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Journal of Applied Mathematics
Volume 2012, Article ID 270104, 14 pages
http://dx.doi.org/10.1155/2012/270104
Research Article

Existence of Almost-Periodic Solutions for Lotka-Volterra Cooperative Systems with Time Delay

Department of Applied Mathematics, Kunming University of Science and Technology, Kunming, Yunnan 650093, China

Received 13 December 2011; Accepted 29 February 2012

Academic Editor: Hongyong Zhao

Copyright © 2012 Kaihong Zhao. 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

This paper considers the existence of positive almost-periodic solutions for almost-periodic Lotka-Volterra cooperative system with time delay. By using Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost-periodic solutions are obtained. An example and its simulation figure are given to illustrate the effectiveness of our results.

1. Introduction

Lotka-Volterra system is one of the most celebrated models in mathematical biology and population dynamics. In recent years, it has also been found with successful and interesting applications in epidemiology, physics, chemistry, economics, biological science, and other areas (see [14]). Moreover, in [5], it was shown that the continuous-time recurrent neural networks can be embedded into Lotka-Volterra models by changing coordinates, which suggests that the existing techniques in the analysis of Lotak-Volterra systems can also be applied to recurrent neural networks.

Owing to its theoretical and practical significance, Lotka-Volterra system have been studied extensively (see [616] and the cites therein). Since biological and environmental parameters are naturally subject to fluctuation in time, the effects of a periodically varying environment (e.g., seasonal effects of weather, food supplies, mating habits, etc.) are considered as important selective forces on systems in a fluctuating environment. Therefore, on the one hand, models should take into account both the periodically changing environment and the effects of time delays. However, on the other hand, in fact, it is more realistic and reasonable to study almost-periodic system than periodic system. Recently, there are two main approaches to obtain sufficient conditions for the existence and stability of the almost-periodic solutions of biological models: one is using the fixed point theorem, Lyapunov functional method, and differential inequality techniques (see [1719]); the other is using functional hull theory and Lyapunov functional method (see [1416]). However, to the best of our knowledge, there are very few published letters considering the almost-periodic solutions for nonautonomous Lotka-Volterra cooperative system with time delay by applying the method of coincidence degree theory. Motivated by this, in this letter, we apply the coincidence theory to study the existence of positive almost-periodic solutions for Lotka-Volterra cooperative system with time delay as follows: where stands for the th species population density at time , is the natural reproduction rate, represents the inner-specific competition, stands for the interspecific cooperation, and are all continuous almost-periodic functions on . Throughout this paper, we always assume that , , and are all nonegative almost periodic functions with respect to .

The initial condition of (1.1) is of the form where is positive bounded continuous function on and .

The organization of the rest of this paper is as follows. In Section 2, we introduce some preliminary results which are needed in later sections. In Section 3, we establish our main results for the existence of almost-periodic solutions of (1.1). Finally, an example and its simulation figure are given to illustrate the effectiveness of our results in Section 4.

2. Preliminaries

To obtain the existence of an almost-periodic solution of system (1.1), we first make the following preparations.

Definition 2.1 (see [20]). Let be continuous in is said to be almost-periodic on , if, for any , the set is relatively dense, that is, for any , it is possible to find a real number , for any interval with length , there exists a number in this interval such that , for any .

Definition 2.2. A solution of (1.1) is called an almost periodic solution if and only if for each , is almost periodic.

For convenience, we denote the set of all real-valued, almost-periodic functions on and for each , let be the set of Fourier exponents and the module of , respectively, where is almost periodic. Suppose is almost periodic in , uniformly with respect to . denote the set of -almost periods for uniformly with respect to . denote the length of inclusion interval. Let be the mean value of on interval , where is a constant. clearly, depends on .

Lemma 2.3 (see [20]). Suppose that and are almost periodic. Then the following statements are equivalent.

(i),(ii)for any sequence , if for each , then there exists a subsequence such that for each .

Lemma 2.4 (see [21]). Let . Then is almost periodic.

Let and be Banach spaces. A linear mapping is called Fredholm if its kernel, denoted by , has finite dimension and its range, denoted by , is closed and has finite codimension. The index of is defined by the integer . If is a Fredholm mapping with index 0, then there exist continuous projections and such that and . Then is bijective, and its inverse mapping is denoted by . Since is isomorphic to , there exists a bijection . Let be a bounded open subset of and let be a continuous mapping. If is bounded and is compact, then is called -compact on , where is the identity.

Let be a Fredholm linear mapping with index 0 and let be a -compact mapping on . Define mapping by . If for all , then by using , , , defined above, the coincidence degree of in with respect to is defined by where is the Leray-Schauder degree of at relative to .

Then The Mawhin’s continuous theorem is given as follows.

Lemma 2.5 (see [22]). Let be an open bounded set and let be a continuous operator which is -compact on . Assume(a)for each , ;(b)for each , ;(c). Then has at least one solution in .

In this paper, since we need some related properties of -matrix we introduce them as follows. In addition, A matrix means that each elements .

Definition 2.6 (see [23]). If a real matrix satisfies the following conditions (1) and (2):(1), , , , ,(2) is a positive-definite matrix,then is called a -matrix.

Lemma 2.7 (see [23]). If matrix is a -matrix, then exists and its every element is nonnegative.

Lemma 2.8. Suppose that matrix is a -matrix, then implies .

Proof. In fact, there exists a nonnegative positive vector such that which imply that . According to Lemma 2.4, there exists at least one positive element in the every row of , which imply . Thus, we obtain .

3. Main Result

In this section, we state and prove our main results of our this paper. By making the substitution Equation (1.1) can be reformulated as The initial condition (1.2) can be rewritten as follows: Set , where and is defined as (3.3), is a given constant. For , define .

Lemma 3.1. is a Banach space equipped with the norm .

Proof. If and converges to , that is, , as . Then it is easy to show that and . For any , we have that therefore, which implies . Then it is not difficult to see that is a Banach space equipped with the norm . Thus, we can easily verify that and are Banach spaces equipped with the norm . The proof of Lemma 3.1 is complete.

Lemma 3.2. Let , , then is a Fredholm mapping of index 0.

Proof. Clearly, is a linear operator and . We claim that . Firstly, we suppose that . Then there exist and constant vector such that that is, From the definition of and , we can easily see that and are almost-periodic functions. So we have , then , which implies , that is .
On the other hand, if , then we have , . If , then we obtain It follows that hence Note that is the primitive of in , we have , that is, . Therefore, .
Furthermore, one can easily show that is closed in and therefore, is a Fredholm mapping of index 0. The proof of Lemma 3.2 is complete.

Lemma 3.3. Let , , where Set Then is -compact on , where is an open bounded subset of .

Proof. Obviously, and are continuous projectors such that It is clear that , . Hence Then in view of we obtain that the inverse of exists and is given by Thus, where
Clearly, and are continuous. Now we will show that is also continuous. By assumptions, for any and any compact set , let be the length of the inclusion interval of , . Suppose that and uniformly converges to , that is , as , . Because of , , there exists such that , . Let be the length of the inclusion interval of and It is easy to see that is the length of the inclusion interval of and , . Hence, for any , there exists such that , . Hence, by the definition of almost periodic function we have From this inequality, we can conclude that is continuous, where . Consequently, and are continuous.
From (3.22), we also have and also are uniformly bounded in . Further, it is not difficult to verify that is bounded and is equicontinuous in . By the Arzela-Ascoli theorem, we have immediately concluded that is compact. Thus is -compact on . The proof of Lemma 3.3 is complete.

Theorem 3.4. Assume that the following conditions and hold:; is a positive-definite matrix, where , , , , Then (1.1) has at least one positive almost periodic solution.

Proof. To use the continuation theorem of coincidence degree theorem to establish the existence of a solution of (3.2), we set Banach space and the same as those in Lemma 3.1 and set mappings , , , the same as those in Lemmas 3.2 and 3.3, respectively. Then we can obtain that is a Fredholm mapping of index 0 and is a continuous operator which is -compact on .
Now, we are in the position of searching for an appropriate open, bounded subset for the application of the continuation theorem. Corresponding to the operator equation we obtain Assume that is a solution of (3.25) for some . Denote , .
On the one hand, by (3.25), we derive On the both sides of (3.26), integrating from 0 to and applying the mean value theorem of integral calculus, we have where , , . In the light of (3.27), we get for , On the both sides of (3.28), taking the supremum with respect to and letting , we obtain that is, Equation (3.30) can be written by the following matrix form By Lemma 2.8 and assumption, we obtain which imply that On the two sides of (3.28), taking the infimum with respect to , , and letting , we obtain
On the other hand, according to (3.25), we derive On the both sides of (3.35), integrating from 0 to and using the mean value theorem of integral calculus, we get where , . On the both sides of (3.36), take the supremum and infimum with respect to , respectively, and let , then we have for , namely, which imply that Combining with (3.33), (3.34), and (3.39), we derive for all , Denote . Clearly, is independent of . Take It is clear that satisfies the requirement (a) in Lemma 2.5. When is a constant vector in with . Then where If , then we have which imply that , . Thus, , this contradicts the fact that . Therefore, , which implies that the requirement (b) in Lemma 2.5 is satisfied. If necessary, we can let be greater such that , for any . Furthermore, take the isomorphism , and let , then for any , , we have So, the requirement (c) in Lemma 2.5 is satisfied. Hence, (3.2) has at least one almost-periodic solution in , that is, (1.1) has at least one positive almost periodic solution. The proof is complete.

4. An Example and Simulation

Consider the following two species cooperative system with time delay: where , , , , , , , , , . Since then, the matrix is positive definite, and Therefore, all conditions of Theorem 3.4 are satisfied. By Theorem 3.4, system (4.1) has one positive almost-periodic solution. The resulting numerical simulation is depicted in Figure 1.

270104.fig.001
Figure 1

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant (no. 11161025), Yunnan Province natural scientific Research Fund Project (no. 2011FZ058), and Yunnan Province education department scientific Research Fund Project (no. 2001Z001).

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