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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
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)
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)
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
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
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
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
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
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)
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)
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
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
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
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
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)
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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