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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 830386, 11 pages
A Hopf Bifurcation in a Three-Component Reaction-Diffusion System with a Chemoattraction
1Department of Mathematics, Kyonggi University, Suwon 443-760, Republic of Korea
2Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
3Department of Mathematics, Ohio University, Athens, OH 45701, USA
4Vietnam Institute of Advanced Study in Mathematics, Hanoi, Vietnam
Received 21 August 2013; Accepted 23 October 2013
Academic Editor: Abdelghani Bellouquid
Copyright © 2013 YoonMee Ham et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a three-component reaction-diffusion system with a chemoattraction. The purpose of this work is to analyze the chemotactic effects due to the gradient of the chemotactic sensitivity and the shape of the interface. Conditions for existence of stationary solutions and the Hopf bifurcation in the interfacial problem as the bifurcation parameters vary are obtained analytically.
We are interested in the effects of diffusivity and chemotaxis on the competition of several species for limited resources. Chemotaxis is an oriented movement of cells in response to a concentration gradient of chemical substances in their environment. It was observed that diffusivity and chemotaxis of cells play a dominant role in cell growth; when several species of cells compete for limited resources, the species with a smaller diffusion rate and larger chemotaxis rate grow better, even when other species have superior growth kinetics.
Mathematical modeling on chemotaxis was initiated in 1970 by Keller and Segel (see ) with the use of the following system of PDEs: where is a diffusion coefficient, and are positive constants, and is the chemotaxis coefficient. In many biological processes, cells often interact with combinations of repulsive and attractive signalling chemicals to produce various interesting biological patterns. In this paper, we consider the attraction chemotaxis system [2–4]: where , , , and are positive constants, is a Heaviside step function, and for all . Here, is the gradient operator, is the chemical sensitivity function of the chemical repulsion satisfying for , and is a positive constant.
Chemotaxis describes the direct migration of cells along the concentration gradient of a specific chemical produced by the cells. The prototype of the population-based chemotaxis model was described in the above mentioned work of Keller and Segal .
Schaaf  discussed the existence of nonconstant equilibrium solutions which exhibit aggregating patterns in a bounded domain. In [4, 6], equations describing the dynamics of the interfaces near equilibrium and the stability of the planar standing pulse solutions in the channel domain are obtained for sufficiently small. Results for several versions of the Keller-Segel system and its related models are discussed in Horstmann [7, 8] and Ward . The effect of chemotaxis or that of lateral inhibition on an activator in reaction-diffusion systems has been studied by several authors (see [10–13]).
In the present work, chemotaxis growth under the influence of lateral inhibition in a three-component reaction-diffusion system is considered. We derive a free boundary problem of this system when and then find conditions which are necessary for occurrence of the Hopf bifurcation of chemotaxis and the lateral inhibition on an activator. We derive an evolutional equation of interfaces that is controlled by the two inhibitors and .
Suppose that there is only one interfacial curve in in such a way that , where and . Let lie on this curve; that is, . Using a stretching transformation at we make the following substitutions: Then, the system (2) at becomes and the boundary conditions are when tends to zero, where and . We put the equation into a traveling coordinate system by setting with velocity . Thus, satisfies the following conditions:
The existence of a solution is given in [12, 14] and satisfies . Hence, the velocity of the one-dimensional interface is given by where is the value of on the interface and is a continuously differentiable function defined on an interval , which is given by [14–16]
Hence, a free boundary problem of (2) when is equal to zero is given by
In this paper, we establish the existence of the Hopf bifurcation described above by an application of the implicit function theorem along the lines of the results in . In order to apply the implicit function theorem, we require more regularity of the solution than that obtained in the papers [4, 6, 13]. Our approach to the problem of well-posedness and to the Hopf bifurcation is to write (9) in the form of an abstract evolution equation on a Banach space, which is the product of a function space and an interval of real numbers. Once we have done this, we are able to apply standard results from the theory of nonlinear evolution equations (see for instance, ) to show the well-posedness of the problem and, more importantly, to give an analysis of the Hopf bifurcation.
The organization of the paper is as follows. In Section 2, a change of variables is given which regularizes problem (9) in such a way that results from the theory of nonlinear evolution equations can be applied. In this way, we obtain a regularity of the solution which is sufficient for an analysis of the bifurcation. In Section 3, we show the existence of equilibrium solutions for (9) and obtain the linearization of problem (9). In the last section, we investigate the conditions to obtain the periodic solutions and the bifurcation of the interface problem as the parameter varies.
2. Regularization of the Interface Equation
Now, we consider the existence problem of (9):
Let . Let be an operator defined by with domain . Let with domain . In order to apply semigroup theory to (10), we choose the space with norm .
To get differential dependence on initial conditions, we decompose in (10) into two parts: which is a solution to a more regular problem and which is less regular but explicitly known in terms of the Green function of the operator . Namely, we define by where is a Green’s function of satisfying the Neumann boundary conditions, and is given by If we take a transformation , we have . Since is discontinuous at , we cannot obtain one step more regular than that of (10).
To overcome this difficulty, let . Then satisfies , where with domain . Define where is a Green’s function of satisfying the Dirichlet boundary conditions, and is given by We define , and where is a Green’s function of satisfying the boundary conditions.
Applying the transformations , , and to (10), we get Thus, we obtain an abstract evolution equation equivalent to (10): where is a matrix with the main diagonal entries being the operators , , , and (the zero operator), and all the other terms are zero. The nonlinear forcing term is where , , , , , , , = , and , and , , .
The well-posedness of solutions of (18) is shown in [4, 10, 11], using the fractional powers of degree of , , and and the methods of the theory of semigroups of operators. Moreover, the nonlinear term is a continuously differentiable function from to , where , , , and .
The velocity ofis denoted by where .
The derivative ofcan be obtained following .
Lemma 1. The functions , , , , and are continuously differentiable with derivatives given by
3. Equilibrium Solutions and Linearization of the Interface Equation
In this section, we will examine the existence of equilibrium solutions of (18). We look for satisfying the following equations:
Theorem 2. Suppose that and ; + for all . Then (18) has at least one equilibrium solution for , where is a solution of
The linearization of at the stationary solution is where − . The pair corresponds to a unique steady state of (10) for with , , and .
Proof. From the system of (22), we have , , and . In order to show existence of , we define
Since and for all , is solvable with if , , and , which means that ; , + , and .
Letbe a solution of Then with . Hence, exists for .
The formula for follows from the relation , and the corresponding steady state for (10) is obtained by using Theorem 2.1 in .
4. A Hopf Bifurcation
In this section, we show that there exists a Hopf bifurcation from the curve of the equilibrium solution. First, let us introduce the following relevant definition.
Definition 3. Under the assumptions of Theorem 2, define (for ) the linear operator from to by We then define to be a Hopf point for (18) if and only if there exist an and a -curve ( denotes the complexification of the real space ) of eigendata for with (i), ,(ii) with ,(iii) for all in the spectrum of ,(iv) (transversality),where .
Next, we check (18) for the Hopf points. For this, we solve the eigenvalue problem: where is a identity matrix. This is equivalent to where , , and .
In the following theorem, we show that an equilibrium solution is a Hopf point.
Theorem 4. Suppose that and + for all . Assume that . Additionally, suppose that the operator has a unique pair , of purely imaginary eigenvalues for some . Then, is a Hopf point for (18).
Proof. We assume, without loss of generality, that , and is the (normalized) eigenfunction of with eigenvalue . We have to show that can be extended to a -curve of eigendata for with .
For this, let . First, we note that . Otherwise, by (31), and , which is not possible because is symmetric. So, without loss of generality, let . Then by (31), where
The equation is equivalent to being an eigenvalue of with eigenfunction . We will apply the implicit function theorem to . For this, we check that is of -class and that is an isomorphism. In addition, the mappingis a compact perturbation of the mapping which is invertible. Thus, is a Fredholm operator of index. Therefore, in order to verify (33), it suffices to show that the system of equations
which is equivalent to necessarily implies that , , , and . If we define , , and , then (37) becomes On the other hand, since , , and are solutions to the equations, we have
Multiplying (39) and (43) by and (38) and (42) by and subtracting one from the other, we obtain Multiplying (38) by , (39) by , and (40) by and adding the resultants to each, we have Multiplying (42) by , (43) by , and (44) by and adding the resultants to each, we obtain From (45), we get and thus (47) implies that
Now, multiplying (38) by , (42) by , and (40) by and adding the resultants to each, we have From (50), we get Multiplying (42) by and (43) by , we get and applying (46) to the above equation, we have Now, multiplying (38) by and (42) by and subtracting the resultants to each other, we obtain Applying (54) to the above equation, we have and thus (52) implies that Since and , we have and , and so, and .
Proof. By the implicit differentiation of , we find
This means that the functions , , , and satisfy the equations By letting , , and as before, we obtain Multiplying (60) by , (61) by, and (62) by and adding the resultants to each, we obtain From (49) and (63), the above equation implies that
Multiplying (60) by, (61) by and (62) by and adding the resultants to each, we have From (65), we have and the real part is Now, multiplying (60) by and (61) by and applying (54) to resultants, we obtain and thus (68) implies that which is positive since and . We have for , and thus, by the Hopf-bifurcation theorem in , there exists a family of periodic solutions which bifurcates from the stationary solution as passes .
Now, we show that there exists a unique such that is a Hopf point; thus is the origin of a branch of nontrivial periodic orbits.
Proof. First, we have . So, if , then . Moreover, as and as , which results in the corresponding limiting behavior for .
To show that is strictly increasing, we define . Then (in the weak sense initially) As a result, and is differentiable with = , and therefore Thus, we get It follows that