Abstract and Applied Analysis

Volume 2014 (2014), Article ID 252579, 8 pages

http://dx.doi.org/10.1155/2014/252579

Research Article

Almost Periodic Solution of a Modified Leslie-Gower Predator-Prey Model with Beddington-DeAngelis Functional Response and Feedback Controls

Kerong Zhang,¹ Jianli Li,¹ and Aiwen Yu²

¹Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China

²Department of Mathematics, Beiya Middle School, Changsha, Hunan 410008, China

Received 30 January 2014; Revised 19 February 2014; Accepted 26 February 2014; Published 22 April 2014

Academic Editor: Bingwen Liu

Copyright © 2014 Kerong Zhang 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 consider a modified Leslie-Gower predator-prey model with the Beddington-DeAngelis functional response and feedback controls as follows: , , , and . Sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained.

1. Introduction

In recent years, the modified predator-prey systems with periodic or almost periodic coefficients have been studied extensively.

Leslie [1] proposed the famous Leslie predator-prey system as follows:

where and stand for the population of the prey and the predator at time , respectively, and is the so-called predator functional response to the prey. The term is the Leslie-Gower term which measures the loss in the predator population due to rarity of its favorite food.

Global stability of the positive locally asymptotically stable equilibrium in a class of predator-prey systems has been introduced by Hsu and Huang [2], and the system is as follows:

When the functional response equals , then (2) turns into a Leslie-Gower system [3].

On the other hand, the periodic solution (almost periodic solution) and some other properties of Leslie-Gower predator-prey models were studied (see [4–9]). In particular, Zhang [10] discussed the almost periodic solution of a modified Leslie-Gower predator-prey model with the Beddington-DeAngelis function response as follows:

where is the size of prey population and is the size of predator population.

Stimulated by the above reasons, in this paper, we incorporate the feedback control into model (3) and consider the following model:

where and all the coefficients , , , , , , , , , , and are all continuous, almost periodic functions on .

Associated with (4), we consider a group of initial conditions with the following form (we assume, without loss of generality, that the initial time ):

Let be a continuous bounded function on and we set

Throughout this paper, we assume that the coefficients of the almost periodic system (4) satisfy

By constructing a suitable Lyapunov functional, we obtain some sufficient conditions for the existence of a globally attractive positive almost periodic solution of system (4) with initial conditions (5).

2. Permanence

In this section, we give some definitions and results that we will use in the rest of the paper.

Lemma 1 (see [11]). If , , and , when and , one has

Lemma 2 (see [11]). If , , and , when and , one has

Set the following:

Theorem 3. Suppose that system (4) with initial condition (5) satisfies the following condition:

Then system (4) is permanent; that is, any positive solution of the system (4) satisfies

Proof. From the first equation of (4), we have the following:

Applying Lemma 1 to (13) leads to

From (14), we know that there exists an enough large such that

so there exists an enough large such that

It follows from (16) and the second equation of system (4) that, for ,

Applying Lemma 2 to (17) leads to

By using a similar argument as that in the proof of (14) and (18), we can get the following:

From (18) and the first equation of system (4) we know

Applying Lemma 1 and (11) to the above leads to

Therefore, we know that there exists an enough large such that

From the second equation of system (4) we have the following:

Applying Lemma 2 to the above, we obtain the following:

By using a similar method as that in the proof of (21) and (24), it follows that

This completes the proof.

We denote by the set of all solutions of system (4) satisfying , , , and for all .

Theorem 4. Consider the following: .

Proof. From the properties of almost periodic function there exists a sequence with as such that

as uniformly on . Let be a solution of system (4) satisfying , , , and for . Clearly, the sequence is uniformly bounded and equicontinuous on each bounded subset of . Therefore, by the Arzelà-Ascoli theorem, there exists a subsequence which converges to a continuous function as uniformly on each bounded subset of . Let be given. We may assume that for all . For , we have the following:

Applying Lebesgue’s dominated convergence theorem and letting in (27), we obtain the following:

Since is arbitrarily given, is a solution of system (4) on . It is clear that , , , for . Thus . This completes the proof.

3. Existence of a Unique Almost Periodic Solution

Now let us state several definitions and lemmas which will be useful in the proving of the main result of this section.

Definition 5 (see [12]). A function , where is an vector, is a real scalar, and is an vector, is said to be almost periodic in uniformly with respect to , if is continuous in and and if, for any , there is a constant such that in any interval of length there exists a such that the inequality

is satisfied for all , . The number is called an number of .

Definition 6 (see [12]). A function is said to be asymptotically almost periodic function, if there exists an almost periodic function and a continuous function such that , and as .

Lemma 7 (see [13]). Let be a nonnegative, integral, and uniformly continuous function defined on ; then .

Theorem 8. Suppose that all conditions of Theorem 3 hold; furthermore assume that

(H), where ,

Then system (4) with initial conditions (5) is globally attractive.

Proof. Let , , and then system (4) is transformed into

Suppose that and are any two positive solutions of (31).

Let , where

Calculating the right derivative of along the solution of (31), we have the following:

Further, it follows that

Therefore, we have the following:

Integrating the above inequality on internal , it follows that, for ,

Then, for , we obtain that

By Lemma 7, we obtain

Then the solution of systems (4) and (5) is globally attractive.

Theorem 9. Suppose that all conditions of Theorem 8 hold; then there exists a unique almost periodic solution of systems (4) and (5).

Proof. According to Theorem 4, there exists a bounded positive solution of (4) and (5). Then there exists a sequence , as , such that is a solution of the following system:

According to Theorem 3, we get that not only but also are uniformly bounded and equicontinuous. By Ascoli’s theorem there exists a uniformly convergent subsequence such that, for any , there exists a with the property that if , then

This is to say, are asymptotically almost periodic functions; hence there exist four almost periodic functions and four continuous functions such that

where

are an almost periodic function.

Therefore,

On the other hand,

So exist. Now we will prove that is an almost periodic solution of system (4).

From properties of almost periodic function, there exits a sequence , as , such that

as uniformly on .

It is easy to know that as , and then we have the following:

By using a similar argument as that in the above, we have the following:

This proves that is a nonnegative almost periodic solution of systems (4) and (5); by Theorem 8, it follows that there exists a globally asymptotically stable nonnegative almost periodic solution of system (4). The proof is complete.

4. An Example

Consider the following system:

By a simple calculation, we check that all conditions in Theorems 8 and 9 are fulfilled. Therefore, by Theorems 8 and 9, system (48) has a unique globally asymptotically stable nonnegative almost periodic solution (see Figure 1).

Figure 1: Dynamic behavior of system (48) with the initial , for , . From the figure, we could easily see that the solution is asymptotic to the unique almost periodic solution of the system (48).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the NNSF of China (no. 11171085) and Hunan Provincial Natural Science Foundation of China (no. 10JJ6002).

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