# Positive Almost Periodic Solutions for a Discrete Competitive System Subject to Feedback Controls

Authors: Qinglong Wang; Zhijun Liu; Zuxiong Li
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper concerns a discrete competitive system subject to feedback
controls. By using Lyapunov function and some preliminary lemmas, the existence
and uniformly asymptotic stability of unique positive almost periodic solution of
the system are investigated. Numerical simulations suggest the feasibility of our
theoretical results.


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## Body

## 1. Introduction

Many real world phenomena are studied through discrete mathematical models involving difference equations which are more suitable than the continuous ones when the populations have nonoverlapping generations. On the other hand, discrete models can also provide efficient computational models of continuous models for numerical stimulations; therefore, the studies of dynamic systems governed by difference equations have attracted more attention from scholars. Many good results concerned with discrete systems are deliberated (see [1–7] in detail).Recently, in [1] we consider the following discrete two-species competitive almost system
(1)x1(n+1)=x1(n)exp[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)],x2(n+1)=x2(n)exp[r2(n)-a2(n)x2(n)-c1(n)x1(n)1+x1(n)],n=0,1,2,….
Here xi(n) stand for the densities of species xi at the nth generation, ri(n) represent the natural growth rates of species xi at the nth generation, ai(n) are the intraspecific effects of the nth generation of species xi on own population, and ci(n) measure the interspecific effects of the nth generation of species xi on species xj(i,j=1,2;i≠j). The coefficients {ri(n)},{ai(n)}, and {ci(n)} are bounded positive almost periodic sequences. We established a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of system (1) (see [1]).Note that ecosystems in the real world are often disturbed by outside continuous forces. In the language of control, we call the disturbance functions control variables and they can be regarded as feedback controls. For more discussions on this direction, we can refer to [8–14] and the references cited therein. Motivated by the above ideas we can establish the discrete two-species competitive almost system with feedback controls
(2)x1(n+1)=x1(n)exp[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)-e1(n)u1(n)c2(n)x2(n)1+x2(n)],x2(n+1)=x2(n)exp[r2(n)-a2(n)x2(n)-c1(n)x1(n)1+x1(n)-e2(n)u2(n)c1(n)x1(n)1+x1(n)],Δu1(n)=-b1(n)u1(n)+d1(n)x1(n),Δu2(n)=-b2(n)u2(n)+d2(n)x2(n),n=0,1,2,…,
where u1 and u2 represent control variables and Δu1(n)=u1(n+1)-u1(n),Δu2(n)=u2(n+1)-u2(n) are the forward difference operators. {bi(n)},{di(n)} and {ei(n)} are bounded positive almost periodic sequences, where 0<bi(n)<1,i=1,2. To belong to the direction of [1], in this contribution, we continue to discuss the effect of feedback controls and establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of system (2).The rest of this paper is organized as follows. In the next section, we introduce some notations, definitions, and lemmas which are available for our main results. Sufficient conditions for the existence and uniformly asymptotic stability of unique positive almost periodic solution of system (2) are established in Section 3. In Section 4, we carry out numerical simulations to substantiate our analytical results.


## 2. Preliminaries

In this section, we give some notations, definitions, and lemmas which will be useful for the later sections.ℝ
,
ℝ
+
,
ℤ, and ℤ+ denote the sets of real numbers, nonnegative real numbers, integers, and nonnegative integers, respectively. ℝ4 and ℝk denote the cone of 4-dimensional and k-dimensional real Euclidean space, respectively. For an almost periodic sequence {g(n)} defined on ℤ+, the notations below will be used
(3)gU=supn∈ℤ+{g(n)},gL=infn∈ℤ+{g(n)}.Definition 1 (see [4]).
A sequencex:ℤ→ℝk is called an almost periodic sequence if the ε-translation set of x,
(4)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 discrete interval of length l(ε) contains a τ=τ(ε)∈E{ε,x} such that
(5)|x(n+τ)-x(n)|<ε,∀n∈ℤ.τ is called the ε-translation number of x(n).Definition 2 (see [4]).
Letf:ℤ×𝔻→ℝk, where 𝔻 is an open set in ℝk. f(n,x) is said to be almost periodic in n uniformly for x∈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 a integer τ for which
(6)|f(n+τ,x)-f(n,x)|<ε,
for all n∈ℤ and all x∈𝕊. τ is called the ε-translation number of f(n,x).Lemma 3 (see [4]).
{
x
(
n
)
} is an almost periodic sequence if and only if for any sequence {tk′}⊂ℤ there exists a subsequence {tk}⊂{tk′} such that x(n+tk) converges uniformly on n∈ℤ as k→∞. Furthermore, the limit sequence is also an almost periodic sequence.Consider the following almost periodic difference system(7)x(n+1)=f(n,x(n)),n∈ℤ+,
where f:ℤ+×𝕊B→ℝk,𝕊B={x∈ℝk:∥x∥<B}, and f(n,x) is almost periodic in n uniformly for x∈𝕊B and is continuous in x. The product system of (7) is the following system:
(8)x(n+1)=f(n,x(n)),y(n+1)=f(n,y(n)),
and Zhang [5] established the following result.Lemma 4 (see [5]).
Suppose that there exists a Lyapunov functionV(n,x,y) defined for n∈ℤ+, ∥x∥<B,∥y∥<B satisfying the following conditions:(i)
a(∥x-y∥)≤V(n,x,y)≤b(∥x-y∥), where a,b∈K with K={α∈C(ℝ+,ℝ+):α(0)=0 and α is increasing};(ii)
|V(n,x1,y1)-V(n,x2,y2)|≤L(∥x1-x2∥+∥y1-y2∥), where L>0 is a constant;(iii)
ΔV(8)(n,x,y)≤-γV(n,x,y), where 0<γ<1 is a constant and
(9)ΔV(8)(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 system (7) such that ∥φ(n)∥≤B*<B for n∈ℤ+, then there exists a unique uniformly asymptotically stable almost periodic solution p(n) of system (7) which satisfies ∥p(n)∥≤B*. In particular, if f(n,x) is periodic of period ω, then there exists a unique uniformly asymptotically stable periodic solution of system (7) of periodic ω.


## 3. Main Results

We first give the following two propositions which are useful for our main results.Proposition 5.
Any positive solution(x1(n),x2(n),u1(n),u2(n)) of system (2) satisfies
(10)limsupn→+∞x1(n)≤x1*,limsupn→+∞x2(n)≤x2*,limsupn→+∞u1(n)≤u1*,limsupn→+∞u2(n)≤u2*,
where
(11)x1*=exp(r1U-1)a1L,x2*=exp(r2U-1)a2L,u1*=x1*d1Ub1L,u2*=x2*d2Ub2L.Proof.
We first prove thatlimsupn→+∞x1(n)≤x1*; to do so, we consider Cases (1) and (2).
Case (1). Suppose that there exists an n0∈ℤ+ such that x1(n0+1)≥x1(n0); it follows from the first equation of system (2) that
(12)r1(n0)-a1(n0)x1(n0)-c2(n0)x2(n0)1+x2(n0)-e1(n0)u1(n0)≥0,
which implies
(13)x1(n0)≤r1(n0)a1(n0)≤r1Ua1L≤exp(r1U-1)a1L=x1*.
Hence,
(14)x1(n0+1)=x1(n0)exp[-c2(n0)x2(n0)1+x2(n0)r1(n0)-a1(n0)x1(n0)-c2(n0)x2(n0)1+x2(n0)-e1(n0)u1(n0)]≤x1(n0)exp[r1U-a1Lx1(n0)]=r1Ua1L·a1Lr1Ux1(n0)exp[r1U(1-a1Lr1Ux1(n0))]≤r1Ua1L·exp(r1U-1)r1U=exp(r1U-1)a1L=x1*,
where we use the fact that maxx∈ℝ+xexp[r(1-x)]=exp(r-1)/r for r>0.
It is claimed thatx1(n)≤x1* for all n≥n0. By way of contradiction, assume that there is a q0>n0 such that x1(q0)>x1*, then q0≥n0+2. Set q~0=min{q0:q0≥n0+2,x1(q0)>x1*}; that is to say, x1(q~0)>x1* and q~0≥n0+2; then x1(q~0)>x1*≥x1(q~0-1). It is easy to see that x1(q~0)≤x1* from the above argument, which is a contradiction. Therefore, x1(n)≤x1* for all n≥n0, then limsupn→+∞x1(n)≤x1*. This proves the claim.
Case (2). Suppose that x1(n)>x1(n+1) for all n∈ℤ+. In particular, limn→+∞x1(n) exists, denoted by x-1. We will prove that x-1≤x1* by way of contradiction, if x-1>x1*, then by taking limit in the first equation in system (2) we have
(15)limn→+∞[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)-e1(n)u1(n)]=0.
Notice that r1U/a1L≤exp(r1U-1)/a1L=x1*<x-1, so we have
(16)limn→+∞[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)-e1(n)u1(n)]≤limn→+∞[r1(n)-a1(n)x1(n)]≤r1U-a1Lx-1<0,
which is a contradiction. This proves the claim. Hence, limsupn→+∞x1(n)≤x1*.
Similar tox1(n), we can prove that limsupn→+∞x2(n)≤x2*.
Next, we provelimsupn→+∞u1(t)≤u1*. For any ε>0, there exists a large enough integer l0∈ℤ+ such that xi(n)≤xi*+ε,i=1,2 for all n≥l0. We have from the third equation of system (2) that
(17)u1(n)=∏i=0n-1‍(1-b1(i))[u1(0)+∑i=0n-1‍d1(i)x1(i)∏j=0i(1-b1(j))]≤(1-b1L)n(u1(0)+ϑ1)+d1U(x1*+ε)×∑i=l0n-1‍∏j=i+1n-1‍(1-b1(j))≤(1-b1L)n(u1(0)+ϑ1)+d1U(x1*+ε)∑i=l0n-1‍(1-b1L)n-i-1,
where ϑ1=∑i=0l0-1(d1(i)x1(i)/∏j=0i‍(1-b1(j)))‍. Since 0<b1L<1, we can find a positive number s such that 1-b1L=e-s, then by using Stolz’s theorem, we obtain that
(18)limn→+∞∑i=l0n-1‍(1-b1L)n-i-1=limn→+∞∑i=l0n-1es(i+1)esn=11-e-s=1b1L.
Thus limsupn→+∞u1(n)≤(x1*+ε)d1U/b1L. Since ε is arbitrary, limsupn→+∞u1(n)≤u1* is valid. Analogously, we can prove limsupn→+∞u2(n)≤u2*. Hence, the proof of Proposition 5 is complete.Proposition 6.
If the following inequalities(19)r1L-c2U-e1Uu1*>0,r2L-c1U-e2Uu2*>0
hold. Then any positive solution (x1(n),x2(n),u1(n),u2(n)) of system (2) satisfies
(20)liminfn→+∞x1(n)≥x1*,liminfn→+∞x2(n)≥x2*,liminfn→+∞u1(n)≥u1*,liminfn→+∞u2(n)≥u2*,
where
(21)x1*=r1L-c2U-e1Uu1*a1Uexp[r1L-a1Ux1*-c2U-e1Uu1*],x2*=r2L-c1U-e2Uu2*a2Uexp[r2L-a2Ux2*-c1U-e2Uu2*],u1*=x1*d1Lb1U,u2*=x2*d2Lb2U.Proof.
For any small enoughε>0, which satisfies r1L-c2U-e1U(u1*+ε)>0, according to Proposition 5, there exists n*∈ℤ+ such that
(22)x1(n)≤x1*+ε,x2(n)≤x2*+ε,u1(n)≤u1*+ε,u2(n)≤u2*+ε,forn≥n*.
We first present Cases (1) and (2) to prove that liminfn→+∞x1(n)≥x1*.
Case (1). Assuming that there exists a positive integer n0≥n* such that x1(n0+1)≤x1(n0), we have from the first equation of system (2) that
(23)x1(n0+1)=x1(n0)exp[c2(n0)x2(n0)1+x2(n0)r1(n0)-a1(n0)x1(n0)-c2(n0)x2(n0)1+x2(n0)-e1(n0)u1(n0)]≥x1(n0)exp[r1L-a1Ux1(n0)-c2U-e1U(u1*+ε)].
Therefore,
(24)r1L-a1Ux1(n0)-c2U-e1U(u1*+ε)≤0,
which implies that
(25)x1(n0)≥r1L-c2U-e1U(u1*+ε)a1U>0.
Then
(26)x1(n0+1)=x1(n0)exp[-c2(n0)x2(n0)1+x2(n0)r1(n0)-a1(n0)x1(n0)-c2(n0)x2(n0)1+x2(n0)-e1(n0)u1(n0)]≥r1L-c2U-e1U(u1*+ε)a1U×exp[r1L-a1U(x1*+ε)-c2U-e1U(u1*+ε)].
Hence x1(n0+1)≥x1ε, where
(27)x1ε=r1L-c2U-e1U(u1*+ε)a1Uexp[r1L-a1U(x1*+ε)-c2U-e1U(u1*+ε)].
We claim thatx1(n)≥x1ε for n≥n0. By way of contradiction, assume that there exists a q0≥n0 such that x1(q0)<x1ε, then q0≥n0+2. Let q-0=min{q0:q0≥n0+2,x1(q0)<x1ε}, that is, x1(q-0)<x1ε and q-0≥n0+2, then x1(q-0)<x1ε≤x1(q-0-1), and the above argument produces that x1(q-0)≥x1ε, which is a contradiction. Thus, x1(n)≥x1ε for all n≥n0; since ε can be sufficiently small, it obtains that liminfn→+∞x1(n)≥x1*. This proves the claim.
Case (2). We assume that x1(n+1)>x1(n) for all n∈ℤ+. Then limn→+∞x1(n) exists, denoted by x_1. We claim that
(28)x_1≥r1L-c2U-e1U(u1*+ε)a1U.
By way of contradiction, assume that x_1<(r1L-c2U-e1U(u1*+ε))/a1U. Taking limit in the first equation in system (2) yields
(29)limn→+∞[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)-e1(n)u1(n)]=0.
However,
(30)limn→+∞[r1(n)-a1(n)x1(n)-c2(n)x2(n)1+x2(n)-e1(n)u1(n)]≥limn→+∞[r1(n)-a1(n)x1(n)-c2(n)-e1(n)u1(n)]≥r1L-a1Ux_1-c2U-e1U(u1*+ε)>0,
which is a contradiction. It implies that x_1≥(r1L-c2U-e1U(u1*+ε))/a1U. By the fact that minx∈ℝ+{[exp(x-1)]/x}=1, we obtain that x1*=exp(r1U-1)/a1L≥r1U/a1L≥r1L/a1U, which means x1*+ε>r1L/a1U. From (27), we know that x_1≥(r1L-c2U-e1U(u1*+ε))/a1U≥x1ε. Therefore, liminfn→+∞x1(n)≥x1ε.
Sinceε can be sufficiently small, we have
(31)liminfn→+∞x1(n)≥x1*=r1L-c2U-e1Uu1*a1Uexp[r1L-a1Ux1*-c2U-e1Uu1*].
By a similar argument, we can prove that
(32)liminfn→+∞x2(n)≥x2*=r2L-c1U-e2Uu2*a2Uexp[r2L-a2Ux2*-c1U-e2Uu2*].
Now we prove thatliminfn→+∞u1(n)≥u1*. For any small enough ε>0, there exists a positive integer l0 such that xi(n)≥xi*-ε>0 for n≥l0.
By the third equation of system (2), we obtain that
(33)u1(n)=∏i=0n-1‍(1-b1(i))[u1(0)+∑i=0n-1‍d1(i)x1(i)∏j=0i(1-b1(j))]≥(1-b1U)n(u1(0)+ϱ1)+d1L(x1*-ε)×∑i=l0n-1‍∏j=i+1n-1‍(1-b1(j))≥(1-b1U)n(u1(0)+ϱ1)+d1L(x1*-ε)∑i=l0n-1‍(1-b1U)n-i-1,
where ϱ1=∑i=0l0-1(d1(i)x1(i)/∏j=0i(1-b1(j)))‍. Since 0<b1U<1, we can find a positive number t such that 1-b1U=e-t, then by Stolz’s theorem, we have
(34)limn→+∞∑i=l0n-1‍(1-b1U)n-i-1=limn→+∞∑i=l0n-1et(i+1)etn=11-e-t=1b1U.
Thus liminfn→+∞u1(n)≥(x1*-ε)d1L/b1U, by the arbitrary of ε, liminfn→+∞u1(n)≥u1* is valid. The conclusion about u2(n) can be obtained in a similar way. Thus the proof of Proposition 6 is complete.Denote(35)Ω={(x1,x2,u1,u2)∣xi*≤xi(n)≤xi*,ui*≤ui(n)≤ui*(i=1,2)}.
From the proofs of Propositions 5 and 6, we know that the set Ω is an invariant set of system (2) under the assumptions in (19).Theorem 7.
If the assumptions in (19) are satisfied, then Ω≠ϕ.Proof.
We have from system (2) that
(36)xi(n)=xi(0)exp∑m=0n-1[-cj(m)xj(m)1+xj(m)ri(m)-ai(m)xi(m)‍-cj(m)xj(m)1+xj(m)-ei(m)ui(m)],ui(n)=ui(0)-∑m=0n-1‍[bi(m)ui(m)-di(m)xi(m)],
for i,j=1,2,i≠j. Based on Propositions 5 and 6,  any solution (x1(n),x2(n),u1(n),u2(n)) of system (2) satisfies (10) and (20). Thus, for any ε>0, there exists p0 large enough such that
(37)xi*-ε≤xi(n)≤xi*+ε,ui*-ε≤ui(n)≤ui*+ε,∀n≥p0,i=1,2.
Setting to {tk} be any positive integer valued sequence such that tk→+∞ as k→+∞, it is easy to show that there exists a subsequence of {tk} still denoted by {tk}, such that xi(n+tk)→xi*(n),ui(n+tk)→ui*(n) uniformly in n on any finite subset D of ℤ+ as k→+∞, where D={α1,α2,…,αl},αh∈ℤ+(h=1,2,…,l) and l is a finite number.
In reality, for any finite subsetD⊂ℤ+, tk+αh>p0,h=1,2,…,l, when k is large enough. Thus, xi*-ε≤xi(n+tk)≤xi*+ε,ui*-ε≤ui(n+tk)≤ui*+ε, which mean that {xi(n+tk)},{ui(n+tk)} are uniformly bounded for k large enough.
Now, forα1∈D, we can select a subsequence {tk(1)} of {tk} such that {xi(α1+tk(1))},{ui(α1+tk(1))} uniformly converge on ℤ+ for k large enough.
Similarly, forα2∈D, we can also select a subsequence {tk(2)} of {tk(1)} such that {xi(α2+tk(2))},{ui(α2+tk(2))} uniformly converge on ℤ+ for k large enough.
Repeating the above process, forαl∈D, we choose a subsequence {tk(l)} of {tk(l-1)} such that {xi(αl+tk(l))},{ui(αl+tk(l))} uniformly converge on ℤ+ for k large enough.
Then we choose the sequence{tk(l)} which is a subsequence of {tk} still denoted by {tk}; for all n∈D, we have xi(n+tk)→xi*(n),ui(n+tk)→ui*(n) uniformly in n∈D as k→+∞. Therefore, the conclusion is true by the arbitrary of D.
Consider the almost periodicity of{ri(n)},{ai(n)},{bi(n)},{ci(n)},{di(n)}, and {ei(n)},i=1,2, for the above sequence {tk}, tk→+∞ as k→+∞, there exists a subsequence still denoted by {tk} such that
(38)ri(n+tk)⟶ri(n),ai(n+tk)⟶ai(n),bi(n+tk)⟶bi(n),ci(n+tk)⟶ci(n),di(n+tk)⟶di(n),ei(n+tk)⟶ei(n),
as k→+∞ uniformly on ℤ+.
For anyβ∈ℤ+, we can presume that tk+β≥p0 for k large enough. Let n∈ℤ+, by an inductive argument of system (2) from tk+β to n+tk+β  which results in
(39)xi(n+tk+β)=xi(tk+β)×exp∑m=tk+βn+tk+β-1‍[-cj(m)xj(m)1+xj(m)ri(m)-ai(m)xi(m)-cj(m)xj(m)1+xj(m)-ei(m)ui(m)cj(m)xj(m)1+xj(m)],ui(n+tk+β)=ui(tk+β)-∑m=tk+βn+tk+β-1‍[bi(m)ui(m)-di(m)xi(m)],
for i,j=1,2,i≠j, so it derives that
(40)xi(n+tk+β)=xi(tk+β)exp∑m=βn+β-1‍[-cj(m+tk)xj(m+tk)1+xj(m+tk)ri(m+tk)-ai(m+tk)×xi(m+tk)-cj(m+tk)xj(m+tk)1+xj(m+tk)-ei(m+tk)ui(m+tk)-cj(m+tk)xj(m+tk)1+xj(m+tk)],ui(n+tk+β)=ui(tk+β)-∑m=βn+β-1‍[bi(m+tk)ui(m+tk)-di(m+tk)xi(m+tk)].
Letting k→+∞, one has
(41)xi*(n+β)=xi*(β)exp∑m=βn+β-1‍[-cj(m)xj*(m)1+xj*(m)ri(m)-ai(m)xi*(m)-cj(m)xj*(m)1+xj*(m)-ei(m)ui*(m)cj(m)xj*(m)1+xj*(m)],ui*(n+β)=ui*(β)-∑m=βn+β-1‍[bi(m)ui*(m)-di(m)xi*(m)].
By the arbitrary of β, we can easily see that (x1*(n),x2*(n),u1*(n),u2*(n)) is a solution of system (2) on ℤ+, and
(42)0<xi*-ε≤xi*(n)≤xi*+ε,0<ui*-ε≤ui*(n)≤ui*+ε,∀n∈ℤ+,i=1,2.
Since ε is an arbitrarily small positive number, we obtain that
(43)0<xi*≤xi*(n)≤xi*,0<ui*≤ui*(n)≤ui*,∀n∈ℤ+,i=1,2.
So Ω≠ϕ. This completes the proof.The following theorem concerns the existence and uniformly asymptotical stability of unique positive almost periodic solution of system (2).Theorem 8.
Suppose the inequalities in (19) are satisfied; furthermore, 0<γ<1, where γ=min{sij,sij*} and
(44)sij=2aiLxi*-(aiU)2(xi*)2-(diU)2(xi*)2-aiUcjUxi*xj*(1+xj*)2-cjUxj*(1+xj*)2-aiUeiUxi*-eiU-diU(1-biL)xi*-ajUciUxi*xj*(1+xi*)2-ciUxi*(1+xi*)2-ejUciUxi*(1+xi*)2-(ciU)2(xi*)2(1+xi*)4,sij*=2biL-aiUeiUxi*-diU(1-biL)xi*-eiU-eiUcjUxj*(1+xj*)2-(eiU)2-(biU)2,i,j=1,2,i≠j, then there exists a unique uniformly asymptotically stable positive almost periodic solution of system (2) which is bounded by Ω for all n∈ℤ+.Proof.
We first make the change of variables(45)y1(n)=lnx1(n),y2(n)=lnx2(n),
then system (2) can be reformulated as
(46)yi(n+1)=yi(n)+ri(n)-ai(n)eyi(n)-cj(n)eyj(n)1+eyj(n)-ei(n)ui(n),Δui(n)=-bi(n)ui(n)+di(n)eyi(n),i=1,2,i≠j.
By Theorem 7, it is easy to see that there exists a bounded solution (y1(n),y2(n),u1(n),u2(n)) of system (46) satisfying
(47)lnxi*≤yi(n)≤lnxi*,ui*≤ui(n)≤ui*,i=1,2,n∈ℤ+.
Then |yi(n)|≤Ai,|ui(n)|≤Bi, where Ai=max{|lnxi*|,|lnxi*|},Bi=max{|ui*|,|ui*|}, and i=1,2. Let ∥(y1(n),y2(n),u1(n),u2(n))∥=∑i=12‍{|yi(n)|+|ui(n)|}, where (y1(n),y2(n),u1(n),u2(n))∈ℝ4.
The following associate product system of system (46) can be expressed as
(48)yi(n+1)=yi(n)+ri(n)-ai(n)eyi(n)-cj(n)eyj(n)1+eyj(n)-ei(n)ui(n),Δui(n)=-bi(n)ui(n)+di(n)eyi(n),zi(n+1)=zi(n)+ri(n)-ai(n)ezi(n)-cj(n)ezj(n)1+ezj(n)-ei(n)vi(n),Δvi(n)=-bi(n)vi(n)+di(n)ezi(n),i=1,2,i≠j.
Suppose that
(49)Y=(y1(n),y2(n),u1(n),u2(n)),Z=(z1(n),z2(n),v1(n),v2(n))
are any two solutions of system (46) defined on 𝕊, then
(50)∥Y∥≤B,∥Z∥≤B,
where B=∑i=12‍{Ai+Bi}, and
(51)𝕊={(y1(n),y2(n),u1(n),u2(n))∣lnxi*≤yi(n)≤lnxi*,ui*≤ui(n)≤ui*,i=1,2,n∈ℤ+(y1(n),y2(n),u1(n),u2(n))}.
Let us construct the following Lyapunov function defined onℤ+×𝕊×𝕊:
(52)V(n,Y,Z)=∑i=12‍{(yi(n)-zi(n))2+(ui(n)-vi(n))2}.
Obviously, ∥Y-Z∥=∑i=12‍{|yi(n)-zi(n)|+|ui(n)-vi(n)|} is equivalent to ∥Y-Z∥*={∑i=12‍[(yi(n)-zi(n))2+(ui(n)-vi(n))2]}1/2; that is, there exist two constants D1>0,D2>0, such that
(53)D1∥Y-Z∥≤∥Y-Z∥*≤D2∥Y-Z∥.
Consequently,
(54)(D1∥Y-Z∥)2≤V(n,Y,Z)≤(D2∥Y-Z∥)2.
Denote a,b∈C(ℝ+,ℝ+),a(x)=D12x2,b(x)=D22x2, thus the condition (i) of Lemma 4 is satisfied.
In addition, for any(n,Y,Z),(n,Y-,Z-)∈ℤ+×𝕊×𝕊, we find that
(55)|V(n,Y,Z)-V(n,Y-,Z-)|=|∑i=12‍{(yi(n)-zi(n))2+(ui(n)-vi(n))2}-∑i=12‍{(y-i(n)-z-i(n))2+(u-i(n)-v-i(n))2}|≤∑i=12‍|(yi(n)-zi(n))2-(y-i(n)-z-i(n))2|+∑i=12‍|(ui(n)-vi(n))2-(u-i(n)-v-i(n))2|=∑i=12‍{|(yi(n)-zi(n))+(y-i(n)-z-i(n))|·|(yi(n)-zi(n))-(y-i(n)-z-i(n))|}+∑i=12‍{|(ui(n)-vi(n))+(u-i(n)-v-i(n))|·|(ui(n)-vi(n))-(u-i(n)-v-i(n))|}≤∑i=12‍{(|yi(n)|+|zi(n)|+|y-i(n)|+|z-i(n)|)·(|yi(n)-y-i(n)|+|zi(n)-z-i(n)|)}+∑i=12‍{(|ui(n)|+|vi(n)|+|u-i(n)|+|v-i(n)|)·(|ui(n)-u-i(n)|+|vi(n)-v-i(n)|)}≤λ{∑i=12‍{|yi(n)-y-i(n)|+|ui(n)-u-i(n)|}+∑i=12‍{|zi(n)-z-i(n)|+|vi(n)-v-i(n)|}∑i=12‍}=λ{∥Y-Y-∥+∥Z-Z-∥},
where Y-=(y-1(n),y-2(n),u-1(n),u-2(n)),Z-=(z-1(n),z-2(n),v-1(n),v-2(n)), and λ=4max{Ai,Bi}(i=1,2). Hence, the condition (ii) of Lemma 4 is satisfied.
At last, we calculate theΔV(n,Y,Z) of V(n,Y,Z) along the solutions of system (48) and obtain that
(56)ΔV(48)(n,Y,Z)=V(n+1,Y,Z)-V(n,Y,Z)=∑i=12‍{(yi(n+1)-zi(n+1))2+(ui(n+1)-vi(n+1))2}-∑i=12‍{(yi(n)-zi(n))2+(ui(n)-vi(n))2}=∑i=12{(yi(n+1)-zi(n+1))2-(yi(n)-zi(n))2‍+(ui(n+1)-vi(n+1))2-(ui(n)-vi(n))2}=∑i=12‍{[-cj(n)(eyj(n)1+eyj(n)-ezj(n)1+ezj(n))(yi(n)-zi(n))-ai(n)(eyi(n)-ezi(n))-ei(n)(ui(n)-vi(n))-cj(n)(eyj(n)1+eyj(n)-ezj(n)1+ezj(n))]2-(yi(n)-zi(n))2+[(eyi(n)-ezi(n))(1-bi(n))(ui(n)-vi(n))+di(n)(eyi(n)-ezi(n))]2-(ui(n)-vi(n))2[-cj(n)(eyj(n)1+eyj(n)-ezj(n)1+ezj(n))(yi(n)-zi(n))-ai(n)(eyi(n)-ezi(n))}=∑i=12‍{(eyj(n)1+eyj(n)-ezj(n)1+ezj(n))2-2ai(n)(yi(n)-zi(n))(eyi(n)-ezi(n))-2ei(n)(yi(n)-zi(n))(ui(n)-vi(n))-2cj(n)(yi(n)-zi(n))(eyj(n)1+eyj(n)-ezj(n)1+ezj(n))+ai2(n)(eyi(n)-ezi(n))2+2ai(n)ei(n)×(ui(n)-vi(n))(eyi(n)-ezi(n))+2ai(n)cj(n)×(eyj(n)1+eyj(n)-ezj(n)1+ezj(n))(eyi(n)-ezi(n))+ei2(n)(ui(n)-vi(n))2+2ei(n)cj(n)×(eyj(n)1+eyj(n)-ezj(n)1+ezj(n))(ui(n)-vi(n))+cj2(n)(eyj(n)1+eyj(n)-ezj(n)1+ezj(n))2+bi(n)(bi(n)-2)(ui(n)-vi(n))2+2di(n)×(1-bi(n))(eyi(n)-ezi(n))(ui(n)-vi(n))+di2(n)(eyi(n)-ezi(n))2(eyj(n)1+eyj(n)-ezj(n)1+ezj(n))2}.
By the mean-value theorem, one has
(57)eyi(n)-ezi(n)=θi(n)(yi(n)-zi(n)),eyj(n)1+eyj(n)-ezj(n)1+ezj(n)=σj(n)(1+σj(n))2(yj(n)-zj(n)),i,j=1,2,i≠j,
where θi(n) and σj(n) lie between eyi(n) and ezi(n) and eyj(n) and ezj(n), respectively. Then xi*≤θi(n)≤xi*,xj*≤σj(n)≤xj*,i,j=1,2,i≠j,n∈ℤ+. Substituting (57) into (56), one has
(58)ΔV(48)(n,Y,Z)=∑i=12‍{2ai(n)cj(n)θi(n)σj(n)(1+σj(n))2-2ai(n)θi(n)(yi(n)-zi(n))2-2ei(n)(yi(n)-zi(n))(ui(n)-vi(n))-2cj(n)σj(n)(1+σj(n))2(yi(n)-zi(n))(yj(n)-zj(n))+ai2(n)θi2(n)(yi(n)-zi(n))2+2ai(n)ei(n)θi(n)(yi(n)-zi(n))(ui(n)-vi(n))+2ai(n)cj(n)θi(n)σj(n)(1+σj(n))2(yi(n)-zi(n))×(yj(n)-zj(n))+ei2(n)(ui(n)-vi(n))2+2ei(n)cj(n)σj(n)(1+σj(n))2(yj(n)-zj(n))×(ui(n)-vi(n))+cj2(n)σj2(n)(1+σj(n))4(yj(n)-zj(n))2+bi(n)(bi(n)-2)(ui(n)-vi(n))2+2di(n)(1-bi(n))θi(n)(yi(n)-zi(n))×(ui(n)-vi(n))+di2(n)θi2(n)(yi(n)-zi(n))22ai(n)cj(n)θi(n)σj(n)(1+σj(n))2}=∑i=12‍{(ai(n)cj(n)θi(n)σj(n)(1+σj(n))2(-2ai(n)θi(n)+ai2(n)θi2(n)+di2(n)θi2(n))×(yi(n)-zi(n))2+2(ai(n)cj(n)θi(n)σj(n)(1+σj(n))2-cj(n)σj(n)(1+σj(n))2)×(yi(n)-zi(n))(yj(n)-zj(n))+2(ai(n)ei(n)θi(n)+di(n)(1-bi(n))×θi(n)-ei(n))(yi(n)-zi(n))(ui(n)-vi(n))+2ei(n)cj(n)σj(n)(1+σj(n))2(yj(n)-zj(n))×(ui(n)-vi(n))+cj2(n)σj2(n)(1+σj(n))4(yj(n)-zj(n))2+(ei2(n)+bi(n)(bi(n)-2))(ui(n)-vi(n))2(ai(n)cj(n)θi(n)σj(n)(1+σj(n))2}≤∑i=12{|(ai(n)cj(n)θi(n)σj(n)(1+σj(n))2-cj(n)σj(n)(1+σj(n))2)(-2ai(n)θi(n)+ai2(n)θi2(n)+di2(n)θi2(n))‍×(yi(n)-zi(n))2+2|(ai(n)cj(n)θi(n)σj(n)(1+σj(n))2-cj(n)σj(n)(1+σj(n))2)×(yi(n)-zi(n))(yj(n)-zj(n))(ai(n)cj(n)θi(n)σj(n)(1+σj(n))2-cj(n)σj(n)(1+σj(n))2)|+2|(ai(n)ei(n)θi(n)+di(n)(1-bi(n))θi(n)-ei(n))(yi(n)-zi(n))(ui(n)-vi(n))(ai(n)ei(n)θi(n)+di(n)(1-bi(n))θi(n)-ei(n))|+2|ei(n)cj(n)σj(n)(1+σj(n))2(ui(n)-vi(n))×(yj(n)-zj(n))ei(n)cj(n)σj(n)(1+σj(n))2|+cj2(n)σj2(n)(1+σj(n))4×(yj(n)-zj(n))2+(ei2(n)+bi2(n)-2bi(n))×(ui(n)-vi(n))2|(ai(n)cj(n)θi(n)σj(n)(1+σj(n))2-cj(n)σj(n)(1+σj(n))2)}.
Now, we set that
(59)ΔV(48)(n,Y,Z)=∑i=12‍{V1i+V2ij+V3i+V4ij+V5j+V6i},i,j=1,2,i≠j,
where
(60)V1i=(-2ai(n)θi(n)+ai2(n)θi2(n)+di2(n)θi2(n))×(yi(n)-zi(n))2≤(-2aiLxi*+(aiU)2(xi*)2+(diU)2(xi*)2)(yi(n)-zi(n))2,V2ij=2|(ai(n)cj(n)θi(n)σj(n)(1+σj(n))2-cj(n)σj(n)(1+σj(n))2)×(yi(n)-zi(n))(yj(n)-zj(n))(ai(n)cj(n)θi(n)σj(n)(1+σj(n))2-cj(n)σj(n)(1+σj(n))2)|≤(aiUcjUxi*xj*(1+xj*)2+cjUxj*(1+xj*)2)×{(yi(n)-zi(n))2+(yj(n)-zj(n))2},V3i=2|(ai(n)ei(n)θi(n)+di(n)(1-bi(n))θi(n)-ei(n))×(yi(n)-zi(n))(ui(n)-vi(n))|≤(aiUeiUxi*+diU(1-biL)xi*+eiU)×{(yi(n)-zi(n))2+(ui(n)-vi(n))2},V4ij=2|ei(n)cj(n)σj(n)(1+σj(n))2(ui(n)-vi(n))(yj(n)-zj(n))|≤eiUcjUxj*(1+xj*)2{(ui(n)-vi(n))2+(yj(n)-zj(n))2},V5j=cj2(n)σj2(n)(1+σj(n))4(yj(n)-zj(n))2≤(cjU)2(xj*)2(1+xj*)4(yj(n)-zj(n))2,V6i=(ei2(n)+bi2(n)-2bi(n))(ui(n)-vi(n))2≤((eiU)2+(biU)2-2biL)(ui(n)-vi(n))2,
which, together with (58), yields that
(61)ΔV(48)(n,Y,Z)≤∑i=12‍{(aiUcjUxi*xj*(1+xj*)2-2aiLxi*+(aiU)2(xi*)2+(diU)2(xi*)2+aiUcjUxi*xj*(1+xj*)2+cjUxj*(1+xj*)2+aiUeiUxi*+diU(1-biL)xi*+eiUaiUcjUxi*xj*(1+xj*)2)(yi(n)-zi(n))2+(aiUcjUxi*xj*(1+xj*)2+cjUxj*(1+xj*)2+eiUcjUxj*(1+xj*)2+(cjU)2(xj*)2(1+xj*)4)(yj(n)-zj(n))2+(aiUeiUxi*+diU(1-biL)xi*+eiU+eiUcjUxj*(1+xj*)2+(eiU)2+(biU)2-2biLeiUcjUxj*(1+xj*)2)(ui(n)-vi(n))2}=∑i=12‍{(aiUcjUxi*xj*(1+xj*)2-2aiLxi*+(aiU)2(xi*)2+(diU)2(xi*)2+aiUcjUxi*xj*(1+xj*)2+cjUxj*(1+xj*)2+aiUeiUxi*+diU(1-biL)xi*+eiUaiUcjUxi*xj*(1+xj*)2)(yi(n)-zi(n))2+(ajUciUxi*xj*(1+xi*)2+ciUxi*(1+xi*)2+ejUciUxi*(1+xi*)2+(ciU)2(xi*)2(1+xi*)4)(yi(n)-zi(n))2+(aiUeiUxi*+diU(1-biL)xi*+eiU+eiUcjUxj*(1+xj*)2+(eiU)2+(biU)2-2biLeiUcjUxj*(1+xj*)2)(ui(n)-vi(n))2(aiUcjUxi*xj*(1+xj*)2-2aiLxi*+aiU2xi*2+diU2xi*2}=-∑i=12‍{(eiUcjUxj*(1+xj*)2(ciU2xi*2(1+xi*)42aiLxi*-(aiU)2(xi*)2-(diU)2(xi*)2-aiUcjUxi*xj*(1+xj*)2-cjUxj*(1+xj*)2-aiUeiUxi*-diU(1-biL)xi*-eiU-ajUciUxi*xj*(1+xi*)2-ciUxi*(1+xi*)2-ejUciUxi*(1+xi*)2-(ciU)2(xi*)2(1+xi*)4)×(yi(n)-zi(n))2+(eiUcjUxj*(1+xj*)22biL-aiUeiUxi*-diU(1-biL)xi*-eiU-eiUcjUxj*(1+xj*)2-(eiU)2-(biU)2)×(ui(n)-vi(n))2(eiUcjUxj*(1+xj*)2}=-∑i=12‍{sij(yi(n)-zi(n))2+sij*(ui(n)-vi(n))2}≤-γ∑i=12‍{(yi(n)-zi(n))2+(ui(n)-vi(n))2}=-γV(n),
where γ=min{sij,sij*},i,j=1,2,i≠j. It follows from the conditions of Theorem 8 that we have 0<γ<1, then the condition (iii) of Lemma 4 is satisfied. Therefore, we have from Lemma 4 that there exists a unique uniformly asymptotically stable almost periodic solution (y1*(n),y2*(n),u1*(n),u2*(n)) of system (46) which is bounded by 𝕊 for all n∈ℤ+, which implies that there exists a unique uniformly asymptotically stable positive almost periodic solution (x1*(n),x2*(n),u1*(n),u2*(n)) of system (2) which is bounded by Ω for all n∈ℤ+. This completed the proof.Remark 9.
If we neglect the role of feedback controls, that is,bi(n)=0,di(n)=0 and ei(n)=0,i=1,2, then system (2) can be reduced to system (1). Propositions 5 and 6 and Theorem 8 can come down to our corresponding main results (see [1]). The fact shows that the feedback controls have influence on the existence and uniformly asymptotic stability of unique positive almost solution of system (2).


## 4. Numerical Simulations

In this section we give a numerical example in support to our analytical findings.Example 10.
Consider the following discrete system with feedback controls:(62)x1(n+1)=x1(n)exp[(0.025+0.002cos(2nπ))x2(n)1+x2(n)1.20-0.02sin(2nπ)-(1.05+0.01sin(2nπ))x1(n)-(0.025+0.002cos(2nπ))x2(n)1+x2(n)-(0.015+0.001sin(3nπ))u1(n)(0.025+0.002cos(2nπ))x2(n)1+x2(n)],x2(n+1)=x2(n)exp[(0.035+0.005sin(3nπ))x1(n)1+x1(n)1.15-0.02cos(2nπ)-(1.02+0.02cos(2nπ))x2(n)-(0.035+0.005sin(3nπ))x1(n)1+x1(n)-(0.025+0.003cos(3nπ))u2(n)(0.035+0.005sin(3nπ))x1(n)1+x1(n)],Δu1(n)=-(0.93-0.03cos(2nπ))u1(n)+(0.024-0.002cos(2nπ))x1(n),Δu2(n)=-(0.95-0.03sin(3nπ))u2(n)+(0.015+0.003sin(2nπ))x2(n).
A simple computation shows that
(63)x1*≈1.1982,x2*≈1.1853,u1*≈0.0346,u2*≈0.0232,x1*≈0.9665,x2*≈0.9076,u1*≈0.0221,u2*≈0.0111,r1L-e1Uu1*-c2U≈1.1524>0,r2L-e2Uu2*-c1U≈1.0894>0.
Obviously, the assumptions in (19) are satisfied, and moreover, one has
(64)s12≈0.3086,s21≈0.1831,s12*≈0.8386,s21*≈0.8142,
that is, 0<γ=min{s12,s21,s12*,s21*}≈0.1831<1, so the assumptions of Theorem 8 are satisfied. Thus, there exists a unique uniformly asymptotically stable positive almost periodic solution of system (62). From Figure 1, we can easily see that system (62) exists a positive almost periodic solution (x1*(t),x2*(t),u1*(n),u2*(n)), and the 2-dimensional phase portraits of almost periodic system (62) are displayed in Figure 2, respectively. Figure 3 shows that any positive solution (x1(n),x2(n),u1(n),u2(n)) tends to the pervious almost periodic solution (x1*(n),x2*(n),u1*(n),u2*(n)); furthermore, the 2-dimensional phase portraits reflect the fact in Figure 4.Positive almost periodic solution of system (62). (a)–(d) Time-series of x1*(n),x2*(n),u1*(n), and u2*(n) with initial values x1*(0)=1.18,x2*(0)=0.96,u1*(0)=0.024,andu2*(0)=0.023 for n∈[0,100], respectively.
(a)
(b)
(c)
(d)2-dimensional phase portrait. (a) Phase portrait ofx1*(n) and x2*(n) with initial values x1*(0)=1.18,x2*(0)=0.96 for n∈[0,100], respectively. (b) Phase portrait of u1*(n) and u2*(n) with initial values u1*(0)=0.024,u2*(0)=0.023 for n∈[0,100], respectively.
(a)
(b)Uniformly asymptotic stability. (a)–(d) Time-series ofx1*(n),x2*(n),u1*(n) and u2*(n) with initial values x1*(0)=1.18,x2*(0)=0.96,u1*(0)=0.024,andu2*(0)=0.023 and x1(n),x2(n),u1(n), and u2(n) with initial values x1(0)=1.06,x2(0)=1.18,u1(0)=0.034,andu2(0)=0.012 for n∈[0,100], respectively.
(a)
(b)
(c)
(d)2-dimensional phase portrait. (a) Phase portrait ofx1*(n) and x2*(n) with initial values x1*(0)=1.18,x2*(0)=0.96 and x1(n) and x2(n) with initial values x1(0)=1.06,x2(0)=1.18 for n∈[0,100], respectively. (b) Phase portrait of u1*(n) and u2*(n) with initial values u1*(0)=0.024,u2*(0)=0.023 and u1(n) and u2(n) with initial values u1(0)=0.034,u2(0)=0.012 for n∈[0,100], respectively.
(a)
(b)



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