# The Permanence and Extinction of a Discrete Predator-Prey System with Time Delay and Feedback Controls

- Qiuying Li
^{1}Email author, - Hanwu Liu
^{1}and - Fengqin Zhang
^{1}

**2010**:738306

**DOI: **10.1155/2010/738306

© Qiuying Li et al. 2010

**Received: **23 May 2010

**Accepted: **7 September 2010

**Published: **14 September 2010

## Abstract

A discrete predator-prey system with time delay and feedback controls is studied. Sufficient conditions which guarantee the predator and the prey to be permanent are obtained. Moreover, under some suitable conditions, we show that the predator species *y* will be driven to extinction. The results indicate that one can choose suitable controls to make the species coexistence in a long term.

## 1. Introduction

For the general nonautonomous case, they addressed properties such as permanence, extinction, and globally asymptotic stability of the system. For the periodic (almost periodic) case, they established sufficient criteria for the existence, uniqueness, and stability of a positive periodic solution and a boundary periodic solution. At the end of their paper, numerical simulation results that complement their analytical findings were present.

However, we note that ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecosystem is the question of whether an ecosystem can withstand those unpredictable forces which persist for a finite period of time or not. In the language of control variables, we call the disturbance functions as control variables. In 1993, Gopalsamy and Weng [13] introduced a control variable into the delay logistic model and discussed the asymptotic behavior of solution in logistic models with feedback controls, in which the control variables satisfy certain differential equation. In recent years, the population dynamical systems with feedback controls have been studied in many papers, for example, see [13–22] and references cited therein.

where , are the density of the prey species and the predator species at time , respectively. , are the feedback control variables. represent the intrinsic growth rate and density-dependent coefficient of the prey at time , respectively. denote the death rate and density-dependent coefficient of the predator at time , respectively. denotes the capturing rate of the predator; represents the rate of conversion of nutrients into the reproduction of the predator. Further, is a positive integer.

For the simplicity and convenience of exposition, we introduce the following notations. Let , and denote the set of integer satisfying We denote to be the space of all nonnegative and bounded discrete time functions. In addition, for any bounded sequence we denote ,

The main purpose of this paper is to establish a new general criterion for the permanence and extinction of system (1.3), which is dependent on feedback controls. This paper is organized as follows. In Section 2, we will give some assumptions and useful lemmas. In Section 3, some new sufficient conditions which guarantee the permanence of all positive solutions of system (1.3) are obtained. Moreover, under some suitable conditions, we show that the predator species will be driven to extinction.

## 2. Preliminaries

In this section, we present some useful assumptions and state several lemmas which will be useful in the proving of the main results.

Throughout this paper, we will have both of the following assumptions:

Now, we state several lemmas which will be used to prove the main results in this paper.

where functions , are bounded and continuous defined on with , . We have the following result which is given in [23].

Lemma 2.1.

- (a)
there exists a positive constant such that

- (b)
for any two positive solutions and of (2.3).

where functions and are bounded and continuous defined on with and The following Lemma 2.2 is a direct corollary of Theorem of L. Wang and M. Q. Wang [24, page 125].

Lemma 2.2.

- (a)
for any positive solution of (2.5);

- (b)
for any two positive solutions and of (2.5).

where functions and are bounded and continuous defined on with , and The following Lemma 2.3 is a direct corollary of Lemma of Xu and Teng [25].

Lemma 2.3.

where is a positive solution of (2.5) with

where functions are bounded and continuous defined on with and In [25], the following Lemma 2.4 has been proved.

Lemma 2.4.

## 3. Main Results

Theorem 3.1.

for any positive solution of system (1.3).

Proof.

for all where is the initial time.

This completes the proof of Theorem 3.1.

In order to obtain the permanence of system (1.3), we assume that

Theorem 3.2.

for any positive solution of system (1.3).

Proof.

where is the solution of (3.21) with initial condition

for all

for any where Thus, from (3.23) and (3.33), we have which leads to a contradiction. Therefore, (3.24) holds.

for all , and .

which is a contradiction. Therefore, the conclusion of Theorem 3.2 holds. This completes the proof of Theorem 3.2.

For system (3.52), we further introduce the following assumption:

suppose , where , are given in the proof of Lemma 3.3.

For system(3.52), we have the following result.

Lemma 3.3.

- (a)
there exists a constant such that

(3.53)for any positive solution of system (3.52).

- (b)
if assumption holds, then each fixed positive solution of system (3.52) is globally uniformly attractive on

Proof.

Based on assumptions , conclusion (a) can be proved by a similar argument as in Theorems 3.1 and 3.2.

Therefore, as and we can easily obtain that and The proof is completed.

then we have the following result.

Lemma 3.4.

where is the solution of system (3.52) with and

The proof of Lemma 3.4 is similar to Lemma 3.3, one omits it here.

Let be a fixed solution of system (3.52) defined on one assumes that

Theorem 3.5.

for any positive solution of system (1.3).

Proof.

for all where Obviously, we have as which is contradictory to the boundedness of solution of system (1.3). Therefore, (3.65) holds.

for all , , and .

which is a contradiction. Therefore, the conclusion of Theorem 3.5 holds.

Remark 3.6.

In Theorems 3.2 and 3.5, we note that are decided by system(1.3), which is dependent on the feedback control . So, the control variable has impact on the permanence of system (1.3). That is, there is the permanence of the species as long as feedback controls should be kept beyond the range. If not, we have the following result.

Theorem 3.7.

for any positive solution of system (1.3).

Proof.

So, which is a contradiction. Therefor, there exists an such that

which leads to a contradiction. This shows that (3.99) holds. By the arbitrariness of it immediately follows that as This completes the proof of Theorem 3.7.

## Declarations

### Acknowledgments

This work was supported by the National Sciences Foundation of China (no. 11071283) and the Sciences Foundation of Shanxi (no. 2009011005-3).

## Authors’ Affiliations

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