Bifurcation Analysis of Gene Propagation Model Governed by Reaction-Diffusion Equations

As the field of gene technology develops, the gene propagation problems continue to be relevant. Some recent advances and problems include the following: the genetic engineering for improving crop pest and disease resistance; the bacteria have developed a tolerance to widely prescribed antibiotics; the human genome project will enable us to deduce more information on human bodies and to deduce historical patterns of migration by archaeologists. Lots of papers developed equations to describe the changes in the frequency of alleles in a population that has several possible alleles at the locus in question. Fisher [1] proposed a reaction-diffusion equation with quadratic source term that models the spread of a recessive advantageous gene through a population that previously had only one allele at the locus in question. Fisher’s equation is


Introduction
As the field of gene technology develops, the gene propagation problems continue to be relevant.Some recent advances and problems include the following: the genetic engineering for improving crop pest and disease resistance; the bacteria have developed a tolerance to widely prescribed antibiotics; the human genome project will enable us to deduce more information on human bodies and to deduce historical patterns of migration by archaeologists.Lots of papers developed equations to describe the changes in the frequency of alleles in a population that has several possible alleles at the locus in question.Fisher [1] proposed a reaction-diffusion equation with quadratic source term that models the spread of a recessive advantageous gene through a population that previously had only one allele at the locus in question.Fisher's equation is where  is the frequency of the new mutant gene,  is the diffusion coefficient, and  is the intensity of selection in favor of the mutant gene.
In [2][3][4][5], the authors have claimed that a cubic source term was more appropriate than a quadratic source term.Although the cubic source term is implicit as one possibility in the general genetic dispersion equations derived by others, its significance has not been highlighted and the difference between cubic and quadratic source terms has not been examined.Based on the Fitzhugh-Nagumo equation and Huxley equation, by using the methods of a continuum limit of a discrete generation model, direct continuum modelling, and Fick's laws for random motion, Bradshaw-Hajek and Broadbridge [6][7][8] have derived a reaction-diffusion equation describing the spread of a new mutant gene; that is, where  is the frequency of the new mutant gene,  is the diffusion coefficient, and  is the intensity of selection in favor of the mutant gene.
In [9][10][11], the authors have discussed the two possible alleles while some others recently investigate another case in which there are more than two possible alleles at the locus in question.For three possible alleles, Littler [12] has mostly used stochastic models while Bradshaw-Hajek and Broadbridge [6][7][8] have developed the reaction-diffusionconvection models.
In this paper, we will follow the work of Bradshaw-Hajek et al. [7] and investigate the gene propagation model of three possible alleles at the locus.By introducing the spatial two-dimensional domains, we will give a detailed analysis of the dynamical properties for the model and consider the attractor bifurcation to show a complete characterization of the attractors and their basins of attraction in terms of the physical parameters of the problem which is developed by Ma and Wang [13,14].
The paper is organized as follows.In Section 2, we briefly summarize the two-dimensional spatial gene propagation model and give some mathematical settings.Section 3 states principle of exchange of stability for system.Section 4 is the main results of the phase transition theorems based on the attractor bifurcation theory.An example with the computer simulation of the pattern formation is given in the concluding remark section to illustrate our main results.

Modelling Analysis
In order to describe the spread of a new mutant gene, based on Skellam's method, Bradshaw-Hajek and Broadbridge [6] have developed a one-dimensional population genetics model governed by reaction-diffusion equation describing the changes in allelic frequencies.For a population having one new mutant allele  1 and two original alleles  2 ,  3 , there are six possible genotypes: Let   ( 1 ,  2 , ), (,  = 1, 2, 3) denote the frequency of individuals of the genotype     on the spatial two-dimensional domain.We follow the ideas of [6] and write the genotype equations as where Δ =  2 / (, ) is the total population density,  is the common death rate, and   is the reproductive success rate of individuals with genotype     for ,  = 1, 2, 3, respectively.
From (5), we can simplify these above six equations into the following two coupled equations describing the change in frequency of two of alleles: where Assume that the total population density is constant across the range (so that / = 0); system (6) becomes One of the attractions of (8) to mathematicians is to study the diffusion induced instability introduced by Turing in his 1952 seminal paper [15].For showing the diffusion effect on stability, we will consider a modified equation of (8): where  1 =  11 − 3 = 0.  is the diffusion coefficient which measures the dispersal rate of allele  2 .On the other hand, diffusive terms can be considered as describing the ability of the allele   to occupy different zones in 2dimensional space either through the action of small-scale mechanism or by some native transport device.
In the present paper, we focus on the bifurcation from the constant solution (, ).Let Omitting the primes, then system (9) becomes where We assume that system ( 13) is satisfied on an open bounded domain Ω ⊂  2 .There are two types of biologically sound boundary conditions: the Dirichlet boundary condition, which means that the frequency is extinct in the boundary of range and the Neumann boundary condition: which means that the frequency is invariant in the boundary of range in biological significant.Define the function spaces It is clear that  and  1 are two Hilbert space and  1 →  is dense and compact inclusion.Later, we choose the bifurcation parameter  to be the diffusion coefficient ; that is,  = .Let   :  1 →  be defined by where where  = ( 1 ,  Furthermore, let ( 1 ,  2 ) be given by where Then ( 13) can be written in the following operator form:

Principle of Exchange of Stability
From the theoretical ecology, it is interesting to study the bifurcation of system (9) at steady state (, ).Bifurcation means that a change in the stability or in the types of steady state which occurs as a parameter is varied in a dissipative dynamic system; that is, the state changes during the biology conditions.The classical bifurcation types are Hopf bifurcation and Turing bifurcation.Ma and Wang [13,14] have developed new methods to study bifurcations and transitions which are called attractor bifurcations.This theory yields complete information about bifurcations, transitions, stability, and persistence, including information about transient states, in terms of the physical parameters of the system.Therefore, in this section, we consider the attractor bifurcation of system (9) at (, ), and from the transformation, we only need to discuss system (13) at (0, 0).Firstly, we consider the linear system of (13), and its eigenvalue problem with the Dirichlet boundary condition (15) or the Neumann boundary condition (16).
Let   be defined by It is easy to see that any eigenvector   and eigenvalue   of (24) can be expressed as where   is as in (25) and   is also the eigenvalue of    .By (24),   can be written as It is clear that  2 () <  1 () = 0 if and only if We define a parameter where   =   such that   () attains its minimum values: In the absence of diffusion, system (23) becomes the spatial homogeneous system From Theorem 1, we can infer that if condition (34) and  ∈ Λ −  hold, then the homogeneous attracting equilibrium loses stability due to the interaction of diffusion processes and system (23) undergoes a Turing bifurcation.

Phase Transition on Homogeneous State
Hereafter, we always assume that the eigenvalue  1 () in ( 24) is simple.Based on Theorem 1, as  ∈ Λ −  the transition of (22) occurs at  =  0 , which is from real eigenvalues.
The following is the main theorem in this paper, which provides not only a precise criterion for the transition types of (22) but also globally dynamical behaviors.Theorem 2. Let   be defined in Theorem 1, and   is the corresponding eigenvector to   of (25) satisfying For system (22), we have the following assertions.
(41) Substituting (41) into (40), we get the bifurcation equation of (22) as follows: By ( 27),   is written as with ( 1 ,  2 ) satisfying from which we get Likewise,  *  is with which yields By ( 22), the nonlinear operator  is Then, in view of ( 45) and ( 48), by direct computation we derive that By ( 45) and (48), we have Hence, by (50) the reduced equation ( 13) is expressed as where  is the parameter as in (39).Based on Theorem A.2 in [16], this theorem follows from (52).The proof is complete.
(4) The bifurcated singular points V  1 and V  2 in the above cases can be expressed in the following form: Proof.Assertion (1) follows from (32).To prove assertions ( 2) and (3), we need to get the reduced equation of ( 22) to the center manifold near  =  0 .Let  =  ⋅   + Φ, where   is the eigenvector of (25) corresponding to  1 () at  =  0 and Φ() the center manifold function of (22).Then the reduced equation of (22) takes the following form: Here  *  is the conjugate eigenvector of   .By (27),   is written as with ( 1 ,  2 ) satisfying (45).
Likewise,  *  is with ( * 1 ,  * 2 ) satisfying (46).It is known that the center manifold function By using the formula for center manifold functions in [13,14,16] where From (55), we see that Let It is clear that where   is the matrix given by (26).Then, it follows from (65) and (66) that for  = , Hence, by (73) the reduced equation ( 13) is expressed as Based on Theorem A.1 in [16], this theorem follows from (73).The proof is complete.