Discrete Dynamics in Nature and Society

Volume 2012 (2012), Article ID 742102, 27 pages

doi:10.1155/2012/742102

Research Article

Almost Periodic Solutions of a Discrete Mutualism Model with Variable Delays

Yongzhi Liao¹ and Tianwei Zhang²

¹School of Mathematics and Computer Science, Panzhihua University Sichuan, Panzhihua 617000, China

²City College, Kunming University of Science and Technology, Kunming 650051, China

Received 26 May 2012; Accepted 4 November 2012

Academic Editor: Zhen Jin

Copyright © 2012 Yongzhi Liao and Tianwei Zhang. 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 discuss a discrete mutualism model with variable delays of the forms, . By means of an almost periodic functional hull theory, sufficient conditions are established for the existence and uniqueness of globally attractive almost periodic solution to the previous system. Our results complement and extend some scientific work in recent years. Finally, some examples and numerical simulations are given to illustrate the effectiveness of our main results.

1. Introduction

All species on the earth are closely related to other species. In a simple view, the interaction between a pair of species can be classified into three typical categories: predation (one gains and the other suffers) , competition , and mutualism (see [1]). In recent years, the concern for mutualism is growing, since most of the world’s biomass is dependent on mutualism (see [1, 2]). For example, microbial species influence the abundances and ecological functions of related species (see [3–5]). Many bacterial species coexist in a syntrophic association (obligate mutualism); that is, one species lives off the products of another species. So far, mathematical models for mutualisms have often been neglected in many ecological textbooks.

The variation of the environment plays an important role in many biological and ecological dynamical systems. As pointed out in [6, 7], a periodically varying environment and an almost periodically varying environment are foundations for the theory of natural selection. Compared with periodic effects, almost periodic effects are more frequent. Hence, the effects of the almost periodic environment on the evolutionary theory have been the object of intensive analysis by numerous authors, and some of these results can be found in [8–12]. On the other hand, discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. In the last ten years, the dynamic behavior (the existence of positive periodic or almost periodic solutions, permanence, oscillation, and stability) of discrete biological systems has attracted much attention. We refer the reader to [13–19] and the references cited therein.

In paper [15], Wang and Li considered the following discrete mutualism model:

where are the density of th mutualist species. By using the main result obtained by Zhang [20], they studied the existence and uniformly asymptotically stability of a unique almost periodic solution of system (1.1).

In biological phenomena, the rate of variation in the system state depends on past states. This characteristic is called a delay or a time delay. Time delay phenomena were first discovered in biological systems. They are often a source of instability and poor control performance. Time-delay systems have attracted the attention of many researchers [8, 10, 12, 16, 18, 21–23] because of their importance and widespread occurrence. Specially, in the real world, the delays in differential equations of biological phenomena are usually time-varying. Thus, it is worthwhile continuing to study the existence and stability of a unique almost periodic solution of the discrete mutualism model with time varying delays.

In this paper, we investigate a discrete mutualism model with variable delays of the form

where all coefficients of system (1.2) are almost periodic sequences, and and are two nonnegative integer valued sequences, .

In recent years, there are some research papers on the dynamic behavior (existence, uniqueness, and stability) of almost periodic solution of discrete biological models with constant delays (see [24–26]). However, there are few papers concerning the discrete biological models with variable delays such as system (1.2). Motivated by the previous reason, our purpose of this paper is to establish sufficient conditions for the existence and uniqueness of globally attractive almost periodic solution of system (1.2) by means of an almost periodic functional hull theory.

For any bounded sequence defined on , , . Let, for all .

Throughout this paper, we assume that

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

Let . We consider system (1.2) together with the following initial condition:

One can easily show that the solutions of system (1.2) with initial condition (1.4) are defined and remain positive for .

The organization of this paper is as follows. In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections. In Section 3, global attractivity of system (1.2) is investigated. In Section 4, by means of an almost periodic functional hull theory, some sufficient conditions are established for the existence and uniqueness of almost periodic solution of system (1.2). Three illustrative examples are given in Section 5.

2. Preliminaries

Now, let us state the following definitions and lemmas, which will be useful in proving our main result.

Definition 2.1 (see [27]). 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 such that

is called the -translation number or -almost period.

Definition 2.2 (see [27]). 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

is called the -translation number of .

Definition 2.3 (see [27]). The hull of , denoted by , is defined by

for some sequence , where is any compact set in .

Definition 2.4. Suppose that is any solution of system (1.2). is said to be a strictly positive solution on if for ,

Lemma 2.5 (see [27]). 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.

Let

In paper [28], Chen obtained the permanence of system (1.2) as follows.

Lemma 2.6 (see [28]). Assume that holds; then every solution of system (1.2) satisfies

In this section, we obtain the following permanence result of system (1.2).

Lemma 2.7. Assume that holds; then every solution of system (1.2) satisfies

where

Proof. Let be any positive solution of system (1.2) with initial condition (1.4). From the first equation of system (1.2), it follows that

which yields that

which implies that

First, we present two cases to prove that

Case I. There exists such that . Then, by (2.12), we have

which implies that . From (2.12), we get

We claim that

In fact, if there exists an integer such that , and letting be the least integer between and such that , then and , which implies from the argument as that in (2.15) that

This is impossible. This proves the claim.

Case II. , for all . In particular, exists, denoted by . Taking limit in the first equation of system (1.2) gives

Hence, . This proves the claim.

So, . In view of the second equation of system (1.2), similar to the previous analysis, we can obtain

For arbitrary , there exists such that

For , from the first equation of system (1.2), we have

Here, we use the inequality . So,

which yields from the first equation of system (1.2) that

Next, we also present two cases to prove that

Case I. There exists such that . Then, we have from (2.23) that

which implies that

In view of (2.21), we can easily obtain that

We claim that

By way of contradiction, assume that there exists a such that . Then, . Let be the smallest integer such that . Then . The previous argument produces that , a contradiction. This proves the claim.

Case II. We assume that , for all . Then, exists, denoted by . Taking limit in the first equation of system (1.2) gives

Hence, and . This proves the claim.

So, . In view of the second equation of system (1.2), similar to the previous analysis, we can obtain

So, the proof of Lemma 2.7 is complete.

Example 2.8. Consider the following discrete mutualism model with delays:

Corresponding to system (1.2), , , , , , . By calculation, we obtain

By Lemma 2.6, one has

Further, we could calculate

By Lemma 2.7, one also has

For system (2.31), it is easy to see that Lemma 2.7 gives a more accurate result than Lemma 2.6 (see Figure 1).

Figure 1: System (2.31) with .

By Lemmas 2.6 and 2.7, we can easily show the following.

Theorem 2.9. Assume that holds; then every solution of system (1.2) satisfies

3. Global Attractivity

Define a function as follows:

Let

Theorem 3.1. Assume that holds. Suppose further that

 there exist two positive constants and such that , where

Then, system (1.2) is globally attractive, that is, for any positive solution and of system (1.2),

Proof. In view of condition , there exist small enough positive constants and such that

where

Suppose that and are two positive solutions of system (1.2). By Theorem 2.9, there exists a constant such that

Let

In view of system (1.2), we have

Using the mean value theorem, it follows that

where lies between and , and

where lies between and .

Define

By a similar argument as that in (3.9), we obtain from (3.11) that

where lies between and , and lies between and , .

In view of (3.9), it follows from (3.10)–(3.13) that

Let

So,

Define

It follows from (3.14)–(3.19) that

Let

where

By a similar argument as that in (3.21), we could easily obtain that

We construct a Lyapunov functional as follows:

which implies from (3.21) and (3.24) that

Taking and Summing both sides of inequality (3.26) over , we have

Therefore,

From the previous inequality one could easily deduce that

This completes the proof.

If in system (1.2), then we obtain a discrete mutualism model without delay as follows:

Let

Corollary 3.2. Assume that holds. Suppose further that

 there exist two positive constants and such that

Then, system (3.30) is globally attractive.

Further, we consider the following discrete mutualism model with constant delays:

where and are nonnegative integers. Let

Corollary 3.3. Assume that holds. Suppose further that

 there exist two positive constants and such that

Then, system (3.33) is globally attractive.

4. Almost Periodic Solution

In this section, we investigate the existence and uniqueness of a globally attractive almost periodic solution of system (1.2) by using almost periodic functional hull theory.

Let be any integer valued sequence such that as . According to Lemma 2.5, taking a subsequence if necessary, we have

for , . Then, we get the hull equations of system (1.2) as follows:

By the almost periodic theory, we can conclude that if system (1.2) satisfies –, then the hull equations (4.2) of system (1.2) also satisfies –.

By Theorem  3.4 in [27], it is easy to obtain the following lemma.

Lemma 4.1. If each of the hull equations of system (1.2) has a unique strictly positive solution, then system (1.2) has a unique strictly positive almost periodic solution.

Theorem 4.2. If system (1.2) satisfies -, then system (1.2) admits a unique strictly positive almost periodic solution.

Proof. By Lemma 4.1, in order to prove the existence of a unique strictly positive almost periodic solution of system (1.2), we only need to prove that each hull equations of system (1.2) has a unique strictly positive solution.

Firstly, we prove the existence of a strictly positive solution of hull equations (4.2). By the almost periodicity of , , , , and , , there exists an integer-valued sequence with as such that

Suppose that is any solution of hull equations (4.2). Let be an arbitrary small positive number. It follows from Theorem 2.9 that there exists a positive integer such that

Write for , , . For any positive integer , it is easy to see that there exist sequences and such that the sequences and have subsequences, denoted by and again, converging on any finite interval of as , respectively. Thus, we have sequences and such that

Combined with

gives

We can easily see that is a solution of hull equations (4.2) and for , . Since is an arbitrary small positive number, it follows that for , , which implies that each of the hull equations of system (1.2) has at least one strictly positive solution.

Now, we prove the uniqueness of the strictly positive solution of each of the hull equations (4.2). Suppose that the hull equations of (4.2) have two arbitrary strictly positive solutions and which satisfy

Similar to Theorem 3.1, we define a Lyapunov functional

where

Here,

Similar to the argument as that in (3.26), one has

Summing both sides of the previous inequality from to , we have

Note that is bounded. Hence, we have

which imply that

Let

For arbitrary , there exists a positive integer such that

Hence, for with , one has

which imply that

So,

Note that is a nonincreasing function on and that . That is,

Therefore, each of the hull equations of system (1.2) has a unique strictly positive solution.

In view of the previous discussion, any of the hull equations of system (1.2) has a unique strictly positive solution. By Lemma 4.1, system (1.2) has a unique strictly positive almost periodic solution. The proof is completed.

By Theorems 3.1 and 4.2, we can easily obtain the following.

Theorem 4.3. Suppose that - hold; then system (1.2) admits a unique strictly positive almost periodic solution, which is globally attractive.

By Corollaries 3.2–3.3 and Theorem 4.2, we can show the following.

Theorem 4.4. Suppose that and hold; then system (3.30) admits a unique strictly positive almost periodic solution, which is globally attractive.

Theorem 4.5. Suppose that and hold, then system (3.33) admits a unique strictly positive almost periodic solution, which is globally attractive.

5. Examples

Example 5.1. Consider the following discrete mutualism model without delay:

Then, system (5.1) admits a unique globally attractive almost periodic solution.

Proof. Corresponding to system (3.30), , , , , . By calculation, we obtain

which implies that condition of Corollary 3.2 is satisfied with . It is easy to verify that holds, and the result follows from Theorem 4.4.

In paper [15], Wang and Li studied system (3.30) and obtained the following result.

Theorem 5.2 (see [15]). Assume that holds. Suppose further that

 , , , where

Here, , and , .

Then, system (3.30) admits a unique uniformly asymptotically stable almost periodic solution.

Remark 5.3. In system (5.1), we can easily calculate

which implies that of Theorem 5.2 is invalid. Therefore, it is impossible to obtain the existence of a unique globally stable almost periodic solution of system (5.1) by Theorem 5.2.

Example 5.4. Consider the following discrete mutualism model with constant delays:

Then, system (5.5) admits a unique globally attractive almost periodic solution.

Proof. Corresponding to system (3.33), , , , , , and , . By calculation, we obtain

which implies that condition of Corollary 3.3 is satisfied with . It is easy to verify that holds, and the result follows from Theorem 4.5 (see Figure 2).

(a)

(b)

Figure 2: System (5.5) with .

Example 5.5. Consider the following discrete mutualism model with variable delays:

Then, system (5.7) admits a unique globally attractive almost periodic solution.

Proof. Corresponding to system (1.2), , , , , , , , and , . By calculation, we obtain

which implies that condition of Theorem 4.3 is satisfied with . It is easy to verify that holds, and the result follows from Theorem 4.3 (see Figure 3).

(a)

(b)

Figure 3: System (5.7) with .

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