# Almost Periodic Solutions of a Discrete Mutualism Model with Variable Delays

Authors: Yongzhi Liao; Tianwei Zhang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We discuss a discrete mutualism model with variable delays of the formsN1(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.


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## Body

## 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:
(1.1)N1(n+1)=N1(n)exp{r1(n)[K1(n)+α1(n)N2(n)1+N2(n)-N1(n)]},N2(n+1)=N2(n)exp{r2(n)[K2(n)+α2(n)N1(n)1+N1(n)-N2(n)]},
where Ni(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 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(1.2)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))]},
where all coefficients of system (1.2) are almost periodic sequences, and μi and νi 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, fu=supn∈ℤ{f(n)}, fl=infn∈ℤ{f(n)}. Let[a,b]ℤ=def[a,b]∩ℤ, for all a,b∈ℝ.Throughout this paper, we assume that(
H
1
)
{
r
i
(
n
)
}, {αi(n)}, {Ki(n)}, {μi(n)}, and {νi(n)} are bounded nonnegative almost periodic sequences such that
(1.3)0<ril≤ri(n)≤riu,0<Kil≤Ki(n)≤Kiu,0<αil≤αi(n)≤αiu,i=1,2.Letρ=defmaxi=1,2{μiu,νiu}. We consider system (1.2) together with the following initial condition:
(1.4)Ni(θ)=φi(θ)≥0,θ∈[-ρ,0]ℤ,φi(0)>0,i=1,2.
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 Section2, 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 sequencex:ℤ→ℝ is called an almost periodic sequence if the ϵ-translation set of x(2.1)E{ϵ,x}={τ∈ℤ:|x(n+τ)-x(n)|<ϵ,∀n∈ℤ}
is a relatively dense set in ℤ for all ϵ>0; that is, for any given ϵ>0, there exists an integer l(ϵ)>0 such that each interval of length l(ϵ) contains an integer τ∈E{ϵ,x} such that
(2.2)|x(n+τ)-x(n)|<ϵ,∀n∈ℤ.τ is called the ϵ-translation number or ϵ-almost period.Definition 2.2 (see [27]).
Letf:ℤ×𝔻→ℝ, where 𝔻 is an open set in C:={ϕ:[-τ,0]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 ϵ>0 and any compact set 𝕊 in 𝔻 there exists a positive integer l(ϵ,𝕊) such that any interval of length l(ϵ,𝕊) contains an integer τ for which
(2.3)|f(n+τ,ϕ)-f(n,ϕ)|<ϵ,∀n∈ℤ,ϕ∈𝕊.τ is called the ϵ-translation number of f(n,ϕ).Definition 2.3 (see [27]).
The hull off, denoted by H(f), is defined by
(2.4)H(f)={g(n,x):limk→∞f(n+τk,x)=g(n,x)uniformly onℤ×𝕊}
for some sequence {τk}, where 𝕊 is any compact set in 𝔻.Definition 2.4.
Suppose that(N1,N2) is any solution of system (1.2). (N1,N2) is said to be a strictly positive solution on ℤ if for n∈ℤ,
(2.5)0<infn∈ℤNi(n)≤supn∈ℤNi(n)<∞,i=1,2.Lemma 2.5 (see [27]).
{
x
(
n
)
} is an almost periodic sequence if and only if for any sequence {hk′}⊂ℤ there exists a subsequence {hk}⊂{hk′} such that x(n+hk) converges uniformly on n∈ℤ as k→+∞. Furthermore, the limit sequence is also an almost periodic sequence.Let(2.6)F1=def1r1lexp{r1u(K1u+α1u)(ν1u+1)-1},F2=def1r2lexp{r2u(K2u+α2u)(ν2u+1)-1},A1=defr1lK1lr1u(1+F2)exp{r1lK1lν1u1+F2-r1uF1ν1u},B1=defA1exp{r1lK1l1+F2-r1uF1exp[r1uF1ν1u-r1lK1lν1u1+F2]},A2=defr2lK2lr2u(1+F1)exp{r2lK2lν2u1+F1-r2uF2ν2u},B2=defA2exp{r2lK2l1+F1-r2uF2exp[r2uF2ν2u-r2lK2lν2u1+F1]},f1=defmin{A1,B1},f2=defmin{A2,B2}.In paper [28], Chen obtained the permanence of system (1.2) as follows.Lemma 2.6 (see [28]).
Assume that(H1) holds; then every solution (N1,N2) of system (1.2) satisfies
(2.7)fi≤liminfn→∞Ni(n)≤limsupn→∞Ni(n)≤Fi,i=1,2.In this section, we obtain the following permanence result of system (1.2).Lemma 2.7.
Assume that(H1) holds; then every solution (N1,N2) of system (1.2) satisfies
(2.8)gi≤liminfn→∞Ni(n)≤limsupn→∞Ni(n)≤Gi,
where
(2.9)Gi=def(Kiu+αiu)exp{riu(Kiu+αiu)(νiu+1)},gi=defK1l1+G2exp{r1u[K1l1+G2-G1](ν1u+1)},i=1,2.Proof.
Let(N1,N2) be any positive solution of system (1.2) with initial condition (1.4). From the first equation of system (1.2), it follows that
(2.10)N1(n+1)≤N1(n)exp{r1(n)[K1(n)+α1(n)N2(n-μ2(n))1+N2(n-μ2(n))]}=N1(n)exp{r1(n)[K1(n)1+N2(n-μ2(n))+α1(n)N2(n-μ2(n))1+N2(n-μ2(n))]}≤N1(n)exp{r1(n)[K1(n)+α1(n)]}≤N1(n)exp{r1u(K1u+α1u)},
which yields that
(2.11)N1(n-ν1(n))≥N1(n)exp{-r1u(K1u+α1u)ν1u},
which implies that
(2.12)N1(n+1)≤N1(n)exp{r1(n)[(K1u+α1u)-N1(n)exp{-r1u(K1u+α1u)ν1u}]}.
First, we present two cases to prove that(2.13)limsupn→∞N1(n)≤G1.
Case I. There exists l0∈ℤ+ such that N1(l0+1)≥N1(l0). Then, by (2.12), we have
(2.14)(K1u+α1u)-N1(l0)exp{-r1u(K1u+α1u)ν1u}≥0,
which implies that N1(l0)≤(K1u+α1u)exp{r1u(K1u+α1u)ν1u}≤G1. From (2.12), we get
(2.15)N1(l0+1)≤N1(l0)exp{r1u(K1u+α1u)}≤(K1u+α1u)exp{r1u(K1u+α1u)(ν1u+1)}=defG1.
We claim that(2.16)N1(n)≤G1,∀n≥l0.
In fact, if there exists an integerk0≥l0+2 such that N1(k0)>G1, and letting k1 be the least integer between l0 and k0 such that N1(k1)=maxl0≤n≤k0{N1(n)}, then k1≥l0+2 and N1(k1)>N1(k1-1), which implies from the argument as that in (2.15) that
(2.17)N1(k1)≤G1<N1(k0).
This is impossible. This proves the claim.
Case II.
N
1
(
n
)
≥
N
1
(
n
+
1
), for all n∈ℤ+. In particular, limn→∞N1(n) exists, denoted by N-1. Taking limit in the first equation of system (1.2) gives
(2.18)limn→∞[K1(n)+α1(n)N2(n-μ2(n))1+N2(n-μ2(n))-N1(n-ν1(n))]=0.
Hence,N-1≤(K1u+α1u)≤G1. This proves the claim.
So,limsupn→∞N1(n)≤G1. In view of the second equation of system (1.2), similar to the previous analysis, we can obtain
(2.19)limsupn→∞N2(n)≤G2.
For arbitraryϵ>0, there exists n0∈ℤ+ such that
(2.20)Ni(n)≤Gi+ϵforn≥n0,i=1,2.
For n>n0+ρ, from the first equation of system (1.2), we have
(2.21)N1(n+1)≥N1(n)exp{r1(n)[K1(n)1+N2(n-μ2(n))-N1(n-ν1(n))]}≥N1(n)exp{r1u[K1l1+(G2+ϵ)-(G1+ϵ)]}.
Here, we use the inequality K1l/[1+(G2+ϵ)]-(G1+ϵ)≤K1l-G1≤K1l-K1u≤0. So,
(2.22)N1(n-ν1(n))≤N1(n)exp{-r1u[K1l1+(G2+ϵ)-(G1+ϵ)]ν1u},
which yields from the first equation of system (1.2) that
(2.23)N1(n+1)≥N1(n)exp{r1(n)[K1l1+(G2+ϵ)-N1(n)exp(-r1u[K1l1+(G2+ϵ)-(G1+ϵ)]ν1u)]}∀n≥n0+ρ.
Next, we also present two cases to prove that(2.24)liminfn→∞N1(n)≥g1.Case I. There exists l0≥n0+ρ such that N1(l0+1)≤N1(l0). Then, we have from (2.23) that
(2.25)K1l1+(G2+ϵ)-N1(l0)exp(-r1u[K1l1+(G2+ϵ)-(G1+ϵ)]ν1u)≤0,
which implies that
(2.26)N1(l0)≥K1l1+(G2+ϵ)exp(r1u[K1l1+(G2+ϵ)-(G1+ϵ)]ν1u).
In view of (2.21), we can easily obtain that
(2.27)N1(l0+1)≥K1l1+(G2+ϵ)exp{r1u[K1l1+(G2+ϵ)-(G1+ϵ)](ν1u+1)}=defg1(ϵ).
We claim that(2.28)N1(n)≥g1(ϵ)forn≥l0.
By way of contradiction, assume that there exists ac0≥l0 such that N1(c0)<g1(ϵ). Then, c0≥l0+2. Let c1≥l0+2 be the smallest integer such that N1(c0)<g1(ϵ). Then N1(c1-1)>N1(c1). The previous argument produces that N1(c1)≥g1(ϵ), a contradiction. This proves the claim.
Case II. We assume that N1(n)<N1(n+1), for all n≥n0+ρ. Then, limn→∞N1(n) exists, denoted by N_1. Taking limit in the first equation of system (1.2) gives
(2.29)limn→∞[K1(n)+α1(n)N2(n-μ2(n))1+N2(n-μ2(n))-N1(n-ν1(n))]=0.
Hence,N_1≥K1l/(1+(G2+ϵ))≥g1(ϵ) and limϵ→0g1(ϵ)=g1. This proves the claim.
So,liminfn→∞N1(n)≥g1. In view of the second equation of system (1.2), similar to the previous analysis, we can obtain
(2.30)liminfn→∞N2(n)≥g2.
So, the proof of Lemma 2.7 is complete.Example 2.8.
Consider the following discrete mutualism model with delays:(2.31)N1(n+1)=N1(n)exp{0.1[0.002+0.001N2(n)1+N2(n)-N1(n-1)]},N2(n+1)=N2(n)exp{0.1[0.002+0.001N1(n)1+N1(n)-N2(n-1)]}.Corresponding to system (1.2), ri≡0.1, Ki≡0.002, αi≡0.001, μi=0, νi=1, i=1,2. By calculation, we obtain
(2.32)F1=F2≈3.68,f1=f2≈1.8×10-4.
By Lemma 2.6, one has
(2.33)1.8×10-4≤liminfn→∞Ni(n)≤limsupn→∞Ni(n)≤3.68,i=1,2.Further, we could calculate(2.34)G1=G2≈3×10-3,g1=g2≈1.994×10-3.
By Lemma 2.7, one also has
(2.35)1.994×10-3≤liminfn→∞Ni(n)≤limsupn→∞Ni(n)≤3×10-3,i=1,2.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 N1(-1)=N1(0)=N2(-1)=N2(0)=0.002.By Lemmas2.6 and 2.7, we can easily show the following.Theorem 2.9.
Assume that(H1) holds; then every solution (N1,N2) of system (1.2) satisfies
(2.36)max{fi,gi}=defmi≤liminfn→∞Ni(n)≤limsupn→∞Ni(n)≤Mi=defmin{Fi,Gi},∀n∈ℤ,i=1,2.


## 3. Global Attractivity

Define a functionχ:[0,∞)ℤ→{0,1} as follows:
(3.1)χ(n)=def{0,ifn=0,1,ifn∈[1,∞)ℤ.
Let
(3.2)σ1=defexp{max(r1l[K1u+α1uM21+m2-m1],r1u[K1u+α1uM21+m2-m1])},σ2=defexp{max(r2l[K2u+α2uM11+m1-m2],r2u[K2u+α2uM11+m1-m2])},υ1=defmax{|K1u+α1uM21+m2-m1|,|K1l+α1lm21+M2-M1|},υ2=defmax{|K2u+α2uM11+m1-m2|,|K2l+α2lm11+M1-M2|}.Theorem 3.1.
Assume that(H1) holds. Suppose further that(H2)
there exist two positive constantsλ1 and λ2 such that min{Θ1,Θ2}>0, where
(3.3)Θ1=defλ1min[r1l,2M1-r1u]-λ1σ1(r1u)2χ(ν1u)ν1u(ν1u-ν1l+1)M1-λ1σ1υ1(r1u)2χ(ν1u)ν1u-λ2σ2(r2u)2χ(ν2u)|K2-α2|u(μ2u-μ2l+1)ν2uM2[1+m1]2-λ2r2u(μ1u-μ1l+1)|K2-α2|u[1+m1]2,Θ2=defλ2min[r2l,2M2-r2u]-λ2σ2(r2u)2χ(ν2u)ν2u(ν2u-ν2l+1)M2-λ2σ2υ2(r2u)2χ(ν2u)ν2u-λ1σ1(r1u)2χ(ν1u)|K1-α1|u(μ1u-μ1l+1)ν1uM1[1+m2]2-λ1r1u(μ2u-μ2l+1)|K1-α1|u[1+m2]2.Then, system (1.2) is globally attractive, that is, for any positive solution (N1,N2) and (W1,W2) of system (1.2),
(3.4)limn→+∞|N1(n)-W1(n)|=0,limn→+∞|N2(n)-W2(n)|=0.Proof.
In view of condition(H2), there exist small enough positive constants ϵ and λ such that
(3.5)Θ1(ϵ)=def{{λ2r2u(μ1u-μ1l+1)|K2-α2|u[1+m1-ϵ]2-λ1σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)}λ1min[r1l,2M1+ϵ-r1u]-λ1σ1(ϵ)(r1u)2χ(ν1u)ν1u(ν1u-ν1l+1)(M1+ϵ)-λ2σ2(ϵ)(r2u)2χ(ν2u)|K2-α2|u(μ2u-μ2l+1)ν2u(M2+ϵ)[1+m1-ϵ]2-λ2r2u(μ1u-μ1l+1)|K2-α2|u[1+m1-ϵ]2-λ1σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)ν1u}>λ,Θ2(ϵ)=def{{λ1r1u(μ2u-μ2l+1)|K1-α1|u[1+m2-ϵ]2-λ2σ2(ϵ)υ2(ϵ)(r2u)2χ(ν2u)ν2u}λ2min[r2l,2M2+ϵ-r2u]-λ2σ2(ϵ)(r2u)2χ(ν2u)ν2u(ν2u-ν2l+1)(M2+ϵ)-λ1σ1(ϵ)(r1u)2χ(ν1u)|K1-α1|u(μ1u-μ1l+1)ν1u(M1+ϵ)[1+m2-ϵ]2-λ1r1u(μ2u-μ2l+1)|K1-α1|u[1+m2-ϵ]2-λ2σ2(ϵ)υ2(ϵ)(r2u)2χ(ν2u)ν2u}>λ,
where
(3.6)σ1(ϵ)=defexp{max(r1l[K1u+α1u(M2+ϵ)1+m2-ϵ-(m1-ϵ)],r1u[K1u+α1u(M2+ϵ)1+m2-ϵ-(m1-ϵ)])},σ2(ϵ)=defexp{max(r2l[K2u+α2u(M1+ϵ)1+m1-ϵ-(m2-ϵ)],r2u[K2u+α2u(M1+ϵ)1+m1-ϵ-(m2-ϵ)])},υ1(ϵ)=defmax{|K1u+α1u(M2+ϵ)1+m2-ϵ-(m1-ϵ)|,|K1l+α1l(m2-ϵ)1+M2+ϵ-(M1+ϵ)|},υ2(ϵ)=defmax{|K2u+α2u(M1+ϵ)1+m1-ϵ-(m2-ϵ)|,|K2l+α2l(m1-ϵ)1+M1+ϵ-(M2+ϵ)|}.
Suppose that(N1,N2) and (W1,W2) are two positive solutions of system (1.2). By Theorem 2.9, there exists a constant 𝕂0>0 such that
(3.7)mi-ϵ≤Ni(n),Wi(n)≤Mi+ϵ,∀n>𝕂0,i=1,2.
Let(3.8)V11(n)=|lnN1(n)-lnW1(n)|.
In view of system (1.2), we have
(3.9)V11(n+1)=|lnN1(n+1)-lnW1(n+1)|=|{+r1(n)[K1(n)-α1(n)1+N2(n-μ2(n))-K1(n)-α1(n)1+W2(n-μ2(n))]}[lnN1(n)-lnW1(n)]-r1(n)[N1(n-ν1(n))-W1(n-ν1(n))]+r1(n)[K1(n)-α1(n)1+N2(n-μ2(n))-K1(n)-α1(n)1+W2(n-μ2(n))]|=|{+r1(n)[K1(n)-α1(n)1+N2(n-μ2(n))-K1(n)-α1(n)1+W2(n-μ2(n))]}[lnN1(n)-lnW1(n)]-r1(n)[N1(n)-W1(n)]-r1(n)χ(ν1u){[W1(n)-W1(n-ν1(n))]-[N1(n)-N1(n-ν1(n))]}+r1(n)[K1(n)-α1(n)1+N2(n-μ2(n))-K1(n)-α1(n)1+W2(n-μ2(n))]|≤|[lnN1(n)-lnW1(n)]-r1(n)[N1(n)-W1(n)]|+r1(n)χ(ν1u)|[W1(n)-W1(n-ν1(n))]-[N1(n)-N1(n-ν1(n))]|+r1(n)|K1(n)-α1(n)1+N2(n-μ2(n))-K1(n)-α1(n)1+W2(n-μ2(n))|.
Using the mean value theorem, it follows that
(3.10)N1(n)-W1(n)=exp{lnN1(n)}-exp{lnW1(n)}=θ1(n)[lnN1(n)-lnW1(n)],
where θ1(n) lies between N1(n) and W1(n), and
(3.11)|K1(n)-α1(n)1+N2(n-μ2(n))-K1(n)-α1(n)1+W2(n-μ2(n))|=|-K1(n)-α1(n)[1+θ2(n)]2[N2(n-μ2(n))-W2(n-μ2(n))]|,
where θ2(n) lies between N2(n-μ2(n)) and W2(n-μ2(n)).
Define(3.12)P1(n)=defr1(n)[K1(n)+α1(n)N2(n-μ2(n))1+N2(n-μ2(n))-N1(n-ν1(n))],Q1(n)=defr1(n)[K1(n)+α1(n)W2(n-μ2(n))1+W2(n-μ2(n))-W1(n-ν1(n))].
By a similar argument as that in (3.9), we obtain from (3.11) that
(3.13)|[W1(n)-W1(n-ν1(n))]-[N1(n)-N1(n-ν1(n))]|=|∑s=n-ν1(n)n-1‍[N1(s+1)-W1(s+1)]-∑s=n-ν1(n)n-1‍[N1(s)-W1(s)]|=|∑s=n-ν1(n)n-1‍[N1(s)eP1(s)-W1(s)eQ1(s)]-∑s=n-ν1(n)n-1‍[N1(s)-W1(s)]|=|∑s=n-ν1(n)n-1‍N1(s)[eP1(s)-eQ1(s)]+∑s=n-ν1(n)n-1‍[N1(s)-W1(s)][eQ1(s)-1]|≤∑s=n-ν1un-1‍N1(s)ξ1(s)r1(s)|{-[K1(s)-α1(s)1+N2(s-μ2(s))-K1(s)-α1(s)1+W2(s-μ2(s))]}[N1(s-ν1(s))-W1(s-ν1(s))]≤∑s=n-ν1un-1‍N1(s)ξ1(s)r1(s)-[K1(s)-α1(s)1+N2(s-μ2(s))-K1(s)-α1(s)1+W2(s-μ2(s))]|+∑s=n-ν1un-1‍ξ2(s)r1(s)|K1(s)+α1(s)W2(s-μ2(s))1+W2(s-μ2(s))-W1(s-ν1(s))||N1(s)-W1(s)|≤∑s=n-ν1un-1‍σ1(ϵ)r1u(M1+ϵ)|N1(s-ν1(s))-W1(s-ν1(s))|+∑s=n-ν1un-1‍σ1(ϵ)r1u|K1-α1|u(M1+ϵ)[1+m2-ϵ]2|N2(s-μ2(s))-W2(s-μ2(s))|+∑s=n-ν1un-1‍σ1(ϵ)υ1(ϵ)r1u|N1(s)-W1(s)|≤σ1(ϵ)r1u(M1+ϵ)∑s=n-ν1un-1‍∑k=ν1lν1u‍|N1(s-k)-W1(s-k)|+σ1(ϵ)r1u|K1-α1|u(M1+ϵ)[1+m2-ϵ]2∑s=n-ν1un-1‍∑k=μ1lμ1u‍|N2(s-k)-W2(s-k)|+σ1(ϵ)υ1(ϵ)r1u∑s=n-ν1un-1‍|N1(s)-W1(s)|≤σ1(ϵ)r1u(M1+ϵ)∑k=ν1lν1u‍∑s=n-ν1u-kn-k-1‍|N1(s)-W1(s)|+σ1(ϵ)r1u|K1-α1|u(M1+ϵ)[1+m2-ϵ]2∑k=μ1lμ1u‍∑s=n-ν1u-kn-k-1‍|N2(s)-W2(s)|+σ1(ϵ)υ1(ϵ)r1u∑s=n-ν1un-1‍|N1(s)-W1(s)|,∀n>𝕂0+2ρ,
where ξ1(s) lies between eP1(s) and eQ1(s), and ξ2(s) lies between eQ1(s) and 1, s=n-ν1u,…,n-1.
In view of (3.9), it follows from (3.10)–(3.13) that
(3.14)ΔV11(n)≤-[1θ1(n)-|1θ1(n)-r1(n)|]|N1(n)-W1(n)|+σ1(ϵ)(r1u)2χ(ν1u)(M1+ϵ)∑k=ν1lν1u‍∑s=n-ν1u-kn-k-1‍|N1(s)-W1(s)|+σ1(ϵ)(r1u)2χ(ν1u)|K1-α1|u(M1+ϵ)[1+m2-ϵ]2∑k=μ1lμ1u‍∑s=n-ν1u-kn-k-1‍|N2(s)-W2(s)|+σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)∑s=n-ν1un-1‍|N1(s)-W1(s)|+r1u|K1-α1|u[1+m2-ϵ]2∑s=n-μ2un-μ2l‍|N2(s)-W2(s)|≤-min[r1l,2M1+ϵ-r1u]|N1(n)-W1(n)|+σ1(ϵ)(r1u)2χ(ν1u)(M1+ϵ)∑k=ν1lν1u‍∑s=n-ν1u-kn-k-1‍|N1(s)-W1(s)|+σ1(ϵ)(r1u)2χ(ν1u)|K1-α1|u(M1+ϵ)[1+m2-ϵ]2∑k=μ1lμ1u‍∑s=n-ν1u-kn-k-1‍|N2(s)-W2(s)|+σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)∑s=n-ν1un-1‍|N1(s)-W1(s)|+r1u|K1-α1|u[1+m2-ϵ]2∑s=n-μ2un-μ2l‍|N2(s)-W2(s)|,∀n>𝕂0+2ρ.
Let(3.15)V12(n)=σ1(ϵ)(r1u)2χ(ν1u)(M1+ϵ)∑k=ν1lν1u‍∑t=0ν1u-1‍∑s=n-ν1u-k+tn-1‍|N1(s)-W1(s)|,V13(n)=σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)∑t=0ν1u-1‍∑s=n-ν1u+tn-1‍|N1(s)-W1(s)|,V14(n)=σ1(ϵ)(r1u)2χ(ν1u)|K1-α1|u(M1+ϵ)[1+m2-ϵ]2∑k=μ1lμ1u‍∑t=0ν1u-1‍∑s=n-ν1u-k+tn-1‍|N2(s)-W2(s)|,V15(n)=r1u|K1-α1|u[1+m2-ϵ]2∑t=0μ2u-μ2l‍∑s=n-μ2u+tn-1‍|N2(s)-W2(s)|.
So,
(3.16)ΔV12(n)=σ1(ϵ)(r1u)2χ(ν1u)(M1+ϵ)∑k=ν1lν1u‍∑t=0ν1u-1‍|N1(n)-W1(n)|-σ1(ϵ)(r1u)2χ(ν1u)(M1+ϵ)∑k=ν1lν1u‍∑t=0ν1u-1‍|N1(n-ν1u-k+t)-W1(n-ν1u-k+t)|=σ1(ϵ)(r1u)2χ(ν1u)ν1u(ν1u-ν1l+1)(M1+ϵ)|N1(n)-W1(n)|-σ1(ϵ)(r1u)2χ(ν1u)(M1+ϵ)∑k=ν1lν1u‍∑s=n-ν1u-kn-k-1‍|N1(s)-W1(s)|,(3.17)ΔV13(n)=σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)ν1u|N1(n)-W1(n)|-σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)∑s=n-ν1un-1‍|N1(s)-W1(s)|,(3.18)ΔV14(n)=σ1(ϵ)(r1u)2χ(ν1u)|K1-α1|u(μ1u-μ1l+1)ν1u(M1+ϵ)[1+m2-ϵ]2|N2(n)-W2(n)|-σ1(ϵ)(r1u)2χ(ν1u)|K1-α1|u(M1+ϵ)[1+m2-ϵ]2∑k=μ1lμ1u‍∑s=n-ν1u-kn-k-1‍|N2(s)-W2(s)|,(3.19)ΔV15(n)=r1u(μ2u-μ2l+1)|K1-α1|u[1+m2-ϵ]2|N2(n)-W2(n)|-r1u|K1-α1|u[1+m2-ϵ]2∑s=n-μ2un-μ2l‍|N2(s)-W2(s)|.
Define(3.20)V1(n)=V11(n)+V12(n)+V13(n)+V14(n)+V15(n).
It follows from (3.14)–(3.19) that
(3.21)ΔV1(n)≤-min[r1l,2M1+ϵ-r1u]|N1(n)-W1(n)|+σ1(ϵ)(r1u)2χ(ν1u)ν1u(ν1u-ν1l+1)(M1+ϵ)|N1(n)-W1(n)|+σ1(ϵ)(r1u)2χ(ν1u)|K1-α1|u(μ1u-μ1l+1)ν1u(M1+ϵ)[1+m2-ϵ]2|N2(n)-W2(n)|+σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)ν1u|N1(n)-W1(n)|+r1u(μ2u-μ2l+1)|K1-α1|u[1+m2-ϵ]2|N2(n)-W2(n)|≤-{min[r1l,2M1+ϵ-r1u]-σ1(ϵ)(r1u)2χ(ν1u)ν1u(ν1u-ν1l+1)(M1+ϵ)-σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)ν1u{min[r1l,2M1+ϵ-r1u]-σ1(ϵ)(r1u)2χ(ν1u)ν1u(ν1u-ν1l+1)(M1+ϵ)}}|N1(n)-W1(n)|+{σ1(ϵ)(r1u)2χ(ν1u)|K1-α1|u(μ1u-μ1l+1)ν1u(M1+ϵ)[1+m2-ϵ]2+r1u(μ2u-μ2l+1)|K1-α1|u[1+m2-ϵ]2}|N2(n)-W2(n)|,∀n>𝕂0+2ρ.
Let(3.22)V2(n)=V21(n)+V22(n)+V23(n)+V24(n)+V25(n),
where
(3.23)V21(n)=|lnN2(n)-lnW2(n)|,V22(n)=σ2(ϵ)(r2u)2χ(ν2u)(M2+ϵ)∑k=ν2lν2u‍∑t=0ν2u-1‍∑s=n-ν2u-k+tn-1‍|N2(s)-W2(s)|,V23(n)=σ2(ϵ)υ2(ϵ)(r2u)2χ(ν2u)∑t=0ν2u-1‍∑s=n-ν2u+tn-1‍|N2(s)-W2(s)|,V24(n)=σ2(ϵ)(r2u)2χ(ν2u)|K2-α2|u(M2+ϵ)[1+m1-ϵ]2∑k=μ2lμ2u‍∑t=0ν2u-1‍∑s=n-ν2u-k+tn-1‍|N1(s)-W1(s)|,V25(n)=r2u|K2-α2|u[1+m1-ϵ]2∑t=0μ1u-μ1l‍∑s=n-μ1u+tn-1‍|N1(s)-W1(s)|.
By a similar argument as that in (3.21), we could easily obtain that
(3.24)ΔV2(n)≤-{min[r2l,2M2+ϵ-r2u]-σ2(ϵ)(r2u)2χ(ν2u)ν2u(ν2u-ν2l+1)(M2+ϵ)-σ2(ϵ)υ2(ϵ)(r2u)2χ(ν2u)ν2u{min[r2l,2M2+ϵ-r2u]-σ2(ϵ)(r2u)2χ(ν2u)ν2u(ν2u-ν2l+1)(M2+ϵ)}}|N2(n)-W2(n)|+{σ2(ϵ)(r2u)2χ(ν2u)|K2-α2|u(μ2u-μ2l+1)ν2u(M2+ϵ)[1+m1-ϵ]2+r2u(μ1u-μ1l+1)|K2-α2|u[1+m1-ϵ]2}|N1(n)-W1(n)|,∀n>𝕂0+2ρ.
We construct a Lyapunov functional as follows:(3.25)V(n)=λ1V1(n)+λ2V2(n),
which implies from (3.21) and (3.24) that
(3.26)ΔV(n)≤-{λ2r2u(μ1u-μ1l+1)|K2-α2|u[1+m1-ϵ]2λ1min[r1l,2M1+ϵ-r1u]-λ1σ1(ϵ)(r1u)2χ(ν1u)ν1u(ν1u-ν1l+1)(M1+ϵ)-λ2σ2(ϵ)(r2u)2χ(ν2u)|K2-α2|u(μ2u-μ2l+1)ν2u(M2+ϵ)[1+m1-ϵ]2-λ2r2u(μ1u-μ1l+1)|K2-α2|u[1+m1-ϵ]2-λ1σ1(ϵ)υ1(ϵ)(r1u)2χ(ν1u)ν1u}|N1(n)-W1(n)|-{{-λ1r1u(μ2u-μ2l+1)|K1-α1|u[1+m2-ϵ]2-λ2σ2(ϵ)υ2(ϵ)(r2u)2χ(ν2u)ν2u}λ2min[r2l,2M2+ϵ-r2u]-λ2σ2(ϵ)(r2u)2χ(ν2u)ν2u(ν2u-ν2l+1)(M2+ϵ)-λ1σ1(ϵ)(r1u)2χ(ν1u)|K1-α1|u(μ1u-μ1l+1)ν1u(M1+ϵ)[1+m2-ϵ]2-λ1r1u(μ2u-μ2l+1)|K1-α1|u[1+m2-ϵ]2-λ2σ2(ϵ)υ2(ϵ)(r2u)2χ(ν2u)ν2u}|N2(n)-W2(n)|≤-λ[|N1(n)-W1(n)|+|N2(n)-W2(n)|],∀n>𝕂0+2ρ.
Takingn1∈(𝕂0+2ρ,∞)ℤ  and Summing  both sides of inequality (3.26) over [n1,n]ℤ, we have
(3.27)V(n+1)+λ∑s=n1n‍|N1(s)-W1(s)|+λ∑s=n1n‍|N2(s)-W2(s)|≤V(n1)<∞.
Therefore,
(3.28)limn→+∞sup∑s=n1n‍|N1(s)-W1(s)|+limn→+∞sup∑s=n1n‍|N2(s)-W2(s)|≤V(n1)λ<∞.
From the previous inequality one could easily deduce that
(3.29)limn→+∞|N1(s)-W1(s)|=0,limn→+∞|N2(s)-W2(s)|=0.
This completes the proof.Ifμ1=μ2=ν1=ν2≡0 in system (1.2), then we obtain a discrete mutualism model without delay as follows:
(3.30)N1(n+1)=N1(n)exp{r1(n)[K1(n)+α1(n)N2(n)1+N2(n)-N1(n)]},N2(n+1)=N2(n)exp{r2(n)[K2(n)+α2(n)N1(n)1+N1(n)-N2(n)]}.
Let
(3.31)M-i=defMiwithνiu=0,m-i=defmiwithν1u=ν2u=0,i=1,2.Corollary 3.2.
Assume that(H1) holds. Suppose further that(H3)
there exist two positive constantsλ1 and λ2 such that
(3.32)λ1min[r1l,2M-1-r1u]>λ2r2u|K2-α2|u[1+m-1]2,λ2min[r2l,2M-2-r2u]>λ1r1u|K1-α1|u[1+m-2]2.Then, system (3.30) is globally attractive.Further, we consider the following discrete mutualism model with constant delays:(3.33)N1(n+1)=N1(n)exp{r1(n)[K1(n)+α1(n)N2(n-μ2)1+N2(n-μ2)-N1(n-ν1)]},N2(n+1)=N2(n)exp{r2(n)[K2(n)+α2(n)N1(n-μ1)1+N1(n-μ1)-N2(n-ν2)]},
where μ1,μ2,ν1, and ν2 are nonnegative integers. Let
(3.34)M^i=defMiwithνiu=νi,m^i=defmiwithν1u=ν1,ν2u=ν2,i=1,2,σ^i=defσiwithM1=M^1,M2=M^2,m1=m^1,m2=m^2,i=1,2,υ^i=defυiwithM1=M^1,M2=M^2,m1=m^1,m2=m^2,i=1,2.Corollary 3.3.
Assume that(H1) holds. Suppose further that (H4)
there exist two positive constantsλ1 and λ2 such that
(3.35)λ1min[r1l,2M^1-r1u]>λ1σ^1(r1u)2χ(ν1)ν1M^1+λ1σ^1υ^1(r1u)2χ(ν1)ν1+λ2σ^2(r2u)2χ(ν2)|K2-α2|uν2M^2[1+m^1]2+λ2r2u|K2-α2|u[1+m^1]2,(3.36)λ2min[r2l,2M^2-r2u]>λ2σ^2(r2u)2χ(ν2)ν2M^2+λ2σ^2υ^2(r2u)2χ(ν2)ν2+λ1σ^1(r1u)2χ(ν1)|K1-α1|uν1M^1[1+m^2]2+λ1r1u|K1-α1|u[1+m^2]2.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{τp} be any integer valued sequence such that τp→∞ as p→∞. According to Lemma 2.5, taking a subsequence if necessary, we have
(4.1)ri(n+τp)→ri*(n),Ki(n+τp)→Ki*(n),αi(n+τp)→αi*(n),μi(n+τp)→μi*(n),νi(n+τp)→νi*(n),asp→∞
for n∈ℤ, i=1,2. Then, we get the hull equations of system (1.2) as follows:
(4.2)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 the almost periodic theory, we can conclude that if system (1.2) satisfies (H1)–(H4), then the hull equations (4.2) of system (1.2) also satisfies (H1)–(H4).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 (H1)-(H2), then system (1.2) admits a unique strictly positive almost periodic solution.Proof.
By Lemma4.1, 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 {ri(n)}, {Ki(n)}, {αi(n)}, {μi(n)}, and {νi(n)}, i=1,2, there exists an integer-valued sequence {ηp} with ηp→∞ as p→∞ such that
(4.3)ri*(n+ηp)→ri*(n),Ki*(n+ηp)→Ki*(n),αi*(n+ηp)→αi*(n),μi*(n+ηp)→μi*(n),νi*(n+ηp)→νi*(n),asp→∞,i=1,2.
Suppose that (N1,N2) is any solution of hull equations (4.2). Let ϵ be an arbitrary small positive number. It follows from Theorem 2.9 that there exists a positive integer I0 such that
(4.4)mi-ϵ≤Ni(n)≤Mi+ϵ,∀n>I0,i=1,2.
Write Nip(n)=Ni(n+ηp) for n≥I0-ηp, p=1,2,…, i=1,2. For any positive integer q, it is easy to see that there exist sequences {N1p(n):p≥q} and {N2p(n):p≥q} such that the sequences {N1p(n)} and {N2p(n)} have subsequences, denoted by {N1p(n)} and {N2p(n)} again, converging on any finite interval of ℤ as p→∞, respectively. Thus, we have sequences {W1(n)} and {W2(n)} such that
(4.5)Nip(n)→Wi(n),∀n∈ℤasp→∞,i=1,2.
Combined with
(4.6)N1p(n+1)=N1p(n)exp{r1*(n+ηp)[K1*(n+ηp)+α1*(n+ηp)N2(n+ηp-μ2*(n+ηp))1+N2(n+ηp-μ2*(n+ηp))-N1(n+ηp-ν1*(n+ηp)){K1*(n+ηp)+α1*(n+ηp)N2(n+ηp-μ2*(n+ηp))1+N2(n+ηp-μ2*(n+ηp))}]},N2p(n+1)=N2p(n)exp{r2*(n+ηp)[K2*(n+ηp)+α2*(n+ηp)N1(n+ηp-μ1*(n+ηp))1+N1(n+ηp-μ1*(n+ηp))-N2(n+ηp-ν2*(n+ηp)){K2*(n+ηp)+α2*(n+ηp)N1(n+ηp-μ1*(n+ηp))1+N1(n+ηp-μ1*(n+ηp))}]}
gives
(4.7)W1(n+1)=W1(n)exp{r1*(n)[K1*(n)+α1*(n)W2(n-μ2*(n))1+W2(n-μ2*(n))-W1(n-ν1*(n))]},W2(n+1)=W2(n)exp{r2*(n)[K2*(n)+α2*(n)W1(n-μ1*(n))1+W1(n-μ1*(n))-W2(n-ν2*(n))]}.
We can easily see that (W1,W2) is a solution of hull equations (4.2) and mi-ϵ≤Wi(n)≤Mi+ϵ for n∈ℤ, i=1,2. Since ϵ is an arbitrary small positive number, it follows that mi≤Wi(n)≤Mi 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 solutions (N1*,N2*) and (W1*,W2*) which satisfy
(4.8)mi≤Ni*,Wi*≤Mi,i=1,2.
Similar to Theorem3.1, we define a Lyapunov functional
(4.9)V*(n)=λ1V1*(n)+λ2V2*(n),
where
(4.10)V1*(n)=V11*(n)+V12*(n)+V13*(n)+V14*(n)+V15*(n),V2*(n)=V21*(n)+V22*(n)+V23*(n)+V24*(n)+V25*(n).
Here,
(4.11)V11*(n)=|lnN1*(n)-lnW1*(n)|,V12*(n)=σ1(r1u)2χ(ν1u)M1∑k=ν1lν1u‍∑t=0ν1u-1‍∑s=n-ν1u-k+tn-1‍|N1*(s)-W1*(s)|,V13*(n)=σ1υ1(r1u)2χ(ν1u)∑t=0ν1u-1‍∑s=n-ν1u+tn-1‍|N1*(s)-W1*(s)|,V14*(n)=σ1(r1u)2χ(ν1u)|K1-α1|uM1[1+m2]2∑k=μ1lμ1u‍∑t=0ν1u-1‍∑s=n-ν1u-k+tn-1‍|N2*(s)-W2*(s)|,V15*(n)=r1u|K1-α1|u[1+m2]2∑t=0μ2u-μ2l‍∑s=n-μ2u+tn-1‍|N2*(s)-W2*(s)|,V21*(n)=|lnN2*(n)-lnW2*(n)|,V22*(n)=σ2(r2u)2χ(ν2u)M2∑k=ν2lν2u‍∑t=0ν2u-1‍∑s=n-ν2u-k+tn-1‍|N2*(s)-W2*(s)|,V23*(n)=σ2υ2(r2u)2χ(ν2u)∑t=0ν2u-1‍∑s=n-ν2u+tn-1‍|N2*(s)-W2*(s)|,V24*(n)=σ2(r2u)2χ(ν2u)|K2-α2|uM2[1+m1]2∑k=μ2lμ2u‍∑t=0ν2u-1‍∑s=n-ν2u-k+tn-1‍|N1*(s)-W1*(s)|,V25*(n)=r2u|K2-α2|u[1+m1]2∑t=0μ1u-μ1l‍∑s=n-μ1u+tn-1‍|N1*(s)-W1*(s)|.
Similar to the argument as that in (3.26), one has
(4.12)ΔV*≤-λ|N1*(n)-W1*(n)|-λ|N2*(n)-W2*(n)|,∀n∈ℤ.
Summing both sides of the previous inequality from n to 0, we have
(4.13)λ∑s=n0‍|N1*(s)-W1*(s)|+λ∑s=n0‍|N2*(s)-W2*(s)|≤V*(n)-V*(1),∀n<0.
Note that V* is bounded. Hence, we have
(4.14)∑s=-∞0‍|N1*(s)-W1*(s)|<∞,∑s=-∞0‍|N2*(s)-W2*(s)|<∞,
which imply that
(4.15)limn→-∞|N1*(n)-W1*(n)|=0,limn→-∞|N2*(n)-W2*(n)|=0.
Let(4.16)P0=def2(λ1+λ2)max1≤i,j≤2,i≠j{1mi+2σi(riu)2Mi(νiu)3+σiυi(riu)2(νiu)2+σi(riu)2|Ki-αi|uMi(μiu+1)(νiu)2+riu|Ki-αi|u(μju+1)μju{1mi+2σi(riu)2Mi(νiu)3+σiυi(riu)2(νiu)2}}.
For arbitrary ϵ>0, there exists a positive integer 𝕂1 such that
(4.17)|N1*(n)-W1*(n)|<ϵP0,|N2*(n)-W2*(n)|<ϵP0,∀n<-𝕂1.
Hence, for i,j=1,2 with i≠j, one has
(4.18)Vi1*(n)≤ϵmiP0,∀n<-𝕂1,Vi2*(n)≤2σi(riu)2Mi(νiu)3ϵP0,∀n<-𝕂1,Vi3*(n)≤σiυi(riu)2(νiu)2ϵP0,∀n<-𝕂1,Vi4*(n)≤σi(riu)2|Ki-αi|uMi(μiu+1)(νiu)2ϵP0,∀n<-𝕂1,Vi5*(n)≤riu|Ki-αi|u(μju+1)μjuϵP0,∀n<-𝕂1,
which imply that
(4.19)V*(n)=λ1V1*(n)+λ2V2*(n)<ϵ,∀n<-𝕂1.
So,
(4.20)limn→-∞V*(n)=0.
Note that V*(n) is a nonincreasing function on ℤ and that V*(n)≡0. That is,
(4.21)N1*(n)=W1*(n),N2*(n)=W2*(n),∀n∈ℤ.
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 Theorems3.1 and 4.2, we can easily obtain the following.Theorem 4.3.
Suppose that(H1)-(H2) hold; then system (1.2) admits a unique strictly positive almost periodic solution, which is globally attractive.By Corollaries3.2–3.3 and Theorem 4.2, we can show the following.Theorem 4.4.
Suppose that(H1) and (H3) hold; then system (3.30) admits a unique strictly positive almost periodic solution, which is globally attractive.Theorem 4.5.
Suppose that(H1) and (H4) 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:(5.1)N1(n+1)=N1(n)exp{0.15+0.05cos(2n)+0.4N2(n)1+N2(n)-N1(n)},N2(n+1)=N2(n)exp{0.15+0.05sin(3n)+0.4N1(n)1+N1(n)-N2(n)}.
Then, system (5.1) admits a unique globally attractive almost periodic solution.Proof.
Corresponding to system (3.30), ri≡1, Kiu=0.2, Kil=0.1, αi≡0.4, i=1,2. By calculation, we obtain
(5.2)M-1=M-2≤0.67,
which implies that condition (H3) of Corollary 3.2 is satisfied with λ1=λ2=1. It is easy to verify that (H1) 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(H1) holds. Suppose further that (H5)
αi>Ki, i=1,2,(H6)
0<min{Θ12,Θ21}<1, where
(5.3)Θij=def2rilxi*-[riuxj*+(riu)2xi*xj*](αiu-Kil)-(riuxi*)2-(rjlxi*)2(αju-Kjl)2-[rjuxi*+(rju)2xi*xj*](αju-Kjl),i,j=1,2,i≠j.
Here,xi*=def(αiu/riu)exp[αiu(riu-1)], and xi*=defKilexp[ril(Kil-xi*)], i=1,2.
Then, system (3.30) admits a unique uniformly asymptotically stable almost periodic solution.Remark 5.3.
In system (5.1), we can easily calculate
(5.4)min{Θ12,Θ21}<0,
which implies that (H6) 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:(5.5)N1(n+1)=N1(n)exp{0.2[0.001+0.001|cos(2n)|+0.002N2(n)1+N2(n)-N1(n-1)]},N2(n+1)=N2(n)exp{0.2[0.001+0.001|sin(3n)|+0.002N1(n)1+N1(n)-N2(n-1)]}.
Then, system (5.5) admits a unique globally attractive almost periodic solution.Proof.
Corresponding to system (3.33), ri≡0.2, Kiu=0.002, Kil=0.001, αi≡0.002, μi=0, and νi=1, i=1,2. By calculation, we obtain
(5.6)M^1=M^2≤5×10-3,m^1=m^2≥2×10-4,σ^1=σ^2≤1,υ^1=υ^2≤5×10-3,
which implies that condition (H4) of Corollary 3.3 is satisfied with λ1=λ2=1. It is easy to verify that (H1) holds, and the result follows from Theorem 4.5 (see Figure 2).System (5.5) with N1(-1)=N1(0)=N2(-1)=N2(0)=0.002.
(a)
(b)Example 5.5.
Consider the following discrete mutualism model with variable delays:(5.7)N1(n+1)=N1(n)exp{0.5[0.002+(0.002+0.001cos2n)N2(n)1+N2(n)-N1(n-1+(-1)n2)]},N2(n+1)=N2(n)exp{0.5[0.002+(0.002+0.001sin2n)N1(n)1+N1(n)-N2(n)]}.
Then, system (5.7) admits a unique globally attractive almost periodic solution.Proof.
Corresponding to system (1.2), ri≡0.5, Ki≡0.002, αil=0.002, αiu=0.003, ν1l=0, ν1u=1, ν2≡0, and μi≡0, i=1,2. By calculation, we obtain
(5.8)M2<M1≤5.03×10-3,m2>m1≥1.974×10-3,σ1=σ2≤1,υ1=υ2≤5.03×10-3,
which implies that condition (H2) of Theorem 4.3 is satisfied with λ1=λ2=1. It is easy to verify that (H1) holds, and the result follows from Theorem 4.3 (see Figure 3).System (5.7) with N1(-1)=N1(0)=N2(-1)=N2(0)=0.002.
(a)
(b)



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*Source: 742102-2012-12-16.xml*