# Feedback Control Variables Have No Influence on the Permanence of a DiscreteN-Species Cooperation System

Authors: Liujuan Chen; Xiangdong Xie; Lijuan Chen
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

A new set of
sufficient conditions for the permanence of a
discreteN-species
cooperation system with delays and feedback controls are obtained.
Our result shows that feedback control variables have no influence
on the persistent property of the discrete cooperative system,
thus improves and supplements the main result of F. D. Chen (2007).


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## Body

## 1. Introduction

The aim of this paper is to investigate the permanent property of the following nonautonomous discreten-species cooperation system with time delays and feedback controls of the form:(1.1)xi(k+1)=xi(k)exp{ri(k)[1-xi(k-τii)ai(k)+∑j=1,j≠inbij(k)xj(k-τij)-ci(k)xi(k-τii)]-di(k)ui(k)-ei(k)ui(k-ηi)xi(k-τii)ai(k)+∑j=1,j≠inbij(k)xj(k-τij)},Δui(k)=-αi(k)ui(k)+βi(k)xi(k)+γi(k)xi(k-σi),
where xi(k)(i=1,…,n) is the density of cooperation species xi, ui(k)(i=1,…,n) is the control variable (see [1, 2]).Throughout this paper, we assume the following.(H1)  ri(k),ai(k),bij(k),ci(k),di(k),ei(k),αi(k),βi(k),γi(k),i,j=1,2,…,n are all bounded nonnegative sequences such that(1.2)0<ril≤riu,0<ail≤aiu,0<bijl≤biju,0<cil≤ciu,0<dil≤diu,0≤eil≤eiu,0<αil≤αiu<1,0<βil≤βiu,0<γil≤γiu.
Here, for any bounded sequence {h(k)} and N={0,1,2,…}, hu=supk∈N{h(k)} and hl=infk∈N{h(k)}.(H2)  τij,ηi,σi,i,j=1,2,…,n are all nonnegative integers.Letτ=max{τij,σi,ηi,i,j=1,2,…,n}; we consider (1.1) together with the following initial conditions:(1.3)xi(θ)=φi(θ)≥0,θ∈N[-τ,0]={-τ,-τ+1,…,0},φi(0)>0,ui(θ)=ψi(θ)≥0,θ∈N[-τ,0]={-τ,-τ+1,…,0},ψi(0)>0.
It is not difficult to see that the solutions of (1.1)–(1.3) are well defined for all k≥0 and satisfy(1.4)xi(k)>0,ui(k)>0,fork∈Z,i=1,2,…,n,
where Z is the set of integer numbers.Recently, Chen [3] proposed and studied the permanence of system (1.1). Set(1.5)Mi1=exp{riu(τii+1)-1}cilriu,Mi2=(βiu+γiu)Mi1αil.
Using the comparison theorem, he obtained the following result.Theorem 1 A (see [3]).
Assume that (H1) and (H2) hold, and assume further that
(H3)(1.6)ril>(diu+eiu)Mi2,i=1,2,…,n
holds, then system (1.1) is permanent.However, as was pointed out by Fan and Wang [4], “if we use the method of comparison theorem, then the additional condition, in some extent, is necessary. But for the system itself, this condition may not necessary.’’  In [4], by establishing a new difference inequality, Fan and Wang showed that feedback control has no influence on the permanence of a single species discrete model. Their success motivated us to consider the persistent property of system (1.1). Indeed, in this paper, we will develop the analysis idea of [3] and apply the difference inequality obtained by Fan and Wang [4] to prove the following result.Theorem 1.1.
Assume that (H1) and (H2) hold, then system (1.1) is permanent.Remark 1.2.
Theorem1.1 shows that feedback control variables have no influence on the permanent property of system (1.1). It is natural to ask whether the feedback control variables have the influence on the stability property of the system or not. At present, we had difficulty to give an affirm answer to this problem, and we will leave this in our future study.We will prove Theorem1.1 in the next section. For more works on cooperative system and feedback control ecosystem, one could refer to [1–23] and the references cited therein.


## 2. Proof of Theorem1.1

Now we state several lemmas which will be useful for the proof of our main result.Lemma 2.1 (see [5, page 125]).
Consider the first-order difference equation(2.1)y(k+1)=Ay(k)+B,k=1,2,…,
where A and B are positive constants. Assume that |A|<1, for any initial value y(0), there exist a unique solution y(k) of (2.1) which can be expressed as follows: y(k)=Ak(y(0)-y*)+y*, where y*=B/(1-A). Thus, for any solution {y(k)} of system (2.1), one has
(2.2)limk→+∞y(k)=y*.Lemma 2.2 (see [5, page 241] (Comparison theorem)).
Letk∈Nk0+={k0,k0+1,…,k0+l,…},r≥0. For any fixed k, g(k,r) is a nondecreasing function, and for k≥k0, the following inequalities hold:
(2.3)y(k+1)≤g(k,y(k)),u(k+1)≥g(k,u(k)).
If y(k0)≤u(k0), then y(k)≤u(k) for all k≥k0.Lemma 2.3 (see [6, Theorem  2.1]).
Consider the following single species discrete model:(2.4)N(k+1)=N(k)exp(r(k)(1-N(k)h(k))),
where {r(k)} and {h(k)} are strictly positive sequences of real numbers defined for k∈N={0,1,2,…} and 0<hl≤hu,0<rl≤ru. Any solution of system (2.4) with initial condition N(0)>0 satisfies m≤lim infk→+∞N(k)≤lim supk→+∞N(k)≤M, where M=(hu/ru)exp(ru-1),m=hlexp(ru(1-M/hl)).Lemma 2.4 (see [7]).
Assume that{x(k)} satisfies
(2.5)x(k+1)≥x(k)exp{a(k)-b(k)x(k)},k≥N0,lim supk→+∞x(k)≤x*, and x(N0)>0, where a(k) and b(k) are nonnegative sequences bounded above and below by positive constants and N0∈N. Then
(2.6)lim infk→+∞x(k)≥min{albuexp{al-bux*},albu}.Lemma 2.5 (see [4]).
Assume thatA>0 and y(0)>0. Further suppose that(i)
(2.7)y(n+1)≤Ay(n)+B(n),n=1,2,…,
then for any integer k≤n,  y(n)≤Aky(n-k)+∑i=0k-1AiB(n-i-1). Especially, if A<1 and B is bounded above with respect to M, then lim supt→+∞y(n)≤M/(1-A);(ii)
(2.8)y(n+1)≥Ay(n)+B(n),n=1,2,…,
then for any integer k≤n, y(n)≥Aky(n-k)+∑i=0k-1AiB(n-i-1). Especially, if A<1 and B is bounded below with respect to m*, then lim inft→+∞y(n)≥m*/(1-A).Lemma 2.6.
Let(x(k),u(k))T=(x1(k),…,xn(k),u1(k),…,un(k))T be any positive solution of system (1.1), there exists a positive constant M, which is independent of the solution of system (1.1), such that
(2.9)lim supk→+∞xi(k)≤M;lim supk→+∞ui(k)≤M,i=1,2,…,n.Proof.
Let(x(k),u(k))T=(x1(k),…,xn(k),u1(k),…,un(k))T be any positive solution of system (1.1); similarly to the proof of Theorem 2.1 in [3], we have
(2.10)lim supk→+∞xi(k)≤Mi1,lim supk→+∞ui(k)≤Mi2,
where Mi1,Mi2,i=1,2,…,n are defined by (1.5). In fact, from the ith equation of (1.1), it follows that
(2.11)xi(k+1)≤xi(k)exp{ri(k)}.
Let xi(k)=exp{Ni(k)}, then (2.11) is equivalent to
(2.12)Ni(k+1)-Ni(k)≤ri(k).
Summing both sides of (2.12) from k-τii to k-1 leads to
(2.13)∑j=k-τiik-1(Ni(j+1)-Ni(j))≤∑j=k-τiik-1ri(j)≤riuτii.
We obtain that Ni(k-τii)≥Ni(k)-riuτii and hence,
(2.14)xi(k-τii)≥xi(k)exp{-riuτii}.
Substituting (2.14) to the ith equation of (1.1), it immediately follows that
(2.15)xi(k+1)≤xi(k)exp[ri(k)(1-ci(k)xi(k-τii))]≤xi(k)exp[ri(k)(1-ci(k)xi(k)exp{-riuτii})].
By applying Lemmas 2.2 and 2.3 to (2.15), we have
(2.16)lim supk→+∞xi(k)≤exp{riu(τii+1)-1}cilriu=Mi1.
For any small enough ϵ>0, it follows from (2.16) that there exists enough large K1 such that
(2.17)xi(k)≤Mi1+ϵ,fork≥K1.
This, together with (n+i)th equation of (1.1), leads to
(2.18)Δui(k)≤-αi(k)ui(k)+(βi(k)+γi(k))(Mi1+ϵ),fork≥K1+τ.
And so,
(2.19)ui(k+1)≤(1-αil)ui(k)+(βiu+γiu)(Mi1+ϵ),fork≥K1+τ.
Notice that 0<1-αil<1; it follows from (2.19) and Lemmas 2.1 and  2.2 that lim supk→+∞ui(k)≤(βiu+γiu)(Mi1+ϵ)/αil. Let ϵ→0 in above inequality, then
(2.20)lim supk→+∞ui(k)≤(βiu+γiu)Mi1αil=Mi2.
Set M=maxi{Mi1,Mi2}. The conclusion of Lemma 2.6 holds. The proof is complete.Lemma 2.7.
Let(x(k),u(k))T=(x1(k),…,xn(k),u1(k),…,un(k))T be any positive solution of system (1.1), there exists a positive constant m, which is independent of the solution of system (1.1), such that
(2.21)lim infk→+∞xi(k)≥m;lim infk→+∞ui(k)≥m.Proof.
Let(x(k),u(k))T=(x1(k),…,xn(k),u1(k),…,un(k))T be a solution of system (1.1) satisfying the initial condition (1.3). From Lemma 2.6, there exists a K1 such that for all k≥K1,  xi(t)≤2Mi1,ui(k)≤2Mi2. Thus, for k>K1+τ, from the ith equation of system (1.1), it follows that
(2.22)xi(k+1)≥xi(k)exp{ri(k)(1-xi(k-τii)ail-ciuxi(k-τii))-2(diu+eiu)Mi2}≥xi(k)exp{ril(1-2Mi1ail-2ciuMi1)-2(diu+eiu)Mi2}≥xi(k)exp{-2rilMi1ail-2rilciuMi1-2(diu+eiu)Mi2}≝xi(k)exp{ζi}.
Obviously, ζi is a negative constant. Let xi(k)=exp{Ni(k)}, the above inequality is equivalent to
(2.23)Ni(k+1)-Ni(k)≥ζi.
Summing both sides of (2.23) from k-m to k-1 leads to ∑j=k-mk-1(Ni(j+1)-Ni(j))≥ζim, and so, Ni(k-m)≤Ni(k)-ζim, therefore,
(2.24)xi(k-m)≤xi(k)exp{-ζim}.
Specially, we have
(2.25)xi(k-σi)≤xi(k)exp{-ζiσi}≤xi(k)exp{-ζiτ},xi(k-τii)≤xi(k)exp{-ζiτii}≤xi(k)exp{-ζiτ},xi(k-ηi)≤xi(k)exp{-ζiηi}≤xi(k)exp{-ζiτ}.
Substituting the first inequality into the (n+i)th equation of system (1.1) leads to
(2.26)ui(k+1)≤(1-αi(k))ui(k)+βi(k)xi(k)+γi(k)xi(k)exp{-ζiτ}≤(1-αil)ui(k)+βiuxi(k)+γiuxi(k)exp{-ζiτ}=Aiui(k)+Bixi(k),
where Ai=1-αil,Bi=βiu+γiuexp{-ζiτ}. Then Lemma 2.5 and (2.24) imply that, for any integer s≤k,
(2.27)ui(k)≤Aisui(k-s)+∑j=0s-1Bixi(k-j-1)≤Aisui(k-s)+∑j=0s-1Biexp{-ζi(j+1)}xi(k).
Note that 0<1-αil<1 and for enough large k,s, which satisfy k-s≥K1, then ui(k-s)≤2M and lims→+∞Ais=0. Thus, for k,s→+∞ and k-s≥K1,  0≤Aisui(k-s)≤2AisM→0. Then, there exists a positive integer K2>K1 such that for any positive solution of system (1.1), 2(diu+eiu)AisM≤(1/2)ril, for all s≥K2 and i=1,2,…,n. In fact, we could choose K2=maxi{|lnCi/lnAi|}, where Ci=(1/2)ril/2M(diu+eiu),i=1,2,…,n. Fix K2, for k>K2+K1, we get
(2.28)ui(k)≤AiK2ui(k-K2)+∑j=0K2-1Bixi(k-j-1)≤2AiK2M+∑j=0K2-1Biexp{-ζi(j+1)}xi(k)≝2AiK2M+Dixi(k).
And so, for k>K2+K1+τ, we have
(2.29)ui(k-ηi)≤2AiK2M+Dixi(k-ηi).
Substituting (2.28) and (2.29) into the ith equation of system (1.1), this together with (2.25) leads to (note that 2(diu+eiu)AiK2M≤(1/2)ril)
(2.30)xi(k+1)≥xi(k)exp[ri(k)(1-(1ail+ciu)exp{-ζiτ}xi(k))-di(k)(2AiK2M+Dixi(k))-ei(k)(2AiK2M+Dixi(k-ηi))(1ail+ciu)]≥xi(k)exp[ri(k)(1-(1ail+ciu)exp{-ζiτ}xi(k))-di(k)(2AiK2M+Dixi(k))-ei(k)(2AiK2M+Diexp{-ζiτ}xi(k))(1ail+ciu)]=xi(k)exp[(1ail+ciu)(ri(k)-2(di(k)+ei(k))2AiK2M)-(ri(k)(1ail+ciu)exp{-ζiτ}+di(k)Di+ei(k)Diexp{-ζiτ})xi(k)]≥xi(k)exp[(1ail+ciu)(ril-2(diu+eiu)2AiK2M)-(riu(1ail+ciu)exp{-ζiτ}+diuDi+eiuDiexp{-ζiτ})xi(k)]≥xi(k)exp[12ril-Eixi(k)],
where Ei=riu(1/ail+ciu)exp{-ζiτ}+diuDi+eiuDiexp{-ζiτ}.
By applying Lemma2.4 to (2.30), it immediately follows that
(2.31)lim infk→+∞xi(k)≥mi1.
where mi1=min{(1/2)ril/Ei,((1/2)ril/Ei)exp{(1/2)ril-EiM}}.
From (2.31), we know that there exists enough large K3>K2+K1+τ such that
(2.32)xi(k)≥12mi1,fork≥K3+τ.
This together with the (n+i)th equation of (1.1) leads to
(2.33)Δui(k)≥-αi(k)ui(k)+12(βi(k)+γi(k))mi1,fork≥K3+τ.
And so,
(2.34)ui(k+1)≥(1-αiu)ui(k)+12(βil+γil)mi1,fork≥K3+τ.
Noticing that 0<1-αiu<1 and applying Lemmas 2.1 and 2.2 to (2.34), we have
(2.35)lim infk→+∞ui(k)≥(1/2)(βil+γil)mi1αiu≝mi2.
Setting m=mini{mi1,mi2},  the conclusion of Lemma 2.7 follows. This ends the proof of Lemma 2.7.Proof of Theorem1.1.
Lemmas2.6 and 2.7 show that under the assumptions (H1) and (H2), for any positive solution (x(k),u(k))=(x1(k),…,xn(k),u1(k),…,un(k))T of system (1.1), one has
(2.36)m≤lim infk→+∞xi(k)≤lim supk→+∞xi(k)≤M,m≤lim infk→+∞ui(k)≤lim supk→+∞ui(k)≤M,
where m and M are independent of the solution of system (1.1), thus, system (1.1) is permanent. This ends the proof of Theorem 1.1.


## 3. Conclusions

Stimulated by the works of Fan and Wang [4], in this paper, we revisit the model proposed by Chen [3]. We showed that condition (H3) in [3] is not necessary to ensure the permanence of the system, which means that feedback control variables have no influence on the persistent property of system (1.1).



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