DDNSDiscrete Dynamics in Nature and Society1607-887X1026-0226Hindawi Publishing Corporation28603110.1155/2010/286031286031Research ArticleAlmost Periodic Solutions of a Discrete Mutualism Model with Feedback ControlsWangZhengLiYongkunBerezanskyLeonidDepartment of MathematicsYunnan UniversityKunmingYunnan 650091Chinaynu.edu.cn20101832010201031102009160320102010Copyright © 2010This 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.

We consider a discrete mutualism model with feedback controls. Assuming that the coefficients in the system are almost periodic sequences, we obtain the existence and uniqueness of the almost periodic solution which is uniformly asymptotically stable.

1. Introduction

Two species cohabit a common habitat and each species enhances the average growth rate of the other; this type of ecological interaction is known as facultative mutualism . A two species mutualism model can be described in the following form:

dN1(t)dt=N1(t)f1(N1(t),N2(t)),dN2(t)dt=N1(t)f2(N1(t),N2(t)), where f1,f2 are continuously differentiable such that

f1N20,f2N10. One of the simplest models which satisfies the above assumption is the traditional Lotka-Volterra two species mutualism model, which takes the form

dN1(t)dt=N1(t)(a1-b1N1(t)+c1N2(t)),dN2(t)dt=N2(t)(a2-b2N2(t)+c2N1(t)). Since the above system could exhibit unbounded solutions [2, 3] and it is well known that in nature, with the restriction of resources, it is impossible for one species to survive if its density is too high. Thus, the above model is not so good in describing the mutualism of two species. Gopalsamy  has proposed the following model to describe the mutualism mechanism:

dN1(t)dt=r1N1(t)[K1+α1N2(t)1+N2(t)-N1(t)],dN2(t)dt=r2N2(t)[K2+α2N1(t)1+N1(t)-N2(t)], where ri denotes the intrinsic growth rate of species Ni and αi>Ki,i=1,2. The carrying capacity of species Ni is Ki in the absence of other species, while with the help of the other species, the carrying capacity becomes (Ki+αiN3-i)/(1+N3-i),i=1,2. The above mutualism can be classified as facultative, obligate, or a combination of both. For more details of mutualistic interactions, we refer to . Realistic models require the inclusion of the effect of changing environment. This motivate us to consider the following nonautonomous model:

dN1(t)dt=r1(t)N1(t)[K1(t)+α1(t)N2(t)1+N2(t)-N1(t)],dN2(t)dt=r2(t)N2(t)[K2(t)+α2(t)N1(t)1+N1(t)-N2(t)]. Since many authors [10, 11] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations, then, discrete-timemodels can provide efficient computational types of continuous models for numerical simulations. It is reasonable to study the discrete-time mutualism model governed by difference equations. One of the ways of deriving difference equations modeling the dynamics of populations with nonoverlapping generations is based on appropriate modifications of the corresponding models with overlapping generations [4, 12]. In this approach, differential equations with piecewise constant arguments have been proved to be useful. Following the same idea and the same method in [4, 12], one can easily derive the following discrete analog of (1.5), which takes the form of

x1(n+1)=x1(n)exp{r1(n)[K1(n)+α1(n)x2(n)1+x2(n)-x1(n)-b1(n)u1(n)]},x2(n+1)=x2(n)exp{r2(n)[K2(n)+α2(n)x1(n)1+x1(n)-x2(n)-b2(n)u2(n)]}. The exponential form of (1.6) is more biologically reasonable than that directly derived by replacing the differential by difference in (1.5). Feedback control is the basic mechanism by which systems, whether mechanical, electrical, or biological, maintain their equilibrium or homeostasis. During the last decade, a series of mathematical models have been established to describe the dynamics of feedback control systems .

In this paper, we are concerned with the following discrete mutualism model with feedback controls:

x1(n+1)=x1(n)exp{r1(n)[K1(n)+α1(n)x2(n)1+x2(n)-x1(n)-b1(n)u1(n)]},x2(n+1)=x2(n)exp{r2(n)[K2(n)+α2(n)x1(n)1+x1(n)-x2(n)-b2(n)u2(n)]},Δu1(n)=-a1(n)u1(n)+c1(n)x1(n),Δu2(n)=-a2(n)u2(n)+c2(n)x2(n). To the best of our knowledge, though many works have been done for the population dynamic system with feedback controls, most of the works dealt with the continuous time model. For more results about the existence of almost periodic solutions of a continuous time system, we can refer to  and the references cited therein. There are few works that consider the existence of almost periodic solutions for discrete time population dynamic model with feedback controls. So, our main purpose of this paper is to study the existence and uniqueness of almost periodic solutions for the model (1.7).

Throughout this paper, we assume that

{ri(n)},{Ki(n)},{αi(n)},{ai(n)},{bi(n)}, and {ci(n)} for i=1,2 are bounded non-negative almost periodic sequences such that 0<rilri(n)riu,0<KilKi(n)Kiu,0<αilαi(n)αiu,0<ailai(n)aiu<1,0bilbi(n)biu,0cilci(n)ciu, and αi>ki for i=1,2.

Here, for any bounded sequence {a(n)}, au=supnN{a(n)} and al=infnN{a(n)}. By the biological meaning, we focus our discussion on the positive solution of the system (1.7). So it is assumed that the initial conditions of (1.7) are of the form

xi(0)>0,    ui(0)>0,    i=1,2. One can easily show that the solutions of (1.5) with the initial condition (1.9) are defined and remain positive for all nZ+={0,1,2,}.

2. Preliminaries

In this section, we will introduce two definitions and a useful lemma.

Definition 2.1 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

A sequence x:ZRk is called an almost periodic sequence if the ɛ- translation set of x: E{ɛ,x}:={τZ:|x(n+τ)-x(n)|<ɛ}, for all nZ is a relatively dense set in Z for all ɛ>0, that is, for any given ɛ>0, there exists an integer l>0 such that each discrete interval of length l contains an integer τ=τ(ɛ)E{ɛ,x} such that |x(n+τ)-x(n)|<ɛ, for all nZ, τ is called the ɛ-translation number of x(n).

Definition 2.2 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Let f:Z×DRk, where D is an open set in Rk, f(n,x) is said to be almost periodic in n uniformly for xD, or uniformly almost periodic for short, if for any ɛ>0 and any compact set S in D, there exists a positive integer l(ɛ,S) such that any interval of length l(ɛ,S) contains an integer τ for which |f(n+τ,x)-f(n,x)|<ɛ for all nZ and xS. τ is called the ɛ-translation number of f(n,x).

Lemma 2.3 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

{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 nZ as k. Furthermore, the limit sequence is also an almost periodic sequence.

3. Persistence

In this section, we establish a persistence result for model (1.7).

Proposition 3.1.

Assume that (H) holds. For every solution (x1(n),x2(n),u1(n),u2(n)) of (1.7) lim supnx1(n)<x1*,lim supnx2(n)<x2*,lim supnu1(n)<u1*,lim supnu2(n)<u2*, where x1*=α1ur1uexp[α1u(r1u-1)],x2*=α2ur2uexp[α2u(r2u-1)],u1*=x1*c1ua1l,u2*=x2*c2ua2l.

Proof.

We first present two cases to prove that lim supnx1(n)<x1*.Case 1.

By the first equation of model (1.7), (H) and (1.9), we have x1(n+1)=x1(n)exp{r1(n)[K1(n)+α1(n)x2(n)1+x2(n)-x1(n)-b1(n)u1(n)]}<x1(n)exp{r1(n)[α1(n)-x1(n)-b1(n)u1(n)]}=x1(n)exp{r1(n)α1(n)[1-x1(n)α1(n)-b1(n)u1(n)α1(n)]}. Then there exists an l0N such that x1(l0+1)x1(l0). So, 1-x1(l0)/α1(l0)-(b1(l0)u1(l0)/α1(l0))0. Hence, x1(l0)α1(l0)α1ux1*, and x1(l0+1)<x1(l0)exp{r1(l0)α1(l0)[1-x1(l0)α1(l0)-b1(l0)u1(l0)α1(l0)]}x1(l0)exp[r1uα1u(1-x1(l0)α1(l0))]=α1(l0)x1(l0)α1(l0)exp[r1uα1u(1-x1(l0)α1(l0))]α1ur1uexp[α1u(r1u-1)]=x1*. Here we used maxxRxexp(r(1-x))=exp(r-1)/r for r>0. We claim that x1(n)x1* for nl0.

In fact, if there exists an integer mn0+2 such that x1(m)>x1*, and letting m1 be the least integer between n0 and m such that x1(m1)=maxn0nm1{x1(n)}, then m1n0+2 and x1(m1)>x1(m1-1) which implies x1(m1)x1*<x1(m). This is impossible. The claim is proved.

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M92"><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>≥</mml:mo><mml:msub><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math></inline-formula> for <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M93"><mml:mi>n</mml:mi><mml:mo>∈</mml:mo><mml:mi>N</mml:mi></mml:math></inline-formula>).

In particular, limnx1(n) exists, denoted by x¯1. We claim that x¯1x1*. By way of contradiction, assume that x¯1>x1*. Taking limn(1-x1(n)/α1(n)-b1(n)u1(n)/α1(n))=0. Noting that α1ux*, hence 1-x1(n)α1(n)-b1(n)u1(n)α1(n)1-x1(n)α1(n)1-x1¯α1u<0, for nN, which is a contradiction. This proves the claim.

We can prove that lim supnx2(n)x2* in the similar way. Therefore, for each ɛ>0, there exists a large enough integer n0 such that xi(n)xi*+ɛ, i=1,2 whenever nn0. The proof of lim supnui(n)<ui*(i=1,2) is very similar to that of Proposition 1 in . Here we omit the details here.

Proposition 3.2.

Assume that (H) and (1.6) hold; furthermore, K1l-b1uu1*>0 and K2l-b2uu2*>0, where u1* and u2* are the same as those in Proposition 3.1. Then lim infnx1(n)>x1*,lim infnx2(n)>x2*,lim infnu1(n)>u1*,lim infnu2(n)>u2*, where x1*=(K1l-b1uu1*)exp[r1l(K1l-x1*-b1uu1*)],  x2*  =  (K2l-b2uu2*)exp[r2l(K2l-x2*-b2uu2*)],u1*  =  c1lx1*/a1u,u2*=c2lx2*/a2u.

Proof.

We also present two cases to prove that lim infnx1(n)>x1*.

For any ɛ>0 which satisfies K1l-b1uu1*>0, according to Proposition 3.1, there exists n0N such that

x1(n)x1*+ɛ,x2(n)x2*+ɛ,u1(n)u1*+ɛ,u2(n)u2*+ɛ, for nn0. Case 1.

There exists a positive integer l0n0 such that x1(l0+1)x1(l0). Note that for nl0, x1(n+1)=x1(n)exp{r1(n)[K1(n)+α1(n)x2(n)1+x2(n)-x1(n)-b1(n)u1(n)]}>x1(n)exp{r1(n)[K1(n)-x1(n)-b1(n)u1(n)]}=x1(n)exp{r1(n)K1(n)[1-x1(n)K1(n)-b1(n)u1(n)K1(n)]}x1(n)exp{r1(n)K1(n)[1-x1(n)K1(n)-b1u(u1*+ɛ)K1(n)]}. In particular, with n=l0, we get 1-x1(l0)K1(l0)-b1u(u1*+ɛ)K1(l0)0, which implies that x1(l0)K1l-b1u(u1*+ɛ). Then x1(l0+1)>[K1l-b1u(u1*+ɛ)]exp[r1lK1l(1-x1*+ɛK1l-b1u(u1*+ɛ)K1l)].

Let x1ɛ=[K1l-b1u(u1*+ɛ)]exp[r1lK1l(1-(x1*+ɛ)/K1l-b1u(u1*+ɛ)/K1l)]. We claim that x1(n)x1ɛ for nl0.

By a way of contradiction, assume that there exists a p0l0 such that x1(p0)<x1ɛ. Then p0l0+2, let p1l0+2 be the smallest integer such that x1(p1)<x1ɛ. Then x(p1-1)<x(p1). The above argument produces that x1(p1)x1ɛ, a contradiction. This proves the claim.

Case 2.

We assume that x1(n+1)>x1(n) for all large n. Then limnx1(n) exists, denoted by x̲1. We claim that x̲1K1l-b1u(u1*+ɛ). By way of contradiction, assume that x̲1<K1l-b1u(u1*+ɛ). Taking limn(1-x1(n)/K1(n)b1(n)u1(n)/K1(n))=0, which is a contradiction, since lim infn(1-x1(n)K1(n)-b1(n)u1(n)K1(n))1-x̲1K1l-b1u(u1*+ɛ)K1l>0.

Noting that x1*K1uK1l, we see that K1l-b1u(u1*+ɛ)x1ɛ, and limɛ0x1ɛ=x1*. We can easily see that lim infnx1(n)x1* holds. Similarly, we can prove that lim infnx2(n)x2*. Thus for any ɛ>0 small enough, there exists a positive integer n0, such that xi(n)xi*-ɛ>0 for nn0.

The proof of lim infnui(n)>ui*,i=1,2 is very similar to that of Proposition 2 in . Here we omit the details.

4. Main Results

For our purpose, we first introduce the following results which are given in Persistence.

Lemma 4.1.

Assume that (1.9), (H), K1l-b1uu1*>0, and K2l-b2uu2*>0 hold, then xi*<lim infnxi(n)lim supnxi(n)<xi*,  ui*<lim infnui(n)lim supnui(n)<ui*, where xi*=(αiu/riu)exp[αiu(riu-1)],ui*=xi*ciu/ail,xi*=(Kil-biuui*)exp[ril(Kil-xi*-biuui*)],ui*=cilxi*/aiu,i=1,2.

In , Zhang considered the following almost periodic difference system

x(n+1)=f(n,x(n)),nZ+, where f:Z×SBRk,SB={xRk:x<B}, and f(n,x) is almost periodic in n uniformly for xSB and is continuous in x. Related to system (4.2), he also considered the following product system:

x(n+1)=f(n,x(n)),y(n+1)=f(n,y(n)) and obtained the following theorem.

Theorem 4.2 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

Suppose that there exists a Lyapunov functional V(n,x,y) defined for nZ+,x<B,y<B satisfying the following conditions:

a(x-y)V(n,x,y)b(x-y), where a,bκ with κ={aC(R+,R+):a(0)=0and  a  is increasing},

|V(n,x1,y1)-V(n,x2,y2)|L(x1-x2+y1-y2), where L>0 is a constant,

ΔV(4.3)(n,x,y)-aV(n,x,y), where 0<a<1 is a constant and ΔV(4.3)(n,x,y)=V(n+1,f(n,x),f(n,y))-V(n,x,y).

Moreover, if there exists a solution φ(n) of (4.2) such that φ(n)B*<B for nZ+, then there exists a unique uniformly asymptotically stable almost periodic solution p(n) of system (4.2) which is bounded by B*. In particular, if f(n,x) is periodic of period ω, then there exists a unique uniformly asymptotically stable periodic solution of (4.2) of period ω.

According to Theorem 4.2, we first prove that there exists a bounded solution of (1.7) and then construct an adaptive Lyapunov functional for (1.7).

We denote by Ω the set of all solutions X(n)=(x1(n),x2(n),u1(n),u2(n)) of system (1.7) satisfying xi*xi(n)xi*,ui*ui(n)ui*(i=1,2) for all nZ+.

Lemma 4.3.

Assume that (H) and the conditions of Lemma 4.1 hold, then Ω.

Proof.

It is now possible to show by an inductive argument that system (1.7) leads to xi(n)=xi(0)expl=0n-1{ri(l)[Ki(l)+αi(l)xj(l)1+xj(l)-xi(l)-bi(l)ui(l)]},ui(n)=ui(0)-l=0n-1{ai(l)ui(l)-ci(l)xi(l)}, for i,j=1,2,ij. From Lemma 4.1, for any solution X(n)=(x1(n),x2(n),u1(n),u2(n)) of (1.7) with initial condition (1.9) satisfies (4.2). Hence, for any ɛ>0, there exist n0, if n0 is sufficiently large, we have xi*-ɛxi(n)xi*+ɛ,ui*-ɛui(n)ui*+ɛ,nn0,i=1,2.

Let {τα} be any integer-valued sequence such that τα as α, we claim that there exists a subsequence of {τα}, we still denote by {τα}, such that

xi(n+τα)xi*(n) uniformly in n on any finite subset B of Z as α, where B={a1,a2,,am},ahZ(h=1,2,,m) and m is a finite number.

In fact, for any finite subset BZ, when α is large enough, τα+ah>n0,h=1,2,,m. So

xi*-ɛxi(n+τα)xi*+ɛ,ui*-ɛui(n+τα)ui*+ɛ. That is, {xi(n+τα)},{ui(n+τα)} are uniformly bounded for large enough α.

Now, for a1B, we can choose a subsequence {τα(1)} of {τα} such that {xi(a1+τα(1))}, {ui(a1+τα(1))} uniformly converges on Z+ for α large enough.

Similarly, for a2B, we can choose a subsequence {τα(2)} of {τα(1)} such that {xi(a2+τα(2))}, {ui(a2+τα(2))} uniformly converges on Z+ for α large enough.

Repeating this procedure, for amB, we obtain a subsequence {τα(m)} of {τα(m-1)} such that {xi(am+τα(m))}, {ui(am+τα(m))} uniformly converges on Z+ for α large enough.

Now pick the sequence {τα(m)} which is a subsequence of {τα}, we still denote it as {τα}, then for all nB, we have xi(n+τα)xi*(n), ui(n+τα)ui*(n) uniformly in nB as α.

By the arbitrariness of B, the conclusion is valid.

Since {ri(n)},{Ki(n)},{αi(n)},{ai(n)},{bi(n)}, and {ci(n)} are almost periodic sequences, for above sequence {τα}, τα as α, there exists a subsequence still denote by {τα} (if necessary, we take subsequence), such that

ri(n+τα)ri(n),Ki(n+τα)Ki(n),αi(n+τα)αi(n),ai(n+τα)ai(n),bi(n+τα)bi(n),ci(n+τα)ci(n), as α uniformly on Z+.

For any σZ, we can assume that τα+σn0 for α large enough. Let n0 and nZ+, by an inductive argument of (1.7) from τα+σ to n+τα+σ leads to

xi(n+τα+σ)=xi(τα+σ)expl=τα+σn+τα+σ-1{ri(l)[Ki(l)+αi(l)xj(l)1+xj(l)-xi(l)-bi(l)ui(l)]},ui(n+τα+σ)=ui(τα+σ)-l=τα+σn+τα+σ-1{ai(l)ui(l)+ci(l)xi(l)}. Then, for i,j=1,2,ij, we have xi(n+τα+σ)=xi(τα+σ)expl=σn+σ-1{ri(l+τα)[Ki(l+τα)+αi(l+τα)xj(l+τα)1+xj(l+τα)-xi(l+τα)-bi(l+τα)ui(l+τα)Ki(l+τα)+αi(l+τα)xj(l+τα)1+xj(l+τα)]},ui(n+τα+σ)=ui(τα+σ)-l=σn+σ-1{ai(l+τα)ui(l+τα)-ci(l+τα)xi(l+τα)}. Let α, for any n0, xi*(n+σ)=xi*(σ)expl=σn+σ-1{ri(l)[Ki(l)+αi(l)xj*(l)1+xj*(l)-xi*(l)-bi(l)ui(l)]},ui*(n+σ)=ui*(σ)-l=σn+σ-1{ai(l)ui*(l)-ci(l)xi*(l)}. By the arbitrariness of σ, X*(n)=(x1*(n),x2*(n),u1*(n),u2*(n)) is a solution of model (1.7) on Z+. It is clear that 0<xi*xi*(n)xi*,0<ui*ui*(n)ui*, for all nZ+,i=1,2. So Ω. Lemma 4.3 is valid.

Theorem 4.4.

Suppose that the conditions of Lemma 4.3 are satisfied, moreover, 0<β<1, where β=min{rij,rij*},rij=2rilxi*-(riuxj*+riu2xi*xj*)(αiu-Kil)-riu2xi*2-ciu2xi*2-biuxi*riu2-riubiu-ciuxi*(1-ail)-rjl2xi*2(αju-kjl)2-(rjuxi*+rju2xj*xi*)(αju-kjl)-bjuxi*rju2(αju-kjl),rij*=-biuxj*riu2(αiu-Kil)-riu2biu2-aiu2-2aiu-biuxi*riu2-riubiu-ciuxi*(1-ail),i,j=1,2,ij, then there exists a unique uniformly asymptotically stable almost periodic solution X(n)=(x1(n),x2(n),u1(n),u2(n)) of (1.7) which is bounded by Ω for all nZ+.

Proof.

Let p1(n)=lnx1(n),p2(n)=lnx2(n). From (1.7), we have pi(n+1)=pi(n)+ri(n)[Ki(n)+αi(n)exppj(n)1+exppj(n)-exppi(n)-bi(n)ui(n)],Δui(n)=-ai(n)ui(n)+ci(n)exppi(n), where i,j=1,2,ij. From Lemma 4.3, we know that system (4.13) has bounded solution Y(n)=(p1(n),p2(n),u1(n),u2(n)) satisfying lnxi*pi(n)lnxi*,ui*ui(n)ui*,  i=1,2,nZ+. Hence, |pi(n)|Ai,|ui(n)|Bi, where Ai=max{|lnxi*|,|lnxi*|},Bi=max{ui*,ui*},i=1,2.

For (X,U)R2+2, we define the norm (X,U)=i=12|xi|+i=12|ui|.

Consider the product system of system (4.13)

pi(n+1)=pi(n)+ri(n)[Ki(n)+αi(n)exppj(n)1+exppj(n)-exppi(n)-bi(n)ui(n)],Δui(n)=-ai(n)ui(n)+ci(n)exppi(n),qi(n+1)=qi(n)+ri(n)[Ki(n)+αi(n)expqj(n)1+expqj(n)-expqi(n)-bi(n)ωi(n)],Δωi(n)=-ai(n)ωi(n)+ci(n)exppi(n).

Suppose Z=(p1(n),p2(n),u1(n),u2(n)), W=(q1(n),q2(n),ω1(n),ω2(n)) are any two solutions of system (4.15) defined on Z+×S*×S*, then ZB,WB, where

B=i=12{Ai+Bi},S*={(p1(n),p2(n),u1(n),u2(n))lnxi*pi(n)lnxi*,ui*ui(n)ui*,i=1,2,nZ+}.

Consider a Lyapunov function defined on Z+×S*×S* as follows:

V(n,Z,W)=i=12{(pi(n)-qi(n))2+(ui(n)-ωi(n))2}. It is easy to see that the norm Z-W=i=12{|pi(n)-qi(n)|+|ui(n)-ωi(n)|} and the norm Z-W*={i=12{(pi(n)-qi(n))2+(ui(n)-ωi(n))2}}1/2 are equivalent that is, there exist two constants C1>0,C2>0, such that C1Z-WZ-W*C2Z-W, then (C1Z-W)2V(n,Z,W)(C2Z-W)2. Let aC(R+,R+),    a(x)=C12x2,    bC(R+,R+),b(x)=C22x2, thus condition (i) in Theorem 4.2 is satisfied.

|V(n,Z,W)-V(n,Z̃,W̃)|=|i=12{(pi(n)-qi(n))2+(ui(n)-ωi(n))2}-i=12{(p̃i(n)-q̃i(n))2+(ũi(n)-ω̃i(n))2}|i=12|(pi(n)-qi(n))2-(p̃i(n)-q̃i(n))2|+i=12|(ui(n)-ωi(n))2-(ũi(n)-ω̃i(n))2|=i=12{|(pi(n)-qi(n))+(p̃i(n)-q̃i(n))||(pi(n)-qi(n))-(p̃i(n)-q̃i(n))|}+i=12{|(ui(n)-ωi(n))+(ũi(n)-ω̃i(n))||(ui(n)-ωi(n))-(ũi(n)-ω̃i(n))|}i=12{(|pi(n)|+|qi(n)|+|p̃i(n)|+|q̃i(n)|)(|pi(n)-p̃i(n)|+|qi(n)-q̃i(n)|)}+i=12{(|ui(n)|+|ωi(n)|+|ũi(n)|+|ω̃i(n)|)(|ui(n)-ũi(n)|+|ωi(n)-ω̃i(n)|)}L{i=12{|pi(n)-p̃i(n)|+|ui(n)-ũi(n)|}+i=12{|qi(n)-q̃i(n)|+|ωi(n)-ω̃i(n)|}}=L{Z-Z̃+W-W̃}, where L=4max{Ai,Bi}(i=1,2). Hence the condition (ii) of Theorem 4.2 is satisfied. Finally, calculate the ΔV of V(n) along the solutions of (4.15), we can obtain ΔV(4.15)(n)=V(n+1)-V(n)=i=12{(pi(n+1)-qi(n+1))2+(ui(n+1)-ωi(n+1))2}-i=12{(pi(n)-qi(n))2+(ui(n)-ωi(n))2}=i=12{(pi(n+1)-qi(n+1))2-(pi(n)-qi(n))2+(ui(n+1)-ωi(n+1))2-(ui(n)-ωi(n))2}=i=12{[(pi(n)-qi(n))+ri(n)(αi(n)-Ki(n))(epj(n)-eqj(n))(1+epj(n))(1+eqj(n))-ri(n)(epi(n)-eqi(n))-ri(n)bi(n)(ui(n)-ωi(n))]2-(pi(n)-qi(n))2+[(1-ai(n))(ui(n)-ωi(n))+ci(n)(epi(n)-eqi(n))]2-(ui(n)-ωi(n))2}=i=12{ri2(n)(αi(n)-Ki(n))2(epj(n)-eqj(n))2(1+epj(n))2(1+eqj(n))2+2ri(n)(αi(n)-Ki(n))(epj(n)-eqj(n))(pi(n)-qi(n))(1+epj(n))(1+eqj(n))+ri2(n)(epi(n)-eqi(n))2+ri2(n)bi2(n)(ui(n)-ωi(n))2+2bi(n)ri2(n)(epi(n)-eqi(n))(ui(n)-ωi(n))-2ri(n)×(pi(n)-qi(n))(epi(n)-eqi(n))-2bi(n)ri(n)(pi(n)-qi(n))(ui(n)-ωi(n))-2ri2(n)(αi(n)-Ki(n))(epj(n)-eqj(n))(epi(n)-eqi(n))(1+epj(n))(1+eqj(n))-2bi(n)ri2(n)(αi(n)-Ki(n))(epj(n)-eqj(n))(ui(n)-ωi(n))(1+epj(n))(1+eqj(n))-(1-ai(n))2(ui(n)-ωi(n))2+ci2(n)(epi(n)-eqi(n))2+2ci(n)(1-ai(n))(ui(n)-ωi(n))(epi(n)-eqi(n))-(ui(n)-ωi(n))2(αi(n)-Ki(n))(epj(n)-eqj(n))(ui(n)-ωi(n))(1+epj(n))(1+eqj(n))}. Using the mean value theorem, we get epi(n)-eqi(n)=ξi(n)(pi(n)-qi(n)),i=1,2, where ξi(n) lies between epi(n) and eqi(n), i=1,2. From (4.21), (4.22), we have ΔV(4.15)(n)=i=12{ri2(n)ξj2(n)(αi(n)-Ki(n))2(pj(n)-qj(n))2(1+epj(n))2(1+eqj(n))2+2ri(n)ξj(n)(αi(n)-Ki(n))(pi(n)-qi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))+ri2(n)ξi2(n)(pi(n)-qi(n))2+ri2(n)bi2(n)(ui(n)-ωi(n))2+2bi(n)ξi(n)ri2(n)(pi(n)-qi(n))(ui(n)-ωi(n))-2ri(n)ξi(n)(pi(n)-qi(n))2-2bi(n)ri(n)(pi(n)-qi(n))(ui(n)-ωi(n))-ri2(n)ξi(n)ξj(n)(αi(n)-Ki(n))(pj(n)-qj(n))(pi(n)-qi(n))(1+epj(n))(1+eqj(n))-2bi(n)ξj(n)ri2(n)(αi(n)-Ki(n))(ui(n)-ωi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))+(1-ai(n))2(ui(n)-ωi(n))2+ci2(n)ξi2(n)(pi(n)-qi(n))2+2ci(n)(1-ai(n))(ui(n)-ωi(n))ξi(n)×(pi(n)-qi(n))-(ui(n)-ωi(n))2(pj(n)-qj(n))2(1+epj(n))2(1+eqj(n))2}=i=12{ri2(n)ξj2(n)(αi(n)-Ki(n))2(pj(n)-qj(n))2(1+epj(n))2(1+eqj(n))2+2[ri(n)ξj(n)(αi(n)-Ki(n))-ri2(n)ξi(n)ξj(n)(αi(n)-Ki(n))]×(pi(n)-qi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))-2bi(n)ξj(n)ri2(n)(αi(n)-Ki(n))(ui(n)-ωi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))+[ri2(n)ξi2(n)-2ri(n)ξi(n)+ci2(n)ξi2(n)](pi(n)-qi(n))2+[ri2(n)bi2(n)+(1-ai(n))2-1](ui(n)-ωi(n))2+2[bi(n)ξi(n)ri2(n)-bi(n)ri(n)+ci(n)ξi(n)(1-ai(n))]×(ui(n)-ωi(n))(pi(n)-qi(n))(pj(n)-qj(n))2(1+epj(n))2(1+eqj(n))2}i=12{ri2(n)ξj2(n)(αi(n)-Ki(n))2(pj(n)-qj(n))2(1+epj(n))2(1+eqj(n))2+2|[(pi(n)-qi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))ri(n)ξj(n)(αi(n)-Ki(n))-ri2(n)ξi(n)ξj(n)(αi(n)-Ki(n))](pi(n)-qi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))|+2|bi(n)ξj(n)ri2(n)(αi(n)-Ki(n))(ui(n)-ωi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))|+|[ri2(n)ξi2(n)-2ri(n)ξi(n)+ci2(n)ξi2(n)](pi(n)-qi(n))2|+|[ri2(n)bi2(n)+(1-ai(n))2-1](ui(n)-ωi(n))2|+2|[bi(n)ξi(n)ri2(n)-bi(n)ri(n)+ci(n)ξi(n)(1-ai(n))]×(ui(n)-ωi(n))(pi(n)-qi(n))[ri2(n)bi2(n)+(1-ai(n))2-1]|(ui(n)-ωi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))}, we get ΔV(4.15)(n)i=12{V1ij+V2ij+V3ij+V4ij+V5ij+V6ij},j=1,2, where V1ij=ri2(n)ξj2(n)(αi(n)-Ki(n))2(pj(n)-qj(n))2(1+epj(n))2(1+eqj(n))2riu2(αiu-Kil)2xj*2(pj(n)-qj(n))2,V2ij=2|(pi(n)-qi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))[ri(n)ξj(n)(αi(n)-Ki(n))-ri2(n)ξi(n)ξj(n)(αi(n)-Ki(n))]×(pi(n)-qi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))|[(riuxj*+riu2xi*xj*)(αiu-Kil)][(pi(n)-qi(n))2+(pj(n)-qj(n))2],V3ij=2|bi(n)ξj(n)ri2(n)(αi(n)-Ki(n))(ui(n)-ωi(n))(pj(n)-qj(n))(1+epj(n))(1+eqj(n))|biuxj*riu2(αiu-Kil)[(ui(n)-ωi(n))2+(pj(n)-qj(n))2],V4ij=|[ri2(n)ξi2(n)-2ri(n)ξi(n)+ci2(n)ξi2(n)](pi(n)-qi(n))2|[riu2xi*2-2rilxi*+ciu2xi*2](pi(n)-qi(n))2,V5ij=|[ri2(n)bi2(n)+(1-ai(n))2-1](ui(n)-ωi(n))2|(riu2biu2+aiu2+2aiu)(ui(n)-ωi(n))2,V6ij=2|[bi(n)ξi(n)ri2(n)-bi(n)ri(n)+ci(n)ξi(n)(1-ai(n))]×(ui(n)-ωi(n))(pi(n)-qi(n))|[biuxi*riu2+riubiu+ciuxi*(1-ail),(ui(n)-ωi(n))2+(pi(n)-qi(n))2]. Hence, ΔV(4.15)(n)i=12{[(riuxj*+riu2xi*xj*)(αiu-Kil)+riu2xi*2-2rilxi*+ciu2xi*2+biuxi*riu2+riubiu+ciuxi*(1-ail)](pi(n)-qi(n))2+[riu2(αiu-Kil)2xj*2+(riuxj*+riu2xi*xj*)(αiu-Kil)+biuxj*riu2(αiu-Kil)](pj(n)-qj(n))2+[biuxj*riu2(αiu-Kil)+riu2biu2+aiu2+2aiu+biuxi*riu2+riubiu+ciuxi*(1-ail)](ui(n)-ωi(n))2}=-i=12{[2rilxi*-(riuxj*+riu2xi*xj*)(αiu-Kil)-riu2xi*2-ciu2xi*2-biuxi*riu2-riubiu+ciuxi*(1-ail)-rju2(αju-kjl)2xi*2-(rjuxi*+rju2xi*xj*)(αju-kjl)-bjuxi*rju2(αju-kjl)](pi(n)-qi(n))2+[-biuxj*riu2(αiu-Kil)-riu2biu2-aiu2-2aiu-biuxi*riu2-riubiu-ciuxi*(1-ail)](ui(n)-ωi(n))2}-i=12{rij(pi(n)-qi(n))2+rij*(ui(n)-ωi(n))2}-βi=12{(pi(n)-qi(n))2+(ui(n)-ωi(n))2}=-βV(n), where β={rij,rij*},i,j=1,2,ij. That is, there exists a positive constant 0<β<1 such that ΔV(4.15)(n)-βV(n). From 0<β<1, condition (iii) of Theorem 4.2 is satisfied. So, from Theorem 4.2, there exists a uniqueness uniformly asymptotically stable almost periodic solution X(n)=(p1(n),p2(n),u1(n),u2(n)) of (4.13) which is bounded by S* for all nZ+. Which means that there exists a uniqueness uniformly asymptotically stable almost periodic solution X(n)=(x1(n),x2(n),u1(n),u2(n)) of (1.7) which is bounded by Ω for all nZ+. This completed the proof.

5. An Example

In this section, we give an example to illustrate that our results are feasible.

In system (1.7), if we take i=1,2 and

r1(n)=1.1+0.4cos(n),r2(n)=0.8+0.2sin(n),K1(n)=6+cos(n),K2(n)=5+cos(n),α1(n)=1.2+sin(n),α2(n)=1.6+cos(n),b1(n)=0.5+0.1sin(n),b2(n)=0.3+0.1cos(n),a1(n)=0.9+0.2sin(n),a2(n)=0.6+0.3sin(n),  c1(n)=0.5+0.3sin(n),c2(n)=0.6+0.2cos(n). Then it is easy to see that {ri(n)},{Ki(n)},{αi(n)},{ai(n)},{bi(n)}, and {ci(n)} for i=1,2 are bounded nonnegative almost periodic sequences. By calculation, we get

x1*=4.4062,  u1*=5.0357,K1l-b1uu1*=1.9775>0,x1*=0.3612,u1*=0.0903,x2*=2.6000,u2*=6.9333,K2l-b2uu2*=1.2267>0,x2*=0.4997,u2*=0.2498,r12=0.6313,r21=0.3583,r12*=0.6380,r21*=0.1025,0<β=0.1025<1. Then we can see that all conditions of Theorem 4.4 hold. According to Theorem 4.4, system (1.7) has a unique uniformly asymptotically stable almost periodic solution which is bounded by Ω for all nZ+.

6. Conclusions

In this paper, we consider a discrete mutualism model with feedback controls. Assuming that the coefficients in the system are almost periodic sequences, first, we establish a persistence result for the model under consideration, then based on the persistence result we obtain the existence and uniqueness of the almost periodic solution of the system which is uniformly asymptotically stable. Finally, an example is given to illustrate the feasibility of our results.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 10971183.

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