# Permanence and Almost Periodic Solution for an Enterprise Cluster Model Based on Ecology Theory with Feedback Controls on Time Scales

Authors: Yuanhong Zhi; Zunling Ding; Yongkun Li
Journal: Discrete Dynamics in Nature and Society
 (2013)

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## Abstract

We present a model with feedback controls based on ecology theory, which effectively describes the competition and cooperation of enterprise cluster in real economic
environments. Applying the comparison theorem of dynamic equations on time scales
and constructing a suitable Lyapunov functional, sufficient conditions which guarantee
the permanence and the existence of uniformly asymptotically stable almost periodic
solution of the system are obtained.


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## Body

## 1. Introduction

In recent years, a few researchers have presented some models about enterprise clusters based on ecology theory, which arouse growing interest in applying the methods of ecology and dynamic system theory to study enterprise clusters, for example [1–9] and references cited therein. In [1], two models from biology were given and explained by economic view, and sufficient conditions were obtained to guarantee the coexistence and stability of enterprise clusters. In [3], the developing strategy of enterprise clusters was analyzed based on the logistic model, and the suggestions of constructing cooperative relation and choosing generalization or specialization tactics for commodity were put forward. In addition, based on the theoretical model of ecological population science, Wang and Pan [6] made a detailed analysis to the equilibrium mechanism of enterprise clusters, including net model and center halfback model and drew a conclusion that the relationship of pierce competition and beneficial cooperation among enterprise clusters was the crucial factor for them to keep stability. More related research about enterprises cluster one can refer to the literatures [10–13]. Recently, the literature [5] considered the competition and cooperation system of two enterprises based on ecosystem:
(1)dx1(t)dt=r1x1(t)(1-x1(t)K-α(x2(t)-c2)2K),dx2(t)dt=r2x2(t)(1-x2(t)K-α(x1(t)-c1)2K),
where x1(t), x2(t) represent the output of enterprises A and B, r1, r2 are the intrinsic growth rate, K denotes the carrying capacity of market under nature unlimited conditions, α, β are the competitive coefficients of two enterprises, c1, c2 are the initial production of two enterprises. Accordingly, we consider now the equation with nonconstant coefficients, which can be obtained as a modified system (1) with variable coefficients (Letting a1=r1/K, a2=r2/K, b1=r1α/K, b2=r2β/K in system (1)):
(2)dx1(t)dt=x1(t)(r1(t)-a1(t)x1(t)-b1(t)[x2(t)-c2(t)]2),dx2(t)dt=x2(t)(r2(t)-a2(t)x2(t)-b1(t)[x1(t)-c1(t)]2).In real world, the situation of enterprises is often distributed by unpredictable forces which can result in changes in enterprises' parameters such as intrinsic growth rates. So it is necessary to study models with control variables which are so-called disturbance functions [14–17].As well known, both continuous and discrete systems are very important in implementation and applications, but it is troublesome to study the permanence and the existence of almost periodic solutions for continuous system and discrete system, respectively. It is very important to study that on time scales which can unify the continuous and discrete situations.Motivated by the above statements, we consider the following competitive and cooperation model of a satellite enterprises and a dominant enterprise with feedback controls on time scale𝕋:
(3)xΔ(t)=r1(t)-a1(t)exp{x(t)}-b1(t)[exp{y(t)}-c2(t)]2-e1(t)u(t),uΔ(t)=-α1(t)u(t)+β1(t)exp{x(t)},yΔ(t)=r2(t)-a2(t)exp{y(t)}+b2(t)[exp{x(t)}-c1(t)]2-e2(t)v(t),vΔ(t)=-α2(t)v(t)+β2(t)exp{y(t)}.As pointed in [18, 19], periodic phenomenon and almost periodic phenomenon are widespread in nature and almost periodic phenomena is more frequent than periodic phenomenon. Hence, they have been the object of intensive analysis by numerous authors. In particular, there have been extensive results on existence of almost periodic solutions of differential equations in the literature. Some of these results can be found in [20–24]. Our main purpose of this paper is by using the comparison theorem of dynamic equations on time scales and constructing a suitable Lyapunov functional to study the permanence and the existence of almost periodic solutions of (3).Remark 1.
LetN1(t)=exp{x(t)}, N2(t)=exp{y(t)}; if 𝕋=ℝ (the set of all real numbers), then (3) reduces to
(4)dN1(t)dt=N1(t)(r1(t)-a1(t)N1(t)N1(t)-b1(t)[N2(t)-c2(t)]2N1(t)-e1(t)u(t)),dN2(t)dt=N2(t)(r2(t)-a2(t)N2(t)-b1(t)[N1(t)-c1(t)]2-e2(t)v(t)),du(t)dt=-α1(t)u(t)+β1(t)N1(t),dv(t)dt=-α2(t)v(t)+β2(t)N2(t).
If 𝕋=ℤ (the set of all integers), then (3) reduces to
(5)N1(t+1)=N1(t)exp(r1(t)-a1(t)N1(t)N1(t+1)=N1(t)exp-b1(t)[N2(t)-c2(t)]2N1(t+1)=N1(t)exp-e1(t)u(t)),N2(t+1)=N2(t)exp(r2(t)-a2(t)N2(t)+b2(t)[N1(t)-c1(t)]2=N2(t)exp-e2(t)v(t)[N1(t)-c1(t)]2),Δu(t)=-α1(t)u(t)+β1(t)N1(t),Δv(t)=-α2(t)v(t)+β2(t)N2(t).Therefore, we only consider the permanence and existence of nonnegative almost periodic solution of system (3), then the permanence and almost periodic solution of (4) and (5) can be obtained as direct results of (3).Let𝕋 and 𝕋+ denote the nonempty closed subset (time scales) of ℝ. Let f be a continuous bounded function defined on 𝕋, and we denote
(6)fM=supt∈𝕋{f(t)},fm=inft∈𝕋{f(t)}.
Throughout this paper, we assume that  (H1)
ri(t), ai(t), bi(t), ci(t), ei(t), αi(t), and βi(t) are almost periodic functions on time scale 𝕋 such that
(7)0<rim≤ri(t)≤riM,0<aim≤ai(t)≤aiM,0<bim≤bi(t)≤biM,0<cim≤ci(t)≤ciM,0<eim≤ei(t)≤eiM,0<αim≤αi(t)≤αiM,0<βim≤βi(t)≤βiM,-aim∈ℛ+,-αim∈ℛ+,i=1,2,whereℛ+ is the set of positively regressive functions from 𝕋 to ℝ.


## 2. Preliminaries

In this section, we will recall some definitions and lemmas which will be used in the proof of our main results.Let𝕋 be a nonempty closed subset (time scale) of ℝ. The forward and backward jump operators σ,ρ:𝕋→𝕋 and the graininess μ:𝕋→ℝ+ (the set of all nonnegative real numbers) are defined, respectively, by
(8)σ(t)=inf{s∈𝕋:s>t},ρ(t)=sup{s∈𝕋:s<t},μ(t)=σ(t)-t.A pointt∈𝕋 is called left-dense if t>inf𝕋 and ρ(t)=t, left-scattered if ρ(t)<t, right-dense if t<sup𝕋 and σ(t)=t, and right-scattered if σ(t)>t. If 𝕋 has a left-scattered maximum m, then 𝕋k=𝕋∖{m}; otherwise, 𝕋k=𝕋. If 𝕋 has a right-scattered minimum m, then 𝕋k=𝕋∖{m}; otherwise 𝕋k=𝕋.A functionf:𝕋→ℝ is called rd-continuous provided it is continuous at right-dense points in 𝕋 (i.e., lims∈𝕋,s→tf(s)=f(t)) and its left-sided limits exist at left-dense points in 𝕋. If f is continuous at each right-dense point and each left-dense point, then f is said to be a continuous function on 𝕋. The set of continuous functions f:𝕋→ℝ will be denoted by C(𝕋).Fory:𝕋→ℝ and t∈𝕋k; we define the delta derivative of y(t), yΔ(t) to be the number (if it exists) with the property that, for any ϵ>0, there exists a neighborhood U of t such that
(9)|[y(σ(t))-y(s)]-yΔ(t)[σ(t)-s]|<ϵ|σ(t)-s|
for all s∈U. Let y be right-dense continuous. If YΔ(t)=y(t), then we define the delta integral by
(10)∫at‍y(s)Δ(s)=Y(t)-Y(a).A functionp:𝕋→ℝ is called regressive provided 1+μ(t)p(t)≠0 for all t∈𝕋k. The set of all regressive and rd-continuous functions p:𝕋→ℝ will be denoted by ℛ=ℛ(𝕋)=ℛ(𝕋,R). If 1+μ(t)p(t)>0, ∀t∈𝕋, then p is a positively regressive function from 𝕋 to ℝ. If r is a regressive function, then the generalized exponential function er is defined by
(11)er(t,s)=exp{∫st‍ξμ(τ)(r(τ))Δτ},s,t∈𝕋
with the cylinder transformation
(12)ξh(z)={log(1+hz)h,ifh≠0,z,ifh=0.Definition 2 (see [25, 26]).
Letp,q:𝕋→ℛ be two regressive functions, we define
(13)p⊕q=p+q+μpq,⊖p=-p1+μp,p⊖q=p⊕(⊖q).
Then the generalized exponential function has the following properties.Lemma 3 (see [25, 26]).
Assume thatp,q:𝕋→ℝ are two regressive functions; then  (i)
e0(t,s)≡1 and ep(t,t)≡1;(ii)
ep(σ(t),s)=(1+μ(t)p(t))ep(t,s);(iii)
ep(t,s)=1/ep(s,t)=e⊖p(s,t);(iv)
ep(t,s)ep(s,r)=ep(t,r);(v)
ep(t,s)ep(s,r)=ep(t,r);(vi)
Ifa,b,c∈𝕋, then ∫ab‍ep(c,σ(t))Δ(t)=ep(c,a)-ep(c,b).Definition 4 (see [27]).
A time scale𝕋 is called an almost periodic time scale if
(14)Π≔{τ∈ℝ:t+τ∈𝕋,∀t∈𝕋}≠{0}.Throughout this paper, we restrict our discussion on almost periodic time scales.Definition 5 (see [27]).
A functionf:𝕋→ℝ is called an almost periodic function if the ϵ-translation number set of f:
(15)T(f,ϵ,𝕋)={τ∈𝕋:|f(t+τ)-f(t)|<ϵ,∀t∈𝕋}
is a relatively dense set in 𝕋 for all ϵ>0; that is, for any given ϵ>0, there exists a number l=l(ϵ)>0 such that for any interval with length l(ϵ), there is a number τ=τ(ϵ)∈T(f,ϵ,𝕋)>0 in this interval such that
(16)|f(t+τ)-f(t)|<ϵ,t∈𝕋.Definition 6 (see [27]).
Letf:𝕋×𝔻→ℝ, where 𝔻 is an open set in ℝ and f(t,x) is said to be almost periodic in t uniformly for x∈𝔻 or uniformly almost periodic for short. If for any compact set 𝕊 in 𝔻, there exists a positive number l(ϵ,s) such that any interval of length l(ϵ,s) contains a number τ for which |f(t+τ,x)-f(t,x)|<ϵ for all t∈𝕋 and x∈𝕊, τ is called the ϵ-translation number of f(t,x).Lemma 7 (see [27]).
Af(t) is an almost periodic function if and only if for any sequence {τn′}⊂𝕋 there exists a subsequence {τn}⊂{τn′} such that f(t+τn) converges uniformly on t∈𝕋 as n→∞. Furthermore, the limit function is also an almost periodic function.Definition 8.
System (3) is said to be permanent if there exist positive constants Mi, Ni, mi, ni(i=1,2,3,4) which are independent of the solutions of the system, such that any solution (x(t),y(t),u(t),v(t))T of system (3) satisfies
(17)m1≤liminft→∞x(t)≤limsupt→∞x(t)≤M1,m2≤liminft→∞y(t)≤limsupt→∞y(t)≤M2,m3≤liminft→∞u(t)≤limsupt→∞u(t)≤M3,m4≤liminft→∞v(t)≤limsupt→∞v(t)≤M4.
Consider the following nonlinear almost periodic differential system on time scales:(18)xΔ=f(t,x)
and the product system of (18)
(19)xΔ=f(t,x),yΔ=f(t,y).Lemma 9 (see [28, 29]).
Suppose that there exists a Lyapunov functionalV(t,x,y) defined for V(t,x,y)∈(𝕋+×𝔻×𝔻,ℝ) satisfying the following conditions:  (i)
a(|x-y|)≤V(t,x,y)≤b(|x-y|), where a,b∈k,k={a∈c(ℛ+,ℛ+):a(0)=0 and a is increasing};(ii)
|V(t,x1,y1)-V(t,x2,y2)|≤L[|x1-x2|+|y1-y2|], where L>0 is a constant;(iii)
D+V(19)Δ(t,x,y)≤-cV(t,x,y), where -c∈ℛ+ and c>0.
In addition, if there exists a solutionx(t) of system (18) such that x(t)∈𝕊 for t∈𝕋+, where 𝕊⊂𝔻 is any compact subset of  𝔻, then there exists a unique uniformly asymptotically stable almost periodic solution p(t) of system (18). In particular, if f(t,x) is periodic of period ω, then there exists a unique uniformly asymptotically stable periodic solution of (18) of period ω.


## 3. Persistence

We need the following lemma to obtain the permanence of system (3).Lemma 10 (see [28, 29]).
Let-a∈ℛ+.  (i)
IfxΔ(t)≤b-ax(t), then for t>t0(20)x(t)≤x(t0)e(-a)(t,t0)+ba(1-e(-a)(t,t0)).In particular, ifa>0,  b>0, we have limsupt→+∞x(t)≤b/a.(ii)
IfxΔ(t)≥b-ax(t), then for t>t0(21)x(t)≥x(t0)e(-a)(t,t0)+ba(1-e(-a)(t,t0)).In particular, ifa>0,  b>0, we have liminft→+∞x(t)≥b/a.Proposition 11.
Assume that(H1) holds; then every solution (x(t),y(t),u(t),v(t))T of system (3) satisfies
(22)limsupt→∞x(t)≤x*,limsupt→∞y(t)≤y*,limsupt→∞u(t)≤u*,limsupt→∞v(t)≤v*,
where
(23)x*=r1M-a1ma1m,y*=r2M-a2m+b2Me2x*a2m,u*=β1Mex*α1m,v*=β2Mey*α2m.Proof.
Let(x(t),y(t),u(t),v(t))T be any solution of system (3); it follows from the first equation of system (3) and the Bernoulli inequality ex≥1+x for x∈ℝ, that
(24)xΔ(t)≤r1(t)-a1(t)expx(t)≤r1(t)-a1(t)(1+x(t))≤(r1M-a1m)-a1mx(t).
By applying (i) of Lemma 10 to the differential inequality above, we have
(25)limsupt→∞x(t)≤r1M-a1ma1m≔x*.
For any positive constant ϵ small enough, it follows from (25) that there exists a large enough T0>0 such that x(t)≤x*+ϵ for all t>T0. Then the second equation and the third equation of system (3) lead to
(26)uΔ(t)≤-α1mu(t)+β1Mex*+ϵ,yΔ(t)≤r2(t)-a2(t)expy(t)+b2(t)e2(x*+ϵ)≤r2(t)-a2(t)(y(t)+1)+b2(t)e2(x*+ϵ).
That is,
(27)uΔ(t)≤-α1mu(t)+β1Mex*+ϵ,yΔ(t)≤r2M+b2Me2(x*+ϵ)-a2m-a2my(t).
By applying Lemma 10, it follows from (27) that
(28)limsupt→∞y(t)≤r2M-a2m+b2Me2x*+ϵa2m,limsupt→∞u(t)≤β1Mex*+ϵα1m.
Letting ϵ→0 in the above inequality leads to
(29)limsupt→∞y(t)≤r2M-a2m+b2Me2x*a2m≔y*,limsupt→∞u(t)≤β1Mex*α1m≔u*.
By using similar arguments as those in the proofs of (28), it follows that
(30)limsupt→∞v(t)≤β2Mey*α2m≔v*.
This completes the proof.Proposition 12.
Assume that(H1) holds; suppose further that  (H2)
r1m-b1M(ey*-c2m)2-e1Mu*>0,r2m-e2Mv*>0, then every solution  (x(t),y(t),u(t),v(t))T  of system  (3)  satisfies(31)liminft→∞x(t)≥x*,liminft→∞y(t)≥y*,liminft→∞u(t)≥u*,liminft→∞v(t)≥v*,where(32)x*=ln(r1m-b1M[ey*-c2m]2-e1Mu*a1M),y*=ln(r2m-e2Mv*+b2m[ex*-c1M]2a2M),u*=β1mex*α1M,v*=β2mey*α2M.Proof.
Let(x(t),y(t),u(t),v(t))T be any positive solution of system (3). We first prove that
(33)liminft→∞x(t)≥x*.
For any small enough constant ϵ≥0, there exists a point T1∈𝕋 and T1≥T0 such that
(34)x(t)≤x*+ϵ,y(t)≤y*+ϵ,u(t)≤u*+ϵ,v(t)≤v*+ϵ,t>T1.
By the first equation of system (3) and condition (H1), we have
(35)xΔ(t)≥r1m-a1Mexp{x(t)}-b1M[exp{y*+ϵ}-c2m]2-e1M(u*+ϵ)≥r1m-b1M[ey*+ϵ-c2m]2-e1M(u*+ϵ)-a1Mexp{x(t)},t>T1.
We assert
(36)r1m-b1M[ey*+ϵ-c2m]2-e1M(u*+ϵ)-a1Mexp{x(t)}≤0,t>T1;
otherwise, we assume that there exists t~>T1 such that
(37)r1m-b1M[ey*+ϵ-c2m]2-e1M(u*+ϵ)-a1Mexp{x(t~)}>0,r1m-b1M[ey*+ϵ-c2m]2-e1M(u*+ϵ)-a1Mexp{x(t)}≤0,∀t∈[T1,t~),
then we get
(38)x(t~)<ln(r1m-b1M[ey*+ϵ-c2m]2-e1M(u*+ϵ)a1M),x(t)≥ln(r1m-b1M[ey*+ϵ-c2m]2-e1M(u*+ϵ)a1M),∀t∈[T1,t~).
Equation (38) implies xΔ(t~)≤0, which is a contradiction; therefore (36) holds for ∀t≥T1.
Consequently,(39)x(t)≥ln(r1m-b1M[ey*+ϵ-c2m]2-e1M(u*+ϵ)a1M),∀t≥T1,
then
(40)liminft→+∞x(t)≥ln(r1m-b1M[ey*-c2m]2-e1Mu*a1M)=x*.
Now, for any small enough ϵ>0, there exists a point T2∈𝕋 such that x(t)≥x*-ϵ,t≥T2; from the third equation of system (3), we have
(41)yΔ(t)≥r2m-a2Mexp{y(t)}+b2m[exp{x*-ϵ}-c1M]2-e2M(v*+ϵ);
by using a similar argument as that in (40), we can get
(42)liminft→+∞y(t)≥ln(r2m+b2m[ex*-c1M]2-e2Mv*a2M)≔y*.
Next we prove(43)liminft→∞u(t)≥u*,liminft→∞v(t)≥v*.
Sinceliminft→∞x(t)≥x* then there exists T∈𝕋 such that x(t)≥x*-ϵ for any ϵ>0 and t>T. By the second equation of system (3), we have
(44)uΔ(t)≥-α1Mu(t)+β1mex*-ɛ,
Applying (ii) of Lemma 10, we have
(45)liminft→∞u(t)≥β1mex*α1M≔u*;
similar to the above proof, it is easy to obtain that
(46)liminft→∞v(t)≥β2mey*α2M≔v*.
So the proof of Proposition 12 is complete.Theorem 13.
Assume that(H1)-(H2) hold; then system (3) is persistent.
We denote byΩ the set of all solutions (x(t),y(t),u(t),v(t))T of system (3) satisfying
(47)x*≤x(t)≤x*,y*≤y(t)≤y*,u*≤u(t)≤u*,v*≤v(t)≤v*,
for all t∈𝕋. It should be noticed that, from the proofs of Propositions 11 and 12, we know that the conditions of Theorem 13 hold and Ω is an invariant set of system (3).


## 4. Existence of Uniformly Asymptotically Stable Almost Periodic Solution

According to Lemma9, we first prove that there exists a bounded solution of system (3) and then construct an adaptive Lyapunov functional for system (3). The next result tells us that there exists a bounded solution of (3). To be precise, consider the following.Proposition 14.
Assume that(H1)-(H2) hold, then Ω≠∅.Proof.
By the almost periodicity ofri(t), ai(t), bi(t), ci(t), ei(t), αi(t), βi(t), i=1,2, there exists a sequence {τn}⊆𝕋 with τn→∞ as n→∞ such that
(48)ri(t+τn)⟶ri(t),ai(t+τn)⟶ai(t),bi(t+τn)⟶bi(t),ci(t+τn)⟶ci(t),ei(t+τn)⟶ei(t),αi(t+τn)⟶αi(t),βi(t+τn)⟶βi(t),n⟶∞.
Let ϵ be an arbitrary small number; it follows from Propositions 11 and 12 that there exists a positive number T3 such that
(49)x*-ϵ≤x(t)≤x*+ϵ,y*-ϵ≤y(t)≤y*+ϵ,u*-ϵ≤u(t)≤u*+ϵ,v*-ϵ≤v(t)≤v*+ϵ∀t>T3.
Denote
(50)xn(t)≔x(t+τn),yn(t)≔y(t+τn),un(t)≔u(t+τn),vn(t)≔v(t+τn),t≥T3-τn,n=1,2,….
For any positive integer m, it is easy to see that there exist sequences
(51){xn(t):n≥m},{yn(t):n≥m},{un(t):n≥m},{vn(t):n≥m}
such that the sequences {xn(t)},{yn(t)},{un(t)},{vn(t)} have subsequences, denoted by {xn(t)},{yn(t)},{un(t)},{vn(t)} again, converging on any finite interval of 𝕋 as n→∞, respectively. Thus we have functions x*(t),y*(t),u*(t),v*(t) such that
(52)xn(t)⟶x*(t),yn(t)⟶y*(t),un(t)⟶u*(t),vn(t)⟶v*(t),t∈𝕋,n⟶∞.
Combining with(53)xΔ(t)=r1(t+τn)-a1(t+τn)exp{x(t+τn)}-b1(t+τn)[exp{y(t+τn)}-c2(t+τn)]2-e1(t+τn)u(t+τn),uΔ(t)=-α1(t+τn)u(t+τn)+β1(t+τn)exp{x(t+τn)},yΔ(t)=r2(t+τn)-a2(t+τn)exp{y(t+τn)}+b2(t+τn)[exp{x(t+τn)}-c1(t+τn)]2-e2(t+τn)v(t+τn),vΔ(t)=-α2(t+τn)v(t+τn)+β2(t+τn)exp{y(t+τn)}gives
(54)x*Δ(t)=r1(t)-a1(t)exp{x*(t)}-b1(t)[exp{y*(t)}-c2(t)]2-e1(t)u*(t),u*Δ(t)=-α1(t)u*(t)+β1(t)exp{x*(t)},y*Δ(t)=r2(t)-a2(t)exp{y*(t)}+b2(t)[exp{x*(t)}-c1(t)]2-e2(t)v*(t),v*Δ(t)=-α2(t)v*(t)+β2(t)exp{y*(t)}.
We can easily see that (x*(t),y*(t),u*(t),v*(t))T is a solution of system (3) and
(55)x*-ϵ≤x*(t)≤x*+ϵ,y*-ϵ≤y*(t)≤y*+ϵ,u*-ϵ≤u*(t)≤u*+ϵ,v*-ϵ≤v*(t)≤v*+ϵ,
for t∈𝕋. Since ϵ is an arbitrary small positive number, it follows that
(56)x*≤x*(t)≤x*,y*≤y*(t)≤y*,u*≤u*(t)≤u*,v*≤v*(t)≤v*
for t∈𝕋. This completes the proof.Theorem 15.
Assume that(H1)- (H2) hold. Suppose further that  (H3)
c>0 and -c∈ℛ+, where c=min{P1-Q1μ,P2-Q2μ,R1-S1μ,R2-S2μ},
(57)P1=2a1mξ1m-b1mξ2m(2ey*-2c2M)-e1m-b2Mξ2M(2ex*-2c1m)-β1Mξ1M,Q1=a1M2ξ1M2+b1Ma1Mξ1Mξ2M[2ey*-2c2m]+a1Mξ1Me1M+b2M2ξ1M2[2ex*-2c1m]2+b2Me2Mξ1M[2ex*-2c1m]+β1M2ξ2M2+a2mb2mξ1mξ2m[2ex*-2c1M]+β1mξ1mα1m,P2=2a2mξ2m-b1mξ2m(2ey*-2c2M)-e2m-b2Mξ2M(2ex*-2c1m)-β2Mξ2M,Q2=a2M2ξ2M2+b1Ma1Mξ1Mξ2M[2ey*-2c2m]+b1M2ξ2M2[2ey*-2c2m]2+b1Me1Mξ2M[2ey*-2c2m]+a2mξ2mb2mξ1m[2ex*-2c1M]+a2Mξ2Me2M+β2M2ξ2M2+β2mξ2mα2m,Ri=2αim-βiMξiM-eim,(i=1,2),S1=e1M2+e1Mb1Mξ2M(2ey*-2c2m)+a1Mξ1Me1M+α1M2+β1mξ1mα1m,S2=e2M2+e2Mb2Mξ1M(2ex*-2c1m)+a2Mξ2Me2M+α2M2+β2mξ2mα2m,μ=supt∈𝕋μ(t),ξ1m=ex*,ξ1M=ex*,ξ2m=ey*,ξ2M=ey*;then (3) has a unique almost periodic solution X(t)=(x(t),y(t),u(t),v(t))T that is uniformly asymptotically stable and satisfies
(58)x*≤x(t)≤x*,y*≤y(t)≤y*,u*≤u(t)≤u*,v*≤v(t)≤v*for allt∈𝕋.Proof.
From Proposition14, we know that system (3) has a bounded solution satisfying
(59)x*≤x(t)≤x*,y*≤y(t)≤y*,u*≤u(t)≤u*,v*≤v(t)≤v*,t∈𝕋.
Hence,
(60)|x(t)|<A1,|y(t)|<A2,|u(t)|<A3,|v(t)|<A4,
where
(61)A1=max{x*,x*},A2=max{y*,y*},A3=max{u*,u*},A4=max{v*,v*}.
For all (x,y,u,v)∈ℝ4, we define the norm
(62)∥(x,y,u,v)∥=|x|+|y|+|u|+|v|.
Suppose that X=(x(t),y(t),u(t),v(t)) and Y=(p(t),q(t),w(t),z(t)) are any two solutions of system (3) defined on 𝕋+×Ω2; then ∥X∥≤C, ∥Y∥≤C, where C=∑14‍Ai.
Consider the product system of system (3)
(63)xΔ(t)=r1(t)-a1(t)exp{x(t)}-b1(t)[exp{y(t)}-c2(t)]2-e1(t)u(t),uΔ(t)=-α1(t)u(t)+β1(t)exp{x(t)},yΔ(t)=r2(t)-a2(t)exp{y(t)}+b2(t)[exp{x(t)}-c1(t)]2-e2(t)v(t),vΔ(t)=-α2(t)v(t)+β2(t)exp{y(t)},pΔ(t)=r1(t)-a1(t)exp{p(t)}-b1(t)[exp{q(t)}-c2(t)]2-e1(t)w(t),wΔ(t)=-α1(t)w(t)+β1(t)exp{p(t)},qΔ(t)=r2(t)-a2(t)exp{q(t)}+b2(t)[exp{p(t)}-c1(t)]2-e2(t)z(t),zΔ(t)=-α2(t)z(t)+β2(t)exp{q(t)}.
We construct a Lyapunov function defined on 𝕋+×Ω2 as follows:
(64)V(t,X,Y)=(x(t)-p(t))2+(y(t)-q(t))2+(u(t)-w(t))2+(v(t)-z(t))2;
it is easy to see that the norm
(65)∥X-Y∥=|x(t)-p(t)|+|y(t)-q(t)|+|u(t)-w(t)|+|v(t)-z(t)|
and the norm
(66)∥X-Y∥*={(x(t)-p(t))2+(y(t)-q(t))2+(u(t)-w(t))2+(v(t)-z(t))2}1/2
are equivalent; that is, there exist two constants C1>0,  C2>0 such that
(67)C1∥X-Y∥≤∥X-Y∥*≤C2∥X-Y∥,
thus
(68)(C1∥X-Y∥)2≤V(t,X,Y)≤(C2∥X-Y∥)2.
Let a,b∈C(ℝ+,ℝ+), a(x)=C12x2, b(x)=C22x2; thus condition (i) of Lemma 9 is satisfied. In addition,
(69)|V(t,X,Y)-V(t,X1,Y1)|=|(x(t)-p(t))2+(y(t)-q(t))2+(u(t)-w(t))2+(v(t)-z(t))2-(x1(t)-p1(t))2+(y1(t)-q1(t))2+(u1(t)-w1(t))2+(v1(t)-z1(t))2|≤|(x(t)-p(t))2-(x1(t)-p1(t))2|+|(y(t)-q(t))2-(y1(t)-q1(t))2|+|(u(t)-w(t))2-(u1(t)-w1(t))2|+|(v(t)-z(t))2-(v1(t)-z1(t))2|=|(x(t)-p(t))+(x1(t)-p1(t))|×|(x(t)-p(t))-(x1(t)-p1(t))|+|(y(t)-q(t))+(y1(t)-q1(t))|×|(y(t)-q(t))-(y1(t)-q1(t))|+|(u(t)-w(t))+(u1(t)-w1(t))|×|(u(t)-w(t))-(u1(t)-w1(t))|+|(v(t)-z(t))+(v1(t)-z1(t))|×|(v(t)-z(t))-(v1(t)-z1(t))|≤L{(|x(t)-x1(t)|+|y(t)-y1(t)|+|u(t)-u1(t)|+|v(t)-v1(t)|)+(|p(t)-p1(t)|+|q(t)-q1(t)|+|w(t)-w1(t)|+|z(t)-z1(t)|)}=L{∥X-X1∥+∥Y-Y1∥},
where
(70)X1=(x1(t),y1(t),u1(t),v1(t)),Y1=(p1(t),q1(t),w1(t),z1(t)),L=4max{A1,A2,A3,A4}.
Therefore, condition (ii) of Lemma 9 is satisfied. Finally, calculating the VΔ(t,X,Y) along system (63), we can obtain
(71)VΔ(t,X,Y)=[2(x(t)-p(t))+μ(t)(x(t)-p(t))Δ](x(t)-p(t))Δ+[2(y(t)-q(t))+μ(t)(y(t)-q(t))Δ](y(t)-q(t))Δ+[2(u(t)-w(t))+μ(t)(u(t)-w(t))Δ](u(t)-w(t))Δ+[2(v(t)-z(t))+μ(t)(v(t)-z(t))Δ](v(t)-z(t))Δ.
Let
(72)x¯(t)=x(t)-p(t),y¯(t)=y(t)-q(t),u¯(t)=u(t)-w(t),v¯(t)=v(t)-z(t).
Then
(73)V(t,X,Y)=x¯2(t)+y¯2(t)+u¯2(t)+v¯2(t)
and (71) is transformed into
(74)VΔ(t,X,Y)=[2x¯(t)+μ(t)x¯Δ(t)]x¯Δ(t)+[2y¯(t)+μ(t)y¯Δ(t)]y¯Δ(t)+[2u¯(t)+μ(t)u¯Δ(t)]u¯Δ(t)+[2v¯(t)+μ(t)v¯Δ(t)]v¯Δ(t)=V1(t)+V2(t)+V3(t)+V4(t),
where V1(t)=[2x¯(t)+μ(t)x¯Δ(t)]x¯Δ(t), V2(t)=[2y¯(t)+μ(t)y¯Δ(t)]y¯Δ(t), V3(t)=[2u¯(t)+μ(t)u¯Δ(t)]u¯Δ(t), V4(t)=[2v¯(t)+μ(t)v¯Δ(t)]v¯Δ(t). Using the mean value theorem, we have
(75)exp{x(t)}-exp{p(t)}=ξ1(t)(x(t)-p(t)),exp{y(t)}-exp{q(t)}=ξ2(t)(y(t)-q(t)),
where ξ1(t) lies between exp{x(t)} and exp{p(t)},ξ2(t) lies between exp{y(t)} and exp{q(t)}. Substituting (75) into (63), we have
(76)x¯Δ(t)=-a1(t)ξ1(t)x¯(t)-b1(t)[(exp{y(t)}+exp{q(t)}-2c2(t))ξ2(t)y¯(t)]-e1(t)u¯(t),y¯Δ(t)=-a2(t)ξ2(t)y¯(t)+b2(t)[(exp{x(t)}+exp{p(t)}-2c1(t))ξ1(t)x¯(t)]-e2(t)v¯(t),u¯Δ(t)=-α1(t)u¯(t)+β1(t)ξ1(t)x¯(t),v¯Δ(t)=-α2(t)v¯(t)+β2(t)ξ2(t)y¯(t),
then
(77)V1(t)={-e1(t)u¯(t))2x¯(t)+μ(t)×(-a1(t)ξ1(t)x¯(t)-b1(t)×[(exp{y(t)}+exp{q(t)}-2c2(t))ξ2(t)y¯(t)]-e1(t)u¯(t))}×(-a1(t)ξ1(t)x¯(t)-b1(t)×[(exp{y(t)}+exp{q(t)}-2c2(t))ξ2(t)y¯(t)]a1(t)ξ1(t)x¯(t)-e1(t)u¯(t))=-2a1(t)ξ1(t)x¯2(t)+μ(t)a12(t)ξ12(t)x¯2(t)+μ(t)b1(t)a1(t)ξ1(t)ξ2(t)×(exp{y(t)}+exp{q(t)}-2c2(t))y¯(t)x¯(t)+μ(t)a1(t)e1(t)ξ1(t)x¯(t)u¯(t)-2b1(t)ξ2(t)x¯(t)y¯(t)×(exp{y(t)}+exp{q(t)}-2c2(t))+μ(t)b1(t)a1(t)ξ1(t)ξ2(t)x¯(t)y¯(t)×(exp{y(t)}+exp{q(t)}-2c2(t))+μ(t)b12(t)ξ22(t)y¯2(t)×(exp{y(t)}+exp{q(t)}-2c2(t))2+μ(t)e1(t)b1(t)ξ2(t)y¯(t)u¯(t)×(exp{y(t)}+exp{q(t)}-2c2(t))-2e1(t)x¯(t)u¯(t)+μ(t)a1(t)ξ1(t)e1(t)x¯(t)u¯(t)+μ(t)e1(t)b1(t)ξ2(t)y¯(t)u¯(t)×(exp{y(t)}+exp{q(t)}-2c2(t))+μ(t)e12(t)u¯2(t)=(-2a1(t)ξ1(t)+μ(t)a12(t)ξ12(t))x¯2(t)+μ(t)e12(t)u¯2(t)+μ(t)b12(t)ξ22(t)y¯2(t)×(exp{y(t)}+exp{q(t)}-2c2(t))2+[2μ(t)b1(t)a1(t)ξ1(t)ξ2(t)×(exp{y(t)}+exp{q(t)}-2c2(t))-2b1(t)ξ2(t)×(exp{y(t)}+exp{q(t)}-2c2(t))]x¯(t)y¯(t)+2μ(t)b1(t)e1(t)ξ2(t)×(exp{y(t)}+exp{q(t)}-2c2(t))y¯(t)u¯(t)+(2μ(t)a1(t)ξ1(t)e1(t)-2e1(t))x¯(t)u¯(t)≤(μ(t)a1M2ξ1M2-2a1Mξ1M)x¯2(t)+μ(t)e1M2u¯2(t)+μ(t)b1M2ξ2M2(2ey*-2c2m)2y¯2(t)+|μ(t)b1Ma1Mξ1Mξ2M(2ey*-2c2m)-b1mξ2m(2ey*-2c2M)|(x¯2(t)+y¯2(t))+μ(t)e1Mb1Mξ2M(2ey*-2c2m)×(u¯2(t)+y¯2(t))+|μ(t)a1Mξ1Me1M-e1m|(u¯2(t)+x¯2(t)),V2(t)={2y¯(t)+μ(t)×(-a2(t)ξ2(t)y¯(t)+b2(t)×[(exp{x(t)}+exp{p(t)}-2c1(t))×ξ1(t)x¯(t)]-e2(t)v¯(t))}×(e2(t)v¯(t)-a2(t)ξ2(t)y¯(t)+b2(t)×[(exp{x(t)}+exp{p(t)}-2c1(t))ξ1(t)x¯(t)]-e2(t)v¯(t))=-2a2(t)ξ2(t)y¯2(t)+μ(t)a22(t)ξ22(t)y¯2(t)-μ(t)b2(t)ξ1(t)a2(t)ξ2(t)×(exp{x(t)}+exp{p(t)}-2c1(t))y¯(t)x¯(t)+μ(t)a2(t)e2(t)ξ2(t)y¯(t)v¯(t)+2b2(t)ξ1(t)×(exp{x(t)}+exp{p(t)}-2c1(t))y¯(t)x¯(t)-μ(t)a2(t)b2(t)ξ2(t)ξ1(t)×(exp{x(t)}+exp{p(t)}-2c1(t))y¯(t)x¯(t)+μ(t)b22(t)ξ12(t)×(exp{x(t)}+exp{p(t)}-2c1(t))2x¯2(t)-μ(t)e2(t)b2(t)ξ1(t)×(exp{x(t)}+exp{p(t)}-2c1(t))v¯(t)x¯(t)-2e2(t)v¯(t)y¯(t)+μ(t)a2(t)e2(t)ξ2(t)v¯(t)y¯(t)+μ(t)e22(t)v¯2(t)-μ(t)e2(t)b2(t)ξ1(t)×(exp{x(t)}+exp{p(t)}-2c1(t))v¯(t)x¯(t)=(μ(t)a22(t)ξ22(t)-2a2(t)ξ2(t))y¯2(t)+[2b2(t)ξ2(t)(exp{x(t)}+exp{p(t)}-2c1(t))-2μ(t)b2(t)ξ1(t)a2(t)ξ2(t)×(exp{x(t)}+exp{p(t)}-2c1(t))]x¯(t)y¯(t)+(2μ(t)a2(t)ξ2(t)e2(t)-2e2(t))v¯(t)y¯(t)+μ(t)b22(t)ξ12(t)×(exp{x(t)}+exp{p(t)}-2c1(t))2x¯2(t)-2μ(t)b2(t)ξ1(t)e2(t)×(exp{x(t)}+exp{p(t)}-2c1(t))x¯(t)v¯(t)+μ(t)e22(t)v¯2(t)+(2μ(t)a2(t)ξ2(t)e2(t)-2e2(t))y¯(t)v¯(t)≤(μ(t)a2M2ξ2M2-2a2mξ2m)y¯2(t)+μ(t)e2M2v¯2(t)+μ(t)b2M2ξ1M2×(2ex*-2c1m)2x¯2(t)+μ(t)b2Me2Mξ1M(2ex*-2c1m)(x¯2(t)+v¯2(t))+|b2Mξ2M(2ex*-2c1m)-μ(t)a2mb2mξ1mξ2m(2ex*-2c1M)|×(x¯2(t)+y¯2(t))+|μ(t)a2Mξ2Me2M-e2m|(y¯2(t)+v¯2(t)),V3(t)=(2u¯(t)+μ(t)u¯Δ(t))u¯Δ(t)=(2u¯(t)+μ(t)(-α1(t)u¯(t)+β1(t)ξ1(t)x¯(t)))×(-α1(t)u¯(t)+β1(t)ξ1(t)x¯(t))=-2α1(t)u¯2(t)+μ(t)α12(t)u¯2(t)-μ(t)β1(t)ξ1(t)α1(t)x¯(t)u¯(t)+2β1(t)ξ1(t)x¯(t)u¯(t)-μ(t)β1(t)ξ1(t)α1(t)x¯(t)u¯(t)+μ(t)β12(t)ξ12(t)x¯2(t)=(μ(t)α12(t)-2α1(t))u¯2(t)+μ(t)β12(t)ξ12(t)x¯2(t)+(2β1(t)ξ1(t)-2μ(t)β1(t)ξ1(t)α1(t))x¯(t)u¯(t)≤(μ(t)α1M2-2α1m)u¯2(t)+μ(t)β1M2ξ1M2x¯2(t)+|β1Mξ1M-μ(t)β1mξ1mα1m|(x¯2(t)+u¯2(t)),V4(t)=(2v¯(t)+μ(t)v¯Δ(t))v¯Δ(t)=(2v¯(t)+μ(t)(-α2(t)v¯(t)+β2(t)ξ2(t)y¯(t)))×(-α2(t)v¯(t)+β2(t)ξ2(t)y¯(t))=(μ(t)α22(t)-2α2(t))v¯2(t)+μ(t)β22(t)ξ22(t)y¯2(t)+(2β2(t)ξ2(t)-2μ(t)β2(t)ξ2(t)α2(t))y¯(t)v¯(t)≤(μ(t)α2M2-2α2m)v¯2(t)+μ(t)β2M2ξ2M2y¯2(t)+|β2Mξ2M-μ(t)β2mξ2mα2m|(y¯2(t)+v¯2(t)),
where
(78)ξ1m=ex*,ξ1M=ex*,ξ2m=ey*,ξ2M=ey*;
hence
(79)VΔ=V1+V2+V3+V4≤(μ(t)a1M2ξ1M2-2a1Mξ1M+|μ(t)b1Ma1Mξ1Mξ2M(2ey*-2c2m)-b1mξ2m(2ey*-2c2M)|+|μ(t)a1Mξ1Me1M-e1m|+μ(t)b2M2ξ1M2(2ex*-2c1m)2+μ(t)b2Me2Mξ1M(2ex*-2c1m)+|b2Mξ2M(2ex*-2c1m)-μ(t)a2mb2mξ1mξ2m(2ex*-2c1M)|+μ(t)β1M2ξ1M2+|β1Mξ1M-μ(t)β1mξ1mα1m|a1M2ξ1M2)x¯2(t)+(μ(t)b1M2ξ2M2(2ey*-2c2m)2+|μ(t)b1Ma1Mξ1Mξ2M(2ey*-2c2m)-b1mξ2m(2ey*-2c2M)(2ey*-2c2m)ξ2M(2ey*-2c2m)|+μ(t)e1Mb1Mξ2M(2ey*-2c2m)+μ(t)a2M2ξ2M2-2a2mξ2m+|b2Mξ2M(2ex*-2c1m)-μ(t)a2mb2mξ1mξ2m(2ex*-2c1M)|+|μ(t)a2Mξ2Me2M-e2m|+μ(t)β2M2ξ2M2+|β2Mξ2M-μ(t)β2mξ2mα2m|e2M2)y¯2(t)+(μ(t)e1M2+μ(t)e1Mb1Mξ2M(2ey*-2c2m)+|μ(t)a1Mξ1Me1M-e1m|+(μ(t)α1M2-2α1m)+|β1Mξ1M-μ(t)β1mξ1mα1m|e2M2)u¯2(t)+(μ(t)e2M2+μ(t)b2Me2Mξ1M(2ex*-2c1m)+|μ(t)a2Mξ2Me2M-e2m|+(μ(t)α2M2-2α2m)+|β2Mξ2M-μ(t)β2mξ2mα2m|e2M2)v¯2(t)≤-[(P1-Q1μ(t))x¯2(t)+(P2-Q2μ(t))y¯2(t)+(R1-S1μ(t))u¯2(t)+(R2-S2μ(t))v¯2(t)]≤-[(P1-Q1μ)x¯2(t)+(P2-Q2μ)y¯2(t)+(R1-S1μ)u¯2(t)+(R2-S2μ)v¯2(t)]≤-c(x¯2(t)+y¯2(t)+u¯2(t)+v¯2(t))=-cV(t,X,Y),
where
(80)c=min{P1-Q1μ,P2-Q2μ,R1-S1μ,R2-S2μ},P1=2a1mξ1m-b1mξ2m(2ey*-2c2M)-e1m-b2Mξ2M(2ex*-2c1m)-β1Mξ1M,Q1=a1M2ξ1M2+b1Ma1Mξ1Mξ2M[2ey*-2c2m]+a1Mξ1Me1M+b2M2ξ1M2[2ex*-2c1m]2+b2Me2Mξ1M[2ex*-2c1m]+β1M2ξ2M2+a2mb2mξ1mξ2m[2ex*-2c1M]+β1mξ1mα1m,P2=2a2mξ2m-b1mξ2m(2ey*-2c2M)-e2m-b2Mξ2M(2ex*-2c1m)-β2Mξ2M,Q2=a2M2ξ2M2+b1Ma1Mξ1Mξ2M[2ey*-2c2m]+b1M2ξ2M2[2ey*-2c2m]2+b1Me1Mξ2M[2ey*-2c2m]+a2mξ2mb2mξ1m[2ex*-2c1M]+a2Mξ2Me2M+β2M2ξ2M2+β2mξ2mα2m,Ri=2αim-βiMξiM-eim,(i=1,2),S1=e1M2+e1Mb1Mξ2M(2ey*-2c2m)+a1Mξ1Me1M+α1M2+β1mξ1mα1m,S2=e2M2+e2Mb2Mξ1M(2ex*-2c1m)+a2Mξ2Me2M+α2M2+β2mξ2mα2m.
From the condition (H3), c>0 and -c∈ℛ+, we see that the condition (iii) of Lemma 9 is satisfied. So all the conditions of Lemma 9 are satisfied. By Lemma 9, there exists a unique uniformly asymptotically stable almost periodic solution X(t)=(x(t),y(t),u(t),v(t))T of system (3) which satisfies
(81)x*≤x(t)≤x*,y*≤y(t)≤y*,u*≤u(t)≤u*,v*≤v(t)≤v*
and X(t)∈Ω for all t∈𝕋. This completes the proof.Corollary 16.
Assume that (H1)–(H3) hold. Suppose that the nonnegative coefficients ri(t), ai(t), bi(t), ci(t), ei(t), αi(t), βi(t), i=1,2 are periodic of period ω; then system (3) has a unique uniformly asymptotically stable periodic solution of period ω.


## 5. An Example

In this section, we present an example to illustrate the feasibility and effectiveness of our results.Consider the system (3) with the following coefficients
(82)r1(t)=14.5+0.5sint+0.5cos2t,r2(t)=14.5+0.4cost+0.6sin2t,a1(t)=10.5+0.2sint+0.3cosπt,a2(t)=10.5+0.4cos2t+0.1cost,b1(t)=0.3+0.1sin2t+0.1sin3t,b2(t)=0.3+0.1cost+0.1sin2t,c1(t)=1.05+0.02sin2t+0.03cos2t,c2(t)=1.05+0.03sin2t+0.02cos2t,α1(t)=1.1+0.05sint+0.05cos2t,α2(t)=1.1+0.03sint+0.07cos2t,β1(t)=0.5+0.1cos2t,β2(t)=0.5+0.1sin3t,e1(t)=0.5+0.4cos2t,e2(t)=0.5+0.4sin3t.
Let 𝕋=⋃k=0∞[k,k+0.75]. Then system (3) is persistent and has a unique uniformly asymptotically stable almost periodic solution.Proof.
By calculation, we have that(83)μ(t)={0,t∈⋃k=0∞‍[k,k+0.75),0.25,t∈⋃k=0∞‍{k+0.75},x*=0.5,y*≈0.63591,u*≈0.9892,v*≈1.1332,x*≈0.0961,y*≈0.1195,u*≈0.3670,v*≈0.3756,c≈1.0478,μ=0.25.
So all conditions are satisfied and the results follow from Theorems 13 and 15. Therefore, system (3) is persistent and has a unique uniformly asymptotically stable almost periodic solution. This completes the proof.



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*Source: 639138-2013-10-23.xml*