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

Tursuneli Niyaz¹ and Ahmadjan Muhammadhaji²

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 [1–4], Schauder’s fixed-point theorem [5–8], Krasnoselskii’s fixed-point theorem [9–14], Horn’s fixed-point theorem [15, 16], and Mawhins continuation theorem [17–36], 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 [17–21, 24–34].

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, 17–21, 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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