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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 812357, 8 pages
http://dx.doi.org/10.1155/2013/812357
Research Article

Global Asymptotic Behavior of a Nonautonomous Competitor-Competitor-Mutualist Model

1College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China
2School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China

Received 10 May 2013; Accepted 17 July 2013

Academic Editor: Rodrigo Lopez Pouso

Copyright © 2013 Shengmao Fu and Fei Qu. 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

The global asymptotic behavior of a nonautonomous competitor-competitor-mutualist model is investigated, where all the coefficients are time-dependent and asymptotically approach periodic functions, respectively. Under certain conditions, it is shown that the limit periodic system of this asymptotically periodic model admits two positive periodic solutions such that , and the sector is a global attractor of the asymptotically periodic model. In particular, we derive sufficient conditions that guarantee the existence of a positive periodic solution which is globally asymptotically stable.

1. Introduction

In this paper, we investigate the global asymptotic behavior of solutions for the following competitor-competitor-mutualist diffusion model: where , , and are the densities of a mutualist-competitor, a competitor, and a mutualist population, respectively. is a bounded smooth domain, is an outward normal derivative on .

In 1983, Rai et al. [1] firstly presented and studied a general competitor-competitor-mutualist ordinary differential equation (ODE) model. Zheng [2] studied the problem (1)–(3) in the case where all coefficients are positive constants. He proved the local stability of the unique positive constant steady-state solution under suitable condition on the reaction rates by the method of spectral analysis for linearized operator. Xu [3] investigated the global asymptotic stability of the unique positive constant steady-state solution under some assumptions by the iteration method. Pao [4] considered the model with time delays, and, under a very simple condition on the reaction rates, proved that the time-dependent solution with any nontrivial initial function converges to the positive steady-state solution by the method of upper and lower solutions. Chen and Peng [5] proved some existence results concerning nonconstant positive steady-states for the model with cross-diffusion and demonstrated that the cross-diffusion can create patterns when the corresponding model without cross-diffusion fails. Li et al. [6] proved that this model with cross-diffusion possesses at least one coexistence state if cross-diffusions and cross-reactions are weak by the Schauder fixed point theory and the method of upper and lower solutions and its associated monotone iterations. Fu et al. [7] investigated the global asymptotic behavior and the global existence of time-dependent solutions for the model with cross-diffusion when the space dimension is at most 5. Very recently, Tian and Ling [8] proved that, under some conditions, a corresponding predator-prey-mutualist model with cross-diffusion admits at least a nonhomogeneous stationary solution by the stability analysis for the positive uniform solution and the Leray-Schauder degree theory and carried out numerical simulations for a Turing pattern.

For the model (1) with -periodic coefficients, Tineo [9] studied the asymptotic behavior of positive solutions by the method of upper and lower solutions. Du [10] investigated the existence of positive -periodic solutions by using the degree and bifurcation theories. Pao [11] proved the existence of maximal and minimal -periodic solutions by the method of upper and lower solutions. Wang et al. [12] considered the local asymptotic behavior of the time-dependent solutions and the existence of periodic solutions to the model in an unbounded domain. Zhou and Fu [13] investigated the global asymptotic behavior of the time-dependent solutions and the existence of periodic solutions for the model with discrete delays. Very recently, replacing the usual term by a degenerate elliptic operator as , Wang and Yin [14] proved the existence of maximal and minimal -periodic solutions to the model with time delays by the Schauder fixed point theorem. It is important to note that the uniqueness of positive periodic solution is not considered in the previous references.

When , (1) reduces to the competition diffusion system where , , , , , and . The system (4) is a diffusion extension of the well-known Lotka-Volterra system In the case that , , , , , and are positive -periodic functions, the existence and asymptotic stability of periodic solutions for (5) was studied by Gopalsamy [15], Alvarez and Lazer [16], and Ahmad [17] in the 1980’s. The global asymptotic behavior of (5) was studied by Ahmad and Lazer [18] and Tineo [19]. Denote and for any function . If , , , , , and are positive asymptotically -periodic functions on , Peng and Chen [20] proved that if the conditions are satisfied for a certain sufficiently small , then any positive solutions of (5) asymptotically approach the unique positive periodic solution for the limit periodic system of (5).

It is well known that periodic reaction diffusion equations are of particular interests since they can take into account seasonal fluctuations occurring in the phenomena appearing in the models, and they have been extensively studied by many researchers (see, e.g., [914, 19, 21]). However, so far, the research work on asymptotically periodic systems is much fewer than on the periodic ones. In fact, asymptotically periodic systems describe our world more realistically and more accurately than periodic ones to some extent. Therefore, for asymptotically periodic systems, studying the dynamics behavior is important and necessary (see, e.g., [2227]).

In this paper, we study the global asymptotic behavior of positive solutions for the asymptotically periodic system (1). Under some conditions, it is shown that any positive solutions of the models asymptotically approach the unique strictly positive periodic solutions of the corresponding periodic system. This means that the results in Tineo [9] and the results for ODE model in Peng and Chen [20] can be extended to the asymptotically periodic reaction diffusion system and the 3-species diffusion system, respectively. Furthermore, using the method of the present paper, we note that the corresponding conclusions hold for the time-dependent -species Lotka-Volterra systems. More specifically, we provide a way of how to use the method of upper and lower solutions to study asymptotic behavior of solutions for asymptotically periodic reaction diffusion systems. As one can see, the optimal bounds and uniqueness of positive periodic solutions will play an important role in the study of the global asymptotic behavior of periodic solutions.

2. Permanence and Extinction

For the sake of convenience, we introduce the two signs and for functions . is said to approach asymptotically in notation, , if uniformly for in . Furthermore, if and are vector functions, then if and only if   . We say that is asymptotically smaller than and write if uniformly for . It is clear that if and only, if for any , there exists a corresponding such that on .

Assume the following., and are positive smooth and -periodic functions on ., and are positive smooth functions on , and

By , the limit periodic system of (1), (2) is given as follows:

As a complement, we state the following main result which comes from [9, Theorem 0.3].

Theorem 1. Assume that holds, and Then (8) has the periodic solutions and such that for any positive -periodic solution of (8). Moreover, given that and a solution of (8) with , there exists such that on .

In order to get the conditions for the permanence of (1)–(3), we need to make the following optimal bounds.

Lemma 2. If (9) and (10) hold and is a positive smooth -periodic solution of (8), then where is the unique positive root of and is the unique positive root of , and

Proof. By the maximum principle (see Lemma 1.2 of [18]), we have Hence, . Since and (by (9)), we can see immediately that . Similarly, if is the unique positive root of , then . So, Evidently, . By (9) and (10), we have from which it follows that . This completes the proof.

Corollary 3. Assume that (9) and (10) hold. If , , and are positive constants, then (9) has the unique positive periodic solution , where is the unique positive root of .

The main results in this section are the following theorems.

Theorem 4 (permanence). Assume that , , (9), and (10) hold. Then (8) has the positive -periodic solutions and such that . Moreover, if is the solution of (1)–(3) with smooth initial values , then

Remark 5. Under the assumptions of Theorem 4, the system (1), (2) is permanent, the sector is a global periodic attractor of (1), (2), and its trivial and semitrivial periodic solutions are unstable. Furthermore, if , , and are positive constants, then is the unique globally asymptotically stable solution of (8).

Theorem 6. Assume that and hold. Then one has the following conclusions. (1) (Extinction of ) Assume that (9) holds and that . Then (8) has a -periodic solution such that , , , and uniformly on , for any positive solution of (1)–(3).(2) (Extinction of ) Assume that (10) holds and that Then (8) has a -periodic solution with , , and satisfying (18), where is any positive solution of (1)–(3).

Proof of Theorem 4. By (9) and (10), there exists a sufficiently small such that, for ,
Consider two auxiliary systems as follows:
By (20), (21), and Theorem 1, (22) has the positive -periodic solutions and such that and   , for any positive -periodic solution of (22). Moreover, if is a solution of (22) with nontrivial nonnegative initial values, then, for any , there exists such that for all and . Similarly, (23) has the positive -periodic solutions and such that and , for any positive -periodic solution of (23). Furthermore, if is a solution of (23) with nontrivial nonnegative initial values, then, for the previous , there exists such that, for all and ,
Now we prove where and are positive -periodic solutions of (8) (see Theorem 1). Let , , and be the solutions of (22), (8), and (23), respectively, which all satisfy the same initial conditions. It is easily testified that , are the upper and lower solutions of (8) and (3), respectively. So from [28, Corollary ], we see that
For sufficiently small and , define
Choose . Then and are the ordered upper and lower solutions of (22) and (3) (also of (23) and (3) and of (8) and (3)). Applying the same technique from [18, Theorem 4.1], we can prove that the solution of (22) and (3), the solution of (23) and (3), and the solution of (8) and (3) satisfy, respectively, It follows from (27) that and that . Similarly, choose ; we can prove the other inequalities of (26).
Denote by , and the optimal bounds for the positive periodic solutions of problems (22) and (23), respectively, (see Lemma 2). Then Moreover, for . By using the dominated convergence theorem and the bootstrap arguments (see [18]), we have uniformly for on , and and are the positive -periodic solutions of (8).
From Theorem 1, we see that and . It follows from (26) and (32) that . So,
Therefore, given that , by (32) and (33), there exists such that Since , for the previous , there exists such that, for ,
Denote by the solution of (1)–(3), and denote by , , and the solutions of problems (22), (8), and (23), respectively, which all satisfy the same initial conditions . Analogous to (27), we have for . By (24), there exists such that for . Similarly, by (25), there exists such that for . Hence, by (34)–(37), we have for . This completes the proof.

3. Global Stability

In order to get conditions of global stability for (1)–(3), we need the following result.

Lemma 7. Let be a -periodic solution for the linear problem where , , , , , , , , and are positive smooth -periodic functions on and where , , and are smooth -periodic functions. If then .

Proof. Let be a smooth -periodic solution of (39), and let positive constants , , , and be chosen so that Such choices are obviously possible because (40) holds.
It is easy to verify that and are a pair of ordered upper and lower solutions of (39). Thus, . This implies that because is -periodic in .
Similarly, . This completes the proof.

Lemma 8 (uniqueness). Assume that (9) and (10) hold. If then the problem (8) has a unique positive -periodic solution.

Proof. Let , and . By Theorem 1, we have It follows from Lemmas 2 and 7 and the conditions (9), (10), and (42) that . This completes the proof.

Theorem 9. If all conditions of Theorem 4 and (42) are satisfied, then for any positive solution of (1)–(3).

Proof. By some elementary calculations, we know that Theorem 9 is an immediate corollary of Theorem 4 and Lemma 8.

Example 10. Consider the following asymptotically periodic system: It is not hard to verify that all conditions of Theorem 9 are satisfies. Thus, any positive solution of (45) asymptotically approach the unique positive periodic solution of the limit periodic system of (45).

4. Case

The following results are natural generalizations of the main results in [20] which can be proved in the similar way as to prove Theorems 4 and 9.

Theorem 11. Assume the following., , , , , , , and are positive smooth -periodic functions on ., , , , , and are positive smooth functions on : Then the limit periodic system of (4) has the positive -periodic maxmini solution and minimax solution . Moreover, if is any positive solution of (4) with smooth initial value , then and . In addition, if then (47) has the unique positive -periodic solution and

Acknowledgments

The authors would like to thank the referees for their helpful comments. This work is supported by the China National Natural Science Foundation (Grants nos. 11061031 and 11261053) and the Fundamental Research Funds for the Gansu University.

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