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Discrete Dynamics in Nature and Society
Volume 2008, Article ID 706154, 13 pages
http://dx.doi.org/10.1155/2008/706154
Research Article

Almost Periodic Solution of a Diffusive Mixed System with Time Delay and Type III Functional Response

Department of Mathematics and Computer Science, Guangxi Qinzhou University, Qinzhou, Guangxi 535000, China

Received 21 February 2008; Accepted 2 June 2008

Academic Editor: Manuel De La Sen

Copyright © 2008 Qiong Liu. 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 delayed predator-prey model with diffusion and competition is proposed. Some sufficient conditions on uniform persistence of the model have been obtained. By applying Liapunov-Razumikhin technique, we will point out, under almost periodic circumstances, a set of sufficient conditions that assure the existence and uniqueness of the positive almost periodic solution which is globally asymptotically stable.

1. Introduction

In the nature world, diffusion often occurs in an ecological environment; that is, species can diffuse between patches. The works about autonomous systems in this field were pioneered by Levin, after Levin [1], Kishimoto [2], and Takeuchi [3] studied this kind of model. But all the coefficients in the system they studied are constants. Since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a varying environment are considered as important selective forces on systems in a fluctuating environment. More realistic and interesting models should take into account both the seasonality of the changing environment and the effects of time delays [47]. This motivated Chen et al. [811], and others to consider nonautonomous predator-prey models with almost periodic coefficients and diffusion. In this paper, we study the almost periodic solution of the delayed predator-prey model with diffusion and competition so as to obtain some conditions under which three species are uniformly persistent. In addition, we obtain that for the almost periodic system there exists a unique almost positive periodic solution which is globally asymptotically stable.

The organization of this paper is as follows. In the next section, we develop our model, establish its important properties, and give several lemmas, which will be a key for our proofs and discussions. In Section 3, sufficient conditions are given for uniform persistence of three species. In Section 4, by applying Liapunov-Razumikhin technique, we prove the existence and uniqueness of the positive almost periodic solution which is globally asymptotically stable. Finally, we give a discussion of our results.

2. Model and Preliminaries

It is assumed that the ecosystem is composed of two isolated patches, and the prey population can disperse among the patches instantaneously. The state variables of the models, describe the densities of the prey population in Patch and Patch , respectively. We suppose that the net exchange of the prey population from Patch to Patch is proportional to the difference of the concentration between with . The state variables of the models, describe the densities of the predator population in Patch with competition.

Let us consider the following delayed diffusive predator-prey system with competition and functional response:with the initial condition Here, and represent the intrinsic growth rate and the intraspecific interference coefficient of the prey population , respectively. We then assume that the death rate of the predator population in Patch 1 is proportional to both the existing predator population with the proportional functions and, respectively, and to its square with the proportional functions and, respectively, . The predator consumes the prey according to Holling type III functional response [12, 13], that is, and . is the time to digest food in the predator organism. . .

We introduce some notations and definitions, and state some preliminary lemmas which will be useful for establishing our main results.

Let . . Assume is a subset of . Denote by the map defined by the right-hand side of system (2.1). If is a continuous function, then the upper right derivative of with respect to system (2.1) is defined asObviously, the global existence and uniqueness of solutions of system (2.1) are guaranteed by the smoothness properties of (see [14, 15] for details on fundamental properties of retarded functional differential equations).

For convenience, we introduce the following notations:In this paper, we need all the coefficients to satisfy

Definition 2.1. System (2.1) is said to be uniformly persistent if there exists a compact region such that every solution of system (2.1) with initial conditions (2.2) eventually enters and remains in the region .

For convenience, the set is positive and nondecreasing for

Lemma 2.2 (See [16, 17]). Consider the following almost periodic equation: Let , , or , , and is uniformly almost periodic with respect to . Let . Assume that the following conditions hold:
(i);(ii), where is a positive constant;(iii) there exists a continuous nondecreasing function such that If system (2.6) has a solution , then system (2.6) has a unique positive almost periodic solution which is uniformly asymptotically stable, and . Furthermore, if is -periodic with respect to , then system (2.6) has a positive -periodic solution which is globally asymptotically stable.

Here, denotes the module of which is the set consisting of all real numbers which are finite linear combinations of elements of the setwith integer coefficients.

Lemma 2.3. is a positive invariant set of system (2.1).

Proof. Let be a solution of system (2.1) with initial conditions (2.2). Hence, for and , we can deriveTherefore, we obtain the positive invariance of . This completes the proof.

We will focus our discussion on with respect to a biological meaning. This also ensures the solution with positive initial value to be positive all the time.

3. Uniform Persistence

In what follows, we want to construct an ultimately bounded region of system (2.1).

Theorem 3.1. There exist three constants such that , , and for each positive solution of system (2.1) with large enough, where

Proof. Suppose that is a solution of system (2.1) with initial conditions (2.2). According to the first two equations of (2.1), we haveWe define the following lines in plane: Then, we haveHence, it follows fromthatIfwe only consider what follows. If , from the given condition we getLetNext, we consider the following three cases.
Case 1. , . Then, there exists such that for . We also derive thatHence, if and , we get
Case 2. . Similarly, we could obtain that there exists . If and , we get
Case 3. . We can also find an interval such that or . In the same way, if and , we can obtainNow, we can know that if , g(t) will monotonously decrease by speed . So, there exists . If , we have
According to the third equation of (2.1), we haveHence, it follows from that for .
Ifwe only consider what follows. If , from the given condition we obtainLetWe also derive thatHence, if and , we getNow, we can know that if , will monotonously decrease by speed . So, there exists such that for . Similarly, we also getWe can also choose the same . There exists such that for . This completes the proof.

Theorem 3.2. Suppose that system (2.1) satisfies the following conditions:in whichThen, system (2.1) is uniformly persistent.

Proof. Suppose is a solution of system (2.1) with the initial condition (2.2). According to the first equation of (2.1), we getSo,Then, there exists a such thatSimilarly,Then, there exists a such thatFrom the third equation of (2.1), we obtainSo,Then, there exists a such thatSimilarly, we also getThen, there exists a such that
Finally, letwhere and ; is given in Theorem 3.1. From Theorem 3.1 and the above analysis, we see that is a bounded compact region in which has positive distance from coordinate hyperplanes. Let , then we obtain that if , then every positive solution of system (2.1) with initial conditions (2.2) eventually enters and remains in the region . This completes the proof.

4. Almost Periodic Solution

In this section, we derive sufficient conditions which guarantee that the periodic solution of periodic system (2.2) is globally attractive.

Theorem 4.1. In addition to (2.5), (3.2), and (3.23), assume further that all the coefficients of system (2.1) are continuous and positive almost periodic functions andThen, system (2.1) has a unique positive almost periodic solution which is globally asymptotically stable. Furthermore, if system (2.1) is an -periodic system, then system (2.1) has a positive -periodic solution which is globally asymptotically stable.

Proof. Consider the product system of (2.1):It is easily noted that the existence and uniqueness of the positive almost periodic solution of system (2.1) are equivalent to the existence and uniqueness of the positive almost periodic solution of system (4.2). Then, choose the following function:Obviously, satisfies conditions (i) and (ii) of Lemma 2.2. Next, we will prove that satisfies condition (iii) of Lemma 2.2. It follows thatin whichalso,In this regard, after few computations, it is noted that It follows from (4.1) that Choose . WhenthenHence,whereThis completes the proof.

5. Discussion

In this work, we consider a nonautonomous delayed predator-prey model with competition and diffusion. Some sufficient conditions on uniform persistence of the model have been given. By means of the Liapunov-Razumikhin technique, it is also seen that, under almost periodic circumstances, the existence and uniqueness of the positive almost periodic solution which is globally asymptotically stable are governed by several inequalities.

Acknowledgments

The author is thankful to the learned referees for their valuable comments which have helped to present a better exposition of the paper. This work is supported by the first project proposals of Guangxi education teaching reform in the 11th five-year plan (2005240), and the project of qualified course reform and establishment of the new century teaching reform in the 11th five-year plan (2006072).

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