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Discrete Dynamics in Nature and Society
Volume 2008, Article ID 629620, 17 pages
http://dx.doi.org/10.1155/2008/629620
Research Article

Permanence and Global Attractivity of a Delayed Discrete Predator-Prey System with General Holling-Type Functional Response and Feedback Controls

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, China

Received 8 June 2008; Accepted 11 September 2008

Academic Editor: Leonid Berezansky

Copyright © 2008 Lijuan Chen 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

This paper discusses a delayed discrete predator-prey system with general Holling-type functional response and feedback controls. Firstly, sufficient conditions are obtained for the permanence of the system. After that, under some additional conditions, we show that the periodic solution of the system is global stable.

1. Introduction

The following predator-prey system with Holling-type II functional response and delays and some generalized systems of general Holling-type functional response have been studied by many scholars (see [13] and the references cited therein). It has been found that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have non-overlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations (see [412]). In [4], Yang considered the following delayed discrete predator-prey system with general Holling-type functional response: Sufficient conditions which guarantee the existence of at least one positive periodic solution are obtained by using the continuation theorem of coincidence degree theory. But Yang did not consider the permanence and globally attractivity of system (1.2), which are two of the most important topics in the study of population dynamics.

On the other hand, as was pointed out by Huo and Li [13], ecosystem in the real world is continuously distributed 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 (for more discussion on this section, one could refer to [1216] for more details). Though much works dealt with the continuous time model. However, to the best of the author's knowledge, up to this day, there are still no scholars that propose and study the system (1.2) with feedback control. Therefore, the main purpose of this paper is to study the following delayed discrete predator-prey system with general Holling-type functional response and feedback control: where is the density of prey species at th generation, is the density of predator species at th generation, and are control variables. Also, denote the intrinsic growth rate and density-dependent coefficient of the prey, respectively, denote the death rate and density-dependent coefficient of the predator, denote the capturing rate of the predator, represent the rate of conversion of nutrients into the reproduction of the predator. Further, are nonnegative constants and are positive constants. In this paper, we always assume that are bounded nonnegative sequences and Here, for any bounded sequence , and , where

This paper is organized as follows. In Section 2, we will introduce some definition and establish several useful lemma. The permanence of system (1.3) is then studied in Section 3. In Section 4, based on the permanence result, under the assumption that all the delays are equal to zero and the coefficients of the system are periodic sequences, we obtain a set of sufficient conditions which guarantee the existence and stability of a unique globally attractive positive periodic solution of the system.

By the biological meaning, we will focus our discussion on the positive solution of system (1.3). So it is assumed that the initial conditions of (1.3) are of the form where

One can easily show that the solutions of (1.3) with the initial condition (1.5) are defined and remain positive for all

2. Preliminaries

In this section, we will introduce the definition of permanence and several useful lemmas.

Definition 2.1. System (1.3) is said to be permanent if there exist positive constants which are independent of the solution of the system, such that for any positive solution of system (1.3) satisfies for

Lemma 2.2. Assume that satisfies where and are positive sequences, is a positive constant, and Then one has where

Lemma 2.3. Assume that satisfies where and are positive sequences, is a positive constant, and Also, and Then one has where

Proof. The proofs of Lemmas 2.2 and 2.3 are very similar to those of [6, Propositions 2.1 and 2.2], respectively. So we omit the detail here.

Lemma 2.4. Assume that satisfies where and are positive sequences, and are positive constants, and Then one has where and

Proof. From the above equation, one has Sequently we can easily obtain that So one has By Lemma 2.2, we can complete the proof of Lemma 2.4.

Lemma 2.5. Assume that satisfies where and are positive sequences, and are positive constants, and Also, and where Then one has where

Proof. From the above equation, one has Sequently we can easily obtain that So one has By Lemma 2.3, we can complete the proof of Lemma 2.5.

Lemma 2.6 is a direct corollary of [17, Theorem 6.2, page 125] by L. Wang and M. Q. Wang.

Lemma 2.6. Consider the following first-order difference equation: where are positive constants. Assuming for any solution of the above system, one has

The following comparison theorem for the difference equation is of [17, Theorem 2.1, page 241] by L. Wang and M. Q. Wang.

Lemma 2.7. Let For fixed is a nondecreasing function with respect to and for the following inequalities hold: If then for all

3. Permanence

In this section, we establish a permanent result for system (1.3).

Proposition 3.1. In addition to (1.4), assume further that
for any positive solution of system (1.3), one has where

Proof. Let be any positive solution of system (1.3), from the first equation of (1.3), it follows that By applying Lemmas 2.4 and 2.7, we obtain where Similarly, from the second equation of (1.3), it follows that Under the assumption , by applying Lemmas 2.4 and 2.7, we obtain where
For any positive constant small enough, it follows from (3.5) and (3.8) that there exists large enough such that Then the third equation of (1.3) leads to And so By applying Lemmas 2.6 and 2.7, it follows from (3.12) that Setting in the above inequality leads to where Similarly, we can obtain where Thus we complete the proof of Proposition 3.1.

Proposition 3.2. In addition to (1.4), assume further that
for any positive solution of system (1.3), there exist positive constants such that

Proof. Let be any positive solution of system (1.3). From and , there exists a small enough positive constant such that Also, according to Proposition 3.1, for the above , there exists such that for Then from the first equation of (1.3), one has Let so the above inequality follows that Consequently, let Because one has Here we use the fact that From (3.19) and (3.23), by Lemmas 2.5 and 2.7, one has Setting in the above inequality leads to where and
Similarly, from the second equation of (1.3), one has Let so the above inequality leads to Consequently, let Because one has Here we use the fact that From (3.20) and (3.29), by Lemmas 2.5 and 2.7, one has Setting in the above inequality leads to where Then the third equation of (1.3) leads to And so
By applying Lemmas 2.6 and 2.7, it follows from (3.35) that Setting in the above inequality leads to where Similarly, we can obtain where Thus we complete the proof of Proposition 3.2.

Theorem 3.3. In addition to (1.4), assume further that and hold, then system (1.3) is permanent.

It should be noticed that, from the proofs of Propositions 3.1 and 3.2, we know that under the conditions of Theorem 3.3, the set is an invariant set of system (1.3).

4. Existence and Stability of a Periodic Solution

In this section, we consider the stability property of system (1.3) under the assumption that is, we consider the following system: which are similar to system (1.3) but do not include delays. In this section, we always assume that , , , , , are bounded nonnegative periodic sequences with a common period and satisfy Also it is assumed that the initial conditions of (4.1) are of the form Using a similar way, under some conditions, we can obtain the permanence of system (4.1). As above, still let and be the upper bound of and and let be the lower bound of and where , and are independent of the solution of system (4.1). Our first result concerns with the existence of a periodic solution.

Theorem 4.1. In addition to (4.2), assume further that and hold, then system (4.1) has a periodic solution denoted by .

Proof. Let is an invariant set of system (4.1). Thus, we can define a mapping on by for
Obviously, depends continuously on Thus is continuous and maps a compact set into itself. Therefore, has a fixed point It is easy to see that the solution passing through is a periodic solution of system (4.1). This completes the proof.

Now, we study the globally stability property of the periodic solution obtained in Theorem 4.1.

Theorem 4.2. In addition to the conditions of Theorem 4.1, if system (4.1) satisfies where the definition of can be seen in the following proof, then the -periodic solution obtained in Theorem 4.1 is globally attractive.

Proof. Assume that is any positive solution of system (4.1), let To complete the proof, it suffices to show that Since where and for Because of the boundedness of , , , , , are bounded, where and mean the partial derivation of the function . Let and
Similarly, we get where Because of the boundedness of is bounded, where and means the derivation of the function . Let
Also, one has
In view of (4.5)–(4.8), we can choose a such that Also, from Propositions 3.1 and 3.2, there exist such that Then from (4.11), for one has So from (4.13), for one has Also, for one has Let then In view of (4.18)–(4.21), one has for This implies Therefore This completes the proof.

5. Examples

The following two examples show the feasibility of our main results.

Example 5.1. Consider system (1.3) with for all One can easily see that which means that Also, one has Inequalities (5.3)–(5.5) show that are fulfilled. From Theorem 3.3, system (1.3) is permanent. Figure 1 is the numeric simulation of the solution of system (1.3) with initial condition

629620.fig.001
Figure 1: Dynamics behaviors of system (1.3) with initial condition   .

Example 5.2. Consider system (4.1) with for all One can easily see that which means that Also, one has Inequalities (5.8)–(5.10) show that are fulfilled. We can obtain that which means that So (4.5)–(4.8) are fulfilled. From Theorem 4.2, system (4.1) is globally attractive. Figure 2 is the numeric simulation of the solution of system (4.1) with initial condition and

629620.fig.002
Figure 2: Dynamics behaviors of system (4.1) with initial values = (0.9, 0.7, 0.6, 0.05), (1.2, 0.5, 0.3, 0.2), and (1, 0.2, 0.25, 0.1).

Acknowledgment

This work was supported by the Foundation of Fujian Education Bureau (JB05042).

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