# Permanence and Almost Periodic Solutions of a Discrete Ratio-Dependent Leslie System with Time Delays and Feedback Controls

Authors: Gang Yu; Hongying Lu
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider a discrete almost periodic ratio-dependent Leslie
system with time delays and feedback controls. Sufficient conditions are obtained
for the permanence and global attractivity of the system. Furthermore, by using an
almost periodic functional Hull theory, we show that the almost periodic system
has a unique globally attractive positive almost periodic solution.


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## Body

## 1. Introduction

Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The predator-prey models have been extensively studied by many scholars [1–4]. Under the assumption that reduction in a predator population has a reciprocal relationship with per capita availability of its preferred food, Leslie [5] proposed the famous Leslie-Gower predator-prey model where the carry capacity of the predator’s environment is proportional to the number of prey:
(1.1)x˙1=x1[b-ax1]-cx1x2,x˙2=x2[g-fx2x1],
where x1 and x2 represent prey and predator densities at time t, respectively. cx1 is the predator’s rate of feeding upon prey, that is, the so-called predator’s functional response. If we assume that the predator consumes the prey according to the functional response p(x1), then system (1.1) formulates as the following:
(1.2)x˙1=x1[b-ax1]-p(x1)x2,x˙2=x2[g-fx2x1],
where p(x1) is prey-dependent functional responses. Owing to its theoretical and practical significance, system (1.2) and its various generalized forms have been studied extensively and seen great progress (see, for example, [6–9]).However, in the study of the dynamic behaviors of predator-prey system, many scholars argued that the ratio-dependent predator-prey systems are more realistic [10, 11]. A ratio-dependent predator-prey system with Leslie-Gower term takes the form of
(1.3)x˙1=x1[b-ax1]-p(x1x2)x2,x˙2=x2[g-fx2x1].Though much progress has been seen in the asymptotic convergence of solutions of population systems, such systems are not well studied in the sense that most results are continuous-time cases related. Already, many scholars have paid attention to the nonautonomous discrete population models, since the discrete time models governed by difference equation are more appropriate than the continuous ones when the populations have a short life expectancy, nonoverlapping generations in the real world (see [12–23]). Furthermore, the asymptotic convergence of solutions of difference equations with delay is one of the most important topics in the study of population dynamics, and many excellent results have already been obtained and seen great progress (see [24–26] and the references cited therein).Since time delays occur so often in nature, a number of ecological systems can be described as systems with time delays (see [14, 15, 19, 21–27]). One of the most important problems for this type of system is to analyze the effect of time delays on the stability of the system. Furthermore, as we know, ecological systems in the real world are often distributed by unpredictable forces which can result in changes in biological parameters such as survival rates, so it is reasonable to study models with control variables which are so-called disturbance functions [22, 23].So it is very interesting to study dynamics of the following discrete ratio-dependent Leslie system with time delays and feedback controls:(1.4)x1(k+1)=x1(k)exp[b(k)-a(k)x1(k-τ1)-c(k)x1(k)x2(k)h2x22(k)+x12(k)-d(k)u1(k-σ1)],x2(k+1)=x2(k)exp[g(k)-f(k)x2(k-τ2)x1(k-τ2)-p(k)u2(k-σ2)],Δui(k)=-αi(k)ui(k)+βi(k)xi(k-ρi),i=1,2,
where xi(k), i=1,2 stand for the density of the prey and the predator at time k, respectively. ui(k), i=1,2 are the control variables at time k. h2 is a positive constant, denoting the constant of capturing half-saturation.In this paper, we are concerned with the effects of the almost periodicity of ecological and environmental parameters and time delays on the global dynamics of the discrete ratio-dependent Leslie systems with feedback controls. To do so, for system (1.4) we always assume that for i=1,2, k∈Z(H1)a(k), b(k), c(k), d(k), g(k), f(k), p(k), αi(k), βi(k) are all bounded nonnegative almost periodic sequences such that
(1.5)0<al≤au,0<bl≤bu,0<cl≤cu,0<dl≤du,0<gl≤gu,0<fl≤fu,0<pl≤pu,0<αil≤αiu<1,0<βil≤βiu.Here, we let Z, Z+ denote the sets of all integers, nonnegative integers, respectively, and use the notations: fu=supk∈Z{f(k)}, fl=infk∈Z{f(k)}, for any bounded sequence {f(k)} defined on Z.Letτ=max{τi,σi,ρi,i=1,2}, we consider system (1.4) with the following initial conditions:
(1.6)xi(θ)=ϕi(θ),θ∈[-τ,0]∩Z,ϕi(0)>0,ui(θ)=φi(θ),θ∈[-τ,0]∩Z,φi(0)>0,i=1,2.One can easily show that the solutions of system (1.4) with initial condition (1.5) are defined and remain positive for k∈Z+.The principle aim of this paper is to study the dynamic behaviors of system (1.4), such as permanence, global attractivity, and existence of a unique globally attractive positive almost periodic solution of the system. To the best of our knowledge, no work has been done for the nonautonomous difference system (1.4).The organization of this paper is as follows. In the next section, we introduce some definitions and several useful lemmas. In Section3, we explore the permanent property of system (1.4). We study globally attractive property of system (1.4) in Section 4 and the almost periodic property of system (1.4) in Section 5. Finally, the conclusion ends with brief remarks.


## 2. Preliminaries

In this section, we will introduce some basic definitions and several useful lemmas.Definition 2.1.
System (1.4) is said to be permanent, if there are positive constants mi and Mi, such that for each positive solution (x1(k),x2(k),u1(k),u2(k))T of system (1.4) satisfies
(2.1)mi≤liminfk→+∞xi(k)≤limsupk→+∞xi(k)≤Mi,ni≤liminfk→+∞ui(k)≤limsupk→+∞ui(k)≤Ni,i=1,2.Definition 2.2.
Suppose thatX(k)=(x1(k),x2(k),u1(k),u2(k))T is any solution of system (1.4). X(k) is said to be a strictly positive solution in Z if for k∈Z and i=1,2 such that
(2.2)0<infk∈Zxi(k)≤supk∈Zxi(k)<∞,0<infk∈Zui(k)≤supk∈Zui(k)<∞.Definition 2.3 (see [16]).
A sequencex:Z→R is called an almost periodic sequence if the ε-translation set of x(2.3)E{ε,x}={τ∈Z:|x(k+τ)-x(k)|<ε,∀k∈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} with
(2.4)|x(k+τ)-x(k)|<ε,∀k∈Z,τ is called an ε-translation number of x(k).Definition 2.4 (see [17]).
The hull off, denoted by H(f), is defined by
(2.5)H(f)={g(k,x):limn→∞f(k+τn,x)=g(k,x)uniformly  onZ×S},
for some sequence {τn}, where S is any compact set in D.Lemma 2.5 (see [20]).
Assume that{y(k)} satisfies y(k1)>0 and
(2.6)y(k+1)≤y(k)exp{r(k)(1-ay(k))},
for k∈[k1,+∞), where a is a positive constant and k1∈Z+. Then
(2.7)limsupk→+∞y(k)≤1aruexp(ru-1).Lemma 2.6 (see [20]).
Assume that{y(k)} satisfies y(k2)>0 and
(2.8)y(k+1)≥y(k)exp{r(k)(1-ay(k))},
for k∈[k2,+∞), limsupk→+∞y(k)≤M, where a is a constant such that aM>1 and k2∈Z+. Then
(2.9)liminfk→+∞y(k)≥1aexp(ru(1-aM)).Lemma 2.7 (see [22]).
Assume thatA>0 and y(0)>0. Suppose that
(2.10)y(k+1)≤Ay(k)+B(k),n=1,2,….
Then for any integer m≤k,
(2.11)y(k)≤Amy(k-m)+∑j=0m-1‍AjB(k-j-1).
Especially, if A<1 and B is bounded above with respect to M, then
(2.12)limsupk→+∞y(k)≤M1-A.Lemma 2.8 (see [22]).
Assume thatA>0 and y(0)>0. Suppose that
(2.13)y(k+1)≥Ay(k)+B(k),n=1,2,….
Then for any integer m≤k,
(2.14)y(k)≥Amy(k-m)+∑j=0m-1‍AjB(k-j-1).
Especially, if A<1 and B is bounded below with respect to N, then
(2.15)liminfk→+∞y(k)≥N1-A.


## 3. Permanence

In this section, we establish a permanent result for system (1.4).Theorem 3.1.
Assume that(H1) holds; assume further that(H2)
bl>cu/2|h|holds. Then system (1.4) is permanent.Proof.
LetX(k)=(x1(k),x2(k),u1(k),u2(k))T be any positive solution of system (1.4), from the first equation of system (1.4), it follows that
(3.1)x1(k+1)≤x1(k)exp[b(k)]≤x1(k)exp[bu].
It follows from (3.1) that
(3.2)∏j=k-τ1k-1x1(j+1)x1(j)≤∏j=k-τ1k-1exp[bu]≤exp[buτ1],
which implies that
(3.3)x1(k-τ1)≥x1(k)exp[-buτ1].
Substituting (3.3) into the first equation of (1.4), it immediately follows that
(3.4)x1(k+1)≤x1(k)exp[b(k)-a(k)x1(k-τ1)]≤x1(k)exp[b(k)-a(k)exp[-buτ1]x1(k)].
By applying Lemma 2.5 to (3.4), we have
(3.5)limsupk→+∞x1(k)≤1alexp[bu(τ1+1)-1]=:M1.
For any ε>0 small enough, it follows from (3.5) that there exists enough large K1 such that for k≥K1,
(3.6)x1(k)≤M1+ε.
From the second equation of system (1.4) it follows that
(3.7)x2(k+1)≤x2(k)exp[g(k)]≤x2(k)exp[gu],x2(k+1)≤x2(k)exp[g(k)-f(k)M1+εx2(k-τ2)],fork≥K1+τ.
From (3.7), similar to the argument of (3.1), one has
(3.8)x2(k+1)≤x2(k)exp[g(k)-f(k)M1+εexp[-guτ2]x2(k)].
By applying Lemma 2.5 to (3.8) again, we have
(3.9)limsupk→+∞x2(k)≤M1+εflexp[gu(τ2+1)-1].
Setting ε→0 in the above inequality, we have
(3.10)limsupk→+∞x2(k)≤M1flexp[gu(τ2+1)-1]=:M2.
For anyε>0 small enough, it follows from (3.5) and (3.10) that there exists enough large K2>K1+τ such that for i=1,2 and k≥K2(3.11)xi(k)≤Mi+ε.
For k>K2+τ, (3.11) combining with the third and fourth equations of system (1.4) leads to
(3.12)Δui(k)≤-αi(k)ui(k)+βi(k)(Mi+ε),i=1,2,
that is,
(3.13)ui(k+1)≤(1-αil)ui(k)+βiu(Mi+ε),i=1,2.
By applying Lemma 2.7, it follows from (3.13) that
(3.14)limsupk→+∞ui(k)≤βiu(Mi+ε)αil,i=1,2.
Letting ε→0 in the above inequality, we have
(3.15)limsupk→+∞ui(k)≤βiuMiαil=:Ni,i=1,2.
For anyε>0 small enough, it follows from (3.11) and (3.15) that there exists enough large K3>K2+τ such that for i=1,2 and k≥K3(3.16)xi(k)≤Mi+ε,ui(k)≤Ni+ε.
Thus, from (3.16) and the first equation of system (1.4), it follows that
(3.17)x1(k+1)≥x1(k)exp[bl-au(M1+ε)-cu2|h|-du(N1+ε)]=:x1(k)exp[D1ε],fork≥K3+τ,
where
(3.18)D1ε=bl-au(M1+ε)-cu2|h|-du(N1+ε)≤bl-auM1≤bl-auexp[bu-1]alexp[-buτ1]≤bl-bu≤0.
For any integer η≤k, it follows from (3.17) that
(3.19)∏j=k-ηk-1x1(j+1)x1(j)≥∏j=k-ηk-1exp[D1ε]≥exp[D1εη],
and hence
(3.20)x1(η)≤x1(k)exp[-(k-η)D1ε].
From the third equation of system (1.4), we have
(3.21)u1(k+1)≤(1-α1l)u1(k)+β1ux1(k-ρ1)≤(1-α1l)u1(k)+β1uexp[-ρ1D1ε]x1(k)=A1u1(k)+B1(k),
where A1=1-α1l and B1(k)=β1uexp[-ρ1D1ε]x1(k). Therefore, for any m≤k-τ, by Lemma 2.7, (3.20), and (3.21), one has
(3.22)u1(k)≤A1mu1(k-m)+∑j=0m-1‍A1jB1(k-j-1)≤A1mu1(k-m)+∑j=0m-1‍A1jβ1uexp[-ρ1D1ε]x1(k-j-1)≤A1mu1(k-m)+x1(k)∑j=0m-1‍A1jβ1uexp[-(j+1+ρ1)D1ε].
Since α1(k)∈(0,1), we have 0<A1<1. So
(3.23)0≤A1mu1(k-m)≤A1m(N1+ε)→0,form→∞.
By the conditions(H1), for any solution (x1(k),x2(k),u1(k),u2(k))T of system (1.4), there exists a positive integer M such that duA1mu1(k-m)<(1/2)(bl-cu/2|h|) for m>M. In fact, we can choose M=max{1,[logA1((bl-cu/2|h|)/2duN1)]+1}. Then we get
(3.24)u1(k)≤A1M(N1+ε)+x1(k)∑j=0M-1‍A1jβ1uexp[-(j+1+ρ1)D1ε]=A1M(N1+ε)+F1εx1(k),
where F1ε=∑j=0M-1‍A1jβ1uexp[-(j+1+ρ1)D1ε].
Substituting (3.20) and (3.24) into the first equation of system (1.4), one has
(3.25)x1(k+1)≥x1(k)exp[b(k)-a(k)x1(k-τ1)-c(k)2|h|-d(k)u1(k-σ1)]≥x1(k)exp[b(k)-c(k)2|h|-d(k)A1M(N1+ε)-(a(k)exp[-τ1D1ε]+d(k)F1εexp[-σ1D1ε])x1(k)c(k)2|h|]=:x1(k)exp[S1ε(k)(1-(S2ε(k)S1ε(k))x1(k))],
where
(3.26)S1ε(k)=b(k)-c(k)2|h|-d(k)A1M(N1+ε),S2ε(k)=a(k)exp[-τ1D1ε]+d(k)F1εexp[-σ1D1ε].
In particular, we have
(3.27)S2εuS1εlM1=auexp[-τ1D1ε]+duF1εexp[-σ1D1ε]bl-cu/2|h|-duA1M(N1+ε)M1≥auexp[-τ1D1ε]blexp[bu(τ1+1)-1]al≥auexp[-τ1bl]blexp[bl(τ1+1)-1]au≥exp[bl-1]bl>1,
where we use the inequality exp(x-1)>x for x>0. By applying Lemma 2.6 to (3.25), it follows that
(3.28)liminfk→+∞x1(k)≥S1εlS2εuexp[S1εu(1-S2εuS1εlM1)].
Setting ε→0 in the above inequality, we have
(3.29)liminfk→+∞x1(k)≥S1lS2uexp[S1u(1-S2uS1lM1)]=:m1.
From (3.29) we know that there exists enough large K4>K3+τ such that for k≥K4,
(3.30)x1(k)≥m1-ε.
From (3.30) and the second equation of system (1.4), it follows that
(3.31)x2(k+1)≥x2(k)exp[gl-fu(M2+ε)m1-ε-pu(N2+ε)]=:x2(k)exp[D2ε],fork≥K4+τ,
where D2ε=gl-fu(M2+ε)/(m1-ε)-pu(N2+ε)≤0.
This implies for any integerη≤k(3.32)x2(η)≤x2(k)exp[-(k-η)D2ε].
From (3.32) and the fourth equation of system (1.4), by a procedure similar to the discussion of (3.21)–(3.24), we can verify that
(3.33)u2(k)≤A2N(N2+ε)+F2εx2(k),
where
(3.34)A2=1-α2l,F2ε=∑j=0N-1‍A2jβ2uexp[-(j+1+ρ2)D2ε],N=max{1,[logA2(gl2puN2)]+1}.
Substituting (3.32) and (3.33) into the second equation of system (1.4), one has
(3.35)x2(k+1)≥x2(k)exp[g(k)-f(k)m1-εx2(k-τ2)-p(k)u2(k-σ2)]≥x2(k)exp[f(k)m1g(k)-p(k)A2N(N2+ε)-(f(k)m1exp[-τ2D2ε]+p(k)F2εexp[-σ2D2ε])x2(k)]=:x2(k)exp[S3ε(k)(1-(S4ε(k)S3ε(k))x2(k))],
where
(3.36)S3ε(k)=g(k)-p(k)A2N(N2+ε),S4ε(k)=f(k)m1exp[-τ2D2ε]+p(k)F2εexp[-σ2D2ε].
In particular, we have
(3.37)S4εuS3εlM2=(fu/m1)exp[-τ2D2ε]+puF2exp[-σ2D2ε]gl-puA2N(N2+ε)M2≥(fu/m1)exp[-τ2D2ε]glM1exp[gu(τ2+1)-1]fl≥(fu/m1)exp[-τ2gl]glM1exp[gl(τ2+1)-1]fu≥exp[gl-1]gl>1.
By applying Lemma 2.6 to (3.35) again, it follows that
(3.38)liminfk→+∞x2(k)≥S3εlS4εuexp[S3εu(1-S4εuS3εlM2)].
Setting ε→0 in the above inequality, we have
(3.39)liminfk→+∞x2(k)≥S3lS4uexp[S3u(1-S4uS3lM2)]=:m2.
From (3.29) and (3.39) we know that there exists enough large K5>K4+τ such that for k≥K5,
(3.40)xi(k)≥mi-ε.
For k>K5+τ, (3.40) combining with the third and fourth equations of system (1.4) produces
(3.41)ui(k+1)≥(1-αiu)ui(k)+βil(mi-ε),i=1,2.
By applying Lemma 2.8, it follows from (3.41) that
(3.42)liminfk→+∞ui(k)≥βil(mi-ε)αiu,i=1,2.
Setting ε→0 in the above inequality, we have
(3.43)liminfk→+∞ui(k)≥βilmiαiu=:ni,i=1,2.
Consequently, combining (3.5), (3.10), (3.15), (3.29), (3.39) with (3.43), system (1.4) is permanent. This completes the proof of Theorem 3.1.


## 4. Global Attractivity

Firstly, we prove two lemmas which will be useful to our main result.Lemma 4.1.
For any two positive solutions(x1(k),x2(k),u1(k),u2(k))T and (y1(k),y2(k),v1(k),v2(k))T of system (1.4), one has
(4.1)lnx1(k+1)y1(k+1)=lnx1(k)y1(k)-a(k)[x1(k)-y1(k)]-d(k)[u1(k-σ1)-v1(k-σ1)]+E1(k)[x1(k)-y1(k)]+E2(k)[x2(k)-y2(k)]+a(k)∑s=k-τ1k-1‍{P1(s)[c(s)y1(s)y2(s)h2y22(s)+y12(s)b(s)-a(s)y1(s-τ1)-c(s)y1(s)y2(s)h2y22(s)+y12(s)-d(s)v1(s-σ1)][x1(s)-y1(s)]+x1(s)P2(s)[-a(s)[x1(s-τ1)-y1(s-τ1)]-d(s)[u1(s-σ1)-v1(s-σ1)]+E1(s)[x1(s)-y1(s)]+E2(s)[x2(s)-y2(s)]]c(s)y1(s)y2(s)h2y22(s)+y12(s)},
where
(4.2)E1(s)=c(s)y1(s)y2(s)[x1(s)+y1(s)][h2x22(s)+x12(s)][h2y22(s)+y12(s)]-c(s)x2(s)h2x22(s)+x12(s),E2(s)=h2c(s)y1(s)y2(s)[x2(s)+y2(s)][h2x22(s)+x12(s)][h2y22(s)+y12(s)]-c(s)y1(s)h2x22(s)+x12(s),P1(s)=exp{θ1(s)[b(s)-a(s)y1(s-τ1)-c(s)y1(s)y2(s)h2y22(s)+y12(s)-d(s)v1(s-σ1)]},P2(s)=exp{θ2(s)[b(s)-a(s)x1(s-τ1)-c(s)x1(s)x2(s)h2x22(s)+x12(s)-d(s)u1(s-σ1)]+(1-θ2(s))[b(s)-a(s)y1(s-τ1)-c(s)y1(s)y2(s)h2y22(s)+y12(s)-d(s)v1(s-σ1)]},θ1(s),θ2(s)∈(0,1).Proof.
It follows from the first equation of system (1.4) that
(4.3)lnx1(k+1)y1(k+1)-lnx1(k)y1(k)=lnx1(k+1)x1(k)-lny1(k+1)y1(k)=[b(k)-a(k)x1(k-τ1)-c(k)x1(k)x2(k)h2x22(k)+x12(k)-d(k)u1(k-σ1)]-[b(k)-a(k)y1(k-τ1)-c(k)y1(k)y2(k)h2y22(k)+y12(k)-d(k)v1(k-σ1)]=-a(k)[x1(k-τ1)-y1(k-τ1)]-d(k)[u1(k-σ1)-v1(k-σ1)]-c(k)x1(k)x2(k)h2x22(k)+x12(k)+c(k)y1(k)y2(k)h2y22(k)+y12(k)=-a(k)[x1(k)-y1(k)]-d(k)[u1(k-σ1)-v1(k-σ1)]+E1(k)[x1(k)-y1(k)]+E2(k)[x2(k)-y2(k)]+a(k){[x1(k)-y1(k)]-[x1(k-τ1)-y1(k-τ1)]},
where
(4.4)E1(k)=c(k)y1(k)y2(k)[x1(k)+y1(k)][h2x22(k)+x12(k)][h2y22(k)+y12(k)]-c(k)x2(k)h2x22(k)+x12(k),E2(k)=h2c(k)y1(k)y2(k)[x2(k)+y2(k)][h2x22(k)+x12(k)][h2y22(k)+y12(k)]-c(k)y1(k)h2x22(k)+x12(k).
Hence
(4.5)lnx1(k+1)y1(k+1)=lnx1(k)y1(k)-a(k)[x1(k)-y1(k)]-d(k)[u1(k-σ1)-v1(k-σ1)]+E1(k)[x1(k)-y1(k)]+E2(k)[x2(k)-y2(k)]+a(k){[x1(k)-x1(k-τ1)]-[y1(k)-y1(k-τ1)]}.
Since
(4.6)[x1(k)-x1(k-τ1)]-[y1(k)-y1(k-τ1)]=∑s=k-τ1k-1‍[x1(s+1)-x1(s)]-∑s=k-τ1k-1‍[y1(s+1)-y1(s)]=∑s=k-τ1k-1‍{[x1(s+1)-y1(s+1)]-[x1(s)-y1(s)]},(4.7)[x1(s+1)-y1(s+1)]-[x1(s)-y1(s)]={x1(s)exp[b(s)-a(s)x1(s-τ1)-c(s)x1(s)x2(s)h2x22(s)+x12(s)-d(s)u1(s-σ1)]-y1(s)exp[b(s)-a(s)y1(s-τ1)-c(s)y1(s)y2(s)h2y22(s)+y12(s)-d(s)v1(s-σ1)]}-[x1(s)-y1(s)]=[x1(s)-y1(s)]{exp[b(s)-a(s)y1(s-τ1)-c(s)y1(s)y2(s)h2y22(s)+y12(s)-d(s)v1(s-σ1)]-1}+x1(s){exp[b(s)-a(s)x1(s-τ1)-c(s)x1(s)x2(s)h2x22(s)+x12(s)-d(s)u1(s-σ1)]-exp[b(s)-a(s)y1(s-τ1)-c(s)y1(s)y2(s)h2y22(s)+y12(s)-d(s)v1(s-σ1)]}.
By the mean value theorem, one has
(4.8)[x1(s+1)-y1(s+1)]-[x1(s)-y1(s)]=[x1(s)-y1(s)]P1(s)[b(s)-a(s)y1(s-τ1)-c(s)y1(s)y2(s)h2y22(s)+y12(s)-d(s)v1(s-σ1)]+x1(s)P2(s)[c(s)y1(s)y2(s)h2y22(s)+y12(s)-a(s)[x1(s-τ1)-y1(s-τ1)]-d(s)[u1(s-σ1)-v1(s-σ1)]-c(s)x1(s)x2(s)h2x22(s)+x12(s)+c(s)y1(s)y2(s)h2y22(s)+y12(s)]=[x1(s)-y1(s)]P1(s)[b(s)-a(s)y1(s-τ1)-c(s)y1(s)y2(s)h2y22(s)+y12(s)-d(s)v1(s-σ1)]+x1(s)P2(s)[-a(s)[x1(s-τ1)-y1(s-τ1)]-d(s)[u1(s-σ1)-v1(s-σ1)]+E1(s)[x1(s)-y1(s)]+E2(s)[x2(s)-y2(s)]].
Combining (4.6) with (4.8), we can easily obtain (4.1). The proof of Lemma 4.1 is completed.Lemma 4.2.
For any two positive solutions(x1(k),x2(k),u1(k),u2(k))T and (y1(k),y2(k),v1(k),v2(k))T of system (1.4), one has
(4.9)lnx2(k+1)y2(k+1)=lnx2(k)y2(k)-f(k)x1(k-τ2)[x2(k)-y2(k)]+f(k)y2(k-τ2)x1(k-τ2)y1(k-τ2)[x1(k-τ2)-y1(k-τ2)]-p(k)[u2(k-σ2)-v2(k-σ2)]+f(k)x1(k-τ2)∑s=k-τ2k-1‍{Q1(s)[g(s)-f(s)y2(s-τ2)y1(s-τ2)-p(s)v2(s-σ2)][x2(s)-y2(s)]+x2(s)Q2(s)[-f(s)x1(s-τ2)[x2(s-τ2)-y2(s-τ2)]+f(s)y2(s-τ2)x1(s-τ2)y1(s-τ2)[x1(s-τ2)-y1(s-τ2)]-p(k)[u2(s-σ2)-v2(s-σ2)]y2(s-τ2)y1(s-τ2)]},
where
(4.10)Q1(s)=exp{φ1(s)[g(s)-f(s)y2(s-τ2)y1(s-τ2)-p(s)v2(s-σ2)]},Q2(s)=exp{φ2(s)[g(s)-f(s)x2(s-τ2)x1(s-τ2)-p(s)u2(s-σ2)]+(1-φ2(s))[g(s)-f(s)y2(s-τ2)y1(s-τ2)-p(s)v2(s-σ2)]},φ1(s),φ2(s)∈(0,1).Proof.
The proofs of Lemma4.2 are very similar to those of Lemma 4.1. So we omit the detail here.Now we are in the position of stating the main result on the global attractivity of system (1.4).Theorem 4.3.
In addition to(H1)-(H2), assume further that (H3) there exist positive constants si, ωi, i=1,2 such that
(4.11)α=:min(s1μ1-s2μ2-ω1β1u,s2ν1-s1ν2-ω2β2u,ω1α1l-s1ϱ1,ω2α2l-s2ϱ2)>0holds, where μ1, μ2, ν1, ν2, ϱ1, ϱ2 are defined by (4.37). Then for any two positive solutions (x1(k),x2(k),u1(k),u2(k))T and (y1(k),y2(k),v1(k),v2(k))T of system (1.4), one has
(4.12)limk→+∞|xi(k)-yi(k)|=0,limk→+∞|ui(k)-vi(k)|=0,i=1,2.Proof.
For two arbitrary nontrivial solutions(x1(k),x2(k),u1(k),u2(k))T and (y1(k),y2(k),v1(k),v2(k))T of system (1.4), we have from Theorem 3.1 that there exist positive constants k0 and Mi, Ni, mi, ni(i=1,2) such that for all k≥k0 and i=1,2(4.13)mi≤xi(k)≤Mi,ni≤ui(k)≤Ni,mi≤yi(k)≤Mi,ni≤vi(k)≤Ni.
Firstly, we define
(4.14)V11(k)=|lnx1(k)-lny1(k)|.
From (4.1), we have
(4.15)|lnx1(k+1)y1(k+1)|≤|lnx1(k)y1(k)-a(k)[x1(k)-y1(k)]|+d(k)|u1(k-σ1)-v1(k-σ1)|+E1(k)|x1(k)-y1(k)|+E2(k)|x2(k)-y2(k)|+a(k)∑s=k-τ1k-1‍{[P1(s)G1(s)+x1(s)P2(s)E1(s)]|x1(s)-y1(s)|+x1(s)P2(s)[a(s)|x1(s-τ1)-y1(s-τ1)|+d(s)|u1(s-σ1)-v1(s-σ1)|+E2(s)|x2(s)-y2(s)|]},
where
(4.16)G1(s)=b(s)+a(s)y1(s-τ1)+c(s)y1(s)y2(s)h2y22(s)+y12(s)+d(s)v1(s-σ1).
By the mean value theorem, we have
(4.17)x1(k)-y1(k)=exp[lnx1(k)]-exp[lny1(k)]=ξ1(k)lnx1(k)y1(k),
that is,
(4.18)lnx1(k)y1(k)=1ξ1(k)[x1(k)-y1(k)],
where ξ1(k) lies between x1(k) and y1(k). So, we have
(4.19)|lnx1(k)y1(k)-a(k)[x1(k)-y1(k)]|=|lnx1(k)y1(k)|-|lnx1(k)y1(k)|+|lnx1(k)y1(k)-a(k)[x1(k)-y1(k)]|=|lnx1(k)y1(k)|-[1ξ1(k)-|1ξ1(k)-a(k)|]|x1(k)-y1(k)|.
For k>k0+τ, it follows that
(4.20)ΔV11≤-[1ξ1(k)-|1ξ1(k)-a(k)|]|x1(k)-y1(k)|+d(k)|u1(k-σ1)-v1(k-σ1)|+E1(k)|x1(k)-y1(k)|+E2(k)|x2(k)-y2(k)|+a(k)∑s=k-τ1k-1‍{[P1(s)G1(s)+M1P2(s)E1(s)]|x1(s)-y1(s)|+M1P2(s)[a(s)|x1(s-τ1)-y1(s-τ1)|+d(s)|u1(s-σ1)-v1(s-σ1)|+E2(s)|x2(s)-y2(s)|]}.
Secondly, we define
(4.21)V12(k)=∑s=k-σ1k-1‍d(s+σ1)|u1(s)-v1(s)|+∑s=kk-1+τ1‍a(s)∑u=s-τ1k-1‍{[P1(u)G1(u)+M1P2(u)E1(u)]|x1(u)-y1(u)|+M1P2(u)[a(u)|x1(u-τ1)-y1(u-τ1)|+d(u)|u1(u-σ1)-v1(u-σ1)|+E2(u)|x2(u)-y2(u)|]}.
Then
(4.22)ΔV12=d(k+σ1)|u1(k)-v1(k)|-d(k)|u1(k-σ1)-v1(k-σ1)|+∑s=k+1k+τ1‍a(s){[P1(k)G1(k)+M1P2(k)E1(k)]|x1(k)-y1(k)|+M1P2(k)[a(k)|x1(k-τ1)-y1(k-τ1)|+d(k)|u1(k-σ1)-v1(k-σ1)|+E2(k)|x2(k)-y2(k)|]}-a(k)∑u=s-τ1k-1‍{[P1(u)G1(u)+M1P2(u)E1(u)]|x1(u)-y1(u)|+M1P2(u)[a(u)|x1(u-τ1)-y1(u-τ1)|+d(u)|u1(u-σ1)-v1(u-σ1)|+E2(u)|x2(u)-y2(u)|]}.
Thirdly, we define
(4.23)V13(k)=M1∑l=k-τ1k-1‍P2(l+τ1)a(l+τ1)|x1(l)-y1(l)|∑s=l+τ1+1l+2τ1‍a(s)+M1∑l=k-σ1k-1‍P2(l+σ1)d(l+σ1)|u1(l)-v1(l)|∑s=l+σ1+1l+σ1+τ1‍a(s).
By a simple calculation, it follows that
(4.24)ΔV13=∑s=k+τ1+1k+2τ1‍a(s)M1P2(k+τ1)a(k+τ1)|x1(k)-y1(k)|-∑s=k+1k+τ1‍a(s)M1P2(k)a(k)|x1(k-τ1)-y1(k-τ1)|+∑s=k+σ1+1k+σ1+τ1‍a(s)M1P2(k+σ1)d(k+σ1)|u1(k)-v1(k)|-∑s=k+1k+τ1‍a(s)M1P2(k)d(k)|u1(k-σ1)-v1(k-σ1)|.
We now define
(4.25)V1(k)=V11(k)+V12(k)+V13(k).
Then for all k>k0+τ, it follows from (4.14)–(4.24) that
(4.26)ΔV1≤-[1ξ1(k)-|1ξ1(k)-a(k)|]|x1(k)-y1(k)|+d(k+σ1)|u1(k)-v1(k)|+E1(k)|x1(k)-y1(k)|+E2(k)|x2(k)-y2(k)|+∑s=k+1k+τ1‍a(s){[P1(k)G1(k)+M1P2(k)E1(k)]|x1(k)-y1(k)|+M1P2(k)E2(k)|x2(k)-y2(k)|}+∑s=k+τ1+1k+2τ1‍a(s)M1P2(k+τ1)a(k+τ1)|x1(k)-y1(k)|+∑s=k+σ1+1k+σ1+τ1‍a(s)M1P2(k+σ1)d(k+σ1)|u1(k)-v1(k)|.
Similarly, we define
(4.27)V2(k)=V21(k)+V22(k)+V23(k),
where
(4.28)V21(k)=|lnx2(k)-lny2(k)|,(4.29)V22(k)=∑s=k-τ2k-1‍M2m12f(s+τ2)|x1(s)-y1(s)|+∑s=k-σ2k-1‍p(s+σ2)|u2(s)-v2(s)|+∑s=kk-1+τ2‍f(s)m1∑u=s-τ2k-1‍{f(u)m1Q1(u)G2(u)|x2(u)-y2(u)|+M2Q2(u)[f(u)m1|x2(u-τ2)-y2(u-τ2)|+M2m12f(u)|x1(u-τ2)-y1(u-τ2)|+p(u)|u2(u-σ2)-v2(u-σ2)|f(u)m1]},(4.30)V23(k)=M2m1∑l=k-τ2k-1‍Q2(l+τ2)f(l+τ2)|x2(l)-y2(l)|∑s=l+τ2+1l+2τ2‍f(s)m1+M22m12∑l=k-τ2k-1‍Q2(l+τ2)f(l+τ2)|x1(l)-y1(l)|∑s=l+τ2+1l+2τ2‍f(s)m1+M2∑l=k-σ2k-1‍Q2(l+σ2)p(l+σ2)|u2(l)-v2(l)|∑s=l+σ2+1l+σ2+τ2‍f(s)m1,(4.31)G2(s)=g(s)+f(s)y2(s-τ2)y1(s-τ2)+p(s)v2(s-σ2).
Thus for all k>k0+τ, it follows from (4.27)–(4.30) that
(4.32)ΔV2≤-[1ξ2(k)-|1ξ2(k)-f(k)x1(k-τ2)|]|x2(k)-y2(k)|+M2m12f(k+τ2)|x1(k)-y1(k)|+p(k+σ2)|u2(k)-v2(k)|+∑s=k+1k+τ2‍f(s)m1Q1(k)G2(k)|x2(k)-y2(k)|+∑s=k+τ2+1k+2τ2‍f(s)m1M2m1Q2(k+τ2)f(k+τ2)|x2(k)-y2(k)|+∑s=k+τ2+1k+2τ2‍f(s)m1M22m12Q2(k+τ2)f(k+τ2)|x1(k)-y1(k)|+∑s=k+σ2+1k+σ2+τ2‍f(s)m1M2Q2(k+σ2)p(k+σ2)|u2(k)-v2(k)|.
For i=1,2, we define
(4.33)Vi′(k)=|ui(k)-vi(k)|+∑s=k-ρik-1‍βi(s+ρi)|xi(s)-yi(s)|.
By calculation, it derives that
(4.34)ΔVi′≤-αi(k)|ui(k)-vi(k)|+βi(k+ρi)|xi(k)-yi(k)|.
We now define a Lyapunov function as
(4.35)V(k)=s1V1(k)+s2V2(k)+ω1V1′(k)+ω2V2′(k).
It is easy to see that V(k)>0 and V(k0+τ)<+∞. Calculating the difference of V along the solution of system (1.4), we have that for k≥k0+τ,
(4.36)ΔV≤{+s2M2m12fu[1+τ2M2m1fuQ2u]+ω1β1us1[-min(al,2M1-au)+E1u+τ1au[P1uG1u+M1P2uE1u+M1P2uau]]+s2M2m12fu[1+τ2M2m1fuQ2u]+ω1β1u}|x1(k)-y1(k)|+{s2[-min(flM1,2M2-fum1)+τ2fum1[Q1uG2u+M2m1Q2ufu]]+s1E2u[1+τ1auM1P2u]+ω2β2u(flM1,2M2-fum1)}|x2(k)-y2(k)|+{-ω1α1l+s1du[1+τ1auM1P2u]}|u1(k)-v1(k)|+{-ω2α2l+s2pu[1+τ2fum1M2Q2u]}|u2(k)-v2(k)|≤-[s1μ1-s2μ2-ω1β1u]|x1(k)-y1(k)|-[s2ν1-s1ν2-ω2β2u]|x2(k)-y2(k)|-[ω1α1l-s1ϱ1]|u1(k)-v1(k)|-[ω2α2l-s2ϱ2]|u2(k)-v2(k)|≤-α∑i=12‍(|xi(k)-yi(k)|+|ui(k)-vi(k)|),
where
(4.37)μ1=min(al,2M1-au)-E1u-τ1au[P1uG1u+M1P2uE1u+M1P2uau],μ2=M2m12fu[1+τ2M2m1Q2ufu],ν1=min(flM1,2M2-fum1)-τ2fum1[Q1uG2u+M2m1Q2ufu],ν2=E2u[1+τ1auM1P2u],ϱ1=du[1+τ1auM1P2u],ϱ2=pu[1+τ2fum1M2Q2u].
Summing both sides of (4.36) from k0+τ to k, it derives that
(4.38)∑s=k0+τk‍[V(s+1)-V(s)]≤-α∑s=k0+τk‍∑i=12‍(|xi(s)-yi(s)|+|ui(s)-vi(s)|).
It then follows from (4.38) that for k>k0+τ,
(4.39)V(k+1)+α∑s=k0+τk‍∑i=12‍(|xi(s)-yi(s)|+|ui(s)-vi(s)|)≤V(k0+τ),
that is
(4.40)∑s=k0+τk‍∑i=12‍(|xi(s)-yi(s)|+|ui(s)-vi(s)|)≤V(k0+τ)α.
Then
(4.41)∑k=k0+τ∞‍∑i=12‍(|xi(s)-yi(s)|+|ui(s)-vi(s)|)≤V(k0+τ)α<+∞,
from which we see that
(4.42)limk→+∞|xi(k)-yi(k)|=0,limk→+∞|ui(k)-vi(k)|=0,i=1,2.
This completes the proof of Theorem 4.3.


## 5. Almost Periodic Solutions

In this section, we consider the almost periodic property of system (1.4).We assume thata*(k)∈H(a(k)), b*(k)∈H(b(k)), c*(k)∈H(c(k)), d*(k)∈H(d(k)), f*(k)∈H(f(k)), g*(k)∈H(g(k)), p*(k)∈H(p(k)), αi*(k)∈H(αi(k)), βi*(k)∈H(βi(k)), i=1,2. By Definitions 2.3 and 2.4, there exists an integer valued sequence γn with γn→∞ as n→∞ such that for i=1,2, k∈Z(5.1)a(k+γn)→a*(k),b(k+γn)→b*(k),c(k+γn)→c*(k),d(k+γn)→d*(k),f(k+γn)→f*(k),g(k+γn)→g*(k),p(k+γn)→p*(k),αi(k+γn)→αi*(k),βi(k+γn)→βi*(k).
Then we have the following Hull equation of system (1.4)
(5.2)x1(k+1)=x1(k)exp[b*(k)-a*(k)x1(k-τ1)-c*(k)x1(k)x2(k)h2x22(k)+x12(k)-d*(k)u1(k-σ1)],x2(k+1)=x2(k)exp[g*(k)-f*(k)x2(k-τ2)x1(k-τ2)-p*(k)u2(k-σ2)],Δui(k)=-αi*(k)ui(k)+βi*(k)xi(k-ρi),i=1,2.According to the almost periodic theory, we know that if system (1.4) satisfies (H1)–(H3), then Hull equation (5.2) also satisfies (H1)–(H3).The following Lemma is Lemma 4.1 in [15].Lemma 5.1.
If each hull equation of system (1.4) has a unique strictly positive solution, then the almost periodic difference system (1.4) has a unique strictly positive almost periodic solution.Theorem 5.2.
Assume that(H1)–(H3) holds. Then the almost periodic difference system (1.4) has a unique strictly positive almost periodic solution which is globally attractive.Proof.
We divided the proof into two steps.
Step1. We prove the existence of a strictly positive solution of each Hull equation (5.2).
By the almost periodicity of the parameters of system (1.4), there exists an integer-valued sequence γn with γn→∞ as n→∞ such that for i=1,2,k∈Z(5.3)a*(k+γn)→a*(k),b*(k+γn)→b*(k),c*(k+γn)→c*(k),d*(k+γn)→d*(k),f*(k+γn)→f*(k),g*(k+γn)→g*(k),p*(k+γn)→p*(k),αi*(k+γn)→αi*(k),βi*(k+γn)→βi*(k).
Suppose thatX(k)=(x1(k),x2(k),u1(k),u2(k))T is any positive solution of Hull equation (5.2). Since (H1), (H2) hold, according to Theorem 3.1, we obtain
(5.4)mi≤liminfk→+∞xi(k)≤limsupk→+∞xi(k)≤Mi,ni≤liminfk→+∞ui(k)≤limsupk→+∞ui(k)≤Ni,i=1,2.
Therefore, one has
(5.5)0<infk∈Z+xi(k)≤supk∈Z+xi(k)<∞,0<infk∈Z+ui(k)≤supk∈Z+ui(k)<∞,i=1,2.
For anyε>0 small enough, it follows from Theorem 3.1 that there exists enough large K0>K5+τ such that for i=1,2 and k≥K0(5.6)mi-ε≤xi(k)≤Mi+ε,ni-ε≤ui(k)≤Ni+ε.
Letxn(k)=xn(k+γn) and un(k)=un(k+γn) for k≥K0+τ-γn, n=1,2,…. For any positive integer q and i=1,2, it is not difficult to show that there are sequences {xin(k):n≥q} and {uin(k):n≥q} such that the sequences {xin(k)} and {uin(k)} have subsequences, denoted by {xin(k)} and {uin(k)} again, converging on any finite interval of Z as n→∞, respectively. So we have sequences {yi(k)} and {vi(k)}i=1,2 such that for k∈Z,n→∞(5.7)xin(k)→yi(k),uin(k)→vi(k),x1(k+1)=x1(k)exp[c*(k+γn)x1(k)x2(k)h2x22(k)+x12(k)b*(k+γn)-a*(k+γn)x1(k-τ1)-c*(k+γn)x1(k)x2(k)h2x22(k)+x12(k)-d*(k+γn)u1(k-σ1)],x2(k+1)=x2(k)exp[g*(k+γn)-f*(k+γn)x2(k-τ2)x1(k-τ2)-p*(k+γn)u2(k-σ2)],Δui(k)=-αi*(k+γn)ui(k)+βi*(k+γn)xi(k-ρi),i=1,2,
which implies
(5.8)y1(k+1)=y1(k)exp[b*(k)-a*(k)y1(k-τ1)-c*(k)y1(k)y2(k)h2y22(k)+y12(k)-d*(k)v1(k-σ1)],y2(k+1)=y2(k)exp[g*(k)-f*(k)y2(k-τ2)y1(k-τ2)-p*(k)v2(k-σ2)],Δvi(k)=-αi*(k)vi(k)+βi*(k)yi(k-ρi),i=1,2.
Thus we can easily see thatY(k)=(y1(k),y2(k),v1(k),v2(k))T is a solution of Hull equation (5.2) and mi-ε≤yi(k)≤Mi+ε, ni-ε≤vi(k)≤Ni+ε, fori=1,2, k∈Z. Since ε is an arbitrary small positive number, it follows that
(5.9)0<infk∈Z+yi(k)≤supk∈Z+yi(k)<∞,0<infk∈Z+vi(k)≤supk∈Z+vi(k)<∞,i=1,2.
This completes the proof of Step 1.
Step2. We show the uniqueness of the strictly positive solution of each Hull equation (5.2).
Suppose that the Hull equation (5.2) has two arbitrary strictly positive solutions X*(k)=(x1*(k),x2*(k),u1*(k),u2*(k))T and Y*(k)=(y1*(k),y2*(k),v1*(k),v2*(k))T. Now we define a Lyapunov function V*(k) on k∈Z as follows:
(5.10)V*(k)=∑i=12‍si(Vi1*(k)+Vi2*(k)+Vi3*(k))+∑i=12‍ωiVi'*(k),
where
(5.11)Vi1*(k)=|lnxi*(k)-lnyi*(k)|,i=1,2,V12*(k)=∑s=k-σ1k-1‍d(s+σ1)|u1*(s)-v1*(s)|+∑s=kk-1+τ1‍a(s)∑u=s-τ1k-1‍{[P1(u)G1(u)M1P2(u)E1(u)]|x1*(u)-y1*(u)|+M1P2(u)[a(u)|x1*(u-τ1)-y1*(u-τ1)|+d(u)|u1*(u-σ1)-v1*(u-σ1)|+E2(u)|x2*(u)-y2*(u)|]},V13*(k)=M1∑l=k-τ1k-1‍P2(l+τ1)a(l+τ1)|x1*(l)-y1*(l)|∑s=l+τ1+1l+2τ1‍a(s)+M1∑l=k-σ1k-1‍P2(l+σ1)d(l+σ1)|u1*(l)-v1*(l)|∑s=l+σ1+1l+σ1+τ1‍a(s),V22*(k)=∑s=k-τ2k-1‍M2m12f(s+τ2)|x1*(s)-y1*(s)|+∑s=k-σ2k-1‍p(s+σ2)|u2*(s)-v2*(s)|+∑s=kk-1+τ2‍f(s)m1∑u=s-τ2k-1‍{f(u)m1Q1(u)G2(u)|x2*(u)-y2*(u)|+M2Q2(u)[f(u)m1|x2*(u-τ2)-y2*(u-τ2)|+M2m12f(u)|x1*(u-τ2)-y1*(u-τ2)|+p(u)|u2*(u-σ2)-v2*(u-σ2)|f(u)m1]},V23*(k)=M2m1∑l=k-τ2k-1‍Q2(l+τ2)f(l+τ2)|x2*(l)-y2*(l)|∑s=l+τ2+1l+2τ2‍f(s)m1+M22m12∑l=k-τ2k-1‍Q2(l+τ2)f(l+τ2)|x1*(l)-y1*(l)|∑s=l+τ2+1l+2τ2‍f(s)m1+M2∑l=k-σ2k-1‍Q2(l+σ2)p(l+σ2)|u2*(l)-v2*(l)|∑s=l+σ2+1l+σ2+τ2‍f(s)m1,Vi'*(k)=|ui*(k)-vi*(k)|+∑s=k-ρik-1‍βi(s+ρi)|xi*(s)-yi*(s)|,
where Ei, Gi, Pi, Qi, i=1,2 are defined by Theorem 4.3.
Similar to the discussion of (3.32), calculating the difference of V*(k) along the solution of the Hull equation (5.2), one has
(5.12)ΔV*≤-α∑i=12‍(|xi*(k)-yi*(k)|+|ui*(k)-vi*(k)|),fork∈Z.
It follows from (5.12) we know that V*(k) is a nonincreasing function on Z. Summing both sides of the above inequalities from k to 0, we get
(5.13)α∑s=k0‍∑i=12‍(|xi*(s)-yi*(s)|+|ui*(s)-vi*(s)|)≤V*(k)-V*(0),fork<0.
So we have ∑s=k0‍∑i=12‍(|xi*(s)-yi*(s)|+|ui*(s)-vi*(s)|)<+∞,k→-∞, thus we obtain
(5.14)limk→-∞|xi*(k)-yi*(k)|=0,limk→-∞|ui*(k)-vi*(k)|=0,i=1,2.
From (5.14), for a positive integer K, we have
(5.15)|xi*(k)-yi*(k)|<ε,|ui*(k)-vi*(k)|<ε,fork<-K,i=1,2.
It follows from (5.10) that
(5.16)Vi1*(k)≤1mi|xi*(k)-yi*(k)|≤1miε,V12*(k)=σ1dumaxs≤k|u1*(s)-v1*(s)|+(τ1)2au{[P1uG1u+M1P2uE1u]maxs≤k|x1*(s)-y1*(s)|+M1P2u[aumaxs≤k|x1*(s)-y1*(s)|+dumaxs≤k|u1*(s)-v1*(s)|+E2umaxs≤k|x2*(s)-y2*(s)|]}≤{σ1du+(τ1)2au[(P1uG1u+M1P2uE1u)+M1P2u(au+du+E2u)]}ε,V13*(k)≤τ1M1P2uau(τ1aumaxs≤k|x1*(s)-y1*(s)|+σ1dumaxs≤k|u1*(s)-v1*(s)|)≤τ1M1P2uau(τ1au+σ1du)ε,V22*(k)≤τ2M2m12fumaxs≤k|x1*(s)-y1*(s)|+σ2pumaxs≤k|u2*(s)-v2*(s)|+(τ2)2fum1{[Q1uF2u+M2Q2u(fum1+M2m12fu+pu)]Q1uG2umaxs≤k|x2*(s)-y2*(s)|+M2Q2u[fum1maxs≤k|x2*(s)-y2*(s)|+M2m12fumaxs≤k|x1*(s)-y1*(s)|+pumaxs≤k|u2*(s)-v2*(s)|[Q1uF2u+M2Q2u(fum1+M2m12fu+pu)]]},≤{τ2M2m12fu+σ2pu+(τ2)2fum1[Q1uF2u+M2Q2u(fum1+M2m12fu+pu)]}ε,V23*(k)≤τ2fum1M2Q2u(τ2fum1maxs≤k|x2*(s)-y2*(s)|+τ2M2m12fumaxs≤k|x1*(s)-y1*(s)|+σ2pumaxs≤k|u2*(s)-v2*(s)|M2m12)≤τ2fum1M2Q2u(τ2fum1+τ2M2m12fu+σ2pu)ε,Vi'*(k)≤maxs≤k|ui*(s)-vi*(s)|+ρiβiumaxs≤k|xi*(s)-yi*(s)|≤(1+ρiβiu)ε.
Let(5.17)Q=s1{1m1+σ1du+(τ1)2au[(P1uG1u+M1P2uE1u)+M1P2u(au+du+E2u)]+τ1M1P2uau(τ1au+σ1du)1m1}+s2{1m2+τ2M2m12fu+σ2pu+(τ2)2fum1[Q1uG2u+M2Q2u(fum1+M2m12fu+pu)]+τ2fum1M2Q2u(τ2fum1+τ2M2m12fu+σ2pu)}+∑i=12‍ωi(1+ρiβiu).
It follows from (5.10) and the above inequalities that V*(k)≤Qε for k<-K, hence limk→-∞V*(k)=0. Furthermore, V*(k) is a nonincreasing function on Z, thus V*(k)≡0, that is, xi*(k) = yi*(k), ui*(k) = vi*(k), i = 1,2 for all k∈Z. This ends the proof of Step 2.
By the above discussion and Theorem4.3, we conclude that any hull equation of system (1.4) has a unique strictly positive solution. Then the almost periodic system (1.4) has a unique strictly positive almost periodic solution which is globally attractive. The proof of Theorem 5.2 is completed.


## 6. Concluding Remarks

In this paper, a discrete almost periodic ratio-dependent Leslie system with time delays and feedback controls is considered. By applying the difference inequality, some sufficient conditions are established, which are independent of feedback control variables, to ensure the permanence of system (1.4). By constructing the suitable Lyapunov function, we show that system (1.4) is globally attractive under some appropriate conditions. Furthermore, by using an almost periodic functional hull theory, we show that the almost periodic system (1.4) has a unique strictly positive almost periodic solution which is globally attractive.We would like to mention here the question of whether the feedback control variables have the influence on the stability property of the system or not. It is, in fact, a very challenging problem, and we leave it for our future work.



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*Source: 358594-2012-11-04.xml*