# Feedback Control Variables Have No Influence on the Permanence of a Discreten-Species Schoener Competition System with Time Delays

Authors: Qianqian Su; Na Zhang
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We consider a discreten-species Schoener competition system with time delays and feedback controls. By using difference inequality theory, a set of conditions which guarantee the permanence of system is obtained. The results indicate that feedback control variables have no influence on the persistent property of the system. Numerical simulations show the feasibility of our results.


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## Body

## 1. Introduction

The Schoener's competition  system has been studied by many scholars. Topics such as existence, uniqueness, and global attractivity of positive periodic solutions of the system were extensively investigated and many excellent results have been derived [1–5].Liu et al. [1] propose and study the global stability of the following continuous Schoener's competition model with delays:(1.1)ẋ1(t)=x1(t)[a10(t)x1(t-τ10)+k1(t)-a11(t)x1(t-τ11)-a12(t)x2(t-τ12)-c1(t)],ẋ2(t)=x2(t)[a20(t)x2(t-τ20)+k2(t)-a21(t)x1(t-τ21)-a22(t)x2(t-τ22)-c2(t)],
where {ki(t)}, {aij(t)}, and {ci(t)} are all positive bounded and continuous functions, τij(i=1,2;j=0,1,2) are positive integers, τ=max0≤i,j≤2{τij},xi(s)=ϕi(s)≥0, s∈[-τ,0]∩Z, ϕi(0)>0, i=1,2.As we all know, though most dynamic behaviors of population models are based on the continuous models governed by differential equations, the discrete time models are more appropriate than the continuous ones when the size of the population is rarely small or the population has nonoverlapping generations. It has been found that the dynamics behaviors of the discrete system are rather complex and contain  more rich dynamics than the continuous ones [6]. Recently, more and more scholars paid attention to study the discrete population models (see [4–13] and the references cited therein). For example, [5] considered the permanence and global attractivity of the following discrete Schoener's competition model with delays: (1.2)x1(n+1)=x1(n)exp{a10(n)x1(n-τ10)+k1(n)-a11(n)x1(n-τ11)-a12(n)x2(n-τ12)-c1(n)},x2(n+1)=x2(n)exp{a20(n)x2(n-τ20)+k2(n)-a21(n)x1(n-τ21)-a22(n)x2(n-τ22)-c2(n)},
where {ki(n)}, {aij(n)}, and  {ci(n)} are real positive bounded sequences, τij(i=1,2;j=0,1,2) are positive integers, τ=max0≤i,j≤2{τij},xi(s)=ϕi(s)≥0, s∈[-τ,0]∩Z, ϕi(0)>0, i=1,2.On the other hand, as was pointed out by Huo and Li [14], ecosystem in the real world is continuously disturbed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In the language of control variables, we call the disturbance functions as control variables. During the last decade, many scholars did excellent works on the feedback control ecosystems (see [14–21] and the references cited therein); however, most of those works are concerned with the continuous model and seldom did scholars considered the discrete ecosystem with feedback controls ([13, 15, 21]).Recently, Li and Yang [15] proposed the following discrete n-species Schoener competition system with time delays and feedback  controls: (1.3)xi(k+1)=xi(k)exp{ri(k)xi(k-τi)+ai(k)-∑j=1nbij(k)xj(k-τj)-ci(k)-di(k)ui(k)-ei(k)ui(k-ηi)∑j=1n},Δui(k)=-αi(k)ui(k)+βi(k)xi(k)+γi(k)xi(k-σi),
where xi(k)(i=1,2,…,n) is the density of competitive species at kth generations; ui(k) is the control variable; Δ is the first-order forward difference operator Δui(k)=ui(k+1)-ui(k), i=1,2,…,n.Throughout this paper, we assume the following.(H1)
αi(k), βi(k), γi(k), ai(k), bij(k), ri(k), ci(k), di(k), ei(k), i=1,2,…,n,  are all bounded nonnegative sequences such that(1.4)0<αil≤αiu<1,0<βil≤βiu,0<γil≤γiu,0<ail≤aiu,0<bijl≤biju,0<ril≤riu,0<cil≤ciu,0<dil≤diu,0<eil≤eiu.Here, for any bounded sequence{a(k)}, au=supk∈Na(k), al=infk∈Na(k).(H2)
τi, ηi, σi, i=1,2,…,n, are all nonnegative integers.Letτ=max{τi,ηi,σi,i=1,2,…,n}; we consider (1.3) together with the following initial conditions: (1.5)xi(θ)=φi(θ),θ∈N[-τ,0]={-τ,-τ+1,…,0},φi(0)>0,ui(θ)=ϕi(θ),θ∈N[-τ,0]={-τ,-τ+1,…,0},ϕi(0)>0.
It is not difficult to see that solutions of (1.3) and (1.5) are well defined for all k≥0 and satisfy  (1.6)xi(k)>0,ui(k)>0,k∈Z,i=1,2,…,n.By applying the comparison theorem of difference equation, they obtained a set of sufficient conditions which guarantee the permanence of the system (1.3). Their result shows that feedback control variables play important roles on the persistent property of the system (1.3). But the question is whether or not the feedback control variables have influence on the permanence of the system. The aim of this paper is to apply the analysis technique of Chen et al. [18] to establish sufficient conditions, which is independent of feedback control variables, to ensure the permanence of the system.The organization of this paper is as follow. In Section2, we will introduce several lemmas. The permanence of system (1.3) is then studied in Section 3. In Section 4, a suitable example together with its numerical simulations shows the feasibility of our results.


## 2. Preliminaries

In this section, we will introduce several useful lemmas.Lemma 2.1 (see [11]).
Assume that{x(k)} satisfies x(k)>0 and
(2.1)x(k+1)≤x(k)exp{a(k)-b(k)x(k)}
for k∈N, where a(k) and b(k) are nonnegative sequences bounded above and below by positive constants. Then
(2.2)lim supk→+∞x(k)≤1blexp(au-1).Lemma 2.2 (see [12]).
Assume that{x(k)} satisfies
(2.3)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
(2.4)lim infk→+∞x(k)≥min{albuexp(al-bux*),albu}.Lemma 2.3 (see [13]).
Assume thatA>0, y(0)>0. Suppose that
(2.5)y(k+1)≤Ay(k)+B(k),k=1,2,….
Then for any integer m≤k,(2.6)y(k)≤Amy(k-m)+∑j=0m-1AjB(k-j-1).
If A<1 and B is bounded above with respect to M, then
(2.7)lim supk→+∞y(k)≤M1-A.Lemma 2.4 (see [13]).
Assume thatA>0,  y(0)>0. Suppose that
(2.8)y(k+1)≥Ay(k)+B(k),k=1,2,….
Then for any integer m≤k,(2.9)y(k)≥Amy(k-m)+∑j=0m-1AjB(k-j-1).
If A<1 and B is bounded below with respect to P,  then
(2.10)lim infk→+∞y(k)≥P1-A.


## 3. Permanence

In this section, we establish the following permanence result for system (1.3).Theorem 3.1.
Assume that(H1) and (H2) hold. Then there exist positive constants Mi, Ni which are independent of the solutions of system (1.3) such that
(3.1)lim supk→∞xi(k)≤Mi,i=1,2,…,n,lim supk→∞ui(k)≤Ni,i=1,2,…,n.Proof.
Letx(k)=(x1(k),x2(k)⋯xn(k),u1(k),u2(k)⋯un(k))T be any positive solution of system (1.3) with initial condition (1.5). From the first equation of system (1.3), it follows that
(3.2)xi(k+1)≤xi(k)exp{ri(k)ai(k)-bii(k)xi(k-τi)}≤xi(k)exp{riuail}.
By using (3.2), one can easily obtain that
(3.3)∏j=k-τik-1xi(j+1)≤∏j=k-τik-1xi(j)exp{riuail},
that is,
(3.4)xi(k)≤xi(k-τi)exp{τiriuail}.
Substituting (3.4) into (3.2), it follows that
(3.5)xi(k+1)≤xi(k)exp{ri(k)ai(k)-bii(k)exp{-τiriuail}xi(k)}.
By applying Lemma 2.1, it follows that
(3.6)lim supk→∞xi(k)≤exp{(riu/ail)(1+τi)-1}biil=:Mi,i=1,2,…,n.
Thus, for all ɛ>0, there exists a K1>0, K1∈N, for all k>K1,xi(k)≤Mi+ɛ, i=1,2,…,n, and so
(3.7)ui(k+1)=(1-αi(k))ui(k)+βi(k)xi(k)+γi(k)xi(k-σi)≤(1-αil)ui(k)+(βiu+γiu)(Mi+ɛ)(k>K1+σi).
For 0<1-αil<1, Lemma 2.3 implies that
(3.8)lim supk→∞ui(k)≤(βiu+γiu)(Mi+ɛ)αil.
Letting ɛ→0 in the above inequality leads to
(3.9)lim supk→∞ui(k)≤(βiu+γiu)Miαil:=Ni.
The proof of   Theorem 3.1 is completed.Theorem 3.2.
Assume that(H1) and (H2) hold; assume further that
HRi:=rilMi+aiu-∑j=1,j≠inbijuMj-ciu>0,i=1,2,…,n.
Then there exist positive constants mi, ni which are independent of the solution of system (1.3), such that
(3.10)lim infk→∞xi(k)≥mi,i=1,2,…,n,lim infk→∞ui(k)≥ni,i=1,2,…,n.Proof.
Letx(k)=(x1(k),x2(k)⋯xn(k),u1(k),u2(k)⋯un(k))T be any positive solution of system (1.3) with initial condition (1.5). From Theorem 3.1, for all 0<ɛ<1, there exists a K2>K1, for all k>K2,(3.11)xi(k)≤Mi+ɛ,ui(k)≤Ni+ɛ,i=1,2,…,n,(3.12)Riɛ:=ril(Mj+ɛ)+aiu-∑j=1,j≠inbiju(Mj+ɛ)-ciu>0,i=1,2,…,n.
From (3.12), we have
(3.13)xi(k+1)≥xi(k)exp{ril(Mi+ɛ)+aiu-∑j=1,j≠inbiju(Mj+ɛ)-ciu-diuui(k)-eiuui(k-ηi)-biiuxi(k-τi)∑j=1,j≠in}≥xi(k)exp{-∑j=1nbiju(Mj+ɛ)-ciu-(diu+eiu)(Ni+ɛ)}=:xi(k)exp{Diɛ}.
for all k>K2+τ, where Diɛ=-∑j=1nbiju(Mj+ɛ)-ciu-(diu+eiu)(Ni+ɛ)<0.
Thus, forη<K, by using (3.13) we obtain
(3.14)xi(η)≤xi(k)exp{-(k-η)Diɛ}.
From the second equation of (1.3), we obtain
(3.15)ui(k+1)=(1-αi(k))ui(k)+βi(k)xi(k)+γi(k)xi(k-σi)≤(1-αil)ui(k)+βiuxi(k)+γiuxi(k-σi)≜Aiui(k)+Bi(k),
where 0<Ai=1-αil<1  and Bi(k)=βiuxi(k)+γiuxi(k-σi).
Then, Lemma2.3, (3.14), and (3.15) imply that for any m≤K,
(3.16)ui(k)≤Aimui(k-m)+∑j=0m-1AijBi(k-j-1)=Aimui(k-m)+∑j=0m-1Aij{βiuxi(k-j-1)+γiuxi(k-j-1-σi)}≤Aimui(k-m)+∑j=0m-1Aij{βiuexp{-(j+1)Diɛ}+γiuexp{-(j+1+σi)Diɛ}}xi(k)≤Aimui(k-m)+xi(k)∑j=0m-1Aij(βiu+γiu)exp{-(j+1)Diɛ}.
For 0<Ai<1,limm→∞(diu+eiu)(Ni+1)Aim=0. That is, there exists an M>0, for all m>M(3.17)(Ni+ɛ)(diu+eiu)Aim<(Ni+1)(diu+eiu)Aim≤Riɛ02≤Riɛ2.
For enough small ɛ0>0, we have
(3.18)Riɛ0:=ril(Mi+ɛ0)+aiu-∑j=1,j≠inbiju(Mj+ɛ0)-ciu<Riɛ<Ri,i=1,2,…,n.
We choose M≥max{1,logAi(Riɛ0/2(Ni+1)(diu+eiu))}, i=1,2,…,n. For fixed M, we get
(3.19)ui(k)≤AiMui(k-M)+xi(k)∑j=0M-1Aij(βiu+γiu)exp{-(j+1)Diɛ}≤AiM(Ni+ɛ)+Eiɛxi(k),
for all k≥K2+M where Eiɛ=∑j=0M-1Aij(βiu+γiu)exp{-(j+1)Diɛ}.
Substituting (3.14) (3.17) and (3.19) into (3.13), for k≥K2+M+τ, one has
(3.20)xi(k+1)≥xi(k)exp{Riɛ-biiuxi(k-τi)-diuui(k)-eiuui(k-ηi)}≥xi(k)exp{Riɛ-(eiu+diu)AiM(Ni+ɛ)-biiuxi(k-τi)-diuEiɛxi(k)-eiuEiɛxi(k-ηi)}≥xi(k)exp[Riɛ2-biiuexp{-τiDiɛ}-diuEiɛ-eiuEiɛexp{-ηiDiɛ}]xi(k):=xi(k)exp{Riɛ2-Qiɛxi(k)},
where   Qiɛ:=biiuexp{-τiDiɛ}+diuEiɛ+eiuEiɛexp{-ηiDiɛ}.
By applying Lemma2.2, it follows that
(3.21)lim infk→∞x(k)≥min{Riɛ2Qiɛexp{Riɛ2-QiɛMi},Riɛ2Qiɛ}.
Letting ɛ→0 in the above inequality leads to
(3.22)lim infk→∞x(k)≥min{Ri2Qiexp{Ri2-QiMi},Ri2Qi}:=mi.
For the above ɛ>0, there exists a K3>K2, for all k>K3, we have xi(k)≥mi-ɛ, i=1,2,…,n. Then for k>K3+σi,
(3.23)ui(k+1)=(1-αi(k))ui(k)+βi(k)xi(k)+γi(k)xi(k-σi)≥(1-αiu)ui(k)+(βil+γil)(mi-ɛ).
By applying Lemma 2.4, it follows that
(3.24)lim infk→∞ui(k)≥(βil+γil)(mi-ɛ)αiu.
Letting ɛ→0 in the above inequality leads to
(3.25)lim infk→∞ui(k)≥(βil+γil)miαiu:=ni,i=1,2,…,n.
The proof of   Theorem 3.2 is completed.Remark 3.3.
Theorems3.1 and 3.2 show that under the assumption (H1), (H2) and   (H) hold,  and  system (1.3) is permanent.


## 4. Example and Numeric Simulations

The following example lends credence to the plausibility of Theorem3.2.Example 4.1.
Consider the following system:(4.1)x1(k+1)=x1(k)exp{0.5x1(k-1)+1-x1(k-1)-0.02(sin(k)+2)x2(k-2)-0.01-(cos(k)+2.5)u1(k)-(cos(k)+1.5)u1(k-1)0.5x1(k-1)+1},x2(k+1)=x2(k)exp{1x2(k-2)+3-0.05(cos(k)+2)x1(k-1)-2x2(k-2)-0.05-(sin(k)+2)u2(k)-(sin(k)+1.1)u2(k-1)0.5x1(k-1)+1},Δu1(k)=-0.5u1(k)+0.1(sin(10+k)+1.5)x1(k)+0.1(sin(k)+1.5)x1(k-1),Δu2(k)=-13u2(k)+0.1(cos(10+k)+1.5)x2(k)+0.1(cos(k+5)+1.5)x2(k-1).
Here corresponding to system (1.3), we assume that
(4.2)r1(k)=0.5,a1(k)=1,b11(k)=1,b12(k)=0.02(sin(k)+2),c1(k)=0.01,η1=1,d1(k)=cos(k)+2.5,e1(k)=cos(k)+1.5,α1(k)=0.5,β1(k)=0.1(sin(k+10)+1.5),γ1(k)=0.1(sin(k)+1.5),σ1=1,r2(k)=1,τ2=2,a2(k)=3,b22(k)=2,b21(k)=0.05(cos(k)+2),c2(k)=0.05,η2=1,d2(k)=sin(k)+2,e2(k)=sin(k)+1.1,α2(k)=13,β2(k)=0.1(cos(k+10)+1.5),γ2(k)=0.1(cos(k+5)+1.5),σ2=1.
It is easy to see thatM1=1,M2=0.5, and
(4.3)r1lM1+a1u-b12uM2-c1u=0.21>0,r2lM2+a2u-b21uM1-c2u=335>0.
Inequalities (4.3) show  that all the conditions of Theorem 3.2 are satisfied; thus, system (1.3) is permanent. Numerical simulation from Figure 1 supports this conclusion.Dynamic behaviors of system (4.1) with initial conditions (x1(s),x2(s),u1(s),u2(s))T=(0.73,0.1,0.1,0.1)T, (0.33,0.24,0.25,0.15)T,  (0.52,0.18,0.5,0.2)T, s=-1,0.
(a)(b)(c)(d)


## 5. Discussion

In this paper we have attempted to understand the effect of feedback control variables on the permanence of system (1.3). The present work is an extension of an earlier work by Li and Yang [15]. By developing the analysis technique of  Chen et al. [18], a set of sufficient conditions are established for the permanence of system (1.3). Theorems 3.1-3.2 and the numerical simulations indicate that feedback control variables have no influence on the permanence of system (1.3).We would like to mention here that an interesting but challenging problem associated with the study of system (1.3) should be the global attractivity. We leave this for future  work.



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