Almost Periodic Solutions of a Discrete Mutualism Model with Variable Delays

We discuss a discrete mutualism model with variable delays of the forms N1 n 1 N1 n exp{r1 n K1 n α1 n N2 n − μ2 n /1 N2 n − μ2 n − N1 n − ν1 n }, N2 n 1 N2 n exp{r2 n K2 n α2 n N1 n − μ1 n / 1 N1 n − μ1 n − N2 n − ν2 n }. 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.


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 N i i 1, 2 are the density of ith 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 timevarying.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 μ i and ν i are two nonnegative integer valued sequences, i 1, 2.
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 {f n } defined on Z, Throughout this paper, we assume that H 1 {r i n }, {α i n }, {K i n }, {μ i n }, and {ν i n } are bounded nonnegative almost periodic sequences such that Let ρ def max i 1,2 {μ u i , ν u i }.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 n ∈ Z .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.

Preliminaries
Now, let us state the following definitions and lemmas, which will be useful in proving our main result.Z there exists a subsequence {h k } ⊂ {h k } such that x n h k converges uniformly on n ∈ Z as k → ∞.Furthermore, the limit sequence is also an almost periodic sequence. Let

2.6
In paper 28 , Chen obtained the permanence of system 1.2 as follows.
Lemma 2.6 see 28 .Assume that H 1 holds; then every solution N 1 , N 2 of system 1.2 satisfies In this section, we obtain the following permanence result of system 1.2 .
Lemma 2.7.Assume that H 1 holds; then every solution N 1 , N 2 of system 1.2 satisfies where 2.9 Proof.Let N 1 , N 2 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

2.12
First, we present two cases to prove that lim sup

2.13
Case I.There exists l 0 ∈ Z such that N 1 l 0 1 ≥ N 1 l 0 .Then, by 2.12 , we have 12 , we get

2.15
We claim that In fact, if there exists an integer k 0 ≥ l 0 2 such that N 1 k 0 > G 1 , and letting k 1 be the least integer between l 0 and k 0 such that which implies from the argument as that in 2.15 that 2.17 This is impossible.This proves the claim.
Taking limit in the first equation of system 1.2 gives In view of the second equation of system 1.2 , similar to the previous analysis, we can obtain lim sup

2.19
For arbitrary > 0, there exists n 0 ∈ Z such that

2.20
For n > n 0 ρ, from the first equation of system 1.2 , we have

2.21
Here, we use the inequality which yields from the first equation of system 1.2 that Next, we also present two cases to prove that lim inf

2.24
Case I.There exists l 0 ≥ n 0 ρ such that N 1 l 0 1 ≤ N 1 l 0 .Then, we have from 2.23 that which implies that

2.26
In view of 2.21 , we can easily obtain that

2.27
We claim that By way of contradiction, assume that there exists a c 0 ≥ l 0 such that N 1 c 0 < g 1 .Then, c 0 ≥ l 0 2. Let c 1 ≥ l 0 2 be the smallest integer such that N 1 c 0 < g 1 .Then The previous argument produces that N 1 c 1 ≥ g 1 , a contradiction.This proves the claim.
Case II.We assume that N 1 n < N 1 n 1 , for all n ≥ n 0 ρ.Then, lim n → ∞ N 1 n exists, denoted by N 1 .Taking limit in the first equation of system 1.2 gives In view of the second equation of system 1.2 , similar to the previous analysis, we can obtain lim inf 2.30 So, the proof of Lemma 2.7 is complete.
Example 2.8.Consider the following discrete mutualism model with delays:

2.32
By Lemma 2.6, one has

2.33
Further, we could calculate

2.34
By Lemma 2.7, one also has

2.35
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 .
By Lemmas 2.6 and 2.7, we can easily show the following.

3.3
Then, system 1.2 is globally attractive, that is, for any positive solution Proof.In view of condition H 2 , there exist small enough positive constants and λ such that where

3.6
Suppose that N 1 , N 2 and W 1 , W 2 are two positive solutions of system 1.2 .By Theorem 2.9, there exists a constant K 0 > 0 such that In view of system 1.2 , we have

3.9
Using the mean value theorem, it follows that where θ 1 n lies between N 1 n and W 1 n , and where θ 2 n lies between N 2 n − μ 2 n and W 2 n − μ 2 n .Define

3.12
By a similar argument as that in 3.9 , we obtain from 3.11 that where ξ 1 s lies between e P 1 s and e Q 1 s , and ξ 2 s lies between e Q 1 s and 1, s n−ν u 1 , . . ., n−1.In view of 3.9 , it follows from 3.10 -3.13 that

3.24
We construct a Lyapunov functional as follows: which implies from 3.21 and 3.24 that Taking n 1 ∈ K 0 2ρ, ∞ Z and Summing both sides of inequality 3.26 over n 1 , n Z , we have

3.28
From the previous inequality one could easily deduce that lim This completes the proof.

3.31
Corollary 3.2.Assume that H 1 holds.Suppose further that H 3 there exist two positive constants λ 1 and λ 2 such that
Further, we consider the following discrete mutualism model with constant delays:

3.34
Corollary 3.3.Assume that H 1 holds.Suppose further that H 4 there exist two positive constants λ 1 and λ 2 such that

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 {τ p } be any integer valued sequence such that τ p → ∞ as p → ∞.According to Lemma 2.5, taking a subsequence if necessary, we have Then, we get the hull equations of system 1.2 as follows:

4.2
By the almost periodic theory, we can conclude that if system 1.2 satisfies H 1 -H 4 , then the hull equations 4.2 of system 1.2 also satisfies H 1 -H 4 .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 H 1 -H 2 , 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 {r i n }, {K i n }, {α i n }, {μ i n }, and {ν i n }, i 1, 2, there exists an integer-valued sequence {η p } with η p → ∞ as p → ∞ such that

4.3
Suppose that N 1 , N 2 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 I 0 such that Write N p i n N i n η p for n ≥ I 0 − η p , p 1, 2, . .., i 1, 2. For any positive integer q, it is easy to see that there exist sequences {N 4.5 Combined with gives

4.7
We can easily see that W Similar to Theorem 3.1, we define a Lyapunov functional where

4.16
For arbitrary > 0, there exists a positive integer K 1 such that 4.17 Hence, for i, j 1, 2 with i / j, one has

4.18
which imply that Note that V * n is a nonincreasing function on Z and that V * n ≡ 0. 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.

Examples
Example 5.1.Consider the following discrete mutualism model without delay: 5.1 Then, system 5.1 admits a unique globally attractive almost periodic solution. Proof.

5.3
Here, x * i def α u i /r u i exp α u i r u i − 1 , and x i * def K l i exp r l i K l i − x * i , i 1, 2. Then, system 3.30 admits a unique uniformly asymptotically stable almost periodic solution.5.5 Then, system 5.5 admits a unique globally attractive almost periodic solution.

p 1 n
: p ≥ q} and {N p 2 n : p ≥ q} such that the sequences {N p 1 n } and {N p 2 n } have subsequences, denoted by {N p 1 n } and {N p 2 n } again, converging on any finite interval of Z as p → ∞, respectively.Thus, we have sequences {W 1 n } and {W 2 n } such that

Lemma 2.5 see
Z → R}. f n, φ is said to be almost periodic in n uniformly for φ ∈ D, or uniformly almost periodic for short, if for any > 0 and any compact set S in D there exists a positive integer l , S such that any interval of length l , S contains an integer τ for which 27 .{x n } is an almost periodic sequence if and only if for any sequence {h k } ⊂ Definition 2.1 see 27 .A sequence x : Z → R is called an almost periodic sequence if the -translation set of x E{ , x} {τ ∈ Z : |x n τ − x n | < , ∀n ∈ Z} 2.1 is a relatively dense set in Z for all > 0; that is, for any given > 0, there exists Definition 2.2 see 27 .Let f : Z × D → R, where D is an open set in C : {φ : −τ, 0 1, W 2 is a solution of hull equations 4.2 andm i − ≤ W i n ≤ M i for n ∈ Z, i 1, 2.Since is an arbitrary small positive number, it follows that m i ≤ W i n ≤ M i for n ∈ Z, i 1, 2, 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 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.Suppose that H 1 -H 2 hold; then system 1.2 admits a unique strictly positive almost periodic solution, which is globally attractive.Suppose that H 1 and H 3 hold; then system 3.30 admits a unique strictly positive almost periodic solution, which is globally attractive.Suppose that H 1 and H 4 hold, then system 3.33 admits a unique strictly positive almost periodic solution, which is globally attractive.
condition H 3 of Corollary 3.2 is satisfied with λ 1 λ 2 1.It is easy to verify that H 1 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 H 1 holds.Suppose further that