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Journal of Applied Mathematics
Volume 2008, Article ID 537284, 17 pages
http://dx.doi.org/10.1155/2008/537284
Research Article

Asymptotic Behavior of a Competition-Diffusion System with Variable Coefficients and Time Delays

Facultad de Matemáticas, Universidad Autónoma de Yucatán, Periférico Norte, Tablaje 13615, C.P. 97119, Mérida, Yucatán, Mexico

Received 26 June 2007; Accepted 15 January 2008

Academic Editor: Malgorzata Peszynska

Copyright © 2008 Miguel Uh Zapata 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

A class of time-delay reaction-diffusion systems with variable coefficients which arise from the model of two competing ecological species is discussed. An asymptotic global attractor is established in terms of the variable coefficients, independent of the time delays and the effect of diffusion by the upper-lower solutions and iteration method.

1. Introduction and Main Result

The Lotka-Volterra competition model with diffusion and time delays has been the object of analysis by numerous authors under different approaches. For the case of two species, Ruan and Zhao [1] considered uniform persistence and global extinction; Lu [2] studied global attractivity, and Gourley and Ruan [3] analyzed stability and traveling fronts. The periodic case has also been considered, Feng and Wang [4] studied asymptotic stability and Zhou et al. [5] investigated the Hopf bifurcation. The cases of three and -species have also been analyzed in [68].

In this paper, we consider the asymptotic behavior of solutions for the competition-diffusion system with time delays of the following two species: where () is a bounded domain with smooth boundary , denotes the differentiation outward normal on , , , , , , , is of bounded variation with , is bounded and nonnegative and (), and and are the density functions of two species competing for a shared limited resource.

The functions and denote the intrinsic growth rate of the species, and represent self-limitation rates, and and represent the coefficients of the infinite continuous delay. The constants , represent the competition rates. The distributed time delay should be viewed as the effects of past history.

Let denote the total variation of on and let , for all . Then, and are nonnegative and nondecreasing on . It easy to see that Denote , , and . Then, Our result can be stated as follows.

Theorem 1.1. Assume that ,, and Then, for any with , the solution of (1.1) satisfies where , , , are constants given by the linear system

Remark 1.2. If with , , and , , , then by Theorem 1.1, we getwhere , , , are constants given by the linear system
Solving this system, we have the following result:
which coincides with the result of [9], where the authors considered a system like (1.1) with constant coefficients.

Reaction-diffusion systems with delay have been treated by many authors. There are two ways to approach them. The first one is in the framework of semigroup theory of dynamical systems [1, 10]. The second one is a method of upper and lower solutions, using associated monotone iterations; several authors have studied their dynamic properties [2, 9, 11]. Sometimes the birth and death rates depend on both space and time, so when we consider instantaneous and delayed interference within the species and the diffusive effects of the species, system (1.1) will be the appropriate model.

The way we organize the paper is as follows: we first introduce several results which play an important role in the proof of Theorem 1.1 which we will prove in Section 2. We will provide some numerical simulations in Section 3 in order to illustrate our theory.

The following results are developed in [11]. They considered the Volterra reaction-diffusion equations with variable coefficients: where , , and is bounded and nonnegative and . Let denote the total variation of and define , of the same form as that of , , respectively.

Lemma 1.3. Assume that and , then the solution of (1.10) satisfies where If , then . Thus,

Lemma 1.4. Assume that and , then the solution of (1.13) satisfieswhere

Lemma 1.5. If is the solution of (1.10), then .

Now, we introduce the existence-comparison result for the competition-diffusion system (1.1) which is a particular case of Theorem 2.2 in[12].

Definition 1.6. A pair of smooth functions and are called upper-lower solutions of (1.1) if () in and the following differential inequalities hold:

Lemma 1.7. If there exists a pair of upper-lower solutions and of (1.1), then the problem (1.1) has a unique solution and , .

2. Proof of the Main Result

The method of proof is via successive improvements of upper-lower solutions of suitable systems.

For given as initial conditions for the system (1.1), let , be constants such thatwhere , .

Then, and are a pair of lower-upper solutions of (1.1). By Lemma 1.7, there exists a unique global nonnegative solution of (1.1) and it satisfies , .

Define and by Then, and are lower and upper solutions, and by Lemma 1.7, By Lemma 1.4, we can get where Then from (2.3) and (2.4), we get From (2.6) and the definition of we have that for any sufficiently small there exist Define and byThen, and are a pair of lower-upper solutions of (1.1), and by Lemma 1.7, From for all and (2.8), for , we get It follows from (2.9) that, for , , By the comparison principle, we get, for where and are the solutions of the following problem, respectively: Using the three initial conditions and sufficiently small, we haveBy Lemma 1.4, we get Then from (2.12), (2.14), and sufficiently small, we can conclude that where For any sufficiently small , there exist Define and byThen and are a pair of lower-upper solutions of (1.1). By Lemma 1.7, From (2.19) and (2.20), for , , we get By the comparison principle, for , we get where and are the solutions of the following problem, respectively:From (2.16), (2.18), and sufficiently small , we getBy Lemma 1.4, we getThen from (2.22), (2.24), and sufficiently small ,whereThen,For any sufficiently small , there exist Define and byThen, and are a pair of lower-upper solutions of (1.1). By Lemma 1.7,From (2.30) and (2.31), for , , we get By the comparison principle, for , we get where and are the solutions of the following problem, respectively:From (2.29), (2.30), and sufficiently small, we getBy Lemma 1.4 we getFrom (2.33), (2.35), and for , we conclude thatwhere Then, Define the sequences , , and as follows:

Lemma 2.1. For the above-defined sequences, one has

Proof. For , it has been shown that and . Using induction, we can complete the proof.
Lemma 2.1 implies thatexist, denoted as , , and respectively. From (2.41), we have the following linear system by which we can obtain the numbers , , and :

Lemma 2.2. For the solutions of (1.1), one has

Proof. We have shown that (2.45) and (3.1) are valid for . Using induction and repeating the above process, we can complete the proof.

Combining the above lemmas, we can complete the proof of Theorem 1.1

Remark 2.3. Following the same kind of proof for Theorem 1.1, it can be shown that the same conclusions hold if instead of system (1.1) we work with solutions of as expected since they do not include the coefficients of diffusion. We thank one of the referees for his comments regarding this matter.

3. Numerical Simulations

In this section, we present some numerical results that agree with Theorem 1.1 proved above. We used the method of upper and lower solutions as developed by Pao [12, 13] discretizing the systems into finite difference systems. On both examples, the domain used is

Example 3.1. In this example, we work with coefficients that depend on and initial values that depend on .
Consider the systemIt is easy to see that this system satisfies the conditions of the main theorem (Theorem 1.1), therefore the global attractors are defined by the solutions of the linear systemthat is, the global attractors for and are given by and