 119887 lt 1

If we set Ω = (0 40120587) times (0 60120587) then 120588119870= 1314400

where119870 = (1 1) and then

1205820=814910923617150223684573395768233809936382690140395479526347826117214934665200

262055733351368170238516966160634367981740693955623134829331749178881678669asymp 3109685536 (77)

By Theorem 1 we obtain that the transition conditions aresatisfied

FromTheorem 4 120578 asymp minus4527181405 lt 0 and system (22)has a continuous transition from (0 120582

0) which is an attractor

bifurcationSince120588

119870+11988611lt 0 we infer that if120582 lt 120582

0 then120573

1119870(120582) gt 0

and the Turing instability occurs By numerical simulationwe have shown that the gene population model is able tosustain Turing patterns (Figure 1)

Appendix

The Expressions of 119867(120585120585lowast)

The Expressions of119867(120585 120585lowast) are as follows

119867(120585 120585lowast) = 4119886

3120585lowast

1119887Φ21198701

11205852+ 4119887120585lowast

2Φ21198701

111988651205852

+ 4120585lowast

211988651198871205851Φ21198701

2+ 4120585lowast

211988641198871205851Φ21198702

1

+ 4120585lowast

21198864119887Φ21198702

11205852+ 4120585lowast

211988641198871205851Φ21198702

2

+ 2120585lowast

211988651198871205852Φ2119870

2+ 2120585lowast

11198864119887Φ2119870

11205852

+ 4120585lowast

111988631198861205852Φ21198702

2+ 4120585lowast

111988631198871205852Φ21198701

2

+ 4120585lowast

211988651198871205851Φ21198702

2+ 6120585lowast

111988651198871205851Φ2119870

1

+ 2120585lowast

21198864119886Φ2119870

11205852+ 4120585lowast

111988641198871205851Φ21198701

1

+ 12120585lowast

111988651198871205851Φ21198701

1+ 2120585lowast

111988631198871205852Φ2119870

2

+ 2120585lowast

111988641198871205851Φ2119870

1+ 24120585lowast

111988651198871205851Φ0

1

+ 2120585lowast

111988631198871205851Φ2119870

2+ 121198863119887120585lowast

21205852Φ21198701

2

+ 241198863119887Φ0

2120585lowast

21205852+ 2120585lowast

211988721205851Φ21198702

2

+ 4120585lowast

21198864119887Φ21198701

11205852+ 4120585lowast

111988641198871205851Φ21198701

2

+ 12120585lowast

111988651198871205851Φ21198702

1+ 4120585lowast

11198864119887Φ21198702

11205852

Discrete Dynamics in Nature and Society 9

+ 8120585lowast

111988641198871205851Φ0

1+ 8120585lowast

211988651198871205851Φ0

2

+ 2120585lowast

211988641198871205851Φ2119870

1+ 8120585lowast

211988641198871205851Φ0

2

+ 4120585lowast

111988641198871205851Φ21198702

2+ 41198863Φ21198702

1120585lowast

11198871205852

+ 121198863119887120585lowast

21205852Φ21198702

2+ 4120585lowast

211988641198871205851Φ21198701

2

+ 4120585lowast

111988631198871205851Φ21198701

2+ 4120585lowast

11198864119887Φ21198701

11205852

+ 4Φ21198702

1119887120585lowast

211988651205852+ 4120585lowast

211988651198871205852Φ21198702

2

+ 4120585lowast

111988641198871205851Φ21198702

1+ 6120585lowast

211988631198871205852Φ2119870

2

+ 4120585lowast

111988631198871205851Φ21198702

2+ 4120585lowast

211988651198871205852Φ21198701

2

+ 2Φ2119870

1119887120585lowast

211988651205852+ 4120585lowast

21198872Φ0

11205852

+ 8120585lowast

111988611205851Φ0

1+ 2120585lowast

111988621205851Φ21198702

2

+ 2120585lowast

11198862Φ21198702

11205852+ 21198872Φ21198702

1120585lowast

21205852

+ 4120585lowast

211988721205851Φ0

2+ 2120585lowast

111988621205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198702

2+ 4120585lowast

11198862Φ0

11205852

+ 4120585lowast

111988611205851Φ21198701

1+ 2120585lowast

11198862Φ21198701

11205852

+ Φ2119870

11198872120585lowast

21205852+ 4120585lowast

111988611205851Φ21198702

1

+ 120585lowast

211988721205851Φ2119870

2+ 2120585lowast

211988711205852Φ2119870

2

+ 4120585lowast

111988621205851Φ0

2+ 120585lowast

11198862Φ2119870

11205852

+ 21198872120585lowast

2Φ21198701

11205852+ 2120585lowast

111988611205851Φ2119870

1

+ 8Φ0

2120585lowast

211988711205852+ 2120585lowast

211988721205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198701

2+ 8120585lowast

111988631198871205852Φ0

2

+ 2120585lowast

111988641198871205851Φ2119870

2+ 2120585lowast

211988651198871205851Φ2119870

2

+ 2120585lowast

211988641198871205851Φ2119870

2+ 2Φ2119870

11198863120585lowast

11198871205852

+ 120585lowast

111988621205851Φ2119870

2+ 8120585lowast

11198863119887Φ0

11205852

+ 8120585lowast

111988631198871205851Φ0

2+ 8120585lowast

111988641198871205851Φ0

2

+ 8120585lowast

211988641198871205851Φ0

1+ 8120585lowast

11198864119887Φ0

11205852

+ 8120585lowast

21198865119887Φ0

11205852+ 8120585lowast

21198864119887Φ0

11205852

+ 8120585lowast

211988651198871205852Φ0

2+ 4120585lowast

211988641198871205851Φ21198701

1

(A1)

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was partially supported by National Natu-ral Science Foundation of China (Grants no 11401062and no 11371386) Research Fund for the National Nat-ural Science Foundation of Chongqing CSTC (Grant nocstc2014jcyjA0080) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(Grant no KJ1400937) and the Scientific Research Founda-tion of CQUT (Grant no 2012ZD37)

References

[1] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 no 4 pp 355ndash369 1937

[2] A D Bazykin ldquoHypothetical mechanism of speciatonrdquo Evolu-tion vol 23 no 4 pp 685ndash687 1969

[3] J Pialek andNH Barton ldquoThe spread of an advantageous alleleacross a barrier the effects of randomdrift and selection againstheterozygotesrdquo Genetics vol 145 no 2 pp 493ndash504 1997

[4] T Nagylaki ldquoConditions for the existence of clinesrdquo Geneticsvol 80 no 3 pp 595ndash615 1975

[5] T Nagylaki and J F Crow ldquoContinuous selective modelsrdquoTheoretical Population Biology vol 5 no 2 pp 257ndash283 1974

[6] B H Bradshaw-Hajek and P Broadbridge ldquoA robust cubicreaction-diffusion system for gene propagationrdquo Mathematicaland Computer Modelling vol 39 no 9-10 pp 1151ndash1163 2004

[7] B H Bradshaw-Hajek P Broadbridge and G H WilliamsldquoEvolving gene frequencies in a population with three possiblealleles at a locusrdquo Mathematical and Computer Modelling vol47 no 1-2 pp 210ndash217 2008

[8] P Broadbridge B H Bradshaw G R Fulford and G K AldisldquoHuxley and Fisher equations for gene propagation an exactsolutionrdquoThe Anziam Journal vol 44 no 1 pp 11ndash20 2002

[9] A Kolmogorov I Petrovsky and N Piscounov ldquoEtudes delrsquoequation aved croissance de la quantite de matiere et sonapplication a un probleme biologiquerdquo Moscow UniversityMathematics Bulletin vol 1 pp 1ndash25 1937

[10] M Slatkin ldquoGene flow and selection in a clinerdquoGenetics vol 75no 4 pp 733ndash756 1973

[11] J M SmithMathematical Ideas in Biology Cambridge Univer-sity Press Cambridge UK 1968

[12] R A Littler ldquoLoss of variability at one locus in a finitepopulationrdquo Mathematical Biosciences vol 25 no 1-2 pp 151ndash163 1975

[13] T Ma and S Wang BifurcationTheory and Applications vol 53of Nonlinear Science Series A World Scientific Beijing China2005

[14] T Ma and S Wang Stability and Bifurcation of the NonlinearEvolution Equations Science Press Singapore 2007 (Chinese)

[15] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society of London B vol 237 no 641pp 37ndash52 1952

[16] T Ma and S Wang ldquoDynamic phase transition theory in PVTsystemsrdquo Indiana University Mathematics Journal vol 57 no 6pp 2861ndash2889 2008

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

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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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

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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

System (33) is local asymptotic stability if

11988611+ 11988622lt 0

1198861111988622minus 1198861211988621gt 0

(34)

and the Hopf bifurcation occurs when

11988611+ 11988622= 0

1198861111988622minus 1198861211988621lt 0

(35)

From Theorem 1 we can infer that if condition (34) and120582 isin Λ

minus

119870hold then the homogeneous attracting equilibrium

loses stability due to the interaction of diffusion processes andsystem (23) undergoes a Turing bifurcation

4 Phase Transition on Homogeneous State

Hereafter we always assume that the eigenvalue 1205731119870(120582) in

(24) is simple Based onTheorem 1 as 120582 isin Λminus

119870the transition

of (22) occurs at 120582 = 1205820 which is from real eigenvalues

The following is the main theorem in this paper whichprovides not only a precise criterion for the transition typesof (22) but also globally dynamical behaviors

Theorem 2 Let 120588119870be defined in Theorem 1 and 119890

119870is the

corresponding eigenvector to 120588119870of (25) satisfying

intΩ

1198903

119870119889119909 = 0 (36)

For system (22) we have the following assertions(1) Equation (22) has a mixed transition from (0 120582

0) more

precisely there exists a neighborhood119880 isin 119883 of 119906 = 0 such that119880 is separated into two disjoint open sets 119880120582

1and 119880

120582

2by the

stable manifold Γ120582of 119906 = 0 satisfying the following

(a) 119880 = 119880120582

1+119880120582

2+ Γ120582 (b) The transition in 119880120582

1is jump (c)

The transition in 1198801205822is continuous

(2) Equation (22) bifurcates in 119880120582

2to a unique singular

point V120582 on120582 isin Λ+

119870which is attractor such that for any120601 isin 119880

120582

2

lim119905rarrinfin

10038171003817100381710038171003817119906 (119905 120601) minus V120582

10038171003817100381710038171003817119883= 0 (37)

(3) Equation (22) bifurcates on 120582 isin Λminus

119870to a unique saddle

point with morse index 1(4) The bifurcated singular point V120582 can be expressed by

V120582 = minus1205731119870(120582)

119887120593119870+ 119900 (

10038171003817100381710038171205731119870 (120582)1003817100381710038171003817)

(38)

Here

119887 =1

1198861211988621+ (11988611minus 120588119870)2[minus119887 (119887

2+ 1198864119887 + 2119886

5119886)

sdot (1198861198863(119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (1198862+ 21198863119887 + 2119886

4119886) 119886 (119886

2+ 21198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870) + (119886

4119887 + 3119886

5119886 + 1198861)

sdot 1198862(1198862+ 21198863119887 + 1198864119886)2

) + (119886 (1198861+ 1198864119887 + 2119886

5119886)

minus 120588119870) ((1198871+ 31198863119887 + 1198864119886)

sdot (119886 (1198861+ 1198864119887 + 2119886

5119886) minus 120588

119870)2

minus (21198865119886 + 1198872+ 21198864119887)

sdot 119886 (1198862+ 21198863119887 + 1198864119886) (119886 (119886

1+ 1198864119887 + 2119886

5119886) minus 120588

119870)

+ 11988711988651198862(1198862+ 21198863119887 + 1198864119886)2

)]

(39)

Proof First we need to get the reduced equation of (22) near120582 = 120582

0

Let 119906 = 119909 sdot 120601119870+ 119910 where 119910 = sum

119899

119894=1119910119894120601119894and 120601

119870is the

eigenvector of (24) corresponding to 1205731119870(120582) at 120582 = 120582

0 Then

the reduced equation of (22) reads

119889119909

119889t= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870+ 119910) 120601

lowast

119870⟩ (40)

Here 120601lowast119870is the conjugate eigenvector of 120601

119870

By Implicit FunctionTheorem we can obtain that

119910 = Φ (119909 120582) = 119900 (119909) (41)

Substituting (41) into (40) we get the bifurcation equation of(22) as follows

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120601119870 120601lowast119870⟩⟨119866 (119909 sdot 120601

119870) 120601lowast

119870⟩

+ 119900 (2)

(42)

By (27) 120601119870is written as

120601119870= (1205851119890119870 1205852119890119870)119879

(43)

with (1205851 1205852) satisfying

(11988611minus 120588119870

11988612

11988621

11988622minus 120582120588K

)(1205851

1205852

) = 1205731119870(1205820) (

1205851

1205852

) (44)

from which we get

(1205851 1205852) = (minus119886

12 11988611minus 120588119870) (45)

Likewise 120601lowast119870is

120601lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(46)

6 Discrete Dynamics in Nature and Society

with (120585lowast1 120585lowast

2) satisfying

(11988611minus 120588119870

11988621

11988612

11988622minus 120582120588119870

)(1205851

1205852

) = 1205731119870(1205820) (

120585lowast

1

120585lowast

2

) (47)

which yields

(120585lowast

1 120585lowast

2) = (minus119886

21 11988611minus 120588119870) (48)

By (22) the nonlinear operator 119866 is

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(49)

Then in view of (45) and (48) by direct computation wederive that

⟨119866 (1199091205851119890119870 1199091205852119890119870) 120601lowast

119870⟩ = [(119886

4119887 + 1198861+ 31198865119886) 1205851

2120585lowast

1

+ 120585lowast

211988651198871205851

2+ (1198872+ 21198865119886 + 2119886

4119887) 12058521205851120585lowast

2

+ (1198871+ 1198864119886 + 3119886

3119887) 1205852

2120585lowast

2

+ (21198863119887 + 1198862+ 21198864119886) 12058521205851120585lowast

1+ 120585lowast

111988631198861205852

2]

sdot 1199092intΩ

1198903

119870119889119909 + 119900 (119909

2)

(50)

By (45) and (48) we have

⟨120601 120601lowast⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909 (51)

Hence by (50) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 119887119909

2+ 119900 (119909

2) (52)

where 119887 is the parameter as in (39) Based onTheorem A2 in[16] this theorem follows from (52) The proof is complete

Remark 3 If the domain Ω = (0 119871) times 119863 with 119863 isin 119877

being a bounded open set then condition (36) holds true forboundary conditions (15) and (16)

If the domainΩ = (0 119871)times119863with119863 isin 119877 being a boundedopen set then

intΩ

1198903

119870119889119909 = 0 (53)

holds true for Neumann condition (16) and Dirichlet condi-tion (15) when the number119898 in 120588

1198960= 119898212058721198712+ 1205881015840

1198960is even

Hereafter we consider the Neumann condition (16) andlet the domain Ω = (0 119871

1) times (0 119871

2) with 119871

1= 1198712

The following eigenvalue problem (24) with the equation

minusΔ119890119896= 120588119896119890119896

120597119890119896

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(54)

has eigenfunctions 119890119896and eigenvalues 120588

119896as follows

119890119896= cos11989611205871199091

1198711

cos119896212058711990921198712

119896 = (1198961 1198962) 1198961 1198962= 0 1 2

(55)

120588119896= 1205872(1198962

1

11987121

+1198962

2

11987122

) (56)

Here 1205880= 0 and 119890

0= 1

Denote 119870 = (1198701 1198702) the pair of integers satisfying (30)

and simply denote 1198701= (1198701 0) and 119870

2= (0 119870

2) then we

have the following

Theorem 4 Let 120578 be the parameter defined by

120578 =[(916) (120585 120585

lowast) + (14)119867 (120585 120585

lowast)]

1198861211988621+ (11988611minus 120588119870)2

(57)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

2120585112058522+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix and 120585 120585lowast are

defined in the proof

Assume that the eigenvalue 1205731119870(120582) satisfying (28) is

simple then for system (22) we have the following assertions

(1) This system has a transition from (0 1205820) that is 119906 = 0

is asymptotically stable for 120582 isin Λminus

119870and unstable for

120582 isin Λ+

119870which transits to a stable equilibrium at 120582 =

1205820

(2) If 120578 gt 0 then (22) has a jump transition from (0 1205820)

and bifurcates on 120582 isin Λ+

119870to exactly two saddle points

V1205821and V1205822with the Morse index one

(3) If 120578 lt 0 then (22) has a continuous transition from(0 1205820) which is an attractor bifurcation

(4) The bifurcated singular points V1205821and V1205822in the above

cases can be expressed in the following form

V12058212

= plusmn

10038161003816100381610038161003816100381610038161003816

1205731119870(120582)

120578

10038161003816100381610038161003816100381610038161003816

12

119890119870+ 119900 (

10038171003817100381710038171205731119870100381710038171003817100381712

) (58)

Proof Assertion (1) follows from (32) To prove assertions (2)and (3) we need to get the reduced equation of (22) to thecenter manifold near 120582 = 120582

0

Let 119906 = 119909 sdot 120593119870+ Φ where 120593

119870is the eigenvector of (25)

corresponding to 1205731119870(120582) at 120582 = 120582

0and Φ(119909) the center

manifold function of (22) Then the reduced equation of (22)takes the following form

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120593119870 120593lowast119870⟩⟨119866 (119909 sdot 120593

119870+ Φ) 120593

lowast

119870⟩ (59)

Here 120593lowast119870is the conjugate eigenvector of 120593

119870

By (27) 120593119870is written as

120593119870= (1205851119890119870 1205852119890119870)119879

(60)

with (1205851 1205852) satisfying (45)

Discrete Dynamics in Nature and Society 7

Likewise 120593lowast119870is

120593lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(61)

with (120585lowast1 120585lowast

2) satisfying (46)

It is known that the center manifold function

Φ (119909) = (Φ1(119909) Φ

2(119909)) = 119900 (119909

2) (62)

By using the formula for center manifold functions in [1314 16] Φ = (Φ

1 Φ2) satisfies

1198711205820Φ = minus119909

21198662(1205851119890119896 1205852119890119870) + 119900 (119909

2)

= minus11990921198902

119870(1205781

1205782

) + 119900 (1199092)

(63)

where

1205781= 11988611988631205852

2+ (1198862+ 21198864119886 + 2119886

3119887) 12058511205852

+ (1198864119887 + 3119886

5119886 + 1198861) 1205851

2

1205782= (1198871+ 31198863119887 + 1198864119886) 1205852

2+ (21198865119886 + 2119886

4119887 + 1198872) 12058511205852

+ 11988711988651205851

2

(64)

From (55) we see that

1198902

119870=1

4(1 + 119890

21198701) (1 + 119890

21198702)

=1

4(1198900+ 11989021198701

+ 11989021198702

+ 1198902119870)

(65)

Let

(Φ1

Φ2

) = (Φ0

1

Φ0

2

)1198900+ (

Φ21198701

1

Φ21198701

2

)11989021198701

+ (Φ21198702

1

Φ21198702

2

)11989021198702

+ (Φ2119870

1

Φ2119870

2

)1198902119870

(66)

It is clear that

1198711205820(Φ119896

1

Φ119896

2

)119890119896= 119872119896(Φ119896

1

Φ119896

2

)119890119896 (67)

where 119872119896is the matrix given by (26) Then it follows from

(65) and (66) that

(Φ119896

1

Φ119896

2

) = minus1

4119872minus1

119896(1205781

1205782

) + 119900 (1199092) (68)

for 119896 = 1198701198701 1198702 where

119872minus1

119896=

1

det (119872119896)[minus1205820120588119896+ 11988622

minus11988612

minus11988621

minus120588119896+ 11988611

]

det (119872119896) = 1205820120588119896

2minus 12058811989611988622minus 119886111205820120588119896+ 1198861111988622

minus 1198861211988621

(69)

Direct calculation shows that

(Φ0

1

Φ0

2

) =1199092

4 (1198861111988622minus 1198861211988621)(minus119886221205782minus 119886121205781

minus119886211205782minus 119886111205781

)

(Φ21198701

1

Φ21198701

2

)

=1199092

4det (11987221198701

)((120582012058821198701

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198701

minus 11988611) 1205781

)

(Φ21198702

1

Φ21198702

2

)

=1199092

4det (11987221198702

)((120582012058821198702

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198702

minus 11988611) 1205781

)

(Φ2119870

1

Φ2119870

2

) =1199092

4det (1198722119870)((12058201205882119870

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (1205882119870

minus 11988611) 1205781

)

(70)

Inserting (66) into (59) by (60) and (61) we get

⟨119866 (119909120593119870+ Φ) 120593

lowast

119870⟩ =

911987111198712

641199093 (120585 120585

lowast)

+11987111198712

161199093119867(120585 120585

lowast) + 119900 (119909

3)

(71)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

212058511205852

2+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix

In view of (70) it follows that

⟨120601119870 120601lowast

119870⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909

=11987111198712

4[1198861211988621+ (11988611minus 120588119870)2

]

(72)

Hence by (73) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 120578119909

3+ 119900 (119909

3) (73)

Based onTheoremA1 in [16] this theorem follows from (73)The proof is complete

5 Concluding Remarks and Example

In this paper we have studied the Turing bifurcation intro-duced by Alan Turing and attractor bifurcation developedby Ma and Wang [13] of gene propagation population modelgoverned by reaction-diffusion equation

Theorems 2 and 4 tell us that the critical value 1205820of

diffusion constant given by

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(74)

8 Discrete Dynamics in Nature and Society

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

05228

Time step = 300

(a) 120582 = 13

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

Time step = 1200

(b) 120582 = 1586209390

Figure 1 Turing patterns in two dimensions of system (9) with 120574119894119895

defined by (75)

plays a crucial role in determining the attractor bifurcationand Turing bifurcation when the diffusion parameter 120582

crosses the critical value 1205820 and the uniform stationary state

(119886 119887) loses its stability which yields the Turing or attractorbifurcation

We give some examples to illustrate our main theories

Example 5 In system (9) we let

12057411=2773376069253

499999999994

12057422=3789926287167

499999999994

12057433=810843022017

45454545454

12057423=4789523227559

249999999997

12057412=6952222073553

499999999994

12057413=436265576135

249999999997

(75)

and then the steady state (119886 119887) of system (9) is

119886 =732070686076807741207077

1417067211664436233221844

119887 =380119934758603124485687

708533605832218116610922

(76)

where 0 lt 119886 and 119887 lt 1

If we set Ω = (0 40120587) times (0 60120587) then 120588119870= 1314400

where119870 = (1 1) and then

1205820=814910923617150223684573395768233809936382690140395479526347826117214934665200

262055733351368170238516966160634367981740693955623134829331749178881678669asymp 3109685536 (77)

By Theorem 1 we obtain that the transition conditions aresatisfied

FromTheorem 4 120578 asymp minus4527181405 lt 0 and system (22)has a continuous transition from (0 120582

0) which is an attractor

bifurcationSince120588

119870+11988611lt 0 we infer that if120582 lt 120582

0 then120573

1119870(120582) gt 0

and the Turing instability occurs By numerical simulationwe have shown that the gene population model is able tosustain Turing patterns (Figure 1)

Appendix

The Expressions of 119867(120585120585lowast)

The Expressions of119867(120585 120585lowast) are as follows

119867(120585 120585lowast) = 4119886

3120585lowast

1119887Φ21198701

11205852+ 4119887120585lowast

2Φ21198701

111988651205852

+ 4120585lowast

211988651198871205851Φ21198701

2+ 4120585lowast

211988641198871205851Φ21198702

1

+ 4120585lowast

21198864119887Φ21198702

11205852+ 4120585lowast

211988641198871205851Φ21198702

2

+ 2120585lowast

211988651198871205852Φ2119870

2+ 2120585lowast

11198864119887Φ2119870

11205852

+ 4120585lowast

111988631198861205852Φ21198702

2+ 4120585lowast

111988631198871205852Φ21198701

2

+ 4120585lowast

211988651198871205851Φ21198702

2+ 6120585lowast

111988651198871205851Φ2119870

1

+ 2120585lowast

21198864119886Φ2119870

11205852+ 4120585lowast

111988641198871205851Φ21198701

1

+ 12120585lowast

111988651198871205851Φ21198701

1+ 2120585lowast

111988631198871205852Φ2119870

2

+ 2120585lowast

111988641198871205851Φ2119870

1+ 24120585lowast

111988651198871205851Φ0

1

+ 2120585lowast

111988631198871205851Φ2119870

2+ 121198863119887120585lowast

21205852Φ21198701

2

+ 241198863119887Φ0

2120585lowast

21205852+ 2120585lowast

211988721205851Φ21198702

2

+ 4120585lowast

21198864119887Φ21198701

11205852+ 4120585lowast

111988641198871205851Φ21198701

2

+ 12120585lowast

111988651198871205851Φ21198702

1+ 4120585lowast

11198864119887Φ21198702

11205852

Discrete Dynamics in Nature and Society 9

+ 8120585lowast

111988641198871205851Φ0

1+ 8120585lowast

211988651198871205851Φ0

2

+ 2120585lowast

211988641198871205851Φ2119870

1+ 8120585lowast

211988641198871205851Φ0

2

+ 4120585lowast

111988641198871205851Φ21198702

2+ 41198863Φ21198702

1120585lowast

11198871205852

+ 121198863119887120585lowast

21205852Φ21198702

2+ 4120585lowast

211988641198871205851Φ21198701

2

+ 4120585lowast

111988631198871205851Φ21198701

2+ 4120585lowast

11198864119887Φ21198701

11205852

+ 4Φ21198702

1119887120585lowast

211988651205852+ 4120585lowast

211988651198871205852Φ21198702

2

+ 4120585lowast

111988641198871205851Φ21198702

1+ 6120585lowast

211988631198871205852Φ2119870

2

+ 4120585lowast

111988631198871205851Φ21198702

2+ 4120585lowast

211988651198871205852Φ21198701

2

+ 2Φ2119870

1119887120585lowast

211988651205852+ 4120585lowast

21198872Φ0

11205852

+ 8120585lowast

111988611205851Φ0

1+ 2120585lowast

111988621205851Φ21198702

2

+ 2120585lowast

11198862Φ21198702

11205852+ 21198872Φ21198702

1120585lowast

21205852

+ 4120585lowast

211988721205851Φ0

2+ 2120585lowast

111988621205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198702

2+ 4120585lowast

11198862Φ0

11205852

+ 4120585lowast

111988611205851Φ21198701

1+ 2120585lowast

11198862Φ21198701

11205852

+ Φ2119870

11198872120585lowast

21205852+ 4120585lowast

111988611205851Φ21198702

1

+ 120585lowast

211988721205851Φ2119870

2+ 2120585lowast

211988711205852Φ2119870

2

+ 4120585lowast

111988621205851Φ0

2+ 120585lowast

11198862Φ2119870

11205852

+ 21198872120585lowast

2Φ21198701

11205852+ 2120585lowast

111988611205851Φ2119870

1

+ 8Φ0

2120585lowast

211988711205852+ 2120585lowast

211988721205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198701

2+ 8120585lowast

111988631198871205852Φ0

2

+ 2120585lowast

111988641198871205851Φ2119870

2+ 2120585lowast

211988651198871205851Φ2119870

2

+ 2120585lowast

211988641198871205851Φ2119870

2+ 2Φ2119870

11198863120585lowast

11198871205852

+ 120585lowast

111988621205851Φ2119870

2+ 8120585lowast

11198863119887Φ0

11205852

+ 8120585lowast

111988631198871205851Φ0

2+ 8120585lowast

111988641198871205851Φ0

2

+ 8120585lowast

211988641198871205851Φ0

1+ 8120585lowast

11198864119887Φ0

11205852

+ 8120585lowast

21198865119887Φ0

11205852+ 8120585lowast

21198864119887Φ0

11205852

+ 8120585lowast

211988651198871205852Φ0

2+ 4120585lowast

211988641198871205851Φ21198701

1

(A1)

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was partially supported by National Natu-ral Science Foundation of China (Grants no 11401062and no 11371386) Research Fund for the National Nat-ural Science Foundation of Chongqing CSTC (Grant nocstc2014jcyjA0080) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(Grant no KJ1400937) and the Scientific Research Founda-tion of CQUT (Grant no 2012ZD37)

References

[1] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 no 4 pp 355ndash369 1937

[2] A D Bazykin ldquoHypothetical mechanism of speciatonrdquo Evolu-tion vol 23 no 4 pp 685ndash687 1969

[3] J Pialek andNH Barton ldquoThe spread of an advantageous alleleacross a barrier the effects of randomdrift and selection againstheterozygotesrdquo Genetics vol 145 no 2 pp 493ndash504 1997

[4] T Nagylaki ldquoConditions for the existence of clinesrdquo Geneticsvol 80 no 3 pp 595ndash615 1975

[5] T Nagylaki and J F Crow ldquoContinuous selective modelsrdquoTheoretical Population Biology vol 5 no 2 pp 257ndash283 1974

[6] B H Bradshaw-Hajek and P Broadbridge ldquoA robust cubicreaction-diffusion system for gene propagationrdquo Mathematicaland Computer Modelling vol 39 no 9-10 pp 1151ndash1163 2004

[7] B H Bradshaw-Hajek P Broadbridge and G H WilliamsldquoEvolving gene frequencies in a population with three possiblealleles at a locusrdquo Mathematical and Computer Modelling vol47 no 1-2 pp 210ndash217 2008

[8] P Broadbridge B H Bradshaw G R Fulford and G K AldisldquoHuxley and Fisher equations for gene propagation an exactsolutionrdquoThe Anziam Journal vol 44 no 1 pp 11ndash20 2002

[9] A Kolmogorov I Petrovsky and N Piscounov ldquoEtudes delrsquoequation aved croissance de la quantite de matiere et sonapplication a un probleme biologiquerdquo Moscow UniversityMathematics Bulletin vol 1 pp 1ndash25 1937

[10] M Slatkin ldquoGene flow and selection in a clinerdquoGenetics vol 75no 4 pp 733ndash756 1973

[11] J M SmithMathematical Ideas in Biology Cambridge Univer-sity Press Cambridge UK 1968

[12] R A Littler ldquoLoss of variability at one locus in a finitepopulationrdquo Mathematical Biosciences vol 25 no 1-2 pp 151ndash163 1975

[13] T Ma and S Wang BifurcationTheory and Applications vol 53of Nonlinear Science Series A World Scientific Beijing China2005

[14] T Ma and S Wang Stability and Bifurcation of the NonlinearEvolution Equations Science Press Singapore 2007 (Chinese)

[15] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society of London B vol 237 no 641pp 37ndash52 1952

[16] T Ma and S Wang ldquoDynamic phase transition theory in PVTsystemsrdquo Indiana University Mathematics Journal vol 57 no 6pp 2861ndash2889 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

with (120585lowast1 120585lowast

2) satisfying

(11988611minus 120588119870

11988621

11988612

11988622minus 120582120588119870

)(1205851

1205852

) = 1205731119870(1205820) (

120585lowast

1

120585lowast

2

) (47)

which yields

(120585lowast

1 120585lowast

2) = (minus119886

21 11988611minus 120588119870) (48)

By (22) the nonlinear operator 119866 is

(1198661

2(1199061 1199062)

1198662

2(1199061 1199062))

= ((1198864119887 + 3119886

5119886 + 1198861) 1199061

2+ (1198862+ 21198863119887 + 2119886

4119886) 11990611199062+ 11988611988631199062

2

11988711988651199061

2+ (21198865119886 + 1198872+ 21198864119887) 11990611199062+ (1198871+ 31198863119887 + 1198864119886) 1199062

2)

(49)

Then in view of (45) and (48) by direct computation wederive that

⟨119866 (1199091205851119890119870 1199091205852119890119870) 120601lowast

119870⟩ = [(119886

4119887 + 1198861+ 31198865119886) 1205851

2120585lowast

1

+ 120585lowast

211988651198871205851

2+ (1198872+ 21198865119886 + 2119886

4119887) 12058521205851120585lowast

2

+ (1198871+ 1198864119886 + 3119886

3119887) 1205852

2120585lowast

2

+ (21198863119887 + 1198862+ 21198864119886) 12058521205851120585lowast

1+ 120585lowast

111988631198861205852

2]

sdot 1199092intΩ

1198903

119870119889119909 + 119900 (119909

2)

(50)

By (45) and (48) we have

⟨120601 120601lowast⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909 (51)

Hence by (50) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 119887119909

2+ 119900 (119909

2) (52)

where 119887 is the parameter as in (39) Based onTheorem A2 in[16] this theorem follows from (52) The proof is complete

Remark 3 If the domain Ω = (0 119871) times 119863 with 119863 isin 119877

being a bounded open set then condition (36) holds true forboundary conditions (15) and (16)

If the domainΩ = (0 119871)times119863with119863 isin 119877 being a boundedopen set then

intΩ

1198903

119870119889119909 = 0 (53)

holds true for Neumann condition (16) and Dirichlet condi-tion (15) when the number119898 in 120588

1198960= 119898212058721198712+ 1205881015840

1198960is even

Hereafter we consider the Neumann condition (16) andlet the domain Ω = (0 119871

1) times (0 119871

2) with 119871

1= 1198712

The following eigenvalue problem (24) with the equation

minusΔ119890119896= 120588119896119890119896

120597119890119896

120597119899

10038161003816100381610038161003816100381610038161003816120597Ω= 0

(54)

has eigenfunctions 119890119896and eigenvalues 120588

119896as follows

119890119896= cos11989611205871199091

1198711

cos119896212058711990921198712

119896 = (1198961 1198962) 1198961 1198962= 0 1 2

(55)

120588119896= 1205872(1198962

1

11987121

+1198962

2

11987122

) (56)

Here 1205880= 0 and 119890

0= 1

Denote 119870 = (1198701 1198702) the pair of integers satisfying (30)

and simply denote 1198701= (1198701 0) and 119870

2= (0 119870

2) then we

have the following

Theorem 4 Let 120578 be the parameter defined by

120578 =[(916) (120585 120585

lowast) + (14)119867 (120585 120585

lowast)]

1198861211988621+ (11988611minus 120588119870)2

(57)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

2120585112058522+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix and 120585 120585lowast are

defined in the proof

Assume that the eigenvalue 1205731119870(120582) satisfying (28) is

simple then for system (22) we have the following assertions

(1) This system has a transition from (0 1205820) that is 119906 = 0

is asymptotically stable for 120582 isin Λminus

119870and unstable for

120582 isin Λ+

119870which transits to a stable equilibrium at 120582 =

1205820

(2) If 120578 gt 0 then (22) has a jump transition from (0 1205820)

and bifurcates on 120582 isin Λ+

119870to exactly two saddle points

V1205821and V1205822with the Morse index one

(3) If 120578 lt 0 then (22) has a continuous transition from(0 1205820) which is an attractor bifurcation

(4) The bifurcated singular points V1205821and V1205822in the above

cases can be expressed in the following form

V12058212

= plusmn

10038161003816100381610038161003816100381610038161003816

1205731119870(120582)

120578

10038161003816100381610038161003816100381610038161003816

12

119890119870+ 119900 (

10038171003817100381710038171205731119870100381710038171003817100381712

) (58)

Proof Assertion (1) follows from (32) To prove assertions (2)and (3) we need to get the reduced equation of (22) to thecenter manifold near 120582 = 120582

0

Let 119906 = 119909 sdot 120593119870+ Φ where 120593

119870is the eigenvector of (25)

corresponding to 1205731119870(120582) at 120582 = 120582

0and Φ(119909) the center

manifold function of (22) Then the reduced equation of (22)takes the following form

119889119909

119889119905= 1205731119870(1205820) 119909 +

1

⟨120593119870 120593lowast119870⟩⟨119866 (119909 sdot 120593

119870+ Φ) 120593

lowast

119870⟩ (59)

Here 120593lowast119870is the conjugate eigenvector of 120593

119870

By (27) 120593119870is written as

120593119870= (1205851119890119870 1205852119890119870)119879

(60)

with (1205851 1205852) satisfying (45)

Discrete Dynamics in Nature and Society 7

Likewise 120593lowast119870is

120593lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(61)

with (120585lowast1 120585lowast

2) satisfying (46)

It is known that the center manifold function

Φ (119909) = (Φ1(119909) Φ

2(119909)) = 119900 (119909

2) (62)

By using the formula for center manifold functions in [1314 16] Φ = (Φ

1 Φ2) satisfies

1198711205820Φ = minus119909

21198662(1205851119890119896 1205852119890119870) + 119900 (119909

2)

= minus11990921198902

119870(1205781

1205782

) + 119900 (1199092)

(63)

where

1205781= 11988611988631205852

2+ (1198862+ 21198864119886 + 2119886

3119887) 12058511205852

+ (1198864119887 + 3119886

5119886 + 1198861) 1205851

2

1205782= (1198871+ 31198863119887 + 1198864119886) 1205852

2+ (21198865119886 + 2119886

4119887 + 1198872) 12058511205852

+ 11988711988651205851

2

(64)

From (55) we see that

1198902

119870=1

4(1 + 119890

21198701) (1 + 119890

21198702)

=1

4(1198900+ 11989021198701

+ 11989021198702

+ 1198902119870)

(65)

Let

(Φ1

Φ2

) = (Φ0

1

Φ0

2

)1198900+ (

Φ21198701

1

Φ21198701

2

)11989021198701

+ (Φ21198702

1

Φ21198702

2

)11989021198702

+ (Φ2119870

1

Φ2119870

2

)1198902119870

(66)

It is clear that

1198711205820(Φ119896

1

Φ119896

2

)119890119896= 119872119896(Φ119896

1

Φ119896

2

)119890119896 (67)

where 119872119896is the matrix given by (26) Then it follows from

(65) and (66) that

(Φ119896

1

Φ119896

2

) = minus1

4119872minus1

119896(1205781

1205782

) + 119900 (1199092) (68)

for 119896 = 1198701198701 1198702 where

119872minus1

119896=

1

det (119872119896)[minus1205820120588119896+ 11988622

minus11988612

minus11988621

minus120588119896+ 11988611

]

det (119872119896) = 1205820120588119896

2minus 12058811989611988622minus 119886111205820120588119896+ 1198861111988622

minus 1198861211988621

(69)

Direct calculation shows that

(Φ0

1

Φ0

2

) =1199092

4 (1198861111988622minus 1198861211988621)(minus119886221205782minus 119886121205781

minus119886211205782minus 119886111205781

)

(Φ21198701

1

Φ21198701

2

)

=1199092

4det (11987221198701

)((120582012058821198701

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198701

minus 11988611) 1205781

)

(Φ21198702

1

Φ21198702

2

)

=1199092

4det (11987221198702

)((120582012058821198702

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198702

minus 11988611) 1205781

)

(Φ2119870

1

Φ2119870

2

) =1199092

4det (1198722119870)((12058201205882119870

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (1205882119870

minus 11988611) 1205781

)

(70)

Inserting (66) into (59) by (60) and (61) we get

⟨119866 (119909120593119870+ Φ) 120593

lowast

119870⟩ =

911987111198712

641199093 (120585 120585

lowast)

+11987111198712

161199093119867(120585 120585

lowast) + 119900 (119909

3)

(71)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

212058511205852

2+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix

In view of (70) it follows that

⟨120601119870 120601lowast

119870⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909

=11987111198712

4[1198861211988621+ (11988611minus 120588119870)2

]

(72)

Hence by (73) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 120578119909

3+ 119900 (119909

3) (73)

Based onTheoremA1 in [16] this theorem follows from (73)The proof is complete

5 Concluding Remarks and Example

In this paper we have studied the Turing bifurcation intro-duced by Alan Turing and attractor bifurcation developedby Ma and Wang [13] of gene propagation population modelgoverned by reaction-diffusion equation

Theorems 2 and 4 tell us that the critical value 1205820of

diffusion constant given by

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(74)

8 Discrete Dynamics in Nature and Society

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

05228

Time step = 300

(a) 120582 = 13

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

Time step = 1200

(b) 120582 = 1586209390

Figure 1 Turing patterns in two dimensions of system (9) with 120574119894119895

defined by (75)

plays a crucial role in determining the attractor bifurcationand Turing bifurcation when the diffusion parameter 120582

crosses the critical value 1205820 and the uniform stationary state

(119886 119887) loses its stability which yields the Turing or attractorbifurcation

We give some examples to illustrate our main theories

Example 5 In system (9) we let

12057411=2773376069253

499999999994

12057422=3789926287167

499999999994

12057433=810843022017

45454545454

12057423=4789523227559

249999999997

12057412=6952222073553

499999999994

12057413=436265576135

249999999997

(75)

and then the steady state (119886 119887) of system (9) is

119886 =732070686076807741207077

1417067211664436233221844

119887 =380119934758603124485687

708533605832218116610922

(76)

where 0 lt 119886 and 119887 lt 1

If we set Ω = (0 40120587) times (0 60120587) then 120588119870= 1314400

where119870 = (1 1) and then

1205820=814910923617150223684573395768233809936382690140395479526347826117214934665200

262055733351368170238516966160634367981740693955623134829331749178881678669asymp 3109685536 (77)

By Theorem 1 we obtain that the transition conditions aresatisfied

FromTheorem 4 120578 asymp minus4527181405 lt 0 and system (22)has a continuous transition from (0 120582

0) which is an attractor

bifurcationSince120588

119870+11988611lt 0 we infer that if120582 lt 120582

0 then120573

1119870(120582) gt 0

and the Turing instability occurs By numerical simulationwe have shown that the gene population model is able tosustain Turing patterns (Figure 1)

Appendix

The Expressions of 119867(120585120585lowast)

The Expressions of119867(120585 120585lowast) are as follows

119867(120585 120585lowast) = 4119886

3120585lowast

1119887Φ21198701

11205852+ 4119887120585lowast

2Φ21198701

111988651205852

+ 4120585lowast

211988651198871205851Φ21198701

2+ 4120585lowast

211988641198871205851Φ21198702

1

+ 4120585lowast

21198864119887Φ21198702

11205852+ 4120585lowast

211988641198871205851Φ21198702

2

+ 2120585lowast

211988651198871205852Φ2119870

2+ 2120585lowast

11198864119887Φ2119870

11205852

+ 4120585lowast

111988631198861205852Φ21198702

2+ 4120585lowast

111988631198871205852Φ21198701

2

+ 4120585lowast

211988651198871205851Φ21198702

2+ 6120585lowast

111988651198871205851Φ2119870

1

+ 2120585lowast

21198864119886Φ2119870

11205852+ 4120585lowast

111988641198871205851Φ21198701

1

+ 12120585lowast

111988651198871205851Φ21198701

1+ 2120585lowast

111988631198871205852Φ2119870

2

+ 2120585lowast

111988641198871205851Φ2119870

1+ 24120585lowast

111988651198871205851Φ0

1

+ 2120585lowast

111988631198871205851Φ2119870

2+ 121198863119887120585lowast

21205852Φ21198701

2

+ 241198863119887Φ0

2120585lowast

21205852+ 2120585lowast

211988721205851Φ21198702

2

+ 4120585lowast

21198864119887Φ21198701

11205852+ 4120585lowast

111988641198871205851Φ21198701

2

+ 12120585lowast

111988651198871205851Φ21198702

1+ 4120585lowast

11198864119887Φ21198702

11205852

Discrete Dynamics in Nature and Society 9

+ 8120585lowast

111988641198871205851Φ0

1+ 8120585lowast

211988651198871205851Φ0

2

+ 2120585lowast

211988641198871205851Φ2119870

1+ 8120585lowast

211988641198871205851Φ0

2

+ 4120585lowast

111988641198871205851Φ21198702

2+ 41198863Φ21198702

1120585lowast

11198871205852

+ 121198863119887120585lowast

21205852Φ21198702

2+ 4120585lowast

211988641198871205851Φ21198701

2

+ 4120585lowast

111988631198871205851Φ21198701

2+ 4120585lowast

11198864119887Φ21198701

11205852

+ 4Φ21198702

1119887120585lowast

211988651205852+ 4120585lowast

211988651198871205852Φ21198702

2

+ 4120585lowast

111988641198871205851Φ21198702

1+ 6120585lowast

211988631198871205852Φ2119870

2

+ 4120585lowast

111988631198871205851Φ21198702

2+ 4120585lowast

211988651198871205852Φ21198701

2

+ 2Φ2119870

1119887120585lowast

211988651205852+ 4120585lowast

21198872Φ0

11205852

+ 8120585lowast

111988611205851Φ0

1+ 2120585lowast

111988621205851Φ21198702

2

+ 2120585lowast

11198862Φ21198702

11205852+ 21198872Φ21198702

1120585lowast

21205852

+ 4120585lowast

211988721205851Φ0

2+ 2120585lowast

111988621205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198702

2+ 4120585lowast

11198862Φ0

11205852

+ 4120585lowast

111988611205851Φ21198701

1+ 2120585lowast

11198862Φ21198701

11205852

+ Φ2119870

11198872120585lowast

21205852+ 4120585lowast

111988611205851Φ21198702

1

+ 120585lowast

211988721205851Φ2119870

2+ 2120585lowast

211988711205852Φ2119870

2

+ 4120585lowast

111988621205851Φ0

2+ 120585lowast

11198862Φ2119870

11205852

+ 21198872120585lowast

2Φ21198701

11205852+ 2120585lowast

111988611205851Φ2119870

1

+ 8Φ0

2120585lowast

211988711205852+ 2120585lowast

211988721205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198701

2+ 8120585lowast

111988631198871205852Φ0

2

+ 2120585lowast

111988641198871205851Φ2119870

2+ 2120585lowast

211988651198871205851Φ2119870

2

+ 2120585lowast

211988641198871205851Φ2119870

2+ 2Φ2119870

11198863120585lowast

11198871205852

+ 120585lowast

111988621205851Φ2119870

2+ 8120585lowast

11198863119887Φ0

11205852

+ 8120585lowast

111988631198871205851Φ0

2+ 8120585lowast

111988641198871205851Φ0

2

+ 8120585lowast

211988641198871205851Φ0

1+ 8120585lowast

11198864119887Φ0

11205852

+ 8120585lowast

21198865119887Φ0

11205852+ 8120585lowast

21198864119887Φ0

11205852

+ 8120585lowast

211988651198871205852Φ0

2+ 4120585lowast

211988641198871205851Φ21198701

1

(A1)

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was partially supported by National Natu-ral Science Foundation of China (Grants no 11401062and no 11371386) Research Fund for the National Nat-ural Science Foundation of Chongqing CSTC (Grant nocstc2014jcyjA0080) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(Grant no KJ1400937) and the Scientific Research Founda-tion of CQUT (Grant no 2012ZD37)

References

[1] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 no 4 pp 355ndash369 1937

[2] A D Bazykin ldquoHypothetical mechanism of speciatonrdquo Evolu-tion vol 23 no 4 pp 685ndash687 1969

[3] J Pialek andNH Barton ldquoThe spread of an advantageous alleleacross a barrier the effects of randomdrift and selection againstheterozygotesrdquo Genetics vol 145 no 2 pp 493ndash504 1997

[4] T Nagylaki ldquoConditions for the existence of clinesrdquo Geneticsvol 80 no 3 pp 595ndash615 1975

[5] T Nagylaki and J F Crow ldquoContinuous selective modelsrdquoTheoretical Population Biology vol 5 no 2 pp 257ndash283 1974

[6] B H Bradshaw-Hajek and P Broadbridge ldquoA robust cubicreaction-diffusion system for gene propagationrdquo Mathematicaland Computer Modelling vol 39 no 9-10 pp 1151ndash1163 2004

[7] B H Bradshaw-Hajek P Broadbridge and G H WilliamsldquoEvolving gene frequencies in a population with three possiblealleles at a locusrdquo Mathematical and Computer Modelling vol47 no 1-2 pp 210ndash217 2008

[8] P Broadbridge B H Bradshaw G R Fulford and G K AldisldquoHuxley and Fisher equations for gene propagation an exactsolutionrdquoThe Anziam Journal vol 44 no 1 pp 11ndash20 2002

[9] A Kolmogorov I Petrovsky and N Piscounov ldquoEtudes delrsquoequation aved croissance de la quantite de matiere et sonapplication a un probleme biologiquerdquo Moscow UniversityMathematics Bulletin vol 1 pp 1ndash25 1937

[10] M Slatkin ldquoGene flow and selection in a clinerdquoGenetics vol 75no 4 pp 733ndash756 1973

[11] J M SmithMathematical Ideas in Biology Cambridge Univer-sity Press Cambridge UK 1968

[12] R A Littler ldquoLoss of variability at one locus in a finitepopulationrdquo Mathematical Biosciences vol 25 no 1-2 pp 151ndash163 1975

[13] T Ma and S Wang BifurcationTheory and Applications vol 53of Nonlinear Science Series A World Scientific Beijing China2005

[14] T Ma and S Wang Stability and Bifurcation of the NonlinearEvolution Equations Science Press Singapore 2007 (Chinese)

[15] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society of London B vol 237 no 641pp 37ndash52 1952

[16] T Ma and S Wang ldquoDynamic phase transition theory in PVTsystemsrdquo Indiana University Mathematics Journal vol 57 no 6pp 2861ndash2889 2008

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

Likewise 120593lowast119870is

120593lowast

119870= (120585lowast

1119890119870 120585lowast

2119890119870)119879

(61)

with (120585lowast1 120585lowast

2) satisfying (46)

It is known that the center manifold function

Φ (119909) = (Φ1(119909) Φ

2(119909)) = 119900 (119909

2) (62)

By using the formula for center manifold functions in [1314 16] Φ = (Φ

1 Φ2) satisfies

1198711205820Φ = minus119909

21198662(1205851119890119896 1205852119890119870) + 119900 (119909

2)

= minus11990921198902

119870(1205781

1205782

) + 119900 (1199092)

(63)

where

1205781= 11988611988631205852

2+ (1198862+ 21198864119886 + 2119886

3119887) 12058511205852

+ (1198864119887 + 3119886

5119886 + 1198861) 1205851

2

1205782= (1198871+ 31198863119887 + 1198864119886) 1205852

2+ (21198865119886 + 2119886

4119887 + 1198872) 12058511205852

+ 11988711988651205851

2

(64)

From (55) we see that

1198902

119870=1

4(1 + 119890

21198701) (1 + 119890

21198702)

=1

4(1198900+ 11989021198701

+ 11989021198702

+ 1198902119870)

(65)

Let

(Φ1

Φ2

) = (Φ0

1

Φ0

2

)1198900+ (

Φ21198701

1

Φ21198701

2

)11989021198701

+ (Φ21198702

1

Φ21198702

2

)11989021198702

+ (Φ2119870

1

Φ2119870

2

)1198902119870

(66)

It is clear that

1198711205820(Φ119896

1

Φ119896

2

)119890119896= 119872119896(Φ119896

1

Φ119896

2

)119890119896 (67)

where 119872119896is the matrix given by (26) Then it follows from

(65) and (66) that

(Φ119896

1

Φ119896

2

) = minus1

4119872minus1

119896(1205781

1205782

) + 119900 (1199092) (68)

for 119896 = 1198701198701 1198702 where

119872minus1

119896=

1

det (119872119896)[minus1205820120588119896+ 11988622

minus11988612

minus11988621

minus120588119896+ 11988611

]

det (119872119896) = 1205820120588119896

2minus 12058811989611988622minus 119886111205820120588119896+ 1198861111988622

minus 1198861211988621

(69)

Direct calculation shows that

(Φ0

1

Φ0

2

) =1199092

4 (1198861111988622minus 1198861211988621)(minus119886221205782minus 119886121205781

minus119886211205782minus 119886111205781

)

(Φ21198701

1

Φ21198701

2

)

=1199092

4det (11987221198701

)((120582012058821198701

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198701

minus 11988611) 1205781

)

(Φ21198702

1

Φ21198702

2

)

=1199092

4det (11987221198702

)((120582012058821198702

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (12058821198702

minus 11988611) 1205781

)

(Φ2119870

1

Φ2119870

2

) =1199092

4det (1198722119870)((12058201205882119870

minus 11988622) 1205782minus 119886121205781

minus119886211205782+ (1205882119870

minus 11988611) 1205781

)

(70)

Inserting (66) into (59) by (60) and (61) we get

⟨119866 (119909120593119870+ Φ) 120593

lowast

119870⟩ =

911987111198712

641199093 (120585 120585

lowast)

+11987111198712

161199093119867(120585 120585

lowast) + 119900 (119909

3)

(71)

where (120585 120585lowast) = 1198863120585lowast

112058511205852

2+119886412057811205851

21205852+1198865120585lowast

11205851

3+119887312057821205851

21205852+

1198874120585lowast

212058511205852

2+ 1198875120585lowast

21205852

3 and119867(120585 120585lowast) is in Appendix

In view of (70) it follows that

⟨120601119870 120601lowast

119870⟩ = [119886

1211988621+ (11988611minus 120588119870)2

] intΩ

1198902

119870119889119909

=11987111198712

4[1198861211988621+ (11988611minus 120588119870)2

]

(72)

Hence by (73) the reduced equation (13) is expressed as

119889119909

119889119905= 1205731119870(120582) 119909 + 120578119909

3+ 119900 (119909

3) (73)

Based onTheoremA1 in [16] this theorem follows from (73)The proof is complete

5 Concluding Remarks and Example

In this paper we have studied the Turing bifurcation intro-duced by Alan Turing and attractor bifurcation developedby Ma and Wang [13] of gene propagation population modelgoverned by reaction-diffusion equation

Theorems 2 and 4 tell us that the critical value 1205820of

diffusion constant given by

1205820=11988622120588119870minus (1198861111988622minus 1198861211988621)

(120588119870minus 11988611) 120588119870

(74)

8 Discrete Dynamics in Nature and Society

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

05228

Time step = 300

(a) 120582 = 13

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

Time step = 1200

(b) 120582 = 1586209390

Figure 1 Turing patterns in two dimensions of system (9) with 120574119894119895

defined by (75)

plays a crucial role in determining the attractor bifurcationand Turing bifurcation when the diffusion parameter 120582

crosses the critical value 1205820 and the uniform stationary state

(119886 119887) loses its stability which yields the Turing or attractorbifurcation

We give some examples to illustrate our main theories

Example 5 In system (9) we let

12057411=2773376069253

499999999994

12057422=3789926287167

499999999994

12057433=810843022017

45454545454

12057423=4789523227559

249999999997

12057412=6952222073553

499999999994

12057413=436265576135

249999999997

(75)

and then the steady state (119886 119887) of system (9) is

119886 =732070686076807741207077

1417067211664436233221844

119887 =380119934758603124485687

708533605832218116610922

(76)

where 0 lt 119886 and 119887 lt 1

If we set Ω = (0 40120587) times (0 60120587) then 120588119870= 1314400

where119870 = (1 1) and then

1205820=814910923617150223684573395768233809936382690140395479526347826117214934665200

262055733351368170238516966160634367981740693955623134829331749178881678669asymp 3109685536 (77)

By Theorem 1 we obtain that the transition conditions aresatisfied

FromTheorem 4 120578 asymp minus4527181405 lt 0 and system (22)has a continuous transition from (0 120582

0) which is an attractor

bifurcationSince120588

119870+11988611lt 0 we infer that if120582 lt 120582

0 then120573

1119870(120582) gt 0

and the Turing instability occurs By numerical simulationwe have shown that the gene population model is able tosustain Turing patterns (Figure 1)

Appendix

The Expressions of 119867(120585120585lowast)

The Expressions of119867(120585 120585lowast) are as follows

119867(120585 120585lowast) = 4119886

3120585lowast

1119887Φ21198701

11205852+ 4119887120585lowast

2Φ21198701

111988651205852

+ 4120585lowast

211988651198871205851Φ21198701

2+ 4120585lowast

211988641198871205851Φ21198702

1

+ 4120585lowast

21198864119887Φ21198702

11205852+ 4120585lowast

211988641198871205851Φ21198702

2

+ 2120585lowast

211988651198871205852Φ2119870

2+ 2120585lowast

11198864119887Φ2119870

11205852

+ 4120585lowast

111988631198861205852Φ21198702

2+ 4120585lowast

111988631198871205852Φ21198701

2

+ 4120585lowast

211988651198871205851Φ21198702

2+ 6120585lowast

111988651198871205851Φ2119870

1

+ 2120585lowast

21198864119886Φ2119870

11205852+ 4120585lowast

111988641198871205851Φ21198701

1

+ 12120585lowast

111988651198871205851Φ21198701

1+ 2120585lowast

111988631198871205852Φ2119870

2

+ 2120585lowast

111988641198871205851Φ2119870

1+ 24120585lowast

111988651198871205851Φ0

1

+ 2120585lowast

111988631198871205851Φ2119870

2+ 121198863119887120585lowast

21205852Φ21198701

2

+ 241198863119887Φ0

2120585lowast

21205852+ 2120585lowast

211988721205851Φ21198702

2

+ 4120585lowast

21198864119887Φ21198701

11205852+ 4120585lowast

111988641198871205851Φ21198701

2

+ 12120585lowast

111988651198871205851Φ21198702

1+ 4120585lowast

11198864119887Φ21198702

11205852

Discrete Dynamics in Nature and Society 9

+ 8120585lowast

111988641198871205851Φ0

1+ 8120585lowast

211988651198871205851Φ0

2

+ 2120585lowast

211988641198871205851Φ2119870

1+ 8120585lowast

211988641198871205851Φ0

2

+ 4120585lowast

111988641198871205851Φ21198702

2+ 41198863Φ21198702

1120585lowast

11198871205852

+ 121198863119887120585lowast

21205852Φ21198702

2+ 4120585lowast

211988641198871205851Φ21198701

2

+ 4120585lowast

111988631198871205851Φ21198701

2+ 4120585lowast

11198864119887Φ21198701

11205852

+ 4Φ21198702

1119887120585lowast

211988651205852+ 4120585lowast

211988651198871205852Φ21198702

2

+ 4120585lowast

111988641198871205851Φ21198702

1+ 6120585lowast

211988631198871205852Φ2119870

2

+ 4120585lowast

111988631198871205851Φ21198702

2+ 4120585lowast

211988651198871205852Φ21198701

2

+ 2Φ2119870

1119887120585lowast

211988651205852+ 4120585lowast

21198872Φ0

11205852

+ 8120585lowast

111988611205851Φ0

1+ 2120585lowast

111988621205851Φ21198702

2

+ 2120585lowast

11198862Φ21198702

11205852+ 21198872Φ21198702

1120585lowast

21205852

+ 4120585lowast

211988721205851Φ0

2+ 2120585lowast

111988621205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198702

2+ 4120585lowast

11198862Φ0

11205852

+ 4120585lowast

111988611205851Φ21198701

1+ 2120585lowast

11198862Φ21198701

11205852

+ Φ2119870

11198872120585lowast

21205852+ 4120585lowast

111988611205851Φ21198702

1

+ 120585lowast

211988721205851Φ2119870

2+ 2120585lowast

211988711205852Φ2119870

2

+ 4120585lowast

111988621205851Φ0

2+ 120585lowast

11198862Φ2119870

11205852

+ 21198872120585lowast

2Φ21198701

11205852+ 2120585lowast

111988611205851Φ2119870

1

+ 8Φ0

2120585lowast

211988711205852+ 2120585lowast

211988721205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198701

2+ 8120585lowast

111988631198871205852Φ0

2

+ 2120585lowast

111988641198871205851Φ2119870

2+ 2120585lowast

211988651198871205851Φ2119870

2

+ 2120585lowast

211988641198871205851Φ2119870

2+ 2Φ2119870

11198863120585lowast

11198871205852

+ 120585lowast

111988621205851Φ2119870

2+ 8120585lowast

11198863119887Φ0

11205852

+ 8120585lowast

111988631198871205851Φ0

2+ 8120585lowast

111988641198871205851Φ0

2

+ 8120585lowast

211988641198871205851Φ0

1+ 8120585lowast

11198864119887Φ0

11205852

+ 8120585lowast

21198865119887Φ0

11205852+ 8120585lowast

21198864119887Φ0

11205852

+ 8120585lowast

211988651198871205852Φ0

2+ 4120585lowast

211988641198871205851Φ21198701

1

(A1)

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was partially supported by National Natu-ral Science Foundation of China (Grants no 11401062and no 11371386) Research Fund for the National Nat-ural Science Foundation of Chongqing CSTC (Grant nocstc2014jcyjA0080) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(Grant no KJ1400937) and the Scientific Research Founda-tion of CQUT (Grant no 2012ZD37)

References

[1] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 no 4 pp 355ndash369 1937

[2] A D Bazykin ldquoHypothetical mechanism of speciatonrdquo Evolu-tion vol 23 no 4 pp 685ndash687 1969

[3] J Pialek andNH Barton ldquoThe spread of an advantageous alleleacross a barrier the effects of randomdrift and selection againstheterozygotesrdquo Genetics vol 145 no 2 pp 493ndash504 1997

[4] T Nagylaki ldquoConditions for the existence of clinesrdquo Geneticsvol 80 no 3 pp 595ndash615 1975

[5] T Nagylaki and J F Crow ldquoContinuous selective modelsrdquoTheoretical Population Biology vol 5 no 2 pp 257ndash283 1974

[6] B H Bradshaw-Hajek and P Broadbridge ldquoA robust cubicreaction-diffusion system for gene propagationrdquo Mathematicaland Computer Modelling vol 39 no 9-10 pp 1151ndash1163 2004

[7] B H Bradshaw-Hajek P Broadbridge and G H WilliamsldquoEvolving gene frequencies in a population with three possiblealleles at a locusrdquo Mathematical and Computer Modelling vol47 no 1-2 pp 210ndash217 2008

[8] P Broadbridge B H Bradshaw G R Fulford and G K AldisldquoHuxley and Fisher equations for gene propagation an exactsolutionrdquoThe Anziam Journal vol 44 no 1 pp 11ndash20 2002

[9] A Kolmogorov I Petrovsky and N Piscounov ldquoEtudes delrsquoequation aved croissance de la quantite de matiere et sonapplication a un probleme biologiquerdquo Moscow UniversityMathematics Bulletin vol 1 pp 1ndash25 1937

[10] M Slatkin ldquoGene flow and selection in a clinerdquoGenetics vol 75no 4 pp 733ndash756 1973

[11] J M SmithMathematical Ideas in Biology Cambridge Univer-sity Press Cambridge UK 1968

[12] R A Littler ldquoLoss of variability at one locus in a finitepopulationrdquo Mathematical Biosciences vol 25 no 1-2 pp 151ndash163 1975

[13] T Ma and S Wang BifurcationTheory and Applications vol 53of Nonlinear Science Series A World Scientific Beijing China2005

[14] T Ma and S Wang Stability and Bifurcation of the NonlinearEvolution Equations Science Press Singapore 2007 (Chinese)

[15] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society of London B vol 237 no 641pp 37ndash52 1952

[16] T Ma and S Wang ldquoDynamic phase transition theory in PVTsystemsrdquo Indiana University Mathematics Journal vol 57 no 6pp 2861ndash2889 2008

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

8 Discrete Dynamics in Nature and Society

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

05228

Time step = 300

(a) 120582 = 13

180

160

140

120

100

80

60

40

20

0120100806040200

05228

05228

05228

05228

05228

Time step = 1200

(b) 120582 = 1586209390

Figure 1 Turing patterns in two dimensions of system (9) with 120574119894119895

defined by (75)

plays a crucial role in determining the attractor bifurcationand Turing bifurcation when the diffusion parameter 120582

crosses the critical value 1205820 and the uniform stationary state

(119886 119887) loses its stability which yields the Turing or attractorbifurcation

We give some examples to illustrate our main theories

Example 5 In system (9) we let

12057411=2773376069253

499999999994

12057422=3789926287167

499999999994

12057433=810843022017

45454545454

12057423=4789523227559

249999999997

12057412=6952222073553

499999999994

12057413=436265576135

249999999997

(75)

and then the steady state (119886 119887) of system (9) is

119886 =732070686076807741207077

1417067211664436233221844

119887 =380119934758603124485687

708533605832218116610922

(76)

where 0 lt 119886 and 119887 lt 1

If we set Ω = (0 40120587) times (0 60120587) then 120588119870= 1314400

where119870 = (1 1) and then

1205820=814910923617150223684573395768233809936382690140395479526347826117214934665200

262055733351368170238516966160634367981740693955623134829331749178881678669asymp 3109685536 (77)

By Theorem 1 we obtain that the transition conditions aresatisfied

FromTheorem 4 120578 asymp minus4527181405 lt 0 and system (22)has a continuous transition from (0 120582

0) which is an attractor

bifurcationSince120588

119870+11988611lt 0 we infer that if120582 lt 120582

0 then120573

1119870(120582) gt 0

and the Turing instability occurs By numerical simulationwe have shown that the gene population model is able tosustain Turing patterns (Figure 1)

Appendix

The Expressions of 119867(120585120585lowast)

The Expressions of119867(120585 120585lowast) are as follows

119867(120585 120585lowast) = 4119886

3120585lowast

1119887Φ21198701

11205852+ 4119887120585lowast

2Φ21198701

111988651205852

+ 4120585lowast

211988651198871205851Φ21198701

2+ 4120585lowast

211988641198871205851Φ21198702

1

+ 4120585lowast

21198864119887Φ21198702

11205852+ 4120585lowast

211988641198871205851Φ21198702

2

+ 2120585lowast

211988651198871205852Φ2119870

2+ 2120585lowast

11198864119887Φ2119870

11205852

+ 4120585lowast

111988631198861205852Φ21198702

2+ 4120585lowast

111988631198871205852Φ21198701

2

+ 4120585lowast

211988651198871205851Φ21198702

2+ 6120585lowast

111988651198871205851Φ2119870

1

+ 2120585lowast

21198864119886Φ2119870

11205852+ 4120585lowast

111988641198871205851Φ21198701

1

+ 12120585lowast

111988651198871205851Φ21198701

1+ 2120585lowast

111988631198871205852Φ2119870

2

+ 2120585lowast

111988641198871205851Φ2119870

1+ 24120585lowast

111988651198871205851Φ0

1

+ 2120585lowast

111988631198871205851Φ2119870

2+ 121198863119887120585lowast

21205852Φ21198701

2

+ 241198863119887Φ0

2120585lowast

21205852+ 2120585lowast

211988721205851Φ21198702

2

+ 4120585lowast

21198864119887Φ21198701

11205852+ 4120585lowast

111988641198871205851Φ21198701

2

+ 12120585lowast

111988651198871205851Φ21198702

1+ 4120585lowast

11198864119887Φ21198702

11205852

Discrete Dynamics in Nature and Society 9

+ 8120585lowast

111988641198871205851Φ0

1+ 8120585lowast

211988651198871205851Φ0

2

+ 2120585lowast

211988641198871205851Φ2119870

1+ 8120585lowast

211988641198871205851Φ0

2

+ 4120585lowast

111988641198871205851Φ21198702

2+ 41198863Φ21198702

1120585lowast

11198871205852

+ 121198863119887120585lowast

21205852Φ21198702

2+ 4120585lowast

211988641198871205851Φ21198701

2

+ 4120585lowast

111988631198871205851Φ21198701

2+ 4120585lowast

11198864119887Φ21198701

11205852

+ 4Φ21198702

1119887120585lowast

211988651205852+ 4120585lowast

211988651198871205852Φ21198702

2

+ 4120585lowast

111988641198871205851Φ21198702

1+ 6120585lowast

211988631198871205852Φ2119870

2

+ 4120585lowast

111988631198871205851Φ21198702

2+ 4120585lowast

211988651198871205852Φ21198701

2

+ 2Φ2119870

1119887120585lowast

211988651205852+ 4120585lowast

21198872Φ0

11205852

+ 8120585lowast

111988611205851Φ0

1+ 2120585lowast

111988621205851Φ21198702

2

+ 2120585lowast

11198862Φ21198702

11205852+ 21198872Φ21198702

1120585lowast

21205852

+ 4120585lowast

211988721205851Φ0

2+ 2120585lowast

111988621205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198702

2+ 4120585lowast

11198862Φ0

11205852

+ 4120585lowast

111988611205851Φ21198701

1+ 2120585lowast

11198862Φ21198701

11205852

+ Φ2119870

11198872120585lowast

21205852+ 4120585lowast

111988611205851Φ21198702

1

+ 120585lowast

211988721205851Φ2119870

2+ 2120585lowast

211988711205852Φ2119870

2

+ 4120585lowast

111988621205851Φ0

2+ 120585lowast

11198862Φ2119870

11205852

+ 21198872120585lowast

2Φ21198701

11205852+ 2120585lowast

111988611205851Φ2119870

1

+ 8Φ0

2120585lowast

211988711205852+ 2120585lowast

211988721205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198701

2+ 8120585lowast

111988631198871205852Φ0

2

+ 2120585lowast

111988641198871205851Φ2119870

2+ 2120585lowast

211988651198871205851Φ2119870

2

+ 2120585lowast

211988641198871205851Φ2119870

2+ 2Φ2119870

11198863120585lowast

11198871205852

+ 120585lowast

111988621205851Φ2119870

2+ 8120585lowast

11198863119887Φ0

11205852

+ 8120585lowast

111988631198871205851Φ0

2+ 8120585lowast

111988641198871205851Φ0

2

+ 8120585lowast

211988641198871205851Φ0

1+ 8120585lowast

11198864119887Φ0

11205852

+ 8120585lowast

21198865119887Φ0

11205852+ 8120585lowast

21198864119887Φ0

11205852

+ 8120585lowast

211988651198871205852Φ0

2+ 4120585lowast

211988641198871205851Φ21198701

1

(A1)

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was partially supported by National Natu-ral Science Foundation of China (Grants no 11401062and no 11371386) Research Fund for the National Nat-ural Science Foundation of Chongqing CSTC (Grant nocstc2014jcyjA0080) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(Grant no KJ1400937) and the Scientific Research Founda-tion of CQUT (Grant no 2012ZD37)

References

[1] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 no 4 pp 355ndash369 1937

[2] A D Bazykin ldquoHypothetical mechanism of speciatonrdquo Evolu-tion vol 23 no 4 pp 685ndash687 1969

[3] J Pialek andNH Barton ldquoThe spread of an advantageous alleleacross a barrier the effects of randomdrift and selection againstheterozygotesrdquo Genetics vol 145 no 2 pp 493ndash504 1997

[4] T Nagylaki ldquoConditions for the existence of clinesrdquo Geneticsvol 80 no 3 pp 595ndash615 1975

[5] T Nagylaki and J F Crow ldquoContinuous selective modelsrdquoTheoretical Population Biology vol 5 no 2 pp 257ndash283 1974

[6] B H Bradshaw-Hajek and P Broadbridge ldquoA robust cubicreaction-diffusion system for gene propagationrdquo Mathematicaland Computer Modelling vol 39 no 9-10 pp 1151ndash1163 2004

[7] B H Bradshaw-Hajek P Broadbridge and G H WilliamsldquoEvolving gene frequencies in a population with three possiblealleles at a locusrdquo Mathematical and Computer Modelling vol47 no 1-2 pp 210ndash217 2008

[8] P Broadbridge B H Bradshaw G R Fulford and G K AldisldquoHuxley and Fisher equations for gene propagation an exactsolutionrdquoThe Anziam Journal vol 44 no 1 pp 11ndash20 2002

[9] A Kolmogorov I Petrovsky and N Piscounov ldquoEtudes delrsquoequation aved croissance de la quantite de matiere et sonapplication a un probleme biologiquerdquo Moscow UniversityMathematics Bulletin vol 1 pp 1ndash25 1937

[10] M Slatkin ldquoGene flow and selection in a clinerdquoGenetics vol 75no 4 pp 733ndash756 1973

[11] J M SmithMathematical Ideas in Biology Cambridge Univer-sity Press Cambridge UK 1968

[12] R A Littler ldquoLoss of variability at one locus in a finitepopulationrdquo Mathematical Biosciences vol 25 no 1-2 pp 151ndash163 1975

[13] T Ma and S Wang BifurcationTheory and Applications vol 53of Nonlinear Science Series A World Scientific Beijing China2005

[14] T Ma and S Wang Stability and Bifurcation of the NonlinearEvolution Equations Science Press Singapore 2007 (Chinese)

[15] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society of London B vol 237 no 641pp 37ndash52 1952

[16] T Ma and S Wang ldquoDynamic phase transition theory in PVTsystemsrdquo Indiana University Mathematics Journal vol 57 no 6pp 2861ndash2889 2008

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 9

+ 8120585lowast

111988641198871205851Φ0

1+ 8120585lowast

211988651198871205851Φ0

2

+ 2120585lowast

211988641198871205851Φ2119870

1+ 8120585lowast

211988641198871205851Φ0

2

+ 4120585lowast

111988641198871205851Φ21198702

2+ 41198863Φ21198702

1120585lowast

11198871205852

+ 121198863119887120585lowast

21205852Φ21198702

2+ 4120585lowast

211988641198871205851Φ21198701

2

+ 4120585lowast

111988631198871205851Φ21198701

2+ 4120585lowast

11198864119887Φ21198701

11205852

+ 4Φ21198702

1119887120585lowast

211988651205852+ 4120585lowast

211988651198871205852Φ21198702

2

+ 4120585lowast

111988641198871205851Φ21198702

1+ 6120585lowast

211988631198871205852Φ2119870

2

+ 4120585lowast

111988631198871205851Φ21198702

2+ 4120585lowast

211988651198871205852Φ21198701

2

+ 2Φ2119870

1119887120585lowast

211988651205852+ 4120585lowast

21198872Φ0

11205852

+ 8120585lowast

111988611205851Φ0

1+ 2120585lowast

111988621205851Φ21198702

2

+ 2120585lowast

11198862Φ21198702

11205852+ 21198872Φ21198702

1120585lowast

21205852

+ 4120585lowast

211988721205851Φ0

2+ 2120585lowast

111988621205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198702

2+ 4120585lowast

11198862Φ0

11205852

+ 4120585lowast

111988611205851Φ21198701

1+ 2120585lowast

11198862Φ21198701

11205852

+ Φ2119870

11198872120585lowast

21205852+ 4120585lowast

111988611205851Φ21198702

1

+ 120585lowast

211988721205851Φ2119870

2+ 2120585lowast

211988711205852Φ2119870

2

+ 4120585lowast

111988621205851Φ0

2+ 120585lowast

11198862Φ2119870

11205852

+ 21198872120585lowast

2Φ21198701

11205852+ 2120585lowast

111988611205851Φ2119870

1

+ 8Φ0

2120585lowast

211988711205852+ 2120585lowast

211988721205851Φ21198701

2

+ 4120585lowast

211988711205852Φ21198701

2+ 8120585lowast

111988631198871205852Φ0

2

+ 2120585lowast

111988641198871205851Φ2119870

2+ 2120585lowast

211988651198871205851Φ2119870

2

+ 2120585lowast

211988641198871205851Φ2119870

2+ 2Φ2119870

11198863120585lowast

11198871205852

+ 120585lowast

111988621205851Φ2119870

2+ 8120585lowast

11198863119887Φ0

11205852

+ 8120585lowast

111988631198871205851Φ0

2+ 8120585lowast

111988641198871205851Φ0

2

+ 8120585lowast

211988641198871205851Φ0

1+ 8120585lowast

11198864119887Φ0

11205852

+ 8120585lowast

21198865119887Φ0

11205852+ 8120585lowast

21198864119887Φ0

11205852

+ 8120585lowast

211988651198871205852Φ0

2+ 4120585lowast

211988641198871205851Φ21198701

1

(A1)

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was partially supported by National Natu-ral Science Foundation of China (Grants no 11401062and no 11371386) Research Fund for the National Nat-ural Science Foundation of Chongqing CSTC (Grant nocstc2014jcyjA0080) Scientific and Technological ResearchProgram of Chongqing Municipal Education Commission(Grant no KJ1400937) and the Scientific Research Founda-tion of CQUT (Grant no 2012ZD37)

References

[1] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 no 4 pp 355ndash369 1937

[2] A D Bazykin ldquoHypothetical mechanism of speciatonrdquo Evolu-tion vol 23 no 4 pp 685ndash687 1969

[3] J Pialek andNH Barton ldquoThe spread of an advantageous alleleacross a barrier the effects of randomdrift and selection againstheterozygotesrdquo Genetics vol 145 no 2 pp 493ndash504 1997

[4] T Nagylaki ldquoConditions for the existence of clinesrdquo Geneticsvol 80 no 3 pp 595ndash615 1975

[5] T Nagylaki and J F Crow ldquoContinuous selective modelsrdquoTheoretical Population Biology vol 5 no 2 pp 257ndash283 1974

[6] B H Bradshaw-Hajek and P Broadbridge ldquoA robust cubicreaction-diffusion system for gene propagationrdquo Mathematicaland Computer Modelling vol 39 no 9-10 pp 1151ndash1163 2004

[7] B H Bradshaw-Hajek P Broadbridge and G H WilliamsldquoEvolving gene frequencies in a population with three possiblealleles at a locusrdquo Mathematical and Computer Modelling vol47 no 1-2 pp 210ndash217 2008

[8] P Broadbridge B H Bradshaw G R Fulford and G K AldisldquoHuxley and Fisher equations for gene propagation an exactsolutionrdquoThe Anziam Journal vol 44 no 1 pp 11ndash20 2002

[9] A Kolmogorov I Petrovsky and N Piscounov ldquoEtudes delrsquoequation aved croissance de la quantite de matiere et sonapplication a un probleme biologiquerdquo Moscow UniversityMathematics Bulletin vol 1 pp 1ndash25 1937

[10] M Slatkin ldquoGene flow and selection in a clinerdquoGenetics vol 75no 4 pp 733ndash756 1973

[11] J M SmithMathematical Ideas in Biology Cambridge Univer-sity Press Cambridge UK 1968

[12] R A Littler ldquoLoss of variability at one locus in a finitepopulationrdquo Mathematical Biosciences vol 25 no 1-2 pp 151ndash163 1975

[13] T Ma and S Wang BifurcationTheory and Applications vol 53of Nonlinear Science Series A World Scientific Beijing China2005

[14] T Ma and S Wang Stability and Bifurcation of the NonlinearEvolution Equations Science Press Singapore 2007 (Chinese)

[15] A Turing ldquoThe chemical basis ofmorphogenesisrdquoPhilosophicalTransactions of the Royal Society of London B vol 237 no 641pp 37ndash52 1952

[16] T Ma and S Wang ldquoDynamic phase transition theory in PVTsystemsrdquo Indiana University Mathematics Journal vol 57 no 6pp 2861ndash2889 2008

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