Journal of Difference Equations

Volume 2014 (2014), Article ID 256094, 9 pages

http://dx.doi.org/10.1155/2014/256094

Research Article

Almost Periodic Solution of a Discrete Schoener’s Competition Model with Delays

Hui Zhang, Yingqi Li, Bin Jing, Xiaofeng Fang, and Jing Wang

Mathematics and OR Section, Xi’an Research Institute of High-Tech Hongqing Town, Xi’an, Shaanxi 710025, China

Received 8 June 2014; Accepted 12 July 2014; Published 24 July 2014

Academic Editor: Cengiz Çinar

Copyright © 2014 Hui 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 an almost periodic discrete Schoener’s competition model with delays. By means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, sufficient conditions are obtained for the existence of a unique strictly positive almost periodic solution which is globally attractive. An example together with numerical simulation indicates the feasibility of the main result.

1. Introduction

In 2009, Wu et al. [1] had studied a discrete Schoener’s competition mode with delays:

where , and are real positive bounded sequences and are positive integers, . Sufficient conditions which guarantee the permanence and the global attractivity of positive solutions for system (1) are obtained.

By the biological meaning, the system (1) is considered together with the following initial condition:

where . Let be any solution of system (1) with the initial condition (2). One could easily see that , for all .

Schoener’s competition system has been studied by many scholars. Topics such as existence, uniqueness, and global attractivity of positive periodic solutions of the system were extensively investigated, and many excellent results have been derived (see [1–6] and the references cited therein). Recently, a few papers investigate the global stability of the pure-delay model (see [7–13]).

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 [14–21] and the references cited therein). But to the best of the author’s knowledge, to this day, still no scholars have studied the almost periodic version which is corresponding to system (1). Therefore, with stimulation from the works of [12, 19], the main purpose of this paper is to derive a set of sufficient conditions ensuring the existence of a unique strictly positive almost periodic solution of system (1) which is globally attractive.

Denote by and the set of integers and the set of nonnegative integers, respectively. For any bounded sequence defined on , define .

Throughout this paper, we assume the following.

(H1), and are bounded positive almost periodic sequences such that

The remaining part of this paper is organized as follows. In Section 2, we will introduce some definitions and several useful lemmas. In Section 3, by applying the theory of difference inequality, we present the permanence results for system (1). In Section 4, we establish the sufficient conditions for the existence of a unique globally attractive almost periodic solution of system (1). The main result is illustrated by an example with a numerical simulation in the last section.

2. Preliminaries

In this section, we give the definitions and lemmas of the terminologies involved.

Definition 1 (see [22]). 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 (see [23]). Let be an open subset of . is said to be almost periodic in uniformly for if for any , and any compact set , there exists a positive integer such that any interval of length contains an integer for which

is called an -translation number of .

Definition 3 (see [24]). The hull of , denoted by , is defined by

for some sequence , where is any compact set in .

Definition 4. Suppose that is any solution of system (1). is said to be a strictly positive solution in if

for .

Lemma 5 (see [25]). Assume that satisfies and

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

Lemma 6 (see [25]). Assume that satisfies

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

3. Permanence and Global Attractivity

Now we state several lemmas which will be useful in proving our main result.

Proposition 7 (see [1]). Any solution of system (1) with the initial condition (2) is positive and ultimately bounded; that is,

where

Proposition 8. Assume that

holds; then for any solution of system (1) with the initial condition (2), one has

where and are defined by (24) and (26), respectively.

Proof. For any small positive constant , according to Proposition 7, there exists a such that, for all and ,

It follows from (17) and the first equation of system (1), for ,

where .

Thus, by using (18) we obtain

Substituting (19) into the first equation of system (1), for , it follows that

When is an arbitrary small positive constant, it follows from condition (H2)

Thus, as a direct corollary of Lemma 6, according to (14) and (20), one has

where

Letting , it follows that

where

Similar to the analysis of (18)–(24), by applying (17), from the second equation of system (1), we also have that

where

The proof is completed.

Note that condition (H2) of Proposition 8 is weakened compared to condition (H) in [1].

Theorem 9 (see [1]). Assume that (H2) holds; system (1) is permanent.

Proposition 10 (see [1]). Assume that (H2) holds and further that there exist positive constants , and such that

where

, , . Then system (1) is globally attractive; that is, for any two positive solutions and of system (1), we have

4. Almost Periodic Solution

In this section, by means of an almost periodic functional hull theory and constructing a suitable Lyapunov function, we will study the existence of a globally attractive almost periodic solution of system (1) with initial condition (2) and obtain the sufficient conditions.

Let be any integer valued sequence such that as . According to Lemma 5, taking a subsequence if necessary, we have , as for . Then we get a hull equation of system (1) as follows:

By the almost periodic theory, we can conclude that if system (1) satisfies (H2) and (H3), then the hull equation (31) of system (1) also satisfies (H2) and (H3).

By Theorem 3.4 in [26], we can easily obtain the lemma as follows.

Lemma 11. If each hull equation of system (1) has a unique strictly positive solution, then the almost periodic difference system (1) has a unique strictly positive almost periodic solution.

Now we investigate a globally attractive almost periodic solution of system (1).

Theorem 12. If the almost periodic difference system (1) satisfies (H1), (H2), and (H3), then the almost periodic difference system (1) admits a unique strictly positive almost periodic solution, which is globally attractive.

Proof. By Lemma 11, we only need to prove that each hull equation of system (1) has a unique globally attractive almost periodic sequence solution; hence we firstly prove that each hull equation of system (1) has at least one strictly positive solution (the existence) and then we prove that each hull equation of system (1) has a unique strictly positive solution (the uniqueness).

Now we prove the existence of a strictly positive solution of any hull equation (31). By the almost periodicity of , and , , , there exists an integer valued sequence with as such that , , , as for . Suppose that is any solution of hull equation (31). By the proof of Propositions 7 and 8, we have

And also

Let be an arbitrary small positive number. It follows from (32) that there exists a positive integer such that , , . Write , for all , . We claim that there exists a sequence , and a subsequence of , which we still denote by such that uniformly in on any finite subset of as , where , , and is a finite number.

In fact, for any finite subset , when is large enough, , . So

that is, are uniformly bounded for large enough.

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

Similarly, for , we can choose a subsequence of such that uniformly converges on for large enough.

Repeating this procedure, for , we can choose a subsequence of such that uniformly converges on for large enough.

Now pick the sequence which is a subsequence of , which we still denote by , then for all , we have uniformly in , as .

By the arbitrary of , the conclusion is valid.

Combining with

gives

We can easily see that is a solution of hull equation (31) and , , for . Since is an arbitrary small positive number, it follows that , , for ; that is,

Hence each hull equation of almost periodic difference system (1) has at least one strictly positive solution.

Now we prove the uniqueness of the strictly positive solution of each hull equation (31). Suppose that the hull equation (31) has two arbitrary strictly positive solutions and . We construct a Lyapunov functional

where

Calculating the difference of along the solution of the hull equation (31), one has

From (40), we can see that is a nonincreasing function on . Summing both sides of the above inequalities from to , we have

Note that is bounded. Hence we have

which implies that

Define , where

Let be an arbitrary small positive number. It follows from (43) that there exists a positive integer such that . Therefore, for ,

It follows from (38) and above inequalities that

so . Note that is a nonincreasing function on , and then ; that is , , for all . Therefore, each hull equation of system (1) has a unique strictly positive solution.

In view of the above discussion, any hull equation of system (1) has a unique strictly positive solution. By Propositions 7–10 and Lemma 11, the almost periodic difference system (1) has a unique strictly positive almost periodic solution which is globally attractive. The proof is completed.

Let , , . Like in the proof of Theorem 12, we have the following corollary.

Corollary 13. Let , , . Assume that

hold and further that there exist two positive constants and , such that

where . Then the almost periodic difference system (1) admits a unique strictly positive almost periodic solution, which is globally attractive.

5. Example and Numerical Simulation

In this section, we give the following example to check the feasibility of our result.

Example 1. Consider the following almost periodic discrete Schoener’s competition model with delays

By simple computation, we derive

Then

Also it is easy to see that conditions (H2) and (H3) are verified. Therefore, system (49) has a unique strictly positive almost periodic solution which is globally attractive. Our numerical simulations support our results (see Figures 1, 2, 3, and 4).

Figure 1: Dynamic behavior of the first component of the solution to system (49) with the initial conditions and for , respectively.

Figure 2: Dynamic behavior of the second component of the solution to system (49) with the initial conditions and for , respectively.

Figure 3: Dynamic behavior of the first component of the solution to system (49) with the initial conditions and for , respectively.

Figure 4: Dynamic behavior of the second component of the solution to system (49) with the initial conditions and for , respectively.

Conflict of Interests

There are no financial interest conflicts between the authors and the commercial identity.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 61132008) and Scientific Research Program Funded by Shaanxi Provincial Education Department of China (no. 2013JK1098).

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