Journal of Applied Mathematics

Volume 2013 (2013), Article ID 182158, 9 pages

http://dx.doi.org/10.1155/2013/182158

Research Article

Uniformly Asymptotic Stability of Positive Almost Periodic Solutions for a Discrete Competitive System

Qinglong Wang and Zhijun Liu

Department of Mathematics, Hubei University for Nationalities, Enshi, Hubei 445000, China

Received 11 February 2013; Accepted 26 April 2013

Academic Editor: Yansheng Liu

Copyright © 2013 Qinglong Wang and Zhijun Liu. 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 is devoted to the study of almost periodic solutions of a discrete two-species competitive system. With the help of the methods of the Lyapunov function, some analysis techniques, and preliminary lemmas, we establish a criterion for the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of the system. Numerical simulations are presented to illustrate the analytical results.

1. Introduction

In recent years, many works have been done for the difference system (see [1–14] and the references cited therein) since the discrete time models governed by the difference equation are more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world. In particular, Qin et al. [1] introduced the following discrete Lotka-Volterra competitive system:

where , stand for the densities of species at the th generation, represent the natural growth rates of species at the th generation, are the intraspecific effects of the th generation of species on own population, and measure the interspecific effects of the th generation of species on species . They investigated the permanence and global asymptotic stability of positive periodic solutions of system (1).

Notice that the investigation of almost periodic solutions for difference equations is one of most important topics in the qualitative theory of difference equations due to the applications in biology, ecology, neural network, and so forth (see [10–14] in detail), and few work has been done previously on an almost periodic version which is corresponding to periodic system (1). In this paper, we will further investigate the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of the above almost periodic version. To this end, we assume that the coefficients of system (1) and are bounded nonnegative almost periodic sequences.

For the sake of simplicity and convenience in the following discussion, the notations below will be used throughout this paper:

where is a bounded sequence and .

The remaining part of this paper is organized as follows. In the next section, we introduce some notations, definitions, and lemmas which are available for our main results. In Section 3, sufficient conditions for the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of system (1) are given. Numerical simulations are carried out to substantiate the above analytical results in Section 4. Finally, we give some proofs of theorems in the appendices for convenience in reading this paper.

2. Preliminaries

In this section, we will need some preparations and give some notations, definitions, and lemmas which will be useful for our main results.

Denote by , , , and the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers, respectively. and denote the cone of 2-dimensional and -dimensional real Euclidean space, respectively.

Definition 1 (see [13]). A sequence is called an almost periodic sequence if the following -translation set of

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

is called the -translation number of .

Definition 2 (see [13]). Let , where is an open set in . is said to be almost periodic in uniformly for , or uniformly almost periodic for short, if for any and any compact set in there exists a positive integer such that any interval of length contains an integer for which

for all and all . is called the -translation number of .

Lemma 3 (see [13]). is an almost periodic sequence if and only if for any sequence there exists a subsequence such that converges uniformly on as . Furthermore, the limit sequence is also an almost periodic sequence.

Consider the following almost periodic difference system:

where , , and is almost periodic in uniformly for and is continuous in . The product system of (6) is the following system:

and Zhang [14] obtained the following lemma.

Lemma 4 (see [14]). Suppose that there exists a Lyapunov function defined for , , satisfying the following conditions:

(i), where with and is increasing};(ii), where is a constant;(iii), where is a constant and

Moreover, if there exists a solution of system (6) such that for , then there exists a unique uniformly asymptotically stable almost periodic solution of system (6) which satisfies . In particular, if is periodic of period , then there exists a unique uniformly asymptotically stable periodic solution of system (6) of periodic .

Lemma 5 (see [1]). Any positive solution of system (1) satisfies

Lemma 6 (see [1]). Suppose that system (1) satisfies the following assumptions:

Then, any positive solution of system (1) satisfies

3. Main Result

From (9) and (11), we denote by the set of all solutions of system (1) satisfying , for all . According to Lemma 4, we first prove that there is a bounded solution of system (1), and then structure a suitable Lyapunov function for system (1).

Theorem 7. If the assumptions in (10) hold, then .

The proof of Theorem 7 is given in Appendix A.

Theorem 8. If the assumptions in (10) are satisfied, furthermore, , where , and

then there exists a unique uniformly asymptotically stable almost periodic solution of system (1) which is bounded by for all .

The proof of Theorem 8 is given in Appendix B.

4. Numerical Simulations

In this section, we give the following example to check the feasibility of the assumptions of Theorem 8.

Example 9. Consider the following discrete system:

A computation shows that

and moreover, we have

that is, . It is easy to see that the assumptions of Theorem 8 are satisfied. Hence, in system (13) there exists a unique uniformly asymptotically stable positive almost periodic solution. From Figure 1, it is easy to see that there exists a positive almost periodic solution , and the 2-dimensional and 3-dimensional phase portraits of almost periodic system (13) are revealed in Figure 2, respectively. Figure 3 shows that any positive solution tends to the almost periodic solution .

(a)

(b)

(c)

(d)

Figure 1: Positive almost periodic solution of system (13). (a), (c) Time-series and with initial values , for , respectively. (b), (d) Time-series and with the above initial values for , respectively.

(a)

(b)

(c)

(d)

Figure 2: Phase portrait. (a), (b) 2-dimensional phase portrait of almost periodic system (13). Time-series and with initial values , for and , respectively. (c), (d) 3-dimensional phase portrait of almost periodic system (13). Time-series and with the above initial values for and , respectively.

(a)

(b)

(c)

(d)

Figure 3: Uniformly asymptotic stability. (a), (c) Time-series and with initial values , and and with initial values , for , respectively. (b), (d) Time-series , , , and with the above initial values for , respectively.

Appendices

A. Proof of Theorem 7

Clearly, by an inductive argument we have from system (1) that

According to Lemmas 5 and 6, for any solution of system (1) and an arbitrarily small constant , there exists sufficiently large such that

Set be any positive integer sequence such that as , we can show that there exists a subsequence of still denoted by , such that , uniformly in on any finite subset of as , where , , and is a finite number.

As a matter of fact, for any finite subset , , , when is large enough. Therefore, , ; that is, are uniformly bounded for large enough.

Now, for , we can choose a subsequence of such that and uniformly converge on for large enough.

Analogously, for , we can also choose a subsequence of such that and uniformly converge on for large enough.

Repeating the above process, for , we get a subsequence of such that and uniformly converge on for large enough.

Now, we choose the sequence which is a subsequence of denoted by ; then, for all , we obtain that , uniformly in as . Hence, the conclusion is valid by the arbitrary of .

Recall the almost periodicity of and , , for the above sequence , as , there exists a subsequence denoted by such that

as uniformly on .

For any , we can assume that for large enough. Let , by an inductive argument of system (1) from to , we obtain

Thus, it derives that

Let , we have

Since is arbitrary, we know that is a solution of system (1) on , and

Notice that is an arbitrarily small positive constant; it follows that

Thus, . This completes the proof.

B. Proof of Theorem 8

Denote , . It follows from system (1) that

According to Theorem 7, we can see that the system (B.1) has a bounded solution satisfying

Thus, , , where , . Define the norm , where . Consider the product system of system (B.1) as follow:

We assume that , are any two solutions of system (B.1) defined on ; then, , , where , and .

Let us construct a Lyapunov function defined on as follows:

It is obvious that the norm is equivalent to ; that is, there are two constants , , such that

then,

Let , , ; then, condition (i) of Lemma 4 is satisfied.

Moreover, for any , we have

where , , and . Thus, condition (ii) of Lemma 4 is satisfied.

Finally, calculating the of along the solutions of system (B.3), we have

By the mean value theorem, it derives that

, where and lie between and , respectively. Substituting (B.9) into (B.8), we get

where . By the conditions of Theorem 8, we have , and hence, condition (iii) of Lemma 4 is satisfied. So, it follows from Lemma 4 that there exists a unique uniformly asymptotically stable almost periodic solution of system (B.1) which is bounded by for all ; that is, there exists a unique uniformly asymptotically stable almost periodic solution of system (1) which is bounded by for all . This completed the proof.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11261017), the Key Project of Chinese Ministry of Education (nos. 210134, 212111), the Project of Key Laboratory of Biological Resources Protection and Utilization of Hubei Province, and the Key Subject of Hubei Province (Forestry).

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