Abstract and Applied Analysis

Volume 2013 (2013), Article ID 830386, 11 pages

http://dx.doi.org/10.1155/2013/830386

Research Article

A Hopf Bifurcation in a Three-Component Reaction-Diffusion System with a Chemoattraction

YoonMee Ham,¹ Sang-Gu Lee,² and Quoc Phong Vu3,4

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.

Abstract.

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.

1. Introduction

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 [1]) 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 [1].

Schaaf [5] 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 [9]. 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 [17]. 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, [18]) 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 [19].

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

Then

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 [19].

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 mapping

is 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 .

Theorem 5. Under the same condition as in Theorem 4, satisfies the transversality condition. Hence, this is a Hopf point for (18).

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 [19], 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.

Lemma 6. Suppose that . Let , , and be Green functions of the differential operators , and satisfying (42), (43), and (44), respectively. Then, and are strictly decreasing in with

Moreover, and for .

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

Since , we have for .

In order to show , from (72), we have

which implies that . Since , we have for.

Let . Then we have and for .

Theorem 7. Under the same condition as in Theorem 4, for a unique critical point , there exists a unique, purely imaginary eigenvalue of (31) with .

Proof. We only need to show that the function has a unique zero with and . This means solving the system of (31) with ,  ,  , and ,

The real and imaginary parts of the above equation are given by

Since by Lemma 6, there is a critical point , provided the existence of . We now define

Using Lemma 6, we have for and + if . Moreover,

for and . Hence, there exists a unique .

The following theorem summarizes the results above.

Theorem 8. Suppose that and + for all . Then (18) and (10) have at least one stationary solution , where , and where , and , for all and for , respectively, where is a solution of

Assume that . Then there exists a unique such that the linearization has a purely imaginary pair of eigenvalues. The point is then a Hopf point for (18), and there exists a -curve of nontrivial periodic orbits for (18) and (10), bifurcating from and , respectively.

Acknowledgment

This paper was supported by 63 Research Fund, Sungkyunkwan University, 2012.

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