# Feedback Control in a Periodic Delay Single-Species Difference System

Authors: Yi Yang; Zhijun Liu
Journal: Discrete Dynamics in Nature and Society
 (2011)

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## Abstract

A periodic delay single-species difference system with
feedback control is established. With the help of analysis method and Lyapunov
function, a good understanding of the permanence and global attractivity of the
system is gained. Numerical simulations are presented to verify the validity of the
proposed criteria. Our results show that feedback control has no influence on the
permanence while it has influence on the global attractivity of the system.


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## Body

## 1. Introduction

In 1978, Ludwig et al. [1] considered a single-species system which is modeled by (1.1)x′(t)=x(t)[a-bx(t)]-h(x),
where x(t) is the density of species x at time t, a is the intrinsic growth rate, b is the competing rate, and h(x)-term represents predation. To be specific, Murray [2] took h(x) in the form of cx2(t)/(d+x2(t)), and the dynamic behavior of x(t) is then governed by (1.2)x′(t)=x(t)[a-bx(t)]-cx2(t)d+x2(t),
where cx2(t)/(d+x2(t))is an S-shaped function and a,  b, c,  andd are positive constants. For the relevant ecology sense of system (1.2), we refer the readers to [1, 2] and the references cited therein.It seems reasonable to assume that the reproduction of speciesx will not be instantaneous, but mediated by a delay required for gestation of species x. Thus, a revised version is (1.3)x′(t)=x(t)[a-bx(t-τ)]-cx2(t)d+x2(t),
where the constant delay τ is positive. It is evident that all the coefficients of system (1.3) are assumed to be constant. However, in the real world, the coefficients are not fixed constants owing to the variation of environments. The influence of a varying environment is important for evolutionary theory as the selective forces on systems in such a fluctuating medium differ from those in a stable environment. In addition, as we know, ecosystems are often disturbed by unpredictable forces and, so, species population may experience changes. In ecology, an interesting issue is whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control, we call the disturbance functions control variables. For more discussion on this direction, we refer the readers to [3–8].Considering the possible effects of fluctuating environment and feedback control on system (1.3), we obtain the following periodic system: (1.4)x′(t)=x(t)[a(t)-b(t)x(t-τ)]-c(t)x2(t)d(t)+x2(t)-g(t)x(t)u(t),u′(t)=-e(t)u(t)+h(t)x(t-τ),
where u(t) is the control variable. We assume that coefficients a(t), b(t), c(t), d(t), g(t), e(t), and h(t) are continuous and bounded above and below by positive constants and e(t)∈(0,1).Following the same idea and method in [9–12], one can easily derive the discrete analogue of system (1.4), which takes the form of (1.5)x(k+1)=x(k)exp⁡{a(k)-b(k)x(k-l)-c(k)x(k)d(k)+x2(k)-g(k)u(k)},Δu(k)=-e(k)u(k)+h(k)x(k-l),
where k∈{0,1,2,…}, l is a positive integer, all the coefficients a(k), b(k), c(k), d(k), g(k), e(k), and h(k) are positive bounded sequences and e(k)∈(0,1). Δ is the first forward difference operator Δu(k)=u(k+1)-u(k). For biological reasons, we only consider the solution (x(k),u(k)) of system (1.5) with initial value x(-l),x(-l+1),…,x(-1)≥0, (x(0),u(0))>0. The principle aim of this paper is to explore the permanence and global attractivity of system (1.5). To the best of our knowledge, no work has been done for system (1.5).For the sake of simplicity and convenience, the notations and definitions below will be used through this paper:Z+ and ⌈⌉  denote the set of nonnegative integers and the greatest integer function, respectively. We denote fU=supk∈Z+f(k),  fL=infk∈Z+f(k) for any bounded nonnegative sequence {f(k)}. Meanwhile, we denote the product of f(k) from k=α to k=β by ∏k=αk=βf(k) with the understanding that ∏k=αk=βf(k)=1 for all α>β.Definition 1.1.
System (1.5) is said to be permanent if there exist positive constants  x̲, x¯  and  u̲, u¯  such that
(1.6)x̲⩽lim⁡k→+∞inf⁡x(k)⩽lim⁡k→+∞sup⁡x(k)⩽x¯,u̲⩽lim⁡k→+∞inf⁡u(k)⩽lim⁡k→+∞sup⁡u(k)⩽u¯.Definition 1.2.
The positive solutions of system (1.5) are globally attractive if any two positive solutions (x(k),u(k)) and (x*(k),u*(k)) of system (1.5) satisfy
(1.7)lim⁡k→+∞|x(k)-x*(k)|=0,lim⁡k→+∞|u(k)-u*(k)|=0.The organization of this paper is as follows. In the next Sections2 and 3, two main results on permanence and global attractivity of system (1.5) are given, respectively. Numerical simulations are present to illustrate the validity of our main results in Section 4, and a brief conclusion is provided to summarize the paper in the final section.


## 2. Permanence

This section is concerned with the permanence of system (1.5). We first introduce the following lemmas which are useful for establishing our result.Lemma 2.1 (see [4]).
Assume the constantA>0 and y(0)>0, and further suppose that(1)
if(2.1)y(k+1)≤Ay(k)+B(k),k=1,2,…,
then for any integer p≤k,
(2.2)y(k)≤Apy(k-p)+∑i=0p-1AiB(k-i-1).
Especially, if A<1 and B(k) is bounded above with respect to M, then
(2.3)lim⁡k→+∞⁡sup⁡y(k)≤M1-A.(2)
If(2.4)y(k+1)≥Ay(k)+B(k),k=1,2,…,
then for any integer p≤k,
(2.5)y(k)≥Apy(k-p)+∑i=0p-1AiB(k-i-1).
Especially, if A<1 and B(k) is bounded below with respect to m, then
(2.6)lim⁡k→+∞inf⁡y(k)≥m1-A.Lemma 2.2 (see [10]).
Assume thaty(k) satisfies y(n1)>0 and
(2.7)y(k+1)≤y(k)exp⁡[r(k)(1-αy(k))]
for k∈[n1,+∞), where α is a positive constant and n1∈Z+. Then
(2.8)lim⁡k→+∞sup⁡y(k)≤1αrUexp⁡(rU-1).Lemma 2.3 (see [10]).
Assume thaty(k) satisfies y(n2)>0 and
(2.9)y(k+1)≥y(k)exp⁡[r(k)(1-αy(k))]
for k∈[n2,+∞),limk→+∞supy(k)≤M, where α is a positive constant such that αM>1 and n2∈Z+. Then
(2.10)lim⁡k→+∞inf⁡y(k)≥1αexp⁡[rU(1-αM)].Now, we state the permanence of system (1.5).Theorem 2.4.
System (1.5) is permanent provided that
(2.11)2aLdL>cU.Proof.
It follows from the first equation of system (1.5) that
(2.12)x(k+1)≤x(k)exp⁡{a(k)}.
Let x(k)=exp{y(k)}, then (2.12) can be rewritten as
(2.13)y(k+1)-y(k)≤a(k).
Summing both sides of (2.13) from k-l to k-1 yields
(2.14)∑i=k-lk-1[y(i+1)-y(i)]≤∑i=k-lk-1a(i)≤aUl,
which implies that
(2.15)y(k-l)≥y(k)-aUl,
and hence
(2.16)x(k-l)≥x(k)exp⁡(-aUl).
Following the above result, again from the first equation of system (1.5), we have
(2.17)x(k+1)≤x(k)exp⁡{a(k)-b(k)exp⁡(-aUl)x(k)}.
By applying Lemma 2.2 to (2.17), we can obtain that
(2.18)lim⁡k→+∞sup⁡x(k)≤exp⁡[aU(l+1)-1]bL=defx¯.
For any constant ɛ>0, it follows from (2.18) that there exists a n1∈Z+ large enough such that
(2.19)x(k)≤x¯+ɛ,fork≥n1,
which, together with the second equation of system (1.5), leads to
(2.20)u(k)≤-e(k)u(k)+h(k)(x¯+ɛ),fork≥n1+l,
that is,
(2.21)u(k+1)≤(1-eL)u(k)+hU(x¯+ɛ),fork≥n1+l.
Combing (2.21) with Lemma 2.1(1) and setting ɛ→0 in (2.21), one has
(2.22)lim⁡k→+∞sup⁡u(k)≤hUx¯eL=defu¯.
For any sufficient small constant ɛ>0, it follows from (2.18) and (2.22) that there exists n2>n1+l such that
(2.23)x(k)≤x¯+ɛ,u(k)≤u¯+ɛ,fork≥n2.
Thus, by (2.23) and the first equation of system (1.5), we have
(2.24)x(k+1)=x(k)exp⁡{a(k)-b(k)x(k-l)-c(k)x(k)d(k)+x2(k)-g(k)u(k)}≥x(k)exp⁡{a(k)-b(k)x¯-c(k)d(k)/x(k)+x(k)-g(k)u¯}≥x(k)exp⁡{a(k)-b(k)x¯-c(k)2d(k)-g(k)u¯}
for k≥n2. Let x(k)=exp{y(k)}, then (2.24) is equivalent to
(2.25)y(k+1)-y(k)≥{a(k)-b(k)x¯-c(k)2d(k)-g(k)u¯}.
For convenience of exposition, we set R(k)={a(k)-b(k)x¯-c(k)/2d(k)-g(k)u¯}. Summing both sides of (2.25) from k-l to k-1 leads to
(2.26)∑i=k-lk-1(y(i+1)-y(i))≥∑i=k-lk-1R(i).
This implies that
(2.27)y(k-l)≤y(k)-∑i=k-lk-1R(i),
and hence,
(2.28)x(k-l)≤x(k)exp⁡{-∑i=k-lk-1R(i)}.
From the second equation of system (1.5), we have
(2.29)u(k+1)=(1-e(k))u(k)+h(k)x(k-l)≤(1-eL)u(k)+h(k)x(k-l)=defAu(k)+B(k),
where A=1-eL and B(k)=h(k)x(k-l). Then, for any integer p≤k-l, combing (2.28), (2.29) with Lemma 2.1 (1), we have
(2.30)u(k)≤Apu(k-p)+∑i=0p-1AiB(k-i-1)=Apu(k-p)+∑i=0p-1Aih(k-i-1)x(k-i-1-l)≤Apu(k-p)+∑i=0p-1AihUx(k)exp⁡{-∑j=k-(i+1+l)k-1R(j)}.
Since e(k)∈(0,1), we obtain that 0<A<1. So
(2.31)0≤Apu(k-p)≤u¯Ap⟶0,forp⟶+∞.
By the assumption of Theorem 2.4, for any solution (x(k),u(k)) of system (1.5), there exists an integer q>0 such that gUApu(k-p)<(1/2)(aL-cU/2dL)  for  p>q. In fact, we can choose q=max{1,⌈logA((aL-cU)/(2dL)/2gUu¯)⌉+1}. Then
(2.32)u(k)≤Aqu(k-q)+∑i=0q-1AihUx(k)exp⁡{-∑j=k-(i+1+l)k-1R(j)}≤u¯Aq+{∑i=0q-1AihUexp⁡[-(i+1+l)RL]}x(k),fork>q.
Since ∑i=0q-1AihUexp[-(i+1+l)RL] is bounded above, let K1={∑i=0q-1AihUexp[-(i+1+l)RL]}, then we have
(2.33)u(k)≤u¯Aq+K1x(k),fork>q,
which, together with (2.24) and (2.28), leads to
(2.34)x(k+1)=x(k)exp⁡{a(k)-b(k)x(k-l)-c(k)x(k)d(k)+x2(k)-g(k)u(k)}≥x(k)exp⁡{[a(k)-c(k)(2d(k))-u¯Aqg(k)]-[K1g(k)+exp⁡(-lRL)b(k)]x(k)}≥x(k)exp⁡{S1(k)[1-(S2(k)S1(k))x(k)]},
where S1(k)=a(k)-c(k)/(2d(k))-u¯Aqg(k), S2(k)=K1g(k)+exp(-lRL)b(k). Note that
(2.35)S2US1Lx¯=gUK1+bUexp⁡(-lRL)aL-cU/(2dL)-u¯AqgU×x¯≥bUexp⁡(-lRL)aL×exp⁡(aU(l+1)-1)bL≥bUexp⁡(-laL)aL×exp⁡(aL(l+1)-1)bU≥exp⁡(aL-1)aL>1,
where we use the inequality exp(x-1)>x for x>0. Therefore, by Lemma 2.3, it follows that
(2.36)lim⁡k→+∞inf⁡x(k)≥S1LS2Uexp⁡[S1U(1-S2US1Lx¯)]=defx̲.
For any constant ɛ>0, by (2.36) we know that there exists a sufficiently large integer n3>n2+l such that
(2.37)x(k)≥x̲-ɛ,fork≥n3.
Hence, it follows from (2.37) and the second equation of system (1.5) that
(2.38)u(k+1)≥(1-eU)u(k)+hL(x̲-ɛ),fork≥n3.
Applying Lemma 2.1(2) and setting ɛ→0 in (2.38), we have
(2.39)lim⁡k→+∞inf⁡u(k)≥hLeUx̲=defu̲.
Consequently, combing (2.18), (2.22), and (2.36) with (2.39), system (1.5) is permanent. This completes the proof.


## 3. Global Attractivity

On the basis of permanence, in this section we further provide sufficient conditions that guarantee the positive solutions of system (1.5) are globally attractive. To do so, we first give the following lemma.Lemma 3.1.
For any two positive solutions(x(k),u(k)) and (x*(k),u*(k)) of system (1.5), one has
(3.1)ln⁡x(k+1)x*(k+1)=ln⁡x(k)x*(k)-b(k)[x(k)-x*(k)]-c(k)J(k)-g(k)[u(k)-u*(k)]+b(k)∑s=k-lk-1{P(s){a(s)-b(s)x*(s-l)-c(s)x*(s)d(s)+x*(s)2-g(s)u*(s)}×[x(s)-x*(s)]-Q(s)x(s){b(s)[x(s-l)-x*(s-l)]+c(s)J(s)+g(s)[u(s)-u*(s)]}{a(s)-b(s)x*(s-l)-c(s)x*(s)d(s)+x*(s)2-g(s)u*(s)}},
where
(3.2)J(s)={[d(s)-x(s)x*(s)]×[x(s)-x*(s)]}{[d(s)+x(s)2]×[d(s)+x*(s)2]},P(s)=exp⁡{η(s)[c(s)x*(s)d(s)+x*(s)2a(s)-b(s)x*(s-l)-c(s)x*(s)d(s)+x*(s)2-g(s)u*(s)]}Q(s)=exp⁡{ξ(s)[a(s)-b(s)x(s-l)-c(s)x(s)d(s)+x(s)2-g(s)u(s)]+(1-ξ(s))[a(s)-b(s)x*(s-l)-c(s)x*(s)d(s)+x*(s)2-g(s)u*(s)]}η(s),ξ(s)∈(0,1).Proof.
It follows from system (1.5) that we have
(3.3)ln⁡x(k+1)x*(k+1)-ln⁡x(k)x*(k)=ln⁡x(k+1)x(k)-ln⁡x*(k+1)x*(k)=[a(k)-b(k)x(k-l)-c(k)x(k)d(k)+x(k)2-g(k)u(k)]-[a(k)-b(k)x*(k-l)-c(k)x*(k)d(k)+x*(k)2-g(k)u*(k)]=-b(k)[x(k-l)-x*(k-l)]-c(k)J(k)-g(k)[u(k)-u*(k)].
Hence,
(3.4)ln⁡x(k+1)x*(k+1)=ln⁡x(k)x*(k)-c(k)J(k)-g(k)[u(k)-u*(k)]-b(k){[x(k)-x*(k)]-[x(k)-x(k-l)]+[x*(k)-x*(k-l)]}=ln⁡x(k)x*(k)-c(k)J(k)-g(k)[u(k)-u*(k)]-b(k)[x(k)-x*(k)]+b(k){[x(k)-x(k-l)]-[x*(k)-x*(k-l)]}.
Note that
(3.5)[x(k)-x(k-l)]-[x*(k)-x*(k-l)]=∑s=k-lk-1[x(s+1)-x(s)]-∑s=k-lk-1[x*(s+1)-x*(s)]=∑s=k-lk-1{[x(s+1)-x*(s+1)]-[x(s)-x*(s)]},(3.6)[x(s+1)-x*(s+1)]-[x(s)-x*(s)]=x(s)exp⁡⁡[a(s)-b(s)x(s-l)-c(s)x(s)d(s)+x(s)2-g(s)u(s)]-x*(s)exp⁡[a(s)-b(s)x*(s-l)-c(s)x*(s)d(s)+x*(s)2-g(s)u*(s)]-[x(s)-x*(s)]=x(s){exp⁡⁡[a(s)-b(s)x(s-l)-c(s)x(s)d(s)+x(s)2-g(s)u(s)]-exp⁡⁡[a(s)-b(s)x*(s-l)-c(s)x*(s)d(s)+x*(s)2-g(s)u*(s)]}+[x(s)-x*(s)]{exp⁡⁡[a(s)-b(s)x*(s-l)-c(s)x*(s)d(s)+x*(s)2-g(s)u*(s)]-1}.
By the mean value theorem, one has
(3.7)[x(s+1)-x*(s+1)]-[x(s)-x*(s)]=P(s){a(s)-b(s)x*(s-l)-c(s)x*(s)d(s)+x*(s)2-g(s)u*(s)}[x(s)-x*(s)]-Q(s)x(s){b(s)[x(s-l)-x*(s-l)]+c(s)J(s)+g(s)[u(s)-u*(s)]}.
Then we can easily obtain (3.1) by substituting (3.5) and (3.7) into (3.4). The proof is complete.Now, we state our main result on the global attractivity of system (1.5).Theorem 3.2.
If the assumption of Theorem2.4 holds and, further, suppose there exist positive constants ρ and δ such that
(3.8)λ=defmin⁡{ρϕ-δhU,δeL-ρφ}>0,
where ϕ, φ are defined by (3.27) and (3.28), respectively. Then the positive solutions of system (1.5) are globally attractive.Proof.
Let(x(k),u(k)) and (x*(k),u*(k)) be any two positive solutions of system (1.5). To prove Theorem 3.2, we first consider the following three steps for the first equation of system (1.5).
Step 1.
LetV11(k)=|lnx(k)-lnx*(k)|. It follows from (3.1) that
(3.9)|ln⁡x(k+1)x*(k+1)|≤|ln⁡x(k)x*(k)-b(k)[x(k)-x*(k)]|+c(k)|J(k)|+g(k)|u(k)-u*(k)|+b(k)∑s=k-lk-1{P(s)β(s)|x(s)-x*(s)|+c(s)Q(s)x(s)|J(s)|+Q(s)x(s)[b(s)|x(s-l)-x*(s-l)|+g(s)|u(s)-u*(s)|]}≤|ln⁡x(k)x*(k)-b(k)[x(k)-x*(k)]|+γ(k)|x(k)-x*(k)|+g(k)|u(k)-u*(k)|+b(k)∑s=k-lk-1{[P(s)β(s)+γ(s)Q(s)x(s)]|x(s)-x*(s)|+Q(s)x(s)[b(s)|x(s-l)-x*(s-l)|+g(s)|u(s)-u*(s)|]},
where
(3.10)β(s)=a(s)+b(s)x*(s-l)+c(s)2d(s)+g(s)u*(s),γ(s)=c(s)(d(s)+x(s)x*(s))d(s)2.
By the mean value theorem, we get
(3.11)x(k)-x*(k)=exp⁡[ln⁡x(k)]-exp⁡[ln⁡x*(k)]=θ(k)ln⁡x(k)x*(k),
that is,
(3.12)ln⁡x(k)x*(k)=1θ(k)[x(k)-x*(k)],
where θ(k) lies between x(k) and x*(k), then
(3.13)|ln⁡x(k)x*(k)-b(k)[x(k)-x*(k)]|=|ln⁡x(k)x*(k)|-|ln⁡x(k)x*(k)|+|ln⁡x(k)x*(k)-b(k)[x(k)-x*(k)]|=|ln⁡x(k)x*(k)|-1θ(k)|x(k)-x*(k)|+|1θ(k)[x(k)-x*(k)]-b(k)[x(k)-x*(k)]|=|ln⁡x(k)x*(k)|-1θ(k)|x(k)-x*(k)|+|1θ(k)-b(k)||x(k)-x*(k)|=|ln⁡x(k)x*(k)|-[1θ(k)-|1θ(k)-b(k)|]|x(k)-x*(k)|.
According to Theorem 2.4, there exists a n0∈Z+ such that {x(k),x*(k)}≤x¯  and {u(k),u*(k)}≤u¯  for all k≥n0. Therefore, for all k≥n0, we can obtain that
(3.14)ΔV11=V11(k+1)-V11(k)≤-[1θ(k)-|1θ(k)-b(k)|]|x(k)-x*(k)|+γ(k)|x(k)-x*(k)|+g(k)|u(k)-u*(k)|+b(k)∑s=k-lk-1{[P(s)β(s)+γ(s)Q(s)x¯]|x(s)-x*(s)|+b(s)Q(s)x¯|x(s-l)-x*(s-l)|+g(s)Q(s)x¯|u(s)-u*(s)|}.Step 2.
Let(3.15)V12(k)=∑s=kk-1+lb(s)∑i=s-lk-1{[P(i)β(i)+γ(i)Q(i)x¯]|x(i)-x*(i)|+b(i)Q(i)x¯|x(i-l)-x*(i-l)|+g(i)Q(i)x¯|u(i)-u*(i)|}.
Then
(3.16)ΔV12=V12(k+1)-V12(k)=∑s=k+1k+lb(s){[P(k)β(k)+γ(k)Q(k)x¯]|x(k)-x*(k)|+b(k)Q(k)x¯|x(k-l)-x*(k-l)|+g(k)Q(k)x¯|u(k)-u*(k)|}-b(k)∑i=k-lk-1{[P(i)β(i)+γ(i)Q(i)x¯]|x(i)-x*(i)|+b(i)Q(i)x¯|x(i-l)-x*(i-l)|+g(i)Q(i)x¯|u(i)-u*(i)|}.Step 3.
Let(3.17)V13(k)=x¯∑w=k-lk-1b(w+l)Q(w+l)|x(w)-x*(w)|∑s=w+l+1w+2lb(s).
By a simple calculation, it derives that
(3.18)V13=V13(k+1)-V13(k)=∑s=k+l+1k+2lb(s)b(k+l)Q(k+l)x¯|x(k)-x*(k)|-∑s=k+1k+lb(s)b(k)Q(k)x¯|x(k-l)-x*(k-l)|.
Now, we can define
(3.19)V1(k)=V11(k)+V12(k)+V13(k).
Then for all k≥n0, it follows from (3.14)–(3.18) that
(3.20)V1=V1(k+1)-V1(k)≤-[1θ(k)-|1θ(k)-b(k)|]|x(k)-x*(k)|+γ(k)|x(k)-x*(k)|+g(k)|u(k)-u*(k)|+∑s=k+1k+lb(s)g(k)Q(k)x¯|u(k)-u*(k)|+∑s=k+l+1k+2lb(s)b(k+l)Q(k+l)x¯|x(k)-x*(k)|+∑s=k+1k+lb(s){[P(k)β(k)+γ(k)Q(k)x¯]|x(k)-x*(k)|}.For the second equation of system (1.5), we will consider the following two steps.Step 1.
LetV21(k)=|u(k)-u*(k)|. Then
(3.21)V21=V21(k+1)-V21(k)=|u(k+1)-u*(k+1)|-|u(k)-u*(k)|=|(1-e(k))[u(k)-u*(k)]+h(k)[x(k-l)-x*(k-l)]|-|u(k)-u*(k)|≤(1-e(k))|u(k)-u*(k)|-|u(k)-u*(k)|+h(k)|x(k-l)-x*(k-l)|≤-e(k)|u(k)-u*(k)|+h(k)|x(k-l)-x*(k-l)|.Step 2.
LetV22(k)=∑w=k-lk-1h(w+l)|x(w)-x*(w)|. Then
(3.22)V22=V22(k+1)-V22(k)=∑w=k+1-lkh(w+l)|x(w)-x*(w)|-∑w=k-lk-1h(w+l)|x(w)-x*(w)|=h(k+l)|x(k)-x*(k)|-h(k)|x(k-l)-x*(k-l)|.
So we can define
(3.23)V2(k)=V21(k)+V22(k).
Then it follows from (3.21) and (3.22) that
(3.24)V2=V2(k+1)-V2(k)≤h(k+l)|x(k)-x*(k)|-e(k)|u(k)-u*(k)|.
Now, we could define
(3.25)V(k)=ρV1(k)+δV2(k),
where ρ, δ are mentioned in (3.8).
Obviously,V(k)≥0 for all k∈Z+ and V(n0+l)<+∞. Therefore, combining (3.20) and (3.24), for all k≥n0+l, we have
(3.26)ΔV=ρΔV1+δΔV2≤ρ{-[1θ(k)-|1θ(k)-b(k)|]+γ(k)+∑s=k+l+1k+2lb(s)b(k+l)Q(k+l)x¯+∑s=k+1k+lb(s)[P(k)β(k)+γ(k)Q(k)x¯]∑s=k+l+1k+2lb(s)b(k+l)Q(k+l)x¯}|x(k)-x*(k)|+δh(k+l)|x(k)-x*(k)|+ρ{g(k)+∑s=k+1k+lb(s)g(k)Q(k)x¯}|u(k)-u*(k)|-δe(k)|u(k)-u*(k)|≤-ρ{min⁡(bL,2x¯-bU)-γU-lbUϜ}|x(k)-x*(k)|+δhU|x(k)-x*(k)|-{δeL-ρ(gU+lbUgUQUx¯)}|u(k)-u*(k)|≤-(ρϕ-δhU)|x(k)-x*(k)|-(δeL-ρφ)|u(k)-u*(k)|≤-λ{|x(k)-x*(k)|+|u(k)-u*(k)|},
where
(3.27)Ϝ=bUQUx¯+PUβU+γUQUx¯,ϕ=min⁡(bL,2x¯-bU)-γU-lbUϜ,(3.28)φ=gU+lbUgUQUx¯.
Summing both sides of (3.26) from n0+l to k, it derives that
(3.29)∑s=n0+lk[V(s+1)-V(s)]≤-λ∑s=n0+lk{|x(s)-x*(s)|+|u(s)-u*(s)|},
which implies
(3.30)V(k+1)+λ∑s=n0+lk{|x(s)-x*(s)|+|u(s)-u*(s)|}≤V(n0+l)≤V(0).
It follows from the above inequality that
(3.31)∑s=n0+l+∞{|x(s)-x*(s)|+|u(s)-u*(s)|}≤V(0)λ<+∞,
that is,
(3.32)lim⁡k→+∞{|x(k)-x*(k)|+|u(k)-u*(k)|}=0,
and we can easily obtain that
(3.33)lim⁡k→+∞|x(k)-x*(k)|=0,lim⁡k→+∞|u(k)-u*(k)|=0,
which implies that the positive solutions of system (1.5) are globally attractive, this completes the proof.


## 4. Numerical Simulations

To verify the feasibilities of our main results, we consider a specific example:(4.1)x(k+1)=x(k)exp⁡{[0.0011+0.00011sin(k)]x(k)[4.02+0.402sin(k)]+x2(k)[0.0069+0.00069sin(k)]-[0.25-0.0025sin(k)]x(k-1)-[0.0011+0.00011sin(k)]x(k)[4.02+0.402sin(k)]+x2(k)-[0.034-0.0034sin(k)]u(k)},u(k+1)={1-[0.99+0.0099sin(k)]}u(k)+[0.042-0.0042sin(k)]x(k-1).Obviously,2aLdL≈0.0236, cU≈0.0012, that is, 2aLdL>cU. Then the assumption of Theorem 2.4 is satisfied, which indicates that system (4.1) is permanent (see Figure 1).Permanence of system (4.1) with x(-1)=0.04 and {x(0),u(0)}={0.0265,0.0008}.  (a) and  (b)  show x for k∈[0,1000] and x for k∈[980,1000], respectively.  (c) and  (d) show u for k∈[0,1000] and u for k∈[980,1000], respectively.
(a)(b)(c)(d)We assumeρ=20 and δ=1.5, by a simple computation, we have ϕ≈0.0049, φ≈0.0584, λ≈0.0286, then the sufficient conditions of Theorem 3.2 are satisfied. Thus, the positive solutions of system (4.1) are globally attractive. From Figure 2, we can see that x(k) and u(k) tend to x*(k) and u*(k), respectively.Global attractivity of system (4.1) with different initial values. x with x(-1)=0.04, {x(0),u(0)}={0.0265,0.0008} and x* with x*(-1)=0.05, {x*(0),u*(0)}={0.0274,0.0014}. (a) and (b) show (x,x*) for k∈[0,1000] and for k∈[980,1000], respectively. (c) and (d) show (u, u*) for k∈[0,1000] and for k∈[980,1000], respectively.
(a)(b)(c)(d)


## 5. Conclusion

We conclude with a brief discussion of our results. The sufficient condition required for the result of Theorem2.4 does not depend on the size of the feedback control. However, the sufficient conditions required for the result of Theorem 3.2 show that the feedback control has an effect on the global attractivity of system (1.5).



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*Source: 872738-2011-01-26.xml*