Abstract and Applied Analysis

Volume 2012 (2012), Article ID 358594, 31 pages

doi:10.1155/2012/358594

Research Article

Permanence and Almost Periodic Solutions of a Discrete Ratio-Dependent Leslie System with Time Delays and Feedback Controls

Gang Yu and Hongying Lu

School of Mathematics and Quantitative Economics, Dongbei University of Finance and Economics, Dalian 116025, China

Received 10 July 2012; Accepted 2 September 2012

Academic Editor: Wolfgang Ruess

Copyright © 2012 Gang Yu and Hongying Lu. 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 almost periodic ratio-dependent Leslie system with time delays and feedback controls. Sufficient conditions are obtained for the permanence and global attractivity of the system. Furthermore, by using an almost periodic functional Hull theory, we show that the almost periodic system has a unique globally attractive positive almost periodic solution.

1. Introduction

Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The predator-prey models have been extensively studied by many scholars [1–4]. Under the assumption that reduction in a predator population has a reciprocal relationship with per capita availability of its preferred food, Leslie [5] proposed the famous Leslie-Gower predator-prey model where the carry capacity of the predator’s environment is proportional to the number of prey:

where and represent prey and predator densities at time , respectively. is the predator’s rate of feeding upon prey, that is, the so-called predator’s functional response. If we assume that the predator consumes the prey according to the functional response , then system (1.1) formulates as the following:

where is prey-dependent functional responses. Owing to its theoretical and practical significance, system (1.2) and its various generalized forms have been studied extensively and seen great progress (see, for example, [6–9]).

However, in the study of the dynamic behaviors of predator-prey system, many scholars argued that the ratio-dependent predator-prey systems are more realistic [10, 11]. A ratio-dependent predator-prey system with Leslie-Gower term takes the form of

Though much progress has been seen in the asymptotic convergence of solutions of population systems, such systems are not well studied in the sense that most results are continuous-time cases related. Already, many scholars have paid attention to the nonautonomous discrete population models, since the discrete time models governed by difference equation are more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world (see [12–23]). Furthermore, the asymptotic convergence of solutions of difference equations with delay is one of the most important topics in the study of population dynamics, and many excellent results have already been obtained and seen great progress (see [24–26] and the references cited therein).

Since time delays occur so often in nature, a number of ecological systems can be described as systems with time delays (see [14, 15, 19, 21–27]). One of the most important problems for this type of system is to analyze the effect of time delays on the stability of the system. Furthermore, as we know, ecological systems in the real world are often distributed by unpredictable forces which can result in changes in biological parameters such as survival rates, so it is reasonable to study models with control variables which are so-called disturbance functions [22, 23].

So it is very interesting to study dynamics of the following discrete ratio-dependent Leslie system with time delays and feedback controls:

where , stand for the density of the prey and the predator at time , respectively. , are the control variables at time . is a positive constant, denoting the constant of capturing half-saturation.

In this paper, we are concerned with the effects of the almost periodicity of ecological and environmental parameters and time delays on the global dynamics of the discrete ratio-dependent Leslie systems with feedback controls. To do so, for system (1.4) we always assume that for ,

 , , , , , , , , are all bounded nonnegative almost periodic sequences such that

Here, we let , denote the sets of all integers, nonnegative integers, respectively, and use the notations: , , for any bounded sequence defined on .

Let , we consider system (1.4) with the following initial conditions:

One can easily show that the solutions of system (1.4) with initial condition (1.5) are defined and remain positive for .

The principle aim of this paper is to study the dynamic behaviors of system (1.4), such as permanence, global attractivity, and existence of a unique globally attractive positive almost periodic solution of the system. To the best of our knowledge, no work has been done for the nonautonomous difference system (1.4).

The organization of this paper is as follows. In the next section, we introduce some definitions and several useful lemmas. In Section 3, we explore the permanent property of system (1.4). We study globally attractive property of system (1.4) in Section 4 and the almost periodic property of system (1.4) in Section 5. Finally, the conclusion ends with brief remarks.

2. Preliminaries

In this section, we will introduce some basic definitions and several useful lemmas.

Definition 2.1. System (1.4) is said to be permanent, if there are positive constants and , such that for each positive solution of system (1.4) satisfies

Definition 2.2. Suppose that is any solution of system (1.4). is said to be a strictly positive solution in if for and such that

Definition 2.3 (see [16]). A sequence is called an almost periodic sequence if the -translation set of

is a relatively dense set in for all ; that is, for any given , there exists an integer such that each interval of length contains an integer with

is called an -translation number of .

Definition 2.4 (see [17]). The hull of , denoted by , is defined by

for some sequence , where is any compact set in .

Lemma 2.5 (see [20]). Assume that satisfies and

for , where is a positive constant and . Then

Lemma 2.6 (see [20]). Assume that satisfies and

for , , where is a constant such that and . Then

Lemma 2.7 (see [22]). Assume that and . Suppose that

Then for any integer ,

Especially, if and is bounded above with respect to , then

Lemma 2.8 (see [22]). Assume that and . Suppose that

Then for any integer ,

Especially, if and is bounded below with respect to , then

3. Permanence

In this section, we establish a permanent result for system (1.4).

Theorem 3.1. Assume that holds; assume further that

holds. Then system (1.4) is permanent.

Proof. Let be any positive solution of system (1.4), from the first equation of system (1.4), it follows that

It follows from (3.1) that

which implies that

Substituting (3.3) into the first equation of (1.4), it immediately follows that

By applying Lemma 2.5 to (3.4), we have

For any small enough, it follows from (3.5) that there exists enough large such that for ,

From the second equation of system (1.4) it follows that

From (3.7), similar to the argument of (3.1), one has

By applying Lemma 2.5 to (3.8) again, we have

Setting in the above inequality, we have

For any small enough, it follows from (3.5) and (3.10) that there exists enough large such that for and

For , (3.11) combining with the third and fourth equations of system (1.4) leads to

that is,

By applying Lemma 2.7, it follows from (3.13) that

Letting in the above inequality, we have

For any small enough, it follows from (3.11) and (3.15) that there exists enough large such that for and

Thus, from (3.16) and the first equation of system (1.4), it follows that

where

For any integer , it follows from (3.17) that

and hence

From the third equation of system (1.4), we have

where and . Therefore, for any , by Lemma 2.7, (3.20), and (3.21), one has

Since , we have . So

By the conditions , for any solution of system (1.4), there exists a positive integer such that for . In fact, we can choose . Then we get

where .

Substituting (3.20) and (3.24) into the first equation of system (1.4), one has

where

In particular, we have

where we use the inequality for . By applying Lemma 2.6 to (3.25), it follows that

Setting in the above inequality, we have

From (3.29) we know that there exists enough large such that for ,

From (3.30) and the second equation of system (1.4), it follows that

where .

This implies for any integer

From (3.32) and the fourth equation of system (1.4), by a procedure similar to the discussion of (3.21)–(3.24), we can verify that

where

Substituting (3.32) and (3.33) into the second equation of system (1.4), one has

where

In particular, we have

By applying Lemma 2.6 to (3.35) again, it follows that

Setting in the above inequality, we have

From (3.29) and (3.39) we know that there exists enough large such that for ,

For , (3.40) combining with the third and fourth equations of system (1.4) produces

By applying Lemma 2.8, it follows from (3.41) that

Setting in the above inequality, we have

Consequently, combining (3.5), (3.10), (3.15), (3.29), (3.39) with (3.43), system (1.4) is permanent. This completes the proof of Theorem 3.1.

4. Global Attractivity

Firstly, we prove two lemmas which will be useful to our main result.

Lemma 4.1. For any two positive solutions and of system (1.4), one has

where

Proof. It follows from the first equation of system (1.4) that

where

Hence

Since

By the mean value theorem, one has

Combining (4.6) with (4.8), we can easily obtain (4.1). The proof of Lemma 4.1 is completed.

Lemma 4.2. For any two positive solutions and of system (1.4), one has

where

Proof. The proofs of Lemma 4.2 are very similar to those of Lemma 4.1. So we omit the detail here.

Now we are in the position of stating the main result on the global attractivity of system (1.4).

Theorem 4.3. In addition to , assume further that

  there exist positive constants , , such that

holds, where , , , , , are defined by (4.37). Then for any two positive solutions and of system (1.4), one has

Proof. For two arbitrary nontrivial solutions and of system (1.4), we have from Theorem 3.1 that there exist positive constants and , , , such that for all and

Firstly, we define

From (4.1), we have

where

By the mean value theorem, we have

that is,

where lies between and . So, we have

For , it follows that

Secondly, we define

Then

Thirdly, we define

By a simple calculation, it follows that

We now define

Then for all , it follows from (4.14)–(4.24) that

Similarly, we define

where

Thus for all , it follows from (4.27)–(4.30) that

For , we define

By calculation, it derives that

We now define a Lyapunov function as

It is easy to see that and . Calculating the difference of along the solution of system (1.4), we have that for ,

where

Summing both sides of (4.36) from to , it derives that

It then follows from (4.38) that for ,

that is

Then

from which we see that

This completes the proof of Theorem 4.3.

5. Almost Periodic Solutions

In this section, we consider the almost periodic property of system (1.4).

We assume that , , , , , , , , , . By Definitions 2.3 and 2.4, there exists an integer valued sequence with as such that for ,

Then we have the following Hull equation of system (1.4)

According to the almost periodic theory, we know that if system (1.4) satisfies , then Hull equation (5.2) also satisfies .

The following Lemma is Lemma 4.1 in [15].

Lemma 5.1. If each hull equation of system (1.4) has a unique strictly positive solution, then the almost periodic difference system (1.4) has a unique strictly positive almost periodic solution.

Theorem 5.2. Assume that holds. Then the almost periodic difference system (1.4) has a unique strictly positive almost periodic solution which is globally attractive.

Proof. We divided the proof into two steps.

Step 1. We prove the existence of a strictly positive solution of each Hull equation (5.2).

By the almost periodicity of the parameters of system (1.4), there exists an integer-valued sequence with as such that for

Suppose that is any positive solution of Hull equation (5.2). Since , hold, according to Theorem 3.1, we obtain

Therefore, one has

For any small enough, it follows from Theorem 3.1 that there exists enough large such that for and

Let and for , . For any positive integer and , it is not difficult to show that there are sequences and such that the sequences and have subsequences, denoted by and again, converging on any finite interval of as , respectively. So we have sequences and such that for

which implies

Thus we can easily see that is a solution of Hull equation (5.2) and , , , . Since is an arbitrary small positive number, it follows that

This completes the proof of Step 1.

Step 2. We show the uniqueness of the strictly positive solution of each Hull equation (5.2).

Suppose that the Hull equation (5.2) has two arbitrary strictly positive solutions and . Now we define a Lyapunov function on as follows:

where

where , , , , are defined by Theorem 4.3.

Similar to the discussion of (3.32), calculating the difference of along the solution of the Hull equation (5.2), one has

It follows from (5.12) we know that is a nonincreasing function on . Summing both sides of the above inequalities from to 0, we get

So we have , thus we obtain

From (5.14), for a positive integer , we have

It follows from (5.10) that

Let

It follows from (5.10) and the above inequalities that for , hence . Furthermore, is a nonincreasing function on , thus , that is,   ,   ,    for all . This ends the proof of Step 2.

By the above discussion and Theorem 4.3, we conclude that any hull equation of system (1.4) has a unique strictly positive solution. Then the almost periodic system (1.4) has a unique strictly positive almost periodic solution which is globally attractive. The proof of Theorem 5.2 is completed.

6. Concluding Remarks

In this paper, a discrete almost periodic ratio-dependent Leslie system with time delays and feedback controls is considered. By applying the difference inequality, some sufficient conditions are established, which are independent of feedback control variables, to ensure the permanence of system (1.4). By constructing the suitable Lyapunov function, we show that system (1.4) is globally attractive under some appropriate conditions. Furthermore, by using an almost periodic functional hull theory, we show that the almost periodic system (1.4) has a unique strictly positive almost periodic solution which is globally attractive.

We would like to mention here the question of whether the feedback control variables have the influence on the stability property of the system or not. It is, in fact, a very challenging problem, and we leave it for our future work.

Acknowledgments

The authors are grateful to the editor, Wolfgang Ruess, and referees for a number of helpful suggestions that have greatly improved our original submission. This work is supported by the National Natural Science Foundation of China (nos. 71171035, 71173029), China Postdoctoral Science Foundation (no. 2012M511598), and the Teaching and Research Foundation of Dongbei University of Finance and Economics (nos. YY11001, GW11003).

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