# Dynamic Behaviors of a Discrete Lotka-Volterra Competition System with Infinite Delays and Single Feedback Control

Authors: Liang Zhao; Xiangdong Xie; Liya Yang; Fengde Chen
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

A nonautonomous discrete two-species Lotka-Volterra competition system with infinite delays and single feedback control is considered in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantee the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the global attractivity and extinction of the system are obtained. It is shown that if the the discrete Lotka-Volterra competitive system with infinite delays and without feedback control is permanent, then, by choosing some suitable feedback control variable, the permanent species will be driven to extinction. That is, the feedback control variable, which represents the biological control or some harvesting procedure, is the unstable factor of the system. Such a finding overturns the previous scholars’ recognition on feedback control variables.


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## Body

## 1. Introduction

During the last decade, the study of the dynamic behaviors of discrete time models governed by difference equation has become one of the most important topics in mathematics biology; many interesting results concerned with permanence, extinction, and existence of positive periodic solution (almost periodic solution) and so forth have been extensively studied by many scholars; see [1–30] and the references cited therein.As far as two-species discrete competition model is concerned, Chen and Zhou [1] proposed and studied the following discrete two-species Lotka-Volterra system:
(1)x1(n+1)=x1(n)exp⁡[r1(n)(1-x1(n)K1(n)-μ2(n)x2(n))],x2(n+1)=x2(n)exp⁡[r2(n)(1-μ1(n)x1(n)-x2(n)K2(n))],
where Ki(n), i=1,2 represent the environmental carrying capacity of species x1 and x2, respectively; ri(n), i=1,2 are the intrinsic growth rate of two species; and x1(n), x2(n) represent the density of species x1 and x2 at the nth generation, respectively. The authors obtained a set of sufficient conditions which ensure the persistence of system (1). Also, for the periodic case, they gave a set of sufficient conditions which guarantee the existence of a globally stable periodic solution of the system. Chen [2] argued that it is more realistic to incorporate delays into system (1), and he proposed and investigated the following model:
(2)x1(n+1)=x1(n)exp⁡{a1(n)-b11(n)x1(n)-b12(n)∑s=0∞‍vvvvvvvvvvv×∑s=0∞‍k12(s)x2(n-s)},x2(n+1)=x2(n)exp⁡{a2(n)-b22(n)x2(n)-b21(n)∑s=0∞‍vvvvvvvvvvv×∑s=0∞‍k21(s)x1(n-s)}.
Concerned with the persistent property of system (2), the author obtained the following result.Throughout this paper, given a bounded nonnegative sequencef(n) defined on Z, let fL and fM denote inf⁡n∈Zf(n) and sup⁡n∈Zf(n), respectively.Theorem A.
Assume that(A1)b11Lb21M>exp⁡(a1M-1)a2L,a1Lexp⁡(a2M-1)>b12Mb22L
hold; then system (2) is permanent.However, the author did not investigate the stability and extinction property of the system (2), which are two of the most important topics on the study of population dynamics.On the other hand, it is well known that, in the real world, ecosystems are continuously disturbed by unpredictable forces which can result in some changes of the biological parameters such as survival rates ([3]). For having a more accurate description of such a system, scholars introduced feedback controls into ecosystems and studied a variety of systems with feedback controls. Based on the work of Chen and Zhou [1], X. X. Chen and F. D. Chen [4] proposed and studied the following nonautonomous two-species discrete competitive system with feedback controls:
(3)x1(n+1)=x1(n)exp⁡{r1(n)[1-x1(n)K1(n)-μ2(n)x2(n)vvvvvvvvvvvvvvvvvvv-b1(n)u1(n)1-x1(n)K1(n)]},x2(n+1)=x2(n)exp⁡{r2(n)[x2(n)K2(n)-b2(n)u2(n)]1-μ1(n)x1(n)vvvvvvvvvvvvvvvvvvv-x2(n)K2(n)-b2(n)u2(n)]},u1(n+1)=u1(n)(1-a1(n))+c1(n)x1(n),u2(n+1)=u2(n)(1-a2(n))+c2(n)x2(n).
Some sufficient conditions for the persistence and global stability of system (3) were obtained. Xu et al. [5] further considered the following two-species nonautonomous Lotka-Volterra competitive system with delays and feedback controls:
(4)X1(n+1)=X1(n)exp⁡{a1(n)-b11(n)∑s=0δ1‍vvvvvvvvvvvvvv×∑s=0τ11‍k11(s)X1(n-s)-b12(n)vvvvvvvvvvvvvv×∑s=0τ12‍k12(s)X2(n-s)-d1(n)vvvvvvvvvvvvvv×∑s=0δ1‍h1(s)u1(n-s)},X2(n+1)=X2(n)exp⁡{a2(n)-b22(n)∑s=0δ2‍vvvvvvvvvvvvvv×∑s=0τ22‍k11(s)X2(n-s)-b21(n)vvvvvvvvvvvvvv×∑s=0τ21‍k21(s)X1(n-s)-d2(n)vvvvvvvvvvvvvv×∑s=0δ2‍h2(s)u2(n-s)},u1(n+1)=u1(n)(1-e1(n))+c1(n)∑s=0σ1‍l1(s)X1(n-s),u2(n+1)=u2(n)(1-e2(n))+c2(n)∑s=0σ2‍l2(s)X2(n-s).
By using the comparison theorem of discrete differential equation and constructing a suitable discrete type Lyapunov functional, they obtained new sufficient conditions on the permanence of species and global attractivity for system (4). Their results show that feedback controls are harmless to the permanence of system (4);that is, feedback controls have no influence on the permanence of system (4). X. Chen and F. Chen [29] and Liao et al. [30] also proposed a discrete time periodic n-species Lotka-Volterra competition system with feedback controls and deviating arguments; some sufficient conditions which ensure the existence of unique globally asymptotically stable periodic solution were obtained. Recently, Wu and Zhang [19] proposed a discrete autonomous Lotka-Volterra competition system with infinite delays and feedback controls; by using the iterative method, sufficient conditions which ensure the global attractivity of the system were obtained.As we can see, those models considered in [4, 5, 19, 29, 30] contain two or more feedback control variables, which means that, for the different species, different control strategy is adopted. But, in the real world, the strategy adopted for one species may also affect the other species; in other words, such a strategy has influence on both species. For instance, in the agricultural system, spraying pesticide can reduce the number of weeds, but pesticide can also have a negative impact on the growth of crops or beneficial animals [6, 7]. In the medical system, when doctor takes chemotherapeutic drugs as tools to cure the cancer patients, cancer cells will decrease rapidly, but at the same time, drugs also do harm to normal cells and body's regulatory immune function. Yao et al. [8] studied the effect of chemotherapeutic drugs on cellular immunity in patients with lung cancer; they found that cell immunity is inhibited in patients with lung cancer; moreover, it is impaired considerably by chemotherapy. So how to keep the negative effect caused by the single strategy adopted for the weeds or cancer cells to a minimum?The above phenomenons motivated us to propose and study the discrete Lotka-Volterra competition system with infinite delays and single feedback control variable as follows:(5)x1(n+1)=x1(n)exp⁡{a1(n)-b11(n)x1(n)-b12(n)∑s=0∞‍vvvvvvvvvvv×∑s=0∞‍k12(s)x2(n-s)-d1(n)vvvvvvvvvvv×∑s=0∞‍h1(s)u(n-s)},x2(n+1)=x2(n)exp⁡{a2(n)-b22(n)x2(n)-b21(n)∑s=0∞‍vvvvvvvvvvv×∑s=0∞‍k21(s)x1(n-s)-d2(n)vvvvvvvvvvv×∑s=0∞‍h2(s)u(n-s)},u(n+1)=u(n)(1-e(n))+c1(n)×∑s=0∞‍l1(s)x1(n-s)+c2(n)×∑s=0∞‍l2(s)x2(n-s).In system (5), xi(n) (i=1,2) is the density of xi species at the nth generation and u(n) is the single feedback control variable.Throughout this paper, we assume the following.(H1)
ai(n), bij(n), ci(n), di(n), and ei(n)  (i,j=1,2) are bounded sequences of real numbers defined on Z such that
(6)aiL>0,bijL>0,ciL>0,iiiiidiL>0,0<eL<eM<1,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii,j=1,2.(H2)
kij(s), hi(s), and li(s) (i,j=1,2) are nonnegative bounded sequences such that
(7)∑s=0∞‍kij(s)=1,∑s=0∞‍hi(s)=1,∑s=0∞‍li(s)=1,nnnnnnnnnnnnnnnnnnnnnnnnnnni,j=1,2;Θij=∑s=0∞‍kij(s)s<+∞,Λi=∑s=0∞‍hi(s)s<+∞,Υi=∑s=0∞‍li(s)s<+∞.According to the biological background of system (5), we only consider the solution of system (5) with the following initial conditions:
(8)xi(s)=Ψi(s)≥0,Ψi(0)>0,sup⁡k∈Z-Ψi(k)<+∞,nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnni=1,2;u(s)=Φ(s)≥0,Φ(0)>0,sup⁡k∈Z-Φ(k)<+∞,
where s=⋯,-n,-n+1,…,-1,0. It is easy to prove that the solution of system (5) which satisfies initial conditions (8) is positive.We mention here that this is the first time such kind of model is proposed and studied, and, as far as system (5) is concerned,whether the single feedback control variable has influence on the persistent property of the system or not is an interesting problem. The aim of this paper is to investigate the dynamic behaviors of the system (5); in particular, we will find out the answer to the above problem.The organization of this paper is as follows. We introduce some useful lemmas in the next section and then state and prove the main results in Sections3, 4, and 5, respectively. Three examples together with their numeric simulations are presented to show the feasibility of the main results in Section 6. We end this paper by a brief discussion.


## 2. Lemmas

Now, let us consider the following difference equation:(9)y(k+1)=ay(k)+b,
where a,b are positive constants.Lemma 1 (see [9]).
Assume that|a|<1; for any initial value y(0), there exists a unique solution y(k) of (9), which can be expressed as follows:
(10)y(k)=ak(y(0)-y*)+y*,
where y*=b/(1-a). Thus, for any solution y(k) of the system (10), we have
(11)lim⁡k→+∞y(k)=y*.Lemma 2 (see [9]).
Letk∈Nk0+={k0,k0+1,…,k0+l,…}, r≥0. For any fixed k, g(k,r) is nondecreasing function with respect to r, and, for k≥k0, the following inequalities hold:
(12)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 3 (see [2]).
Assume thatr(n)>0, x(n) satisfies x(n)>0, and
(13)x(n+1)≤x(n)exp⁡{r(n)(1-ax(n))},
for n∈[n1,∞), where a is a positive constant. Then
(14)limsup⁡n→+∞x(n)≤1arMexp⁡(rM-1).Lemma 4 (see [2]).
Assume thatr(n)>0, x(n) satisfies x(n)>0, and
(15)x(n+1)≥x(n)exp⁡{r(n)(1-ax(n))},
for n∈[n1,∞), limsup⁡n→+∞x(n)≤x*, and x(n1)>0, where a and x* are positive constants such that ax*>1. Then
(16)liminf⁡n→+∞x(n)≥1aexp⁡(rM(1-ax*)).Lemma 5 (see [2]).
Letx:Z→R be a nonnegative bounded sequence, and let H:Z+→R be a nonnegative sequence such that ∑n=0∞H(n)=1; then
(17)liminf⁡n→+∞x(n)≤liminf⁡n→+∞∑s=-∞n‍H(n-s)x(s)≤limsup⁡n→+∞∑s=-∞n‍H(n-s)x(s)≤limsup⁡n→+∞x(n).


## 3. Permanence

Concerned with the persistent property of the system (5), we have the following result.Theorem 6.
Assume that(18)aiL-bijMMj-diMB>0,i,j=1,2,i≠j,
holds; then, for any positive solution (x1(n),x2(n),u(n)) of the system (5), we have
(19)mi≤lim⁡inf⁡n→+∞xi(n)<limsup⁡n→+∞xi(n)≤Mi,A≤lim⁡inf⁡n→+∞u(n)<limsup⁡n→+∞u(n)≤B,
where
(20)mi=aiL-bijMMj-diMBbiiMexp⁡(aiL-bijMMj-diMB-biiMMi),Mi=1biiLexp⁡(aiM-1),i,j=1,2,i≠j,A=c1Lm1+c2Lm2eM,B=c1MM1+c2MM2eL.Proof.
From the first and second equations of system (5), we have
(21)xi(n+1)≤xi(n)exp⁡[ai(n)(1-biiLaiMxi(n))].
And so, from Lemma 3, we can obtain
(22)limsup⁡n→+∞xi(n)≤1biiLexp⁡(aiM-1)=defMi.
According to Lemma 5, from the above inequality we have
(23)limsup⁡n→+∞∑s=0∞‍li(s)xi(n-s)=limsup⁡n→+∞∑s=-∞n‍li(n-s)xi(s)≤limsup⁡n→+∞xi(n)≤Mi.
For any ε>0, there exists a positive integer n1 such that
(24)∑s=0∞‍li(s)xi(n-s)≤Mi+ε∀n>n1.
By the third equation of system (5), we have
(25)u(n+1)≤(1-eL)u(n)+c1M(M1+ε)+c2M(M2+ε).
Hence, by applying Lemmas 1 and 2 to (25), we obtain
(26)limsup⁡n→+∞u(n)≤c1M(M1+ε)+c2M(M2+ε)eL.
Setting ε→0, it follows that
(27)limsup⁡n→+∞u(n)≤c1MM1+c2MM2eL=defB.
Condition (18) implies that, for enough small positive constant ε1, the following inequalities hold:
(28)aiL-bijM(Mj+ε1)-diM(B+ε1)>0,i,j=1,2,i≠j.
For the above ε1, it follows from (22) and (27) that there exists a positive integer n2≥n1 such that
(29)∑s=0∞‍hi(s)u(n-s)≤B+ε1,∑s=0∞‍kij(s)xi(n-s)≤Mi+ε1∀n>n2.
Thus, for all n>n2, from (28), (29), and the first two equations of system (5), we have
(30)xi(n+1)≥xi(n)exp⁡{(aiL-bijM(Mj+ε1)diM(B+ε1))vvvvvvvvvvv×(aiL-bijM(Mj+ε1)diM(B+ε1))(1-Diεxi(n))},
where Diε=biiM/(aiL-bijM(Mj+ε1)-diM(B+ε1)) for i,j=1,2, i≠j.
Noticing that(31)biiMbiiL≥1,exp⁡(aiM-1)aiL-bijM(Mj+ε1)-diM(B+ε1)>1,
then
(32)Diε·Mi=biiMaiL-bijM(Mj+ε1)-diM(B+ε1)·1biiLexp⁡(aiM-1)>1.
Hence, according to Lemma 4, we have
(33)liminf⁡n→+∞xi(n)≥1Diε·exp⁡(aiL-bijM(Mj+ε1)-diM(B+ε1)-biiMMi).
Setting ε1→0, it follows that
(34)liminf⁡n→+∞xi(n)≥1Di·exp⁡(aiL-bijMMj-diMB-biiMMi)=defmi,
where Di=biiM/(aiL-bijMMj-diMB), i=1,2.
According to Lemma5, from (34) we have that, for any ε2>0 small enough (without loss of generality, assume that ε<(1/2)min⁡i{mi}), there exists an n3>n2, such that
(35)∑s=0∞‍li(s)xi(n-s)≥mi-ε2∀n≥n3.
For n≥n3, from (35) and the last equation of system (5), we have
(36)u(n+1)≥(1-eM)u(n)+c1L(m1-ε2)+c2L(m2-ε2).
Hence, by applying Lemmas 1 and 2 to (36), we obtain
(37)limsup⁡n→+∞u(n)≥c1L(m1-ε2)+c2L(m2-ε2)eM.
Setting ε2→0, it follows that
(38)limsup⁡n→+∞u(n)≥c1Lm1+c2Lm2eM=defA.
This ends the proof Theorem 6.


## 4. Global Attractivity

Concerned with the stability property of the system (5), we have the following result.Theorem 7.
Assume that there exist positive constantsα1, α2, and α3 such that
(39)α1A11-α2b21M-α3c1M>0,α2A22-α1b12M-α3c2M>0,α3eL-α1d1M-α2d2M>0
hold; then, for any two positive solutions (x1(n),x2(n),u(n)) and (x1*(n),x2*(n),u*(n)) of system (5), we have
(40)lim⁡n→+∞(xi(n)-xi*(n))=0,lim⁡n→+∞(u(n)-u*(n))=0,
where
(41)Aii=min⁡{biiL,2Mi-biiM},i=1,2.Proof.
By (39), we can choose enough small positive constants δ and ε such that
(42)α1A11ε-α2b21M-α3c1M>δ,α2A22ε-α1b12M-α3c2M>δ,α3eL-α1d1M-α2d2M>δ,
where
(43)Aiiε=min⁡{biiL,2Mi+ε-biiM},i=1,2.
Let(x1(n),x2(n),u(n)) be any positive solution of system (5). For the above ε, from (22) and (27), there exists an enough large n*>n1, such that
(44)xi(n)<Mi+ε,u(n)<B+ε∀n≥n*,i=1,2.
Now, let us define a Lyapunov functional(45)V(n)=α1V1(n)+α2V2(n)+α3V3(n),
where αi, i=1,2,3, are positive constants:
(46)V1(n)=|ln⁡x1(n)-ln⁡x1*(n)|+∑s=0∞‍k12(s)×∑q=n-sn-1‍b12(q+s)|x2(q)-x2*(q)|+∑s=0∞‍h1(s)×∑q=n-sn-1‍d1(q+s)|u(q)-u*(q)|;V2(n)=|ln⁡x2(n)-ln⁡x2*(n)|+∑s=0∞‍k21(s)×∑q=n-sn-1‍b21(q+s)|x1(q)-x1*(q)|+∑s=0∞‍h2(s)∑q=n-sn-1‍d2(q+s)|u(q)-u*(q)|;V3(n)=|u(n)-u*(n)|+∑i=12‍∑s=0∞‍li(s)×∑q=n-sn-1‍ci(q+s)|xi(q)-xi*(q)|.
Then, from the definition of Vi(n), i=1,2,3, one could easily see that V(n)≥0 for all n∈Z+. Also, for any fixed n*∈Z+,
(47)V(n*)=α1(|ln⁡x1(n*)-ln⁡x1*(n*)|∑q=n*-sn*-1‍vvvvvvvv+∑s=0∞‍k12(s)∑q=n*-sn*-1‍b12(q+s)|x2(q)-x2*(q)|vvvvvvvv+∑s=0∞‍h1(s)∑q=n*-sn*-1‍d1(q+s)|u(q)-u*(q)|)+α2(|ln⁡x2(n*)-ln⁡x2*(n*)|∑q=n*-sn*-1‍vvvvvvvvvv+∑s=0∞‍k21(s)∑q=n*-sn*-1‍b21(q+s)|x1(q)-x1*(q)|vvvvvvvvvv+∑s=0∞‍h2(s)∑q=n*-sn*-1‍d2(q+s)|u(q)-u*(q)|)+α3(|u(n*)-u*(n*)|∑i=12‍vvvvvvvvvv+∑i=12‍∑s=0∞‍li(s)∑q=n*-sn*-1‍ci(q+s)|xi(q)-xi*(q)|)≤α1(|ln⁡x1(n*)-ln⁡x1*(n*)|∑s=0∞‍vvvvvvvv+b12Msup⁡q∈Z,q≤n*|x2(q)-x2*(q)|∑s=0∞‍k12(s)svvvvvvvv+d1Msup⁡q∈Z,q≤n*|u(q)-u*(q)|∑s=0∞‍h1(s)s)+α2(|ln⁡x2(n*)-ln⁡x2*(n*)|∑s=0∞‍vvvvvvvvv+b21Msup⁡q∈Z,q≤n*|x1(q)-x1*(q)|vvvvvvvvv×∑s=0∞‍k21(s)s+d2Msup⁡q∈Z,q≤n*|u(q)-u*(q)|vvvvvvvvv×∑s=0∞‍h2(s)s)+α3(|u(n*)-u*(n*)|∑s=0∞‍vvvvvvvvv+ciMsup⁡q∈Z,q≤n*|xi(q)-xi*(q)|vvvvvvvvv×∑i=12‍∑s=0∞‍li(s)s)<+∞.
Also, from the first equation of system (5) and using the Mean Value Theorem, we can obtain
(48)ΔV1(n)≤|ln⁡x1(n)-ln⁡x1*(n)-b11(n)(x1(n)-x1*(n))|-|ln⁡x1(n)-ln⁡x1*(n)|+b12(n)×∑s=0∞‍k12(s)|x2(n-s)-x2*(n-s)|+d1(n)∑s=0∞‍h1(s)|u(n-s)-u*(n-s)|+∑s=0∞‍k12(s)(b12(n+s)|x2(n)-x2*(n)|vvvvvvvvvvvv-b12(n)|x2(n-s)-x2*(n-s)|)+∑s=0∞‍h1(s)(d1(n+s)|u(n)-u*(n)|vvvvvvvvvvvv-d1(n)|u(n-s)-u*(n-s)|)≤-(1φ1(n)-|1φ1(n)-b11(n)|)|x1(n)-x1*(n)|+∑s=0∞‍k12(s)b12(n+s)|x2(n)-x2*(n)|+∑s=0∞‍h1(s)d1(n+s)|u(n)-u*(n)|≤-(1φ1(n)-|1φ1(n)-b11(n)|)|x1(n)-x1*(n)|+b12M|x2(n)-x2*(n)|+d1M|u(n)-u*(n)|,
where φ1(n) lies between x1(n) and x1*(n).
Similarly to the analysis of (48), we can obtain
(49)ΔV2(n)≤-(1φ2(n)-|1φ2(n)-b22(n)|)|x2(n)-x2*(n)|+b21M|x1(n)-x1*(n)|+d2M|u(n)-u*(n)|,
where φ2(n) lies between x2(n) and x2*(n):
(50)ΔV3(n)=V3(n+1)-V3(n)≤-e(n)|u(n)-u*(n)|+∑i=12‍ci(n)×∑s=0∞‍li(s)|xi(n-s)-xi*(n-s)|+∑i=12‍∑s=0∞‍li(s)(ci(n+s)vvvvvvvvvvvvv×|xi(n)-xi*(n)|-ci(n)vvvvvvvvvvvvv×|xi(n-s)-xi*(n-s)|)=-e(n)|u(n)-u*(n)|+∑i=12‍∑s=0∞‍li(s)ci(n+s)|xi(n)-xi*(n)|≤-eL|u(n)-u*(n)|+∑i=12‍ciM|xi(n)-xi*(n)|.
From (42)–(44) and (48)–(50), for any n≥n*, we have
(51)ΔV(n)≤α1{-(1φ1(n)-|1φ1(n)-b11(n)|)vvvvvv×|x1(n)-x1*(n)|vvvvvv+b12M|x2(n)-x2*(n)|vvvvvv+(1φ1(n)-|1φ1(n)-b11(n)|)d1M|u(n)-u*(n)|}+α2{-(1φ2(n)-|1φ2(n)-b22(n)|)vvvvvvv×|x2(n)-x2*(n)|vvvvvvv+b21M|x1(n)-x1*(n)|vvvvvvv+(1φ2(n)-|1φ2(n)-b22(n)|)d2M|u(n)-u*(n)|}+α3{∑i=12‍-eL|u(n)-u*(n)|vvvvvvv+∑i=12‍ciM|xi(n)-xi*(n)|}≤-{α1min⁡[b11L,2M1+ε-b11M]-α2b21M-α3c1M}×|x1(n)-x1*(n)|-{α2min⁡[b22L,2M2+ε-b22M]-α1b12M-α3c2M}×|x2(n)-x2*(n)|-{α3eL-α1d1M-α2d2M}|u(n)-u*(n)|=-{α1A11ε-α2b21M-α3c1M}|x1(n)-x1*(n)|-{α2A22ε-α1b12M-α3c2M}|x2(n)-x2*(n)|-{α3eL-α1d1M-α2d2M}|u(n)-u*(n)|≤-δ(∑i=12‍|xi(n)-xi*(n)|+|u(n)-u*(n)|).
Summating both sides of the above inequalities from n* to n, we have
(52)∑p=n*n‍(V(p+1)-V(p))≤-δ∑p=n*n‍(∑i=12‍|xi(p)-xi*(p)|+|u(p)-u*(p)|).
Hence
(53)V(n+1)+δ∑p=n*n‍(∑i=12‍|xi(p)-xi*(p)|+|u(p)-u*(p)|)≤V(n*)<+∞.
Then, we have
(54)∑p=n*n‍(∑i=12‍|xi(p)-xi*(p)|+|u(p)-u*(p)|)≤V(n*)δ.
Therefore
(55)∑p=n*∞‍(∑i=12‍|xi(p)-xi*(p)|+|u(p)-u*(p)|)<+∞,
which means that
(56)lim⁡n→+∞(∑i=12‍|xi(n)-xi*(n)|+|u(n)-u*(n)|)=0.
Consequently
(57)lim⁡n→+∞(xi(n)-xi*(n))=0,i=1,2,lim⁡n→+∞(u(n)-u*(n))=0.
This completes the proof of Theorem 7


## 5. Extinction

Concerned with the extinction property of the system (5), when the coefficients of the third equation are all constants, we could establish the following results.Theorem 8.
Assume that(58)a1Ma2L<b11Le+d1Lc1b21Me+d2Mc1,a1Ma2L<b12Le+d1Lc2b22Me+d2Mc2
hold; let (x1(n),x2(n),u(n)) be any positive solution of system (5); then
(59)lim⁡n→+∞x1(n)=0.Theorem 9.
Assume that(60)a1La2M>b11Me+d1Mc1b21Le+d2Lc1,a1La2M>b12Me+d1Mc2b22Le+d2Lc2
hold; let (x1(n),x2(n),u(n)) be any positive solution of system (5); then
(61)lim⁡n→+∞x2(n)=0.Proof of Theorem8.
By conditions (58), we can choose positive constants αn1, αn2 such that
(62)a1Ma2L<αn2αn1<b11Le+d1Lc1b21Me+d2Mc1,a1Ma2L<αn2αn1<b12Le+d1Lc2b22Me+d2Mc2.
Thus, there exists a positive constant η such that
(63)αn1a1M-αn2a2L<-η<0,αn2b21M-αn1b11L+(αn2d2M-αn1d1L)c1e<0,αn2b22M-αn1b12L+(αn2d2M-αn1d1L)c2e<0.
There exists a constant αn3 such that
(64)αn2d2M-αn1d1Le<αn3⟹αn2d2M-αn1d1L-αn3e<0,αn2b21M-αn1b11L+αn3c1<0,αn2b22M-αn1b12L+αn3c2<0.
Consider the following discrete Lyapunov functional:
(65)V4(n)=x1αn1(n)x2-αn2(n)×exp⁡{αn3u(n)-αn1∑s=0∞‍k12(s)vvvvvvvv×∑q=n-sn-1‍b12(q+s)x2(q)vvvvvvvv-αn1∑s=0∞‍h1(s)∑q=n-sn-1‍d1(q+s)u(q)vvvvvvvv+αn2∑s=0∞‍k21(s)∑q=n-sn-1‍b21(q+s)x1(q)vvvvvvvv+αn2∑s=0∞‍h2(s)∑q=n-sn-1‍d2(q+s)u(q)vvvvvvvv+αn3∑i=12‍∑s=0∞‍li(s)∑q=n-sn-1‍cixi(q)}.
From (65), we obtain
(66)V4(n+1)V4(n)=exp⁡{αn1(a1(n)-b11(n)x1(n)∑s=0∞‍vvvvvvvvvvv-b12(n)∑s=0∞‍k12(s)x2(n-s)vvvvvvvvvvv-d1(n)∑s=0∞‍h1(s)u(n-s))vvvvvv-αn2(a2(n)-b22(n)x2(n)∑s=0∞‍vvvvvvvvvvvv-b21(n)∑s=0∞‍k21(s)x1(n-s)vvvvvvvvvvvv-d2(n)∑s=0∞‍h2(s)u(n-s))vvvvvv+αn3(-eu(n)+c1∑s=0∞‍l1(s)x1(n-s)vvvvvvvvvvvvv+c2∑s=0∞‍l2(s)x2(n-s))vvvvvv-αn1∑s=0∞‍k12(s)(b12(n+s)x2(n)vvvvvvvvvvvvvvvvvvv-b12(n)x2(n-s))vvvvvv-αn1∑s=0∞‍h1(s)(d1(n+s)u(n)vvvvvvvvvvvvvvvvvvv-d1(n)u(n-s))vvvvvv+αn2∑s=0∞‍k21(s)(b21(n+s)x1(n)vvvvvvvvvvvvvvvvvvv-b21(n)x1(n-s))vvvvvv+αn2∑s=0∞‍h2(s)(d2(n+s)u(n)vvvvvvvvvvvvvvvvvvv-d2(n)u(n-s))vvvvvv+αn3∑i=12‍∑s=0∞‍li(s)vvvvvv∑i=12‍×(cixi(n)-cixi(n-s))}≤exp⁡{(αn1a1M-αn2a2L)vvvvvv+(αn2b21M-αn1b11L+αn3c1)x1(n)vvvvvv+(αn2b22M-αn1b12L+αn3c2)x2(n)vvvvvv+(αn2d2M-αn1d1L-αn3e)u(n)}.
From inequalities (63) and (64), we can obtain
(67)V4(n+1)≤V4(n)exp⁡(-η).
Therefore
(68)V4(n)≤V4(0)exp⁡(-nη).
From (22) and (27) we know that there exists an M>0 such that
(69)xi(k)<M,u(k)<M∀k∈Z,
and so
(70)V4(0)=x1αn1(0)x2-αn2(0)×exp⁡{∑q=-s-1‍αn3u(0)-αn1∑s=0∞‍k12(s)vvvvvvvv×∑q=-s-1‍b12(q+s)x2(q)vvvvvvvv-αn1∑s=0∞‍h1(s)∑q=-s-1‍d1(q+s)u(q)vvvvvvvv+αn2∑s=0∞‍k21(s)∑q=-s-1‍b21(q+s)x1(q)vvvvvvvv+αn2∑s=0∞‍h2(s)∑q=n-sn-1‍d2(q+s)u(q)vvvvvvvv+αn3∑i=12‍∑s=0∞‍li(s)∑q=-s-1‍cixi(q)}<x1αn1(0)x2-αn2(0)×exp⁡{∑q=-s-1‍|αn3|M+αn2Mb21M∑s=0∞‍k21(s)svvvvvvvv+αn2Md2M∑s=0∞‍h2(s)svvvvvvvv+αn3Mci∑i=12‍∑s=0∞‍li(s)s∑q=-s-1‍}=x1αn1(0)x2-αn2(0)×exp⁡{∑q=-s-1‍M(|αn3|+αn2b21MΘ21+αn2d2MΛ2∑i=12‍vvvvvvvvvvvvv+αn3∑i=12‍ciΥi)∑q=-s-1‍}<+∞.
On the other hand, we also have
(71)V4(n)≥x1αn1(n)x2-αn2(n)×exp⁡{-αn1∑s=0∞‍k12(s)∑q=n-sn-1‍b12(q+s)x2(q)vvvvvvvv-αn1∑s=0∞‍h1(s)∑q=n-sn-1‍d1(q+s)u(q)}≥x1αn1(n)M-αn2×exp⁡{-αn1Mb12M∑s=0∞‍k12(s)svvvvvvvv-αn1Md1M∑s=0∞‍h1(s)s}=x1αn1(n)M-αn2×exp⁡{-αn1M(b12MΘ12+d1MΛ1)}.
Combining inequalities (68), (70), and (71), we have
(72)x1(n)≤λexp⁡{-ηαn1n},
where
(73)λ=(V4(0))1/αn1Mαn2/αn1exp⁡{M(b12MΘ12+d1MΛ1)}<+∞.
Hence we obtain that
(74)lim⁡n→+∞x1(n)=0.
This ends the proof of Theorem 8.Proof of Theorem9.
By (60), we can choose positive constants βn1, βn2, and βn3 and constant γ such that
(75)a1Ma2L>βn2βn1>b11Me+d1Mc1b21Le+d2Lc1,a1Ma2L>βn2βn1>b12Me+d1Mc2b22Le+d2Lc2.βn2a2M-βn1a1L<-γ<0,βn1b11M-βn2b21L+βn3c1<0,βn1b12M-βn2b22L+βn3c2<0,βn1d1M-βn2d2L-βn3e<0.
Define the following Lyapunov functional:
(76)V5(n)=x1-βn1(n)x2βn2(n)×exp⁡{∑q=n-sn-1‍βn3u(n)+βn1∑s=0∞‍k12(s)vvvvvvvv×∑q=n-sn-1‍b12(q+s)x2(q)vvvvvvvv+βn1∑s=0∞‍h1(s)∑q=n-sn-1‍d1(q+s)u(q)vvvvvvvv-βn2∑s=0∞‍k21(s)∑q=n-sn-1‍b21(q+s)x1(q)vvvvvvvv-βn2∑s=0∞‍h2(s)∑q=n-sn-1‍d2(q+s)u(q)vvvvvvvv+βn3∑i=12‍∑s=0∞‍li(s)∑q=n-sn-1‍cixi(q)}.
From (76), we have
(77)V5(n+1)V5(n)≤exp⁡{(βn2a2M-βn1a1L)vvvvvvvv+(βn1b11M-βn2b21L+βn3c1)x1(n)vvvvvvvv+(βn1b12M-βn2b22L+βn3c2)x2(n)vvvvvvvv+(βn1d1M-βn2d2L-βn3e)u(n)}≤exp⁡{-γ}.
Similarly to the analysis of (68)–(73), we have
(78)lim⁡n→+∞x2(n)=0.
This ends the proof of Theorem 9.From Theorems8 and 9 we know that, under some suitable assumption, one of the species in the system may be driven to extinction; in this case, one interesting problem is to investigate the stability property of the rest of the species.Consider the following discrete equations:(79)x2(n+1)=x2(n)exp⁡{a2(n)-b22(n)x2(n)∑s=0∞‍vvvvvvvvvv-d2(n)∑s=0∞‍h2(s)u(n-s)},u(n+1)=u(n)(1-e)+c2∑s=0∞‍l2(s)x2(n-s).(80)x1(n+1)=x1(n)exp⁡⁡{a1(n)-b11(n)x1(n)∑s=0∞‍vvvvvvvvvv-d1(n)∑s=0∞‍h1(s)u(n-s)},u(n+1)=u(n)(1-e)+c1∑s=0∞‍l1(s)x1(n-s).Theorem 10.
Assume that (58) holds and also
(81)d2M<A22ec2
holds; then, for any positive solution (x1(n),x2(n),u(n)) of system (5) and any positive solution (x2*(n),u*(n)) of system (79), we have
(82)lim⁡n→+∞x1(n)=0,lim⁡n→+∞(x2(n)-x2*(n))=0,lim⁡n→+∞(u(n)-u*(n))=0,
where A22 is defined in Theorem 7.Theorem 11.
Assume that (60) holds and also
(83)d1M<A11ec1
holds; then, for any positive solution (x1(n),x2(n),u(n)) of system (5) and any positive solution (x1*(n),u*(n)) of system (80), we have
(84)lim⁡n→+∞(x1(n)-x1*(n))=0,lim⁡n→+∞x2(n)=0,lim⁡n→+∞(u(n)-u*(n))=0,
where A11 is defined in Theorem 7.Proof of Theorem10.
By conditions (81), we can choose positive constants β1, β2 such that
(85)ed2M>β1β2>c2A22.
Thus, there exist enough small positive constants δ and ε such that
(86)β1A22ε-β2c2>δ,β2e-β1d2M>δ,
where A22ε is defined in(43)
Now, we define a Lyapunov functional(87)Q1(n)=β1G1(n)+β2H1(n),
where
(88)G1(n)=|ln⁡x2(n)-ln⁡x2*(n)|+∑s=0∞‍k21(s)×∑q=n-sn-1‍b21(q+s)x1(q)+∑s=0∞‍h2(s)×∑q=n-sn-1‍d2(q+s)|u(q)-u*(q)|;H1(n)=|u(n)-u*(n)|+∑s=0∞‍l1(s)×∑q=n-sn-1‍c1x1(q)+∑s=0∞‍l2(s)×∑q=n-sn-1‍c2|x2(q)-x2*(q)|.
Then, from the definition of G1(n), H1(n), one could easily see that Q1(n)≥0 for all n∈Z+. Also, for any fixed n*∈Z+, from (69) one could see that
(89)Q1(n*)=β1|ln⁡x2(n*)-ln⁡x2*(n*)|+β1Mb21M∑s=0∞‍k21(s)s+β1d2Msup⁡q∈Z+,q≤n*|u(q)-u*(q)|∑s=0∞‍h2(s)s+β2|u(n*)-u*(n*)|+β2Mc1∑s=0∞‍l1(s)s+β2c2sup⁡q∈Z+,q≤n*|x2(q)-x2*(q)|∑s=0∞‍l2(s)s<+∞.
It follows from the second equation of system (5) and the Mean Value Theorem that
(90)ΔG1(n)≤|ln⁡x2(n)-ln⁡x2*(n)-b22(n)(x2(n)-x2*(n))|-|ln⁡x2(n)-ln⁡x2*(n)|+b21(n)∑s=0∞‍k21(s)x1(n-s)+d2(n)×∑s=0∞‍h2(s)|u(n-s)-u*(n-s)|+∑s=0∞‍k21(s)(b21(n+s)x1(n)vvvvvvvvvvvv-b21(n)x1(n-s))+∑s=0∞‍h2(s)(d2(n+s)|u(n)-u*(n)|vvvvvvvvvvv-d2(n)|u(n-s)-u*(n-s)|)≤-(1φ2(n)-|1φ2(n)-b22(n)|)|x2(n)-x2*(n)|+∑s=0∞‍k21(s)b21(n+s)x1(n)+∑s=0∞‍h2(s)d2(n+s)|u(n)-u*(n)|≤-(1φ2(n)-|1φ2(n)-b22(n)|)|x2(n)-x2*(n)|+b12Mx1(n)+d2M|u(n)-u*(n)|.
Also, from the third equation of system (5) we have
(91)ΔH1(n)=H1(n+1)-H1(n)≤-e|u(n)-u*(n)|+c1∑s=0∞‍l1(s)x1(n-s)+c2∑s=0∞‍l2(s)|x2(n-s)-x2*(n-s)|+∑s=0∞‍l1(s)(c1x1(n)-c1x1(n-s))+∑s=0∞‍l2(s)(c2|x2(n)-x2*(n)|vvvvvvvvvvv-c2|x2(n-s)-x2*(n-s)|)=-e|u(n)-u*(n)|+∑s=0∞‍l1(s)c1x1(n)+∑s=0∞‍l2(s)c2|x2(n)-x2*(n)|≤-e|u(n)-u*(n)|+c1x1(n)+c2|x2(n)-x2*(n)|.
From (70), we have
(92)∑p=n*∞‍x1(p)≤∑p=n*∞‍λexp⁡{-ηαn1p}<+∞.
Therefore, from (44), (88), and (90), for all n>n*, we have
(93)ΔQ1(n)≤β1{-(1φ2(n)-|1φ2(n)-b22(n)|)|x2(n)-x2*(n)|vvvvvv+(1φ2(n)-|1φ2(n)-b22(n)|)b21Mx1(n)+d2M|u(n)-u*(n)|}+β2{-e|u(n)-u*(n)|+c1x1(n)+c2|x2(n)-x2*(n)|}≤-{β1min⁡[b22L,2M2+ε-b22M]-β2c2}|x2(n)-x2*(n)|-{β2e-β1d2M}|u(n)-u*(n)|+(β1b21M+β2c1)x1(n)=-{β1A22ε-β2c2}|x2(n)-x2*(n)|-{β2e-β1d2M}|u(n)-u*(n)|+(β1b21M+β2c1)x1(n)≤-δ(|x2(n)-x2*(n)|+|u(n)-u*(n)|)+ρx1(n),
where ρ=β1b21M+β2c1.
Summating both sides of the above inequality fromn* to n, we have
(94)∑p=n*n‍(Q1(p+1)-Q1(p))≤-δ∑p=n*n‍(|x2(p)-x2*(p)|+|u(p)-u*(p)|)+ρ∑p=n*n‍x1(p).
Hence
(95)Q1(n+1)+δ∑p=n*n‍(|x2(p)-x2*(p)|+|u(p)-u*(p)|)≤Q1(n*)+ρ∑p=n*n‍x1(p).
Then, from (89) and (92) we have
(96)∑p=n*n‍(|x2(p)-x2*(p)|+|u(p)-u*(p)|)≤Q1(n*)+ρ∑p=n*nx1(p)δ<+∞.
Therefore
(97)∑p=n*∞‍(|x2(p)-x2*(p)|+|u(p)-u*(p)|)<+∞,
which means that
(98)lim⁡n→+∞(|x2(n)-x2*(n)|+|u(n)-u*(n)|)=0.
Consequently
(99)lim⁡n→+∞(x2(n)-x2*(n))=0,lim⁡n→+∞(u(n)-u*(n))=0.
This completes the proof of Theorem 10.Proof of Theorem11.
The proof of Theorem11 is similar to that of Theorem 10, and we omit the details here.Remark 12.
One of the purposes of this paper is to find out the influence of feedback control variable on the persistent property of the system. Obviously, the answer lies in the relations among conditions(A1), (58), and (60). Now let us consider conditions (A1) and (60); there may exist the following subcase:
(100)b11Lb21M>exp⁡(a1M-1)a2L>a1Ma2L>a1La2M>b11Me+d1Mc1b21Le+d2Lc1.
If b12M/b22L>d1M/d2L, then we have
(101)a1La2M>a1Lexp⁡(a2M-1)>b12Mb22L>b12Me+d1Mc2b22Le+d2Lc2.
If b12M/b22L<d1M/d2L, then there may exist the following two subcases:
(102)a1La2M>a1Lexp⁡(a2M-1)>b12Me+d1Mc2b22Le+d2Lc2>b12Mb22L,a1La2M>b12Me+d1Mc2b22Le+d2Lc2>a1Lexp⁡(a2M-1)>b12Mb22L.
(101) and (100) (or (102) and (100)) show that conditions (A1) and (60) could be satisfied together; that is, for the original permanent system (2), by choosing suitable feedback control variable, species x2 will be driven to extinction.Remark 13.
Similarly to the above analysis, if system (2) is permanent, by choosing suitable feedback control variable, the first species x1 will be driven to extinction.Remark 14.
From Theorem7, we can find that feedback control variable has influence on the global attractivity of system (5); that is, only the feedback control variable is very low such that the third inequality of (39) holds; in other words, when inequality d=max⁡{d1M,d2M}<α3eL/(α1+α2) holds, the two species can be coexist.


## 6. Examples

In this section we will give three examples to illustrate the feasibility of main results.Example 1.
Consider the following equations:(103)x1(n+1)=x1(n)exp⁡{0.9+0.1sin(2n)∑s=0∞‍vvvvvvvvvvv-(0.25+0.05cos⁡⁡(3n))x1(n)vvvvvvvvvvv-(0.08+0.02cos⁡(3n))vvvvvvvvvvv×∑s=0∞‍e-1ee-sx2(n-s)vvvvvvvvvvv-(0.45+0.03sin(2n))vvvvvvvvvvv×∑s=0∞‍e2-1e2e-2su(n-s)},x2(n+1)=x2(n)exp⁡{0.9+0.1sin(2n)∑s=0∞‍vvvvvvvvvvv-(0.6+0.1cos⁡(3n))x2(n)vvvvvvvvvvv-(0.15+0.05cos⁡(3n))vvvvvvvvvvv×∑s=0∞‍e-1ee-sx1(n-s)vvvvvvvvvvv-(0.42+0.04sin(2n))vvvvvvvvvvv×∑s=0∞‍e2-1e2e-2su(n-s)},u(n+1)=u(n)(1-(0.92+0.03cos⁡(3n)))+(0.01+0.005cos⁡(2n))×∑s=0∞‍e3-1e3e-3sx1(n-s)+(0.008+0.002cos⁡(3n))×∑s=0∞‍e3-1e3e-3sx2(n-s).
One could easily see that conditions (H1) and (H2) are satisfied. Also, by calculating, one has
(104)A11=min⁡⁡{b11L,2M1-b11M}=0.1,A22=min⁡{b22L,2M2-b22M}=0.3.
Now, let us take α1=1.3, α2=0.5, and α3=1; then
(105)α1A11-α2b21M-α3c1M=0.015>0,α2A22-α1b12M-α3c2M=0.01>0,α3eL-α1d1M-α2d2M=0.046>0.
Clearly, condition (39) is satisfied, and so from Theorem 7 we have lim⁡n→+∞(xi(n)-xi*(n))=0, lim⁡n→+∞(u(n)-u*(n))=0, where (x1(n),x2(n),u(n)) and (x1*(n),x2*(n),u*(n)) are any two positive solutions of system (103).Figure1 shows the dynamic behaviors of system (103), which strongly supports the above assertions.Numeric simulations of the solutions(x1(n),x2(n),u(n)) of system (103), with the initial conditions (x1(s),x2(s),u(s))=(1.8,1,0.005), (2.3,1.5,0.01), and (3.7,0.8,0.015), s=⋯,-n,-n+1,…,-1,0, respectively.
(a)
(b)Example 2.
Consider the following equations:(106)x1(n+1)=x1(n)exp⁡⁡{0.8+0.2sin⁡(2n)-3x1(n)∑s=0∞‍vvvvvvvvxvvv-1.5∑s=0∞‍e-1ee-sx2(n-s)vvvvvvvxvvvv-0.1∑s=0∞‍e2-1e2e-2su(n-s)},x2(n+1)=x2(n)exp⁡{1.8+0.2sin(2n)-x2(n)∑s=0∞‍vvvvvvvvxvvv-3∑s=0∞‍e-1ee-sx1(n-s)vvvvvvvvxvvv-0.2∑s=0∞‍e2-1e2e-2su(n-s)},u(n+1)=u(n)(1-0.8)+1.2∑s=0∞‍e3-1e3e-3sx1(n-s)+3∑s=0∞‍e3-1e3e-3sx2(n-s).
One could easily see that conditions (H1) and (H2) are satisfied. Also, by calculating, one has
(107)a1Ma2L=58,b11Le+d1Lc1b21Me+d2Mc1=910,b12Le+d1Lc2b22Me+d2Mc2=1514.A22=min⁡⁡{b22L,2M2-b22M}=1,0.2=d2M<A22ec2≈0.67.
Clearly, conditions (58) and (81) are satisfied, and so from Theorems 8 and 10 we know that species x1 will be driven to extinction, while species x2 is globally attractive.Figure2 shows the dynamic behaviors of system (106), which strongly supports our results.Figure 2
Numeric simulations of the solutions(x1(n),x2(n),u(n)) of system (106), with the initial conditions (x1(s),x2(s),u(s)) = (0.8,1.4,1.2), (0.5,1.2,1.5), and (0.2,1.6,2.1), s=⋯,-n,-n+1,…,-1,0, respectively.Example 3.
Consider the following example:(108)x1(n+1)=x1(n)exp⁡{0.9+0.1sin⁡(n)-0.6x1(n)∑s=0∞‍vvvvvvvvvv-0.4∑s=0∞‍e-1ee-sx2(n-s)},x2(n+1)=x2(n)exp⁡{0.8+0.2sin(n)-0.8x2(n)∑s=0∞‍vvvvvvvvvv-0.3∑s=0∞‍e-1ee-sx1(n-s)}.
By calculating, one has
(109)b11Lb21M=2>exp⁡(a1M-1)a2L=53,a1Lexp⁡(a2M-1)=0.8>b12Mb22L=0.5.
Clearly, condition (A1) is satisfied; thus from Theorem A we know that system (108) is permanent. Figure 3 shows the dynamic behaviors of system (108).Figure 3
Dynamic behaviors of the solution(x1(n),x2(n)) of system (110), with the initial conditions (x1(s),x2(s)) = (0.8,0.4), (1.4,1.2), and (1.2,1.0),  s=⋯,-n,-n+1,…,-1,0, respectively.Now let us further incorporate the feedback control variable into system (108) and consider the following system:
(110)x1(n+1)=x1(n)exp⁡{0.9+0.1sin⁡(n)-0.6x1(n)∑s=0∞‍vvvvvvvvvvv-0.4∑s=0∞‍e-1ee-sx2(n-s)vvvvvvvvvvv-0.2∑s=0∞‍e2-1e2e-2su(n-s)},x2(n+1)=x2(n)exp⁡{0.8+0.2sin⁡(n)-0.8x2(n)∑s=0∞‍vvvvvvvvvvv-0.3∑s=0∞‍e-1ee-sx1(n-s)vvvvvvvvvvv-0.3∑s=0∞‍e2-1e2e-2su(n-s)},u(n+1)=u(n)(1-0.5)+5∑s=0∞‍e3-1e3e-3sx1(n-s)+∑s=0∞‍e3-1e3e-3sx2(n-s).
By calculating, one has
(111)a1La2M=45>b12Me+d1Mc2b22Le+d2Lc2=47,a1La2M=45>b11Me+d1Mc1b21Le+d2Lc1=2633.
Thus, all the conditions of Theorem 9 are satisfied, and so the permanent x2 species in system (108) become extinct in system (110).Figure4 shows the dynamic behaviors of system (110), which supports our assertion.Numeric simulations of the solutions(x1(n),x2(n),u(n)) of system (110), with the initial conditions (x1(s),x2(s),u(s)) = (0.8,0.4,0.5), (1.4,1.2,2), and (1.2,1.0,1.5),  s=⋯,-n,-n+1,…,-1,0, respectively.
(a)
(b)


## 7. Discussion

(1)
Li et al. [3] consider a continuous and autonomous Lotka-Volterra competitive system with infinite delays and feedback controls; if the Lotka-Volterra competitive system is globally stable, then they showed that the feedback controls only change the position of the unique positive equilibrium and retain the stable property. As a consequence of this result, feedback control variables have no influence on the persistent property of the system they considered; Xu et al. [5] proposed and studied the discrete nonautonomous Lotka-Volterra competitive system with delays and feedback controls (model (4)). Theorem 3.2 in [5] showed that feedback controls do not affect the persistent property of system (4). In this paper, we propose and study a discrete Lotka-Volterra competition system with single feedback control variable. Remarks 12 and 13 show that the feedback control variable plays an important role in the persistent property of the system (2). If the Lotka-Volterra competitive system is permanent, then we show that, by choosing suitable feedback control variable, one of the species will be driven to extinction; that is, feedback control variable, which represents the biological control or some harvesting procedure, is an unstable factor of the system. Such a finding overturns the previous recognition on feedback control variables ([3, 5]) and shows that it is better for us to take a single policy to protect the rare species through reducing the species' competitions than to take two different policies.
(2)
In [2], Chen proposed system (2) and investigated the persistent property of the system (see Theorem A in introduction section); however, he did not investigate the stability property and extinction property of the system. As a direct corollary of Theorem 7, concerned with the stability property of system (2), we have the following.Theorem B.
Assume that there exist positive constantsαi, i=1,2, such that
(112)α1A11-α2b21M>0,α2A22-α1b12M>0
hold; then, for any two positive solutions (x1(n),x2(n)) and (x1*(n),x2*(n)) of system (2), one has
(113)lim⁡n→+∞(xi(n)-xi*(n))=0,
where
(114)Aii=min⁡{biiL,2Mi-biiM},i=1,2.Now let us consider the autonomous case of system (2). Assume that ai>1, i=1,2; then
(115)Aii=min⁡{bii,2biiexp⁡⁡{ai-1}-bii}=(2exp⁡⁡{ai-1}-1)bii.
As a corollary of Theorem B, we have the following.Theorem B.'Assume thatai>1,i=1,2, and(116)(2exp⁡⁡{a1-1}-1)b11-b21>0,(2exp⁡⁡{a2-1}-1)b22-b12>0hold; then, for any two positive solutions(x1(n),x2(n))and(x1*(n),x2*(n))of the system (2), one has(117)lim⁡n→+∞(xi(n)-xi*(n))=0.Remark 15.
Wu and Zhang [19] obtained a set of sufficient conditions which ensure the global attractivity of a discrete autonomous Lotka-Volterra competition system with infinite delays and feedback controls; as a direct corollary of their main result, one could also obtain a set of sufficient conditions which ensure the global stability of the autonomous case of system (2). However, their result required the assumption ai≤1, i=1,2. While Theorem B′ solves the case ai>1, i=1,2. It is in this sense that our result Theorem B′ supplements and complements the main result of Wu and Zhang [19].Now, concerned with the extinction property of the system (2), with some minor revision of the proof of Theorems 8 and 9, we could obtain the following results.Theorem C.
Assume that(118)a1Ma2L<b11Lb21M,a1Ma2L<b12Lb22M
hold; let (x1(n),x2(n)) be any positive solution of system (2); then
(119)lim⁡n→+∞x1(n)=0.Theorem D.
Assume that(120)a1La2M>b11Mb21L,a1La2M>b12Mb22L
hold; let (x1(n),x2(n)) be any positive solution of system (2); then
(121)lim⁡n→+∞x2(n)=0.It is interesting to investigate the stability property of the rest of the species when one of the species in system (2) is driven to extinction. In this case, as a direct corollary of Theorems 10 and 11, by simple computation, we have the following.Theorem E.
Assume that (118) holds and also
(122)b22Mb22Lexp⁡{a2M-1}<2
holds; then, for any positive solution (x1(n),x2(n)) of system (2), we have
(123)lim⁡n→+∞x1(n)=0,lim⁡n→+∞(x2(n)-x2*(n))=0,
where x2*(n) is any positive solution of the system x2(n+1)=x2(n)exp⁡(a2(n)-b22(n)x2(n)).Theorem F.
Assume that (120) holds and also
(124)b11Mb11Lexp⁡{a1M-1}<2
holds; then, for any positive solution (x1(n),x2(n)) of system (2), we have
(125)lim⁡n→+∞x2(n)=0,lim⁡n→+∞(x1(n)-x1*(n))=0,
where x1*(n) is any positive solution of the system x1(n+1)=x1(n)exp⁡(a1(n)-b11(n)x1(n)).Obviously, Theorems B–F complement and supplement the main results of Chen [2].



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*Source: 867313-2014-10-14.xml*