Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 326848, 6 pages

http://dx.doi.org/10.1155/2013/326848

Research Article

Permanence for a Discrete Ratio-Dependent Predator-Prey System with Holling Type III Functional Response and Feedback Controls

Jiangbin Chen¹ and Shengbin Yu²

¹Zhicheng College, Fuzhou University, Fuzhou, Fujian 350002, China

²Sunshine College, Fuzhou University, Fuzhou, Fujian 350015, China

Received 6 October 2013; Accepted 2 December 2013

Academic Editor: Candace M. Kent

Copyright © 2013 Jiangbin Chen and Shengbin Yu. 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.

A new set of sufficient conditions for the permanence of a ratio-dependent predator-prey system with Holling type III functional response and feedback controls are obtained. The result shows that feedback control variables have no influence on the persistent property of the system, thus improving and supplementing the main result of Yang (2008).

1. Introduction

The aim of this paper is to investigate the permanent property of the following discrete ratio-dependent predator-prey system with Holling type III and feedback controls:

where and are the densities of the prey population and predator population at time , respectively, for , and are all bounded nonnegative sequences such that

Here, for any bounded sequence , , .

By the biological meaning, we will focus our discussion on the positive solutions of system (1). So, we consider (1) together with the following initial conditions:

It is not difficult to see that the solutions of (1)–(3) are well defined and satisfy

Recently, Yang [1] proposed and studied the permanence of system (1). Set

Using the comparison theorem of difference equation, Yang obtained the following result.

Theorem A (see [1]). Assume that

hold; then system (1) is permanent.

Theorem A shows that feedback control variables play important roles in the persistent property of the system (1). But the question is whether or not the feedback control variables have influence on the permanence of the system. On the other hand, as was pointed out by Fan and Wang [2], “if we use the method of comparison theorem, then the additional condition, in some extent, is necessary. But for the system itself, this condition may not necessary.[sic]” In [2], by establishing a new difference inequality, Fan and Wang showed that feedback control has no influence on the permanence of a single species discrete model. Their success motivated us to consider the persistent property of system (1). Indeed, in this paper, we will apply the analysis technique of Fan and Wang [2] to establish sufficient conditions, which is independent of feedback control variables, to ensure the permanence of the system. We finally obtain the following main results.

Theorem B. Assume that

hold; then system (1) is permanent.

Comparing with Theorem A, it is easy to see that in Theorem B are weaker than in Theorem A and feedback control variables have no influence on the permanent property of system (1), so our results improve the main results in [1]. For more works on this direction, one could refer to [3–10] and the references cited therein.

The remaining part of this paper is organized as follows. In Section 2, we will introduce several lemmas. The permanence of system (1) 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 1 (see [11]). Assume that satisfies

and , where and are nonnegative sequences bounded above and below by positive constants and . Then

Lemma 2 (see [2]). Assume that and . Suppose that

Then, for any integer ,

In particular, if and is bounded above with respect to , then

Lemma 3 (see [2]). Assume that and . Suppose that

Then, for any integer ,

In particular, if and is bounded above with respect to , then

3. Permanence

In this section, we detail the proof of our main result by several lemmas. The following lemma is a direct conclusion of [1].

Lemma 4. There exist two positive constants and such that

where and are defined in (5).

Lemma 5. Assume

holds then there exist two positive constants and such that

where and are defined in the proof.

Proof. According to Lemma 4, for any small enough, there exists enough large , such that, for ,

Thus, it follows from (16) and the first equation of system (1) that

for , where . For any integer , it follows from (17) that

Thus

From the third equation of system (1), we have

where and . Then, for any , according to Lemma 2, (19), and (20)

Note that ; hence . Therefore,

Then, there exists a positive integer such that, for any positive solution of system (1), for all . In fact, we could choose , where . Fixing , for , we get

where .

Substituting (23) into the first equation of system (1), we can get

where and .

By applying Lemma 1 to (24), it immediately follows that

Setting in the above inequality, we obtain

It follows from (26) that there exists large enough such that

This together with the third equation of system (1) leads to

Hence,

By applying Lemma 3, it follows from (29) that

This completes the proof of Lemma 5.

Lemma 6. Assume that

holds; then there exist two positive constants and such that

where and are defined in the proof.

Proof. According to Lemmas 4 and 5, for any small enough, there exists enough large , such that, for ,

Thus, it follows from (32) and the second equation of system (1) that

for , where . For any integer , it follows from (33) that

Thus

From the fourth equation of system (1), we can get

where and . Then, for any , according to Lemma 2, (35) and (36), that

Note that ; hence . Therefore,

Then, there exists a positive integer such that, for any positive solution of system (1), for all . In fact, we could choose , where . Fixing , for , we get

where .

Substituting (39) into the second equation of system (1), we can get

for all , where and .

By applying Lemma 1 to (40), it immediately follows that

Setting in the above inequality, we obtain

It follows from (42) that there exists large enough such that

This together with the fourth equation of system (1) leads to

Hence,

By applying Lemma 3, it follows from (45) that

This completes the proof of Lemma 6.

Lemmas 4–6 show that the conclusion of Theorem B holds.

4. Example and Numerical Simulation

Consider the following system:

In this case, we have

Equation (48) means that all conditions of Theorem B are satisfied in system (47). Thus, the system (47) is permanent. Our numerical simulation supports our result (see Figure 1). However,

that is to say, does not hold and we could not obtain the result of the permanence from Theorem A. Thus our results improve the main results in [1].

Figure 1: Dynamic behavior of the system (47) with the initial condition and , respectively.

Acknowledgment

This research is supported by the Foundation of Fujian Education Bureau (JA11294 and JA13365).

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