Abstract
We present a theoretical analysis of the attractor bifurcation for gene propagation model governed by reaction-diffusion equations. We investigate the dynamical transition problems of the model under the homogeneous boundary conditions. By using the dynamical transition theory, we give a complete characterization of the bifurcated objects in terms of the biological parameters of the problem.
1. 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 iswhere 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 [2–5], 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 [6–8] 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 [9–11], 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 [6–8] have developed the reaction-diffusion-convection 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.
2. 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 and two original alleles , there are six possible genotypes:Let 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 aswhere is the frequency of allele , which can be expressed as is the total population density, is the common death rate, and is the reproductive success rate of individuals with genotype for , 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 ); 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 , , , , , , , and . is the diffusion coefficient which measures the dispersal rate of allele On the other hand, diffusive terms can be considered as describing the ability of the allele to occupy different zones in 2-dimensional space either through the action of small-scale mechanism or by some native transport device.
System (9) has seven constant solutions: , where are complicated expressions of the reproductive success rate and are given by where Considering the biological context, we assume that the steady state solution is positive and
In the present paper, we focus on the bifurcation from the constant solution Let Omitting the primes, then system (9) becomeswhere
We assume that system (13) is satisfied on an open bounded domain . 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 are two Hilbert space and is dense and compact inclusion.
Later, we choose the bifurcation parameter to be the diffusion coefficient ; that is, Let be defined by where where , and
Furthermore, let be given by where
Then (13) can be written in the following operator form:
3. 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
Firstly, we consider the linear system of (13),and its eigenvalue problemwith 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 aswhere is as in (25) and is also the eigenvalue of . By (24), can be written as
It is clear that if and only if
We define a parameterwhere such that attains its minimum values:
Theorem 1. Let be the number given in (30) such that (29) is satisfied and let be the integer such that the minimum is achieved at . Then is the first real eigenvalue of near satisfying thatHere and
In the absence of diffusion, system (23) becomes the spatial homogeneous systemSystem (33) is local asymptotic stability ifand the Hopf bifurcation occurs when
From Theorem 1, we can infer that if condition (34) and hold, then the homogeneous attracting equilibrium loses stability due to the interaction of diffusion processes and system (23) undergoes a Turing bifurcation.
4. Phase Transition on Homogeneous State
Hereafter, we always assume that the eigenvalue in (24) is simple. Based on Theorem 1, as the transition of (22) occurs at , 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) satisfyingFor system (22), we have the following assertions.
(1) Equation (22) has a mixed transition from ; more precisely, there exists a neighborhood of , such that is separated into two disjoint open sets and by the stable manifold of , satisfying the following.
(a) (b) The transition in is jump. (c) The transition in is continuous.
(2) Equation (22) bifurcates in to a unique singular point on which is attractor such that, for any , (3) Equation (22) bifurcates on to a unique saddle point with morse index 1.
(4) The bifurcated singular point can be expressed by Here
Proof. First, we need to get the reduced equation of (22) near .
Let , where and is the eigenvector of (24) corresponding to at . Then the reduced equation of (22) readsHere is the conjugate eigenvector of
By Implicit Function Theorem, we can obtain thatSubstituting (41) into (40), we get the bifurcation equation of (22) as follows:By (27), is written aswith satisfying from which we getLikewise, iswith satisfying which yieldsBy (22), the nonlinear operator isThen, in view of (45) and (48), by direct computation we derive thatBy (45) and (48), we have Hence, by (50) the reduced equation (13) is expressed aswhere is the parameter as in (39). Based on Theorem in [16], this theorem follows from (52). The proof is complete.
Remark 3. If the domain with being a bounded open set, then condition (36) holds true for boundary conditions (15) and (16).
If the domain with being a bounded open set, thenholds true for Neumann condition (16) and Dirichlet condition (15) when the number in is even.
Hereafter, we consider the Neumann condition (16) and let the domain with
The following eigenvalue problem (24) with the equationhas eigenfunctions and eigenvalues as follows:Here .
Denote , the pair of integers satisfying (30), and simply denote ; then we have the following.
Theorem 4. Let be the parameter defined bywhere and is in Appendix and are defined in the proof.
Assume that the eigenvalue satisfying (28) is simple; then for system (22), we have the following assertions:(1)This system has a transition from ; that is, is asymptotically stable for and unstable for which transits to a stable equilibrium at .(2)If , then (22) has a jump transition from and bifurcates on to exactly two saddle points and with the Morse index one.(3)If , then (22) has a continuous transition from , which is an attractor bifurcation.(4)The bifurcated singular points and 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 .
Let , where is the eigenvector of (25) corresponding to at and 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 aswith satisfying (45).
Likewise, iswith satisfying (46).
It is known that the center manifold functionBy using the formula for center manifold functions in [13, 14, 16], satisfieswhereFrom (55), we see thatLetIt is clear that where is the matrix given by (26). Then, it follows from (65) and (66) that for , whereDirect calculation shows thatInserting (66) into (59), by (60) and (61) we get where and is in Appendix.
In view of (70), it follows that Hence, by (73) the reduced equation (13) is expressed asBased on Theorem 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 introduced by Alan Turing and attractor bifurcation developed by Ma and Wang [13] of gene propagation population model governed by reaction-diffusion equation.
Theorems 2 and 4 tell us that the critical value of diffusion constant given by plays a crucial role in determining the attractor bifurcation and Turing bifurcation when the diffusion parameter crosses the critical value , and the uniform stationary state loses its stability, which yields the Turing or attractor bifurcation.
We give some examples to illustrate our main theories.
Example 5. In system (9), we let and then the steady state of system (9) is where
If we set , then , where , and then By Theorem 1, we obtain that the transition conditions are satisfied.
From Theorem 4, , and system (22) has a continuous transition from , which is an attractor bifurcation.
Since , we infer that if , then , and the Turing instability occurs. By numerical simulation, we have shown that the gene population model is able to sustain Turing patterns (Figure 1).
(a)
(b)
Appendix
The Expressions of
The Expressions of are as follows:
Competing Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This paper was partially supported by National Natural Science Foundation of China (Grants no. 11401062 and no. 11371386), Research Fund for the National Natural Science Foundation of Chongqing CSTC (Grant no. cstc2014jcyjA0080), Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ1400937), and the Scientific Research Foundation of CQUT (Grant no. 2012ZD37).