Abstract and Applied Analysis

Volume 2012 (2012), Article ID 867364, 14 pages

doi:10.1155/2012/867364

Research Article

Permanence of a Single Species System with Distributed Time Delay and Feedback Control

Yali Shen,¹ Fengqin Zhang,¹ and Kai Wang²

¹Department of Mathematics, Yuncheng University, Yuncheng 044000, China

²Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi 830011, China

Received 29 February 2012; Accepted 30 April 2012

Academic Editor: Jinhu Lü

Copyright © 2012 Yali Shen et al. 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.

We study the permanence of a classofsingle species system with distributed time delay and feedback controls. General criteria on permanence are established in this paper. A very important fact is found in our results; that is, the feedback control is harmless to the permanence of species.

1. Introduction

Ecosystems in the real world are continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. During the last decade, many scholars did works on the feedback control ecosystems. Motivated by those work especially [1], we consider the following single species system with distributed time delay and feedback control:

where is the density of the species at time , is the control variable; , ,,,,,, and are defined on and are bounded and continuous functions; ,,,,,, , and are nonnegative for all . Further, we assume that the delay kernels are nonnegative integral functions defined on such that and .

We denote by the space of all bounded continuous functions with norm . In this paper, we always assume that all solutions of system (1.1) satisfy the following initial condition:

where and .

Let be the solution of system (1.1) satisfying initial condition (1.2). We easily prove and in maximal interval of the existence of the solution. For the sake of convenience, the solution of system (1.1) with initial condition (1.2) is said to be positive.

In addition, for a function defined on set , we denote

In the theory of mathematical biology, systems such as (1.1) are very important in a single species system in time-fluctuating environments, the effect of time delays and feedback controls. We see that there has been a series of articles which deal with the dynamical behaviors of the autonomous, periodic, and general nonautonomous population growth systems with feedback controls, for example [1–12] and reference cited therein. In [1] the authors proposed the following single species model with feedback regulation and distributed time delay of the form:

By using the continuation theorem of coincidence theory, a criterion which guarantees the existence of positive periodic solution of system (1.4) is obtained. In [2], the authors obtain sufficient condition which guarantees the global attractivity of the positive solution of system (1.4) by constructing a suitable Lyapunov functional. The aim of this paper is to establish new sufficient conditions on the permanence for all positive solutions of system (1.1) by improving the method given in [13–15].

2. Preliminaries

Throughout this paper, we will introduce the following assumptions:

(H₁) there exists a constant , such that

(H₂) there exists a constant , such that

(H₃) there exists a constant such that

(H₄) there exists a constant such that

(H₅) there exists a constant such that

First, we consider the following nonautonomous logistic equation

where functions are bounded and continuous on . Furthermore, for all . We have the following result which is given in [15] by Teng and Li.

Lemma 2.1. Suppose that assumptions (H₁)-(H₂) hold. Then,

(a)there exist positive constants and such that

for any positive solution of (2.6);(b) for any two positive solutions and of (2.6).

Further, we consider the following nonautonomous linear equation

where functions and are bounded continuous defined on , and for all . We have the following result.

Lemma 2.2. Suppose that assumptions (H₄) hold. Then,

(a)there exists a positive constant such that for any positive solution of (2.8);(b) for any two positive solutions and of (2.8).

The proof of Lemma 2.2 is very simple, we hence omit it here.

Lemma 2.3. Suppose that assumption (H₄) holds. Then for any constants and there exist constants and such that for any and with , when for all , one has

where is the solution of (2.8) with initial condition .

The proof of Lemma 2.3 can be found as Lemma  2.4 in [16] by Wang et al.

Lemma 2.4. Let be a nonnegative and bounded continuous function, and let be an integral function satisfying . Then one has

Lemma 2.4 is given in [17] by Montes de Oca and Vivas.

3. Main Results

Theorem 3.1. Suppose that assumption (H₁) holds and (H₂) or (H₃) holds. Then there exists a positive constant such that

for any positive solution of system (1.1).

Proof. Let be a positive solution of the first equation system (1.1). Since

for all as long as the solution exists and the function is bounded and continuous on . We can obtain that the solution exists for all . For any , and , by integrating (3.2) from to , we have

For any , where , by (3.3) we can directly from the first equation of (1.1)

where . Since for any and

We have

Let

Since , we have

Hence, assumption (H₂) or (H₃) implies

where the constant or . Since for all , we finally obtain

We consider the following auxiliary equation:

then by conclusion (a) of Lemma 2.1 and inequality (3.10) we obtain that there exist a constant such that for any solution of (3.11) with initial condition . Let be the solution of (3.11) with initial condition , then by the comparison theorem, we have from (3.4) that

Thus, we finally obtain that

Moreover, by Lemma 2.4, for any positive constant , there exists a constant such that

Considering the second equation of the system (1.1), we have

We consider the following auxiliary equations:

then, by assumption (H₄) and conclusion (a) of the Lemma 2.2, we obtain that there is a constant such that for any solution of (3.16) with initial condition . Let be the solution of (3.16) with initial condition and , then, by comparison theorem, we have from (3.15) that

Thus, we finally obtain that

Choose the constant , then we finally obtain

This completes the proof.

Theorem 3.2. Suppose that assumptions (H₁)–(H₅) hold. Then there exists a constant which is independent of the solution of system (1.1) such that

for any positive solutions of system (1.1).

Proof. Let be a solution of system (1.1); from Theorem 3.1 there exists a such that for all we have . According to the assumption (H₁) we can choose positive constants and such that, for all , we have

Consider the following equation:

where is a parameter. By Lemma 2.3, for given in above and positive constant (Theorem 3.1), there exist constants and such that for any and , when , for all , we have

where is the solution of (3.22) with initial condition . Hence, we can choose a positive constant such that, for all ,

We first prove that

In fact, if (3.25) is not true, then there exists a constant such that for all . Choose constants , such that

where, then for we have

Let be the solution of (3.22) with initial condition , then by the comparison theorem we have

In (3.23), we choose and , since for all , we have

Hence, we further have

For any , from the first equation of (1.1) we have

Integrating (3.31) from to we obtain

Obviously, inequality (3.24) implies that as , which leads to a contradiction.

Now, we prove the conclusion of Theorem 3.2 to hold. Assume that it is not true, then there exists a sequence of initial functions of system (1.1) such that

where is the solution of system (1.1) with initial condition

From (3.25) and (3.33), we obtain that for every there are two time sequences and , satisfying

Let . For each we can choose a constant such that

By Theorem 3.1, there exists a constant such that for each there exists a such that

Further, from (3.36) there is an integer such that for all for all . Hence, for any and by (3.39) and (3.40) we have

where . Therefore, for any and integrating the above inequality from to , we further have

Consequently,

For any and we can obtain

 (3.20)For each by (3.36) there exists a and constant such that

By (3.43) there exists a large enough such that

Hence, for any and , by (3.38), (3.20), and (3.45) we have

Assume that is the solution of (3.22) satisfying initial condition , then we have

In (3.23), we choose and , since for all , we have

Hence, we further have

for all and .

For any , and , from (3.38)–(3.40), we have

Integrating the above inequality from to , then by (3.24), (3.37) and (3.38) we obtain

which leads to a contradiction. Therefore, this contradiction shows that there exists constant such that

for any positive solutions of system (1.1). This completes the proof.

Applying Theorem 3.2 to system (1.4), we have the following corollary.

Corollary 3.3. Suppose that assumptions (H₁)–(H₅) hold, then system (1.4) is permanent.

Obviously, we will first consider system (1.1) which is more general than system (1.4). Moreover, Corollary 3.3 is a very good improvement of Theorem  2.1 in [2]. From Corollary 3.3, we find that it is established that a very weak sufficient condition for the permanent of system (1.4).

Remark 3.4. From Theorem 3.2 we directly see that for system (1.1) the feedback control is harmless to the permanence of system (1.1).

Remark 3.5. In this paper, the biological model about a single species is considered. However, complex networks have attracted increasing attention from various fields of science and engineering in recent years. Meanwhile, in some sense, the coupled species systems can be treated as a typical complex networks. The real world biological systems have more complex structures and relationships. Motivated by above work [18–21], our future work is that how to apply the current complex networks theory to improve the current work.

Acknowledgments

This work is supported by the National Sciences Foundation of China (11071283), the Sciences Foundation of Shanxi (2009011005-3), the Young Foundation of Shanxi Province (no. 2011021001-1), Research Project supported by Shanxi Scholarship Council of China (2011-093) and the Major Subject Foundation of Shanxi and Doctoral Scientific Research Fund of Xinjiang Medical University.

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