Research Article | Open Access
Almost Periodic Solutions of a Discrete Mutualism Model with Variable Delays
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.
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 ). 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 , 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 , 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
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.
Now, let us state the following definitions and lemmas, which will be useful in proving our main result.
Definition 2.1 (see ). 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 ). 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 ). 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 ). 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.
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:
Further, we could calculate By Lemma 2.7, one also has
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
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 .
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