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International Journal of Differential Equations
Volume 2013 (2013), Article ID 502963, 9 pages
http://dx.doi.org/10.1155/2013/502963
Research Article

Positive Periodic Solutions of Cooperative Systems with Delays and Feedback Controls

1College of Applied Mathematics, Xinjiang University of Finance and Economics, Ürümqi 830012, China
2College of Mathematical Sciences, Xinjiang Normal University, Urumqi 830054, China

Received 23 April 2013; Accepted 13 May 2013

Academic Editor: Norio Yoshida

Copyright © 2013 Tursuneli Niyaz and Ahmadjan Muhammadhaji. 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 studies a class of periodic n species cooperative Lotka-Volterra systems with continuous time delays and feedback controls. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin, some new sufficient conditions on the existence of positive periodic solutions are established.

1. Introduction

Mathematical ecological system has become one of the most important topics in the study of modern applied mathematics. Its dynamical behavior includes persistence, permanence and extinction of species, global stability of systems, the existence of positive periodic solutions, positive almost periodic solutions, and strictly positive solutions. The existence of positive periodic solutions has already become one of the most interesting subjects for scholars. In the recent years, the application of fixed-point theorems to the existence of positive periodic solutions in mathematical ecology has been studied extensively, for example, Brouwer’s fixed point theorem [14], Schauder’s fixed-point theorem [58], Krasnoselskii’s fixed-point theorem [914], Horn’s fixed-point theorem [15, 16], and Mawhins continuation theorem [1736], and so forth. In particular, Mawhins continuation theorem is a powerful tool for studying the existence of periodic solutions of periodic high-dimensional time-delayed problems. When dealing with a time-delayed problem, it is very convenient and the result is relatively simple [30]. Recently, a considerable number of mathematical models with delays have been proposed in the study of population dynamics. One of the most celebrated models for population is the Lotka-Volterra system. Subsequently, a lot of the literature related to the study of the existence of positive periodic solutions for various Lotka-Volterra-type population dynamical systems with delays by using the method of continuation theorem was published and extensive research results were obtained [1721, 2434].

On the other hand, in some situations, people may wish to change the position of the existing periodic solution but to keep its stability. This is of significance in the control of ecology balance. One of the methods for the realization of it is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at another periodic solution. The realization of the feedback control mechanism might be implemented by means of some biological control scheme or by harvesting procedure [21]. In fact, during the last decade, the existence of positive periodic solutions for the population dynamics with feedback control has been studied extensively [8, 14, 1721, 24, 29]. To the best of our knowledge, studies on the existence of positive periodic solutions for cooperative systems with delays and feedback controls are fairly rare.

In [21], the authors studied the following neutral Lotka-Volterra system with feedback controls: By using Mawhin’s continuation theorem, the sufficient conditions on the existence of positive periodic solutions are established. In [24], the authors considered the following delay differential system with feedback control: A set of natural and easily verifiable sufficient conditions of the existence of positive periodic solutions are established, by using Mawhin’s continuation theorem. In [29], the authors considered the following single-species periodic logistic systems with feedback regulation and infinite distributed delay: The sufficient conditions for the existence of positive periodic solutions are established, based on Mawhin’s continuation theorem.

Motivated by the above works, in this paper, we investigate the following species periodic Lotka-Volterra-type cooperative systems with continuous time delays and feedback controls: By using the technique of coincidence degree developed by Gaines and Mawhin in [36], we will establish some new sufficient conditions which guarantee that the system has at least one positive periodic solution.

2. Preliminaries

In system (4), we have that represent the density of cooperative species at time , respectively; represent the intrinsic growth rate of species at time , respectively; , represent the intrapatch restriction density of species at time , respectively; , , represent the cooperative coefficients between species at time , respectively. represent the indirect feedback control variables [21] at time , respectively. , , , , and represent the feedback control coefficients at time , respectively. In this paper, we always assume that, ,    are continuous -periodic functions with and . , , , , , , and are continuous, positive -periodic functions.

From the viewpoint of mathematical biology, in this paper for system (4) we only consider the solution with the following initial conditions: where are nonnegative continuous functions defined on satisfying , with , .

In this paper, for any -periodic continuous function we denote

In order to obtain the existence of positive -periodic solutions of system (4), we will use the continuation theorem developed by Gaines and Mawhin in [36]. For the reader’s convenience, we will introduce the continuation theorem in the following.

Let and be two normed vector spaces. Let be a linear operator and let be a continuous operator. The operator is called a Fredholm operator of index zero, if Ker codimIm and Im is a closed set in . If is a Fredholm operator of index zero, then there exist continuous projectors and such that Im Ker  and Im Ker Im. It follows that DomKer DomKer Im is invertible and its inverse is denoted by ; denote by an isomorphism of onto . Let be a bounded open subset of ; we say that the operator is -compact on , where denotes the closure of in , if is bounded and is compact.

Lemma 1 (see [35]). Suppose with and , . Then the function has a unique inverse function satisfying , .

Lemma 2 (see [36]). Let be a Fredholm operator of index zero and let be -compact on . If(a)for each and , ;(b)for each , ;(c),then the operator equation has at least one solution lying in .

3. Main Results

In order to obtain the existence of positive periodic solutions of system (4), firstly, we introduce the following lemma.

Lemma 3. Suppose that ,, , ,   is an -periodic solution of (4) and (5); then ,  ,   satisfies the system where The converse is also true.

Proof. By (4), (5), and the variation constants formula in ordinary differential equations, we have From (9), we obtain Considering that ,  , ,   ,   , is an -periodic solution of system (4) and (5), we obtain Then which implies That is, Hence On the other hand, assume that , , , is an -periodic solution of system (7), then By a direct calculation, we have This completes the proof.

It is easy to see that system (7) is equivalent to the following system: where It is clear that in order to prove that systems (4) and (5) have at least one -periodic solution, we only need to prove that system (18) has at least one -periodic solution.

Now, for convenience of statements we denote the functions The following theorem is about the existence of positive periodic solutions of system (4).

Theorem 4. Suppose that assumption (H1) holds and there exists a constant , , , such that where and the algebraic equation where has a unique positive solution. Then system (4) has at least one positive -periodic solution.

Proof. For system (18) we introduce new variables such that Then system (18) is rewritten in the following form: where In order to apply Lemma 2 to system (26), we introduce the normed vector spaces and as follows. Let denote the space of all continuous functions  . We take with norm It is obvious that and are the Banach spaces. We define a linear operator Dom and a continuous operator as follows: where Further, we define continuous projectors and as follows: We easily see and . It is obvious that Im is closed in and dimKer. Since for any there are unique and with such that , we have codimIm. Therefore, is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) Im is given in the following form: For convenience, we denote as follows: Thus, we have From formulas (36), we easily see that and are continuous operators. Furthermore, it can be verified that is compact for any open bounded set by using the Arzela-Ascoli theorem and is bounded. Therefore, is -compact on for any open bounded subset .
Now, we reach the position to search for an appropriate open bounded subset for the application of the continuation theorem (Lemma 2) to system (26).
Corresponding to the operator equation with parameter , we have where are given in (35).
Assume that is a solution of system (37) for some parameter . By integrating system (37) over the interval , we obtain Consequently, From the continuity of , there exist constants such that By (39) and (40) we obtain where Therefore, we further have Let ,  ; then from Lemma 1 and (H1) we get that function has a unique periodic inverse function ; then, for every , we have One can see that are periodic functions. Then, for every , we have From (39) and (46) we further obtain From the above equality we have From the assumptions of Theorem 4 and (48) we can obtain Consequently, From (40) and (50), we further obtain On the other hand, directly from system (26) we have where From (43), (51), and (52), we have for any Therefore, from (54) and (55), we have It can be seen that the constants are independent of parameter . For any , from (31) we can obtain We consider the following algebraic equation:
From the assumption of Theorem 4, the equation has a unique positive solution . Hence, the equation has a unique solution .
Choosing constant large enough such that and , we define a bounded open set as follows: It is clear that satisfies conditions and of Lemma 2. On the other hand, by directly calculating we can obtain where From the assumption of Theorem 4, we have From this, we finally have This shows that satisfies condition of Lemma 2. Therefore, system (26) has an -periodic solution . Finally, we have system that (4) has a positive -periodic solution. This completes the proof.

Acknowledgments

This work was supported by the Natural Science Foundation of Xinjiang (Grants no. 2012223177, 200721104) and the National Natural Science Foundation of China (Grants no. 10661010, 11261061).

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