# Extinction in Nonautonomous Discrete Lotka-Volterra Competitive System with Pure Delays and Feedback Controls

Authors: Ling Zhang; Zhidong Teng; Tailei  Zhang; Shujing Gao
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

The paper discusses a nonautonomous discrete time Lotka-Volterra competitive system with pure delays and feedback controls. New sufficient conditions for which a part of then-species is driven to extinction are established by using the method of multiple discrete Lyapunov functionals.


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## Body

## 1. Introduction

The coexistence and global stability of population models are the interesting subjects in mathematical biology. Many authors have argued that the discrete time models are governed by differential equations which are more appropriate than the continuous ones to describe the dynamics of population when the population has nonoverlapping generations, a lot has been done on discrete Lotka-Volterra systems.May in [1] firstly considered the following autonomous discrete two-species Lotka-Volterra competitive system:(1.1)x(n+1)=x(n)exp{r1-a11x(n)-a12y(n)},y(n+1)=y(n)exp{r2-a21x(n)-a22y(n)}
and studied the stable points, stable cycles, and the chaos behaivor. Further, Lu and Wang  [2] studied the permanence and global attractivity of this system.Chen and Zhou in [3] considered the following periodic discrete two-species Lotka-Volterra competitive system:(1.2)x(n+1)=x(n)exp{r1(1-x(n)K1(n)-μ2(n)y(n))},y(n+1)=y(n)exp{r2(1-μ1x(n)-y(n)K2(n))}
and studied the permanence and existence of a periodic solution, and further, sufficient conditions are established on the global stability of the periodic solution.Zhang and Zhou in [4] investigated the following nonautonomous discrete two-species Lotka-Volterra competitive system:(1.3)x(n+1)=x(n)exp{r1(n)-a11(n)x(n-k1)-a12(n)y(n-k2)},y(n+1)=y(n)exp{r2(n)-a21(n)x(n-l1)-a22(n)y(n-l2)}.
Some sufficient conditions were obtained for the permanence of the system.Wang et al. in [5] studied the following general nonautonomous discrete n-species Lotka-Volterra systems:(1.4)xi(k+1)=xi(k)exp{ri(k)-∑j=1n∑l=0maijl(k)xj(k-l)-a12(n)y(n-k2)},i=1,2,…,n.
By applying the linear method and constructing the appropriate Lyapunov functionals, the author established the sufficient conditions which guarantee that any positive solution of this system is stable and attracts others, and obtained some applications of main results.Muroya in [6, 7] considered the following general nonautonomous discrete n-species Lotka-Volterra systems:(1.5)Ni(p+1)=Ni(p)exp{ci(p)-ai(p)Ni(p)-∑j=1n∑l=0maijl(p)Nj(p-kl)},p=0,1,2,…,Ni(s)=Nis≥0∀s≤0,Ni0>0,i=1,2,…,n,
and related pure delays models, that is, ai(p)=0 for all p≥0. The author obtained the permanence and the global asymptotically stable by applying mean-value conditions and the method of constructing discrete Lyapunov functionals.Liao et al. in [8] discussed the following general discrete nonautonomous n-species competitive system with feedback controls:(1.6)xi(k+1)=xi(k)exp{bi(k)-∑j=1naij(k)xj(k)-∑j=1,j≠incij(k)xi(k)xj(k)-di(k)ui(k)},k=0,1,2,…,Δui(k)=ri(k)-ei(k)ui(k)+fi(k)xi(k),i=1,2,…,n.
Some sufficient conditions are established on the permanence and the global stability of the system.Recently, we see that in [9, 10] the authors studied the following nonautonomous continuous Lotka-Volterra competitive system with pure delays and feedback controls:(1.7)ẋ(t)=xi(t)[ri(t)-∑j=1naij(t)xj(t-τij(t))-∑j=1n∫-σij0bij(t,s)xj(t+s)ds-ci(t)ui(t)-di(t)ui(t-τi(t))∑j=1n],u̇(t)=-ei(t)ui(t)+fi(t)xi(t)+gi(t)xi(t-δi(t)),i=1,2,…,n.
The sufficient conditions for which a part of the n-species is driven to extinction and the surplus part of the n-species remains permanence are established.However, we see that for general discreten-species population systems the results for which a part of the n-species is driven to extinction and the surplus part of the n-species remains the permanence, up to now, are still not obtained. Therefore, motivated by the above works, in this paper we study the following discrete nonautonomous Lotka-Volterra competitive system with pure delays and feedback controls(1.8)xi(k+1)=xi(k)exp{ri(k)-∑j=1naij(k)xj(k-τij)-∑j=1n∑l=0σijbijl(k)xj(k-l)-di(k)ui(k-τi)},ui(k+1)=(1-ei(k))ui(k)+gi(k)xi(k-δi),i=1,2,…,n.
The main purpose is to establish a criterion for which guarantee the part species xr+1,xr+2,…,xn in system (1.8) is driven to extinction. The method used in this paper is to constructing the multiple discrete Lyapunov functions. On the permanence of the surplus species x1,x2,…,xr, owing to the length of this paper, we will give the discussion in another paper.This paper is organized as follows. In next section, as preliminaries, some assumptions and useful lemmas are introduced. In Section3, the main results of this paper on the extinction of a part of the n-species of system (1.8) are established. In Section 4, an example is presented to illustrate the feasibility of our results.


## 2. Preliminaries

LetZ denote the set of all nonnegative integers. For any bounded sequence y(k), we denote y¯=supk∈Z{y(k)}, y̲=infk∈Z{y(k)}. Throughout this paper, we introduce the following assumptions.(H1)
ri(k) is a bounded sequence defined on Z; aij(k), di(k), gi(k) and bijl(k)(i,j=1,2,…,n,l=0,1,…,σij) are nonnegative bounded sequences defined on Z; τij, τi, δi, and σij(i,j=1,2,…,n) are nonnegative integers.(H2)
Sequencesei(k)(i=1,2,…,n) satisfy 0<ei(k)≤1 for all k∈Z.(H3)
There exist positive integersω and λ such that for each i=1,2,…,n(2.1)lim infn→∞∑s=nn+ω-1ri(s)>0,lim infn→∞∑s=nn+λ-1[aii(s)+∑l=0σiibiil(s)]>0.(H4)
There exists positive integerβ such that for each i=1,2,…,n(2.2)lim supn→∞∏nn+β-1(1-ei(s))<1.LetR+=[0,∞) and R+n={x=(x1,x2,…,xn):xi∈R+,i=1,2,…,n}. We denote by intR+n the interior of R+n. For any nonnegative constants a and b with a<b, we denote by [a,b]Z the set of all integers in the interval [a,b]. For some integer m≥0, we denote by D+[-m,0]Z the space of all nonnegative discrete time function ϕ:[-m,0]Z→R+n with norm ∥ϕ∥=sup{|ϕ(s)|:s∈[-m,0]Z}.Letτ=max{τij,τi,δi,σij:i,j=1,2,…,n}. Motivated by the biological background of system (1.8), in this paper we only consider all solutions of system (1.8) that satisfy the following initial conditions:(2.3)xi(s)=ϕi(s)≥0,ui(s)=ψi(s)≥0,s∈[-m,0]Z,i=1,2,…,n,
where ϕ=(ϕ1,ϕ2,…,ϕn)∈D+[-m,0]Z and ψ=(ψ1,ψ2,…,ψn)∈D+[-m,0]Z. For any ϕ,ψ∈D+[-m,0]Z, let z=(ϕ,ψ), by the fundamental theory of difference equations, system (1.8) has a unique solution (x(s,z),(s,z)) satisfying the initial condition (2.3), where x(s,z)=(x1(s,z),x2(s,z),…,xn(s,z)) and u(s,z)=(u1(s,z),u2(s,z),…,un(s,z)). It is obvious that solution (x(s,z),(s,z)) is positive, that is, xi(s,z)>0 and ui(s,z)>0(i=1,2,…,n) for all s∈Z.We first consider the following nonautonomous difference inequality system:(2.4)y(n+1)≤y(n)exp{α(n)-β(n)y(n)},n∈Z,
where α(n) and β(n) are bounded sequences defined on Z and β(n)≥0 for all n∈Z. We have the following result.Lemma 2.1 (see [11]).
Assume that there exists an integerλ>0 such that
(2.5)lim infn→∞∑k=nn+λβ(k)>0,
then there exists a constant M>0 such that for any nonnegative solution y(n) of system (2.4) with initial value y(n0)=y0≥0, where n0∈Z is some integer,
(2.6)lim supn→∞y(n)<M.Next, we consider the following nonautonomous linear difference equation:(2.7)v(n+1)=γ(n)v(n)+ω(n),
where γ(n) and ω(n) are nonnegative bounded sequences defined on Z. We have the following results.Lemma 2.2 (see [11]).
Assume that there exists an integerλ>0 such that
(2.8)lim supn→∞∏k=nn+λγ(k)<1,
then there exists a constant M>0 such that for any nonnegative solution v(n) of system (2.7) with initial value v(n0)=v0≥0, where n0∈Z is some integer,
(2.9)lim supn→∞v(n)<M.Lemma 2.3 (see [11]).
Assume that the conditions of Lemma2.2 hold, then for any constants ε>0 and M1>0 there exist positive constants δ̂=δ̂(ε) and n̂=n̂(ε,M1) such that for any n̂0∈Z and 0≤v0≤M1, when ω(n)<δ̂ for all n≥n̂0, one has
(2.10)v(n,n̂0,v0)<ε∀n≥n̂0+n̂,
where v(n,n̂0,v0) is the solution of (2.7) with initial value v(n̂0)=v0.Lemma 2.4.
Assume that assumptions(H1)–(H4) hold, then there exists a constant M0>0 such that
(2.11)lim supk→∞xi(k)<M0,lim supk→∞ui(k)<M0,i=1,2,…,n
for any positive solution (x1(k),x2(k),…,xn(k),u1(k),u2(k),…,un(k)) of system (1.8).Proof.
Let(x1(k),x2(k),…,xn(k),u1(k),u2(k),…,un(k)) be any positive solution of system (1.8). For each i∈{1,2,…,n}, we have
(2.12)xi(k+1)=xi(k)exp{ri(k)-∑j=1naij(k)xj(k-τij)-∑j=1n∑l=0σijbijl(k)xj(k-l)-di(k)ui(k-τi)}≤xi(k)exp{ri(k)},
then, for any integer k≥0 and θ≤0 with k+θ≥0, summing inequality (2.12) from k+θ to k-1, we obtain
(2.13)xi(k+θ)≥xi(k)exp{-∑s=k+θk-1ri(s)}.
Therefore, for any integer k≥τ, from (2.13) and the first equation of system (1.8) we obtain
(2.14)xi(k+1)≤xi(k)exp{ri(k)-aii(k)xi(k-τii)-∑l=0σiibiil(k)xi(k-l)}≤xi(k)exp{ri(k)-aii(k)xi(k)exp{-∑s=k+θk-1ri(s)}-∑l=0σiibiil(k)xi(k)exp{-∑s=k-lk-1ri(s)}}.
Since for any k≥τ and l∈[0,τ]Z(2.15)∑s=k+θk-1ri(s)≤τriu,
where riu=supk∈Zri(k), we have from (2.14),
(2.16)xi(k+i)≤xi(k)exp{ri(k)-hi(aii(k)+∑l=0σiibiil(k))xi(k)}
for any k≥τ, where hi=exp{-τriu}.
We consider the following auxiliary equation:
(2.17)yi(k+1)≤yi(k)exp{ri(k)-βi(k)yi(k)},k∈Z,
where βi(k)=hi(aii(k)+∑l=0σiibiil(k)), then by assumption (H3) and applying Lemma 2.1 there exists a constant Ni>0 such that
(2.18)lim supk→∞xi(k)<Ni
for any positive solution yi(k) of (2.17). Therefore, from the comparison theorem of difference equation, we finally obtain
(2.19)lim supk→∞xi(k)<Ni.
Further form inequality (2.19), there exists a positive constant ki≥τ such that
(2.20)xi(k)<Ni,xi(k-δi)<Ni∀k≥ki.
Thus, from the second equation of system (1.8), we obtain
(2.21)ui(k+1)≤(1-ei(k))ui(k)+(fi(k)+gi(k))Ni
for all k≥ki. We consider the following auxiliary equation:
(2.22)vi(k+1)=(1-ei(k))vi(k)+(fi(k)+gi(k))Ni,
then by assumption (H4) and applying Lemma 2.2, there exists a constant Ni* such that
(2.23)lim supk→∞vi(k)<Ni*
for any positive solution vi(k) of (2.22). Let vi*(k) be the solution of (2.22) with initial value vi*(ki)=ui(ki), then from the comparison theorem of difference equation, we have ui(k)≤vi*(ki) for all k≥ki. Thus, we finally obtain
(2.24)lim supk→∞ui(k)<Ni*.
Let N=max1≤i≤n{Ni,Ni*}, then from (2.19) and (2.24) we finally see that the conclusions of Lemma 2.4 hold.


## 3. Main Results

In this section, we discuss the extinction of the part of speciesxr+1,…,xn of system (1.8). Define functions as follows:(3.1)Aij(k)=aij(k+τij)+∑l=0σijbijl(k+l),Di(k)=di(k+τi),Ri(k)=gi(k+δi),
where i,j=1,2,…,n.Theorem 3.1.
Assume that assumptions(H1)–(H4) hold and there exists an integer 1≤r<n such that for any h>r there exists an integer ih<h such that
(3.2)lim supk→∞∑s=kk+ω-1rh(s)∑s=kk+ω-1rih(s)<lim infk→∞Ahj(k)Aihj(k)∀j≤h,lim infk→∞Dh(k)eh(k)>lim supk→∞(Aihh(k)Rh(k)lim supk→∞∑s=kk+ω-1rh(s)∑s=kk+ω-1rih(s)-Ahh(k)Rh(k)),lim supk→∞Dih(k)eih(k)<lim infk→∞(Ahih(k)Rih(k)lim infk→∞∑s=kk+ω-1rih(s)∑s=kk+ω-1rh(s)-Aihih(k)Rih(k)),
then for each i=r+1,…,n one has
(3.3)limk→∞xi(k)=0,limk→∞ui(k)=0,∑k=0∞xi(k)<∞
for any positive solution (x1(k),x2(k),…,xn(k),u1(k),u2(k),…,un(k)) of system (1.8).Proof.
From assumption(H2), there exist constant η0>0 and integer K0>0 such that
(3.4)∑s=kk+ω-1ri(s)≥η0∀k≥K0.
We first prove that limk→∞xn(k)=0. Let h=n and ih=p. From conditions (3.2) we can find positive constants αn1,αn2,αn3,αn4,εn and integer Kn≥K0 such that
(3.5)∑s=kk+ω-1rn(s)∑s=kk+ω-1rp(s)<αn1αn2-εn<αn1αn2<Anj(k)Apj(k),Dn(k)en(k)>αn4αn2>αn1Apn(k)-αn2Ann(k)αn2Rn(k),Dp(k)ep(k)<αn3αn1<αn2Anp(k)-αn1App(k)αn1Rp(k)
for all k≥Kn and j=1,2,…,n. Consequently,
(3.6)∑s=kk+ω-1(-αn1rp(s)+αn2rn(s))<-εnαn2η0,(3.7)αn1Apj(k)-αn2Anj(k)<0,(3.8)αn4en(k)-αn2Dn(k)<0,(3.9)αn1App(k)-αn2Anp(k)+αn3Rp(k)<0,(3.10)-αn3ep(k)+αn1Dp(k)<0(3.11)αn1Apn(k)-αn2Ann(k)-αn4Rn(k)<0
for all k≥Kn and j=1,2,…,n.
Let(x(k),u(k))=(x1(k),x2(k),…,xn(k),u1(k),u2(k),…,un(k)) be any positive solution of system (1.8). Constructing the following discrete Lyapunov functional
(3.12)Vn(k)=xp-αn1(k)xnαn2(k)×exp{∑j=1n∑s=k-τpjk-1αn3up(k)-αn4un(k)+αn1∑j=1n∑s=k-τpjk-1apj(s+τpj)xj(s)+αn1∑j=1n∑l=0σpj∑s=k-lk-1bpjl(s+l)xj(s)+αn1∑s=k-τpk-1dp(s+τp)up(s)-αn2∑j=1n∑s=k-τnjk-1anj(s+τnj)xj(s)-αn2∑j=1n∑l=0σnj∑s=k-lk-1bnjl(s+l)xj(s)-αn2∑s=k-τnk-1dn(s+τn)un(s)+αn3∑s=k-δpk-1gp(s+δp)xp(s)-αn4∑s=k-δnk-1gn(s+δn)xn(s)}.
By calculating, we obtain
(3.13)Vn(k+1)Vn(k)=exp{∑l=0σpj-αn1[rp(k)-∑j=1napj(k)xj(k-τpj)-∑j=1n∑l=0σpjbpjl(k)xj(k+l)-dp(k)up(k-τp)]+αn2[rn(k)-∑j=1nanj(k)xj(k-τnj)-∑j=1n∑l=0σnjbnjl(k)xj(k+l)-dn(k)un(k-τn)]+αn3[-ep(k)up(k)+gp(k)xp(k-δp)]-αn4[-en(k)un(k)+gn(k)xn(k-δn)]+αn1∑j=1n[apj(k+τpj)xj(k)-apj(k)xj(k-τpj)]+αn1∑j=1n∑l=0σpj[bpjl(k+l)xj(k)-bpjl(k)xj(k-l)]+αn1[dp(k+τp)up(k)-dp(k)up(k-τp)]-αn2∑j=1n[anj(k+τnj)xj(k)-anj(k)xj(k-τnj)]-αn2∑j=1n∑l=0σnj[bnjl(k+l)xj(k)-bnjl(k)xj(k-l)]-αn2[dn(k+τn)un(k)-dn(k)un(k-τn)]+αn3[gp(k+δp)xp(k)-gp(k)xp(k-δp)]-αn4[gn(k+δn)xn(k)-gn(k)xn(k-δn)]∑j=1n∑l=0σpj}=exp{∑j≠pn-1-αn1rp(k)+αn2rn(k)-(αn3ep(k)-αn1Dp(k))up(k)-(αn2Dn(k)-αn4en(k))un(k)+∑j≠pn-1(αn1Apj(k)-αn2Anj(k))xj(k)+(αn1App(k)-αn2Anp(k)-αn3Rp(k))xp(k)+(αn1Apn(k)-αn2Ann(k)-αn4Rn(k))xn(k)∑j≠pn-1}.
From inequalities (3.7)–(3.11), we can obtain
(3.14)Vn(k+1)≤Vn(k)exp{-αn1rp(k)+αn2rn(k)}∀k≥Kn.
For any k>Kn, we choose an integer m≥0, such that k∈[Kn+mω,Kn+(m+1)ω), then from (3.6) and (3.14) we further have
(3.15)Vn(k)≤Vn(Kn)exp{∑s=Knk-1(-αn1rp(s)+αn2rn(s))}≤Vn(Kn)exp{(∑s=KnKn+mω-1+∑s=Kn+mωk-1)(-αn1rp(s)+αn2rn(s))}≤Vn(Kn)exp{-εnαn2η0m+Mn}≤Vn(Kn)exp{-λnk+Mn*},
where
(3.16)λn=εnαn2η0ω,Mn*=Mn+εnαn2η0(1+Knω),Mn=supk∈Z(-αn1rp(k)+αn2rn(k))ω.
On the other hand, from assumptions (H1), we have
(3.17)Vn(k)≥xp-αn1(k)xnαn2(k)×exp{-αn4un(k)-∑j=1nαn2∑s=k-τnjk-1anj(s+τnj)xj(s)-∑j=1nαn2∑l=0σnj∑s=k-lk-1bnjl(s+l)xj(s)-αn2∑s=k-τnk-1dn(s+τn)un(s)-αn4∑s=k-δnk-1gn(s+δn)xn(s)}≥xp-αn1(k)xnαn2(k)×exp{-αn4M0-αn2τ∑j=1nanj¯M0-αn2∑j=1n∑l=0σnjbnjl¯lM0-αn2τdn¯M0-αn4τgn¯M0}
for all k≥τ, where constant M0 is given in Lemma 2.4. Hence, there exist a positive constant Qn>0 such that
(3.18)Vn(k)≥xp-αn1(k)xnαn2(k)Qn∀k≥τ.
From (3.15) and (3.18), we obtain
(3.19)xn(k)≤[Qn-1xpαn1(k)Vn(Kn)exp{-λnk+Mn*}]1/αn2≤Qn*exp{-λn*k}
for all k≥Kn+τ, where
(3.20)Qn*=supk∈Z[xpαn1(k)Vn(Kn)Qn-1expMn*]1/αn2,λn*=λnαn2.
From (3.19), we finally obtain
(3.21)limk→∞xn(k)=0,∑k=0∞xn(k)<∞.
Next, we consider the second equation of system (1.8), applying Lemma 2.3 we can easily obtain un(k)→0 as k→∞.
Now, we suppose that for anyt>r, we have obtained
(3.22)limk→∞xi(k)=0,limk→∞ui(k)=0,∑k=0∞xn(k)<∞
for all i>t. We further will prove
(3.23)limk→∞xt(k)=0,∑k=0∞xt(k)<∞.
From conditions (3.2), we can choose positive constants αt1, αt2, αt3, αt4, εt and integer Kt≥Kn such that
(3.24)∑s=kk+ω-1rt(s)∑s=kk+ω-1rq(s)<αt1αt2-εt<αt1αt2<Atj(k)Aqj(k),Dt(k)et(k)>αt4αt2>αt1Aqn(k)-αt2Att(k)αt2Rt(k),Dq(k)eq(k)<αt3αt1>αt2Atq(k)-αt1Aqq(k)αt1Rq(k)
for all k≥Kt and j=1,2,…,t, where ih=q. Consequently,
(3.25)∑s=kk+ω-1(-αt1rq(s)+αt2rt(s))<-εtαt2η0,(3.26)αt1Aqj(k)-αt2Atj(k)<0,(3.27)αt4et(k)-αt2Dt(k)<0,(3.28)αt1Aqq(k)-αt2Atq(k)+αt3Rq(k)<0,(3.29)-αt3eq(k)+αt1Dq(k)<0(3.30)αt1Aqt(k)-αt2Att(k)-αt4Rt(k)<0
for all k≥Kt and j=1,2,…,t. Constructing the following discrete Lyapunov functional
(3.31)Vt(k)=xq-αt1(k)xtαt2(k)×exp{∑j=1nαt3uq(k)-αt4ut(k)+αt1∑j=1n∑s=k-τqjk-1aqj(k+τqj)xj(s)+αt1∑j=1n∑l=0σqj∑s=k-lk-1bqjl(k+l)xj(s)+αt1∑s=k-τqk-1dq(k+τq)uq(s)-αt2∑j=1n∑s=k-τtjk-1atj(k+τtj)xj(s)-αt2∑j=1n∑l=0σtj∑s=k-lk-1btjl(k+l)xj(s)-αt2∑s=k-τtk-1dt(k+τt)ut(s)+αt3∑s=k-δqk-1gq(k+δq)xq(s)-αt4∑s=k-δtk-1gt(k+δt)xt(s)}.
Calculating Vt(k+1)/Vt(k), similarly to Vn(k), we can obtain
(3.32)Vt(k+1)Vt(k)=exp{∑j≠q,tn-1-αt1rq(k)+αt2rt(k)-(αt3eq(k)-αt1Dq(k))uq(k)+(-αt2Dt(k)+αt4et(k))ut(k)+∑j≠q,tn-1(αt1Aqj(k)-αt2Atj(k))xj(k)+(αt1Aqq(k)-αt2Atp(k)-αt2Rq(k))xq(k)+(αt1Aqt(k)-αt2Att(k)-αt4Rt(k))xt(k)∑j≠q,tn-1}
for all k≥Kt. From inequalities (3.26)–(3.30), we further obtain
(3.33)Vt(k+1)≤Vt(k)exp{-αt1rq(k)+αt2rt(k)+P(k)}
for all k≥Kt, where
(3.34)P(k)=∑j=t+1n(αt1Aqj(k)+αt2Atj(k))xj(k).
From (3.19) we have P(k)→0 as k→∞. Hence,
(3.35)limk→∞∑s=kk+ω-1P(s)=0.
Thus, from (3.22) we can obtain that there exists an integer Kt*≥Kt such that
(3.36)∑s=kk+ω-1(-αt1rq(s)+αt2rt(s)+P(s))<-12εtαn2η0
for all k>Kt*. By calculating, from (3.33) we obtain
(3.37)Vt(k)≤Vt(Kt*)exp{∑s=Kt*k-1(-αt1rq(s)+αt2rt(s)+P(s))}.
From this, a similar argument as in the proof of (3.15)–(3.19), we further can obtain
(3.38)xt(k)≤Qt*exp{-λt*k}∀k≥Kt*,
where Qt* and λt* are two positive constants. From (3.38), we finally obtain
(3.39)limk→∞xt(k)=0,∑k=0∞xt(k)<∞.
Next, we consider the second equation of system (1.8), applying Lemma 2.3 we can easily obtain ut(k)→0 as k→∞.
Finally, according to the induction principle, we have
(3.40)limk→∞xi(k)=0,limk→∞ui(k)=0,∑k=0∞xi(k)<∞∀i>r.
This completes the proof of Theorem 3.1.As consequences of Theorem3.1, we consider the following two special cases of system (1.8).Case 1.
Nondelayed nonautonomous discreten-species Lotka-Volterra competitive systems with feedback controls
(3.41)xi(k+1)=xi(k)exp{ri(k)-∑j=1naij(k)xj(k)-di(k)ui(k)},ui(k+1)=(1-ei(k))ui(k)+gi(k)xi(k),i=1,2,…,n.
For system (3.41), assumptions (H1) and (H3) become into the following form(H1′)
ri(k) is a bounded sequence defined on Z; aij(k), di(k) and gi(k)(i,j=1,2,…,n) are nonnegative bounded sequences defined on Z.(H3′)
There exist positive integersω and λ such that for each i=1,2,…,n(3.42)lim infn→∞∑s=nn+ω-1ri(s)>0,lim infn→∞∑s=nn+λ-1aii(s)>0.Directly from Theorem3.1, we have the following corollary.Corollary 3.2.
Assume that assumptions(H1′), (H2), (H3′) and (H4) hold and there exists a integer 1≤r<n such that for any h>r there exists an integer ih<h such that
(3.43)lim supk→∞∑s=kk+ω-1rh(s)∑s=kk+ω-1rih(s)<lim infk→∞ahj(k)aihj(k)∀j≤h,lim infk→∞dh(k)eh(k)>lim supk→∞(aihh(k)gh(k)lim supk→∞∑s=kk+ω-1rh(s)∑s=kk+ω-1rih(s)-ahh(k)gh(k)),lim supk→∞dih(k)eih(k)<lim infk→∞(ahih(k)gih(k)lim infk→∞∑s=kk+ω-1rih(s)∑s=kk+ω-1rh(s)-aihih(k)gih(k)),
then for each i=r+1,…,n one has
(3.44)limk→∞xi(k)=0,limk→∞ui(k)=0,∑k=0∞xi(k)<∞
for any positive solution (x1(k),x2(k),…,xn(k),u1(k),u2(k),…,un(k)) of system (3.41).Case 2.
Pure delayed nonautonomous discreten-species Lotka-Volterra competitive systems without feedback controls
(3.45)xi(k+1)=xi(k)exp{ri(k)-∑j=1naij(k)xj(k-τij)-∑j=1n‍∑l=0σijbijl(k)xj(k-l)},i=1,2,…,n.
For system (3.41), assumption (H1) becomes into the following form.(H1′′)
ri(k) is a bounded sequence defined on Z; aij(k) and bijl(k)(i,j=1,2,…,n,l=0,1,…,σij) are nonnegative bounded sequences defined on Z; τij and σij(i,j=1,2,…,n) are nonnegative integers.Directly from Theorem3.1, we have the following corollary.Corollary 3.3.
Assume that assumptions(H1′′) and (H3) hold and there exists an integer 1≤r<n such that for any h>r there exists an integer ih<h such that
(3.46)lim supk→∞∑s=kk+ω-1rh(s)∑s=kk+ω-1rih(s)<lim infk→∞Ahj(k)Aihj(k)∀j≤h,
then for each i=r+1,…,n one has
(3.47)limk→∞xi(k)=0,∑k=0∞xi(k)<∞
for any positive solution (x1(k),x2(k),…,xn(k)) of system (3.45).Remark 3.4.
By comparison, we easily see that the results obtained in this paper are a very good extension of the corresponding results obtained in [9] on the extinction of species for nonautonomous continuous Lotka-Volterra competitive system with pure-delays and feedback controls to discrete ones.


## 4. An Example

We consider the following periodic discrete three-species Lotka-Volterra competitive system with pure-delays and feedback controls:(4.1)xi(k+1)=xi(k)exp{ri(k)-∑j=13aij(k)xj(k-τij)-di(k)ui(k-τi)},ui(k+1)=(1-ei(k))ui(k)+gi(k)xi(k-δi),i=1,2,3.
where(4.2)r1(k)=12+sin(13+π2k),r2(k)=12+sin(14+π2k),r3(k)=3(14+sin(15+π2k)),a11(k)=13(1+cos2(13+π2k)),a12(k)=16+sin2(13+π2k),a13(k)=14+cos2(13+π2k),a21(k)=1+cos2(13+π2k),a22(k)=12+cos2(13+π2k),a31(k)=4+sin2(13+π2k),a32(k)=92+sin2(13+π2k),a33(k)=8+sin2(13+π2k),d1(k)=110(1+sin2(13+π2k)),d2(k)=12+sin2(13+π2k),d3(k)=2+cos2(13+π2k),e1(k)=13(1+cos2(13+π2k)),e2(k)=14(1+sin2(13+π2k)),e3(k)=13(1+sin2(13+π2k)),g1(k)=13(1+cos2(13+π2k)),g2(k)=1+cos2(13+π2k),g3(k)=1+cos2(13+π2k).Clearly, in system (4.2), r=2,h=3,andih=1. By calculating, we obtain(4.3)lim supk→∞∑s=kk+3r3(s)∑s=kk+3r1(s)=lim supk→∞∑s=kk+33(1/4+sin(1/5+(π/2)s))∑s=kk+3[1/2+sin(1/3+(π/2)s)]=32,lim infk→∞∑s=kk+3r1(s)∑s=kk+3r3(s)=lim infk→∞∑s=kk+3[1/2+sin(1/3+(π/2)s)]∑s=kk+33(1/4+sin(1/5+(π/2)s))=23,lim infk→∞A33(k)A13(k)=lim infk→∞8+sin2(1/3+(π/2)(k+τ33))1/4+cos2(1/3+(π/2)(k+τ13))≥325,lim infk→∞A32(k)A12(k)=lim infk→∞9/2+sin2(1/3+(π/2)(k+τ32))1/6+sin2(1/3+(π/2)(k+τ13))≥5414,lim infk→∞A31(k)A11(k)=lim infk→∞4+sin2(1/3+(π/2)(k+τ31))1/3(1+cos2(1/3+(π/2)(k+τ11)))≥6,lim supk→∞(A13(k)R3(k)lim supk→∞∑s=kk+3r3(s)∑s=kk+3r1(s)-A33(k)R3(k))=lim supk→∞3/2(1/4+cos2(1/3+(π/2)(k+τ13)))-(8+sin2(1/3+(π/2)(k+τ33)))1+cos2(1/3+(π/2)(k+δ3))≤3/2(1/4+1)-82<0,lim infk→∞(A31(k)R1(k)lim infk→∞∑s=kk+3r1(s)∑s=kk+3r3(s)-A11(k)R1(k))=lim infk→∞2/3(4+sin2(1/3+(π/2)(k+τ31)))-(1/3(1+cos2(1/3+(π/2)(k+τ11))))1/3(1+cos2(1/3+(π/2)(k+δ1)))≥(2/3)4-(1/3)22/3=3,lim infk→∞D3(k)e3(k)=lim infk→∞2+cos2(1/3+(π/2)(k+τ3))1/3(1+sin2(1/3+(π/2)k))≥3,lim supk→∞D1(k)e1(k)=lim supk→∞1/10(1+sin2(1/3+(π/2)(k+τ1)))1/3(1+cos2(1/3+(π/2)k))≤35.
From these inequalities we see that all conditions (3.2) in Theorem 3.1 hold. In addition, we also see that assumptions (H1)–(H4) obviously hold. Therefore, by Theorem 3.1 we obtain that species x3 in system (4.3) is extinct.



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*Source: 656549-2009-11-08.xml*