Discrete Dynamics in Nature and Society

Volume 2010, Article ID 583203, 12 pages

http://dx.doi.org/10.1155/2010/583203

## Feedback Control Variables Have No Influence on the Permanence of a Discrete -Species Schoener Competition System with Time Delays

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350108, China

Received 1 May 2010; Accepted 15 July 2010

Academic Editor: Antonia Vecchio

Copyright © 2010 Qianqian Su and Na Zhang. 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 consider a discrete -species Schoener competition system with time delays and feedback controls. By using difference inequality theory, a set of conditions which guarantee the permanence of system is obtained. The results indicate that feedback control variables have no influence on the persistent property of the system. Numerical simulations show the feasibility of our results.

#### 1. Introduction

The Schoener's competition system has been studied by many scholars. Topics such as existence, uniqueness, and global attractivity of positive periodic solutions of the system were extensively investigated and many excellent results have been derived [1–5].

Liu et al. [1] propose and study the global stability of the following continuous Schoener's competition model with delays: where , , and are all positive bounded and continuous functions, are positive integers, , , , .

As we all know, though most dynamic behaviors of population models are based on the continuous models governed by differential equations, the discrete time models are more appropriate than the continuous ones when the size of the population is rarely small or the population has nonoverlapping generations. It has been found that the dynamics behaviors of the discrete system are rather complex and contain more rich dynamics than the continuous ones [6]. Recently, more and more scholars paid attention to study the discrete population models (see [4–13] and the references cited therein). For example, [5] considered the permanence and global attractivity of the following discrete Schoener's competition model with delays: where , , and are real positive bounded sequences, are positive integers, , , ,

On the other hand, as was pointed out by Huo and Li [14], ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. 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 excellent works on the feedback control ecosystems (see [14–21] and the references cited therein); however, most of those works are concerned with the continuous model and seldom did scholars considered the discrete ecosystem with feedback controls ([13, 15, 21]).

Recently, Li and Yang [15] proposed the following discrete -species Schoener competition system with time delays and feedback controls: where is the density of competitive species at th generations; is the control variable; is the first-order forward difference operator ,

Throughout this paper, we assume the following., , , , , , , , , , are all bounded nonnegative sequences such that

Here, for any bounded sequence , , , , , , are all nonnegative integers.

Let we consider (1.3) together with the following initial conditions: It is not difficult to see that solutions of (1.3) and (1.5) are well defined for all and satisfy

By applying the comparison theorem of difference equation, they obtained a set of sufficient conditions which guarantee the permanence of the system (1.3). Their result shows that feedback control variables play important roles on the persistent property of the system (1.3). But the question is whether or not the feedback control variables have influence on the permanence of the system. The aim of this paper is to apply the analysis technique of Chen et al. [18] to establish sufficient conditions, which is independent of feedback control variables, to ensure the permanence of the system.

The organization of this paper is as follow. In Section 2, we will introduce several lemmas. The permanence of system (1.3) is then studied in Section 3. In Section 4, a suitable example together with its numerical simulations shows the feasibility of our results.

#### 2. Preliminaries

In this section, we will introduce several useful lemmas.

Lemma 2.1 (see [11]). *Assume that satisfies and
**
for where and are nonnegative sequences bounded above and below by positive constants. Then
*

Lemma 2.2 (see [12]). *Assume that satisfies
** and where and are nonnegative sequences bounded above and below by positive constants and . Then
*

Lemma 2.3 (see [13]). *Assume that , Suppose that
**
Then for any integer **
If and is bounded above with respect to then
*

Lemma 2.4 (see [13]). *Assume that , Suppose that
**
Then for any integer **
If and is bounded below with respect to then
*

#### 3. Permanence

In this section, we establish the following permanence result for system (1.3).

Theorem 3.1. *Assume that and hold. Then there exist positive constants , which are independent of the solutions of system (1.3) such that
*

*Proof. *Let be any positive solution of system (1.3) with initial condition (1.5). From the first equation of system (1.3), it follows that
By using (3.2), one can easily obtain that
that is,
Substituting (3.4) into (3.2), it follows that
By applying Lemma 2.1, it follows that
Thus, for all there exists a , for all , and so
For Lemma 2.3 implies that
Letting in the above inequality leads to
The proof of Theorem 3.1 is completed.

Theorem 3.2. *Assume that and hold; assume further that
**
Then there exist positive constants , which are independent of the solution of system (1.3), such that
*

*Proof. *Let be any positive solution of system (1.3) with initial condition (1.5). From Theorem 3.1, for all there exists a for all
From (3.12), we have
for all where

Thus, for by using (3.13) we obtain
From the second equation of (1.3), we obtain
where and

Then, Lemma 2.3, (3.14), and (3.15) imply that for any ,
For That is, there exists an for all
For enough small we have
We choose , For fixed we get
for all where

Substituting (3.14) (3.17) and (3.19) into (3.13), for one has
where

By applying Lemma 2.2, it follows that
Letting in the above inequality leads to
For the above there exists a for all , we have , Then for ,
By applying Lemma 2.4, it follows that
Letting in the above inequality leads to
The proof of Theorem 3.2 is completed.

*Remark 3.3. *Theorems 3.1 and 3.2 show that under the assumption , and () hold, and system (1.3) is permanent.

#### 4. Example and Numeric Simulations

The following example lends credence to the plausibility of Theorem 3.2.

*Example 4.1. *Consider the following system:
Here corresponding to system (1.3), we assume that

It is easy to see that and
Inequalities (4.3) show that all the conditions of Theorem 3.2 are satisfied; thus, system (1.3) is permanent. Numerical simulation from Figure 1 supports this conclusion.

#### 5. Discussion

In this paper we have attempted to understand the effect of feedback control variables on the permanence of system (1.3). The present work is an extension of an earlier work by Li and Yang [15]. By developing the analysis technique of Chen et al. [18], a set of sufficient conditions are established for the permanence of system (1.3). Theorems 3.1-3.2 and the numerical simulations indicate that feedback control variables have no influence on the permanence of system (1.3).

We would like to mention here that an interesting but challenging problem associated with the study of system (1.3) should be the global attractivity. We leave this for future work.

#### Acknowledgments

The authors would like to thank the two anonymous referees for their constructive suggestions on improving the presentation of the paper. Also, this work is supported by the Technology Innovation Platform Project of Fujian Province (2009J1007).

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