# Dynamics of a Discrete Allelopathic Phytoplankton Model with Infinite Delays and Feedback Controls

Authors: Liang Zhao; Bin Qin; Fengde Chen
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

A discrete allelopathic phytoplankton model with infinite delays and feedback controls is studied in this paper. By applying the discrete comparison theorem, a set of sufficient conditions which guarantees the permanence of the system is obtained. Also, by constructing some suitable discrete Lyapunov functionals, some sufficient conditions for the extinction of the system are obtained. Our results extend and supplement some known results and show that the feedback controls and toxic substances play a crucial role on the permanence and extinction of the system.


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## Body

## 1. Introduction

Given a bounded sequence of real numbersfk, let fu and fl denote supk∈Zfk and infk∈Zfk, respectively.Many real-world phenomena are studied through discrete mathematical models governed by difference equations which are more appropriate than the continuous ones when the populations have nonoverlapping generations; the study of the dynamic behaviors of discrete time models becomes the subject of intense research in mathematics biology, such topics as permanence and extinction, and existence of positive periodic solution (almost periodic solution), have been extensively studied by many scholars (see [1–23] and the references cited therein).Recently, some scholars believed that a more appropriate competition model should be considered with nonlinear interinhibition terms. Qin et al. [1] and Wang et al. [2] considered the following discrete two species competitive system with nonlinear interinhibition terms:(1)x1k+1=x1kexpr1k−a1kx1k−c2kx2k1+x2k,x2k+1=x2kexpr2k−a2kx2k−c1kx1k1+x1k,where rik,aik,ciki=1,2 are assumed to be bounded positive sequences, and x1kandx2k represent the density of species x1 and x2 at the kth generation, respectively, riki=1,2 is the intrinsic growth rate of two species. aiki=1,2 is the rate of intraspecific competition of the first and second species, respectively. cik,i=1,2 is the rate of interspecific competition of the first and second species, respectively. Qin et al. [1] obtained sufficient conditions which ensure the permanence, existence, and global stability of positive periodic solutions of system (1). As for the almost periodic case, Wang and Liu [2] further investigated the existence, uniqueness, and uniformly asymptotic stability of positive almost periodic solution of system (1).In recent years, many scholars have made great achievements in the competition system with the effect of toxic substance [3, 4, 24–26]. Yue [3] investigated system (1) with the second species which could be toxic, while the other one is nontoxic. He studied system (2) and gave the sufficient conditions of the extinction of one species and the global attractive of the other one.(2)x1k+1=x1kexpr1k−a1kx1k−c2kx2k1+x2k−b1kx1kx2k,x2k+1=x2kexpr2k−a2kx2k−c1kx1k1+x1k.Note that ecosystems are easily disturbed by human activities, such as planting and harvesting, which can give rise to changes of population density. In order to give a better description of such a system, scholars introduced feedback control variables into ecosystems. Many researchers have done research on the systems with feedback control variables [6–8, 11, 14, 16, 17]. Recently, Wang and Liu [6] studied the competitive system with feedback controls as follows:(3)x1n+1=x1nexpr1n−a1nx1n−b1nx2n1+x2n−e1nu1n,x2n+1=x2nexpr2n−b2nx1n1+x1n−a2nx2n−e2nu2n,Δu1n=−b1nu1n+d1nx1n,Δu2n=−b2nu2n+d2nx2n,where uin,i=1,2, denotes the feedback control variables. Wang and Liu [6] studied the existence and uniformly asymptotic stability of unique positive almost periodic solution of system (3). Yu [7] further investigated the influence of feedback control variables on the extinction of the system. He gave a set of sufficient conditions for the extinction of one species.As we all know, due to seasonal fluctuations in the environment and hereditary factors, time delays have been introduced into the biological system (see [11, 14, 21, 27–36]). Zhao et al. [8] further considered a discrete Lotka–Volterra competition system with infinite delays and single feedback control variable as follows:(4)x1n+1=x1nexpa1n−b11nx1n−b12n∑s=0∞k12sx2n−s−d1n∑s=0∞h1sun−s,x2n+1=x2nexpa2n−b22nx2n−b21n∑s=0∞k21sx1n−s−d2n∑s=0∞h2sun−s,un+1=un1−en+c1n∑s=0∞l1sx1n−s+c2n∑s=0∞l2sx2n−s.By applying the discrete comparison theorem, a set of sufficient conditions which guarantees 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.Through the above discussion, we find that, based on system (2), we can further consider the infinite delay and feedback control. So, we propose and study the following discrete competition system with infinite delays and feedback control variables:(5)x1k+1=x1kexpr1k−a1kx1k−b1k∑s=0∞K12sx2k−s1+∑s=0∞K12sx2k−s−γkx1kx22k−c1k∑s=0∞H1su1k−s,x2k+1=x2kexpr2k−b2k∑s=0∞K21sx1k−s1+∑s=0∞K21sx1k−s−a2kx2k−c2k∑s=0∞H2su2k−s,u1k+1=u1k1−e1k+d1k∑s=0∞L1sx1k−s,u2k+1=u2k1−e2k+d2k∑s=0∞L2sx2k−s.In system (4), xik i=1,2 is the density of xi species at the kth generation, and uik,i=1,2, is the feedback control variable.Throughout this paper, we assume thatC
1 rik,aik,bik,cik,dik,and eik (i=1,2) are bounded sequences of real numbers defined on Z such that(6)ril>0,ail>0,bil>0,cil>0,dil>0,0<eil<eiu<1,i=1,2.C
2
K
i
j
s
,
H
i
s, and Lis (i,j=1,2) are nonnegative bounded sequences such that(7)∑s=0∞Kijs=1,∑s=0∞His=1,∑s=0∞Lis=1,i,j=1,2,Θij=∑s=0∞Kijs,s<+∞,Λi=∑s=0∞His,s<+∞,ϒi=∑s=0∞Lis,s<+∞.According to the biological background of system (5), we only consider the solution of system (5) with the following initial conditions:(8)xis=Ψis≥0,Ψi0>0,supk∈Z−Ψik<+∞,uis=Φs≥0,Φi0>0,supk∈Z−Φik<+∞,i=1,2,where s=…,−n,−n+1,…,−1,0. It is easy to prove that the solution of system (5) which satisfies the initial conditions (8) is positive.Here, we mention that this is the first time such kind of model be proposed and studied, and as far as system (5) is concerned, whether the feedback control variables and toxic substances have influence on the permanence and extinction of the system or not is an interesting problem. The aim of this paper is to investigate the permanence and extinction of system (5); specially, we will find out the answer to the above problem.The paper is structured in the following way. Some useful Lemmas are presented in Section2. In Sections 3 and 4, by using the methods of Zhao et al. [8], we investigate the permanence and extinction of system (5). Three examples are presented to show the feasibility of the main results in Section 5. We end this paper by a brief discussion.


## 2. Lemmas

Now, let us consider the following difference equation:(9)yk+1=ayk+b,where aandb are positive constants.Lemma 1 (see [37]).
Assume thata<1, for any initial value y0, there exists a unique solution yk of equation (9), which can be expressed as follows:(10)yk=aky0−y∗+y∗,where y∗=b/1−a. Thus, for any solution y(k) of system (9), we have(11)limk⟶+∞yk=y∗.Lemma 2 (see [37]).
Letk∈Nk0+=k0,k0+1,…,k0+l,…, r≥0. For any fixed k, gk,r is a nondecreasing function with respect to r, and for k≥k0, the following inequalities hold:(12)yk+1≤gk,yk,uk+1≥gk,uk.
Ifyk0≤uk0, then yk≤uk for all k≥k0.Lemma 3 (see [38]).
Assume thatrn>0, x(n) satisfies xn>0 and(13)xn+1≤xnexprn1−axn,for n∈n1,∞, where a is a positive constant. Then,(14)limsupn⟶+∞xn≤1aruexpru−1.Lemma 4 (see [38]).
Assume thatrn>0, xn satisfies xn>0 and(15)xn+1≥xnexprn1−axn,for n∈n1,∞, lim supn⟶+∞xn≤x∗ and xn1>0, where a and x∗ are positive constants such that ax∗>1. Then,(16)lim infn⟶+∞xn≥1aexpru1−ax∗.Lemma 5 (see [38]).
Letx: Z⟶R be a nonnegative bounded sequence, and let H: Z+⟶R be a nonnegative sequence such that ∑n=0∞Hn=1, then(17)lim infn⟶+∞xn≤lim infn⟶+∞∑ns=−∞Hn−sxs≤lim supn⟶+∞∑ns=−∞Hn−sxs≤lim supn⟶+∞xn.


## 3. Permanence

Concerned with the permanence of system (5), we have the following result.Theorem 1.
Assume that(18)ril−biuMj−ciuUi>0,i,j=1,2,i≠j,γu<r1l−b1uM2−c1uU1M1M22,hold; then, for any positive solution x1k,x2k,u1k,u2k of system (5), we have(19)mi≤lim infk⟶+∞xik<lim supk⟶+∞xik≤Mi,Bi≤lim infk⟶+∞uik<lim supk⟶+∞uik≤Ui,where(20)Mi=1ailexpriu−1,Ui=diuMieil,m1=r1l−b1uM2−γuM1M22−c1uU1a1u⋅expr1l−b1uM2−γuM1M22−c1uU1−a1uM1,m2=r2l−b2uM1−c2uU2a2u⋅expr2l−b2uM1−c2uU2−a2uM2,Bi=dilmieiu,i=1,2.Proof.
From the first and second equations of system (5), we have(21)xik+1≤xikexprik1−ailriuxik,i=1,2.Hence, from Lemma 3, we can obtain(22)lim supk⟶+∞xik≤1ailexpriu−1=defMi.
According to Lemma5 and the above inequality, we have(23)lim supk⟶+∞∑s=0∞Lisxik−s=lim supk⟶+∞∑s=0∞Lik−sxis≤lim supk⟶+∞xik≤Mi.
For anyε>0, there exists a positive integer N1 such that(24)∑s=0∞Lisxik−s≤Mi+ε,for alln>N1.
By the third and fourth equations of system (5), we have(25)uik+1≤1−eiluik+diuMi+ε.
Hence, by applying Lemmas1 and 2 to (25), we obtain(26)lim supk⟶+∞uik≤diuMi+εeil.
Settingε⟶0, it follows that(27)lim supk⟶+∞uik≤diuMieil=defUi.
Condition (18) implies that, for enough small positive constant ε1, the following inequalities hold:(28)r1l−b1uM2+ε1−γuM1+ε1M2+ε12−c1uU1+ε1>0,r2l−b2uM1+ε1−c2uU2+ε1>0.
For aboveε1, it follows from (22) and (27) that there exists a positive integer N2≥N1 such that(29)∑s=0∞Hisuik−s≤Ui+ε1,∑s=0∞Kjisxik−s≤Mi+ε1,xik≤Mi+ε1for alln>N2.
Thus, for alln>N2, from (28), (29), and the first two equations of system (5), we have(30)x1k+1≥x1kexpr1l−b1uM2+ε1−γuM1+ε1M2+ε12−c1uU1+ε11−D1εx1k,x2k+1≥x2kexpr2l−b2uM1+ε1−c2uU2+ε11−D2εx2k,where(31)D1ε=a1ur1l−b1uM2+ε1−γuM1+ε1M2+ε12−c1uU1+ε1,D2ε=a2ur2l−b2uM1+ε1−c2uU2+ε1.
Notice that(32)aiuail≥1,expr1u−1r1l−b1uM2+ε1−γuM1+ε1M2+ε12−c1uU1+ε1>1,expr2u−1r2l−b2uM1+ε1−c2uU2+ε1>1,then(33)D1ε⋅M1=a1ur1l−b1uM2+ε1−γuM1+ε1M2+ε12−c1uU1+ε1⋅1a1lexpr1u−1>1,D2ε⋅M2=a2ur2l−b2uM1+ε1−c2uU2+ε1⋅1a2lexpr2u−1>1.
Hence, according to Lemma4, we have(34)liminfk⟶+∞x1k≥1D1ε⋅expr1l−b1uM2+ε1−γuM1+ε1M2+ε12−c1uU1+ε1−a1uM1,liminfk⟶+∞x2k≥1D2ε⋅expr2l−b2uM1+ε1−c2uU2+ε1−a2uM2.
Settingε1⟶0, it follows that(35)lim infk⟶+∞x1k≥1D1⋅expr1l−b1uM2−γuM1M22−c1uU1−a1uM1=defm1,(36)lim infk⟶+∞x2k≥1D2⋅expr2l−b2uM1−c2uU2−a2uM2=defm2,where(37)D1=a1ur1l−b1uM2−γuM1M22−c1uU1,D2=a2ur2l−b2uM1−c2uU2.
According to Lemma5, from (35) and (36), we have, for any ε2>0 small enough (without loss of generality, assume that ε2<1/2minimi, i = 1, 2), there exists N3>N2, such that(38)∑s=0∞Lisxik−s≥mi−ε2,for alln≥N3.
Forn≥N3, from (38) and the third and fourth equations of system (5), we have(39)uik+1≥1−eiuuik+dilmi−ε2.
Hence, by applying Lemmas1 and 2 to (39), we obtain(40)lim supk⟶+∞uik≥dilmi−ε2eiu.
Settingε2⟶0, it follows that(41)lim supk⟶+∞uik≥dilmieiu=defBi.
This ends the proof of Theorem 3.1.


## 4. Extinction

Concerned with the extinction property of system (5), we could establish the following results.Theorem 2.
Assume that(42)r1ur2l<mina1le1u+c1ld1lb2ue1u,b1le2l1+M2a2ue2l+c2ud2uH1,holds. Let x1k,x2k,u1k,u2k be any positive solution of system (5), then(43)limk⟶+∞x1k=0,limk⟶+∞u1k=0.Theorem 3.
Assume that(44)r1lr2u>max1+M1a1ue1l+c1ud1ub2le1l,b1ue2ua2le2u+c2ld2lH2,γu<min1M1M22r1l−1+M1r2ua1ue1l+c1ud1ub2le1l,r1l−r2ub1ue2ua2le2u+c2ld2lH3hold. Let x1k,x2k,u1k,u2k be any positive solution of system (5), then(45)limk⟶+∞x2k=0,limk⟶+∞u2k=0.Proof of Theorem 2.
ConditionH1 is equivalent to(46)c1le1u>r1ur2lb2ud1l−a1ld1l,c2ue2l<r2lb1l1+M2r1ud2u−a2ud2u.
From (46), one could choose positive constants α,β, η1,andη2 and enough small positive ε such that(47)r1ur2l<βα,c1le1u>η1α>βb2u−αa1lαd1l>r1ur2lb2ud1l−a1ld1l,c2ue2l<η2β<αb1l−1+M2+εβa2u1+M2+εβd2u<r2lb1l1+M2+εr1ud2u−a2ud2u.That is,(48)αr1u−βr2l=−λ1<0,η1e1u−αc1l<0,βc2u−η2e2l<0,βb2u−αa1l−η1d1l<0,η2d2u+βa2u−αb1l1+M2+ε<0.
Letx1k,x2k,u1k,u2kT be a positive solution of system (5). For above ε, from Theorem 1, there exists a enough large T1, such that(49)xit<Mi+ε,uit<Ui+ε,t≥T1,i=1,2.
Consider the following discrete Lyapunov functional(50)V1k=x1αkx2−βkexpη2u2k−η1u1k−α∑s=0∞K12s∑q=k−sk−1b1q+sx2q1+M2−α∑∞s=0H1s∑q=k−sk−1c1q+su1q+β∑s=0∞K21s∑q=k−sk−1b2q+sx1q1+m1+β∑s=0∞H2s∑q=k−sk−1c2q+su2q+η2∑s=0∞L2s∑q=k−sk−1d2q+sx2q−η1∑s=0∞L1s∑q=k−sk−1d1q+sx1q.
By calculating, we obtain(51)V1k+1V1k=expαr1k−a1kx1k−b1k∑s=0∞K12sx2k−s1+∑s=0∞K12sx2k−s−γkx1kx22k−c1k∑s=0∞H1su1k−s−βr2k−b2k∑s=0∞K21sx1k−s1+∑s=0∞K21sx1k−s−a2kx2k−c2k∑s=0∞H2su2k−s−η1−e1ku1k+d1k∑s=0∞L1sx1k−s+η2−e2ku2k+d2k∑s=0∞L2sx2k−s−α1+M2∑s=0∞K12sb1k+sx2k−b1kx2k−s−α∑s=0∞H1sc1k+su1k−c1ku1k−s+β1+m1∑s=0∞K21sb2k+sx1k−b2kx1k−s+β∑s=0∞H2sc2k+su2k−c2ku2k−s−η1∑s=0∞L1sd1k+sx1k−d1kx1k−s+η2∑s=0∞L2sd2k+sx2k−d2kx2k−s≤expαr1k−βr2k+−αa1k+β∑s=0∞K21sb2k+s−η1∑s=0∞L1sd1k+sx1k+βa2k+η2∑s=0∞L2sd2k+s−α∑s=0∞K12sb1k+s1+M2+εx2k+η1e1k−α∑s=0∞H1sc1k+su1k+β∑s=0∞H2sc2k+s−η2e2ku2k≤expαr1u−βr2l+βb2u−αa1l−η1d1lx1k+η2d2u+βa2u−αb1l1+M2+εx2k+η1e1u−αc1lu1k+βc2u−η2e2lu2k.
From inequalities (48) and (51), we can obtain(52)V1k+1≤V1kexp−λ1.
Therefore,(53)V1k≤V10exp−kλ1.
From (22) and (27), we know that there exists a M>0 such that(54)xik<M,uik<M,for allk∈Z,i=1,2.
Hence,(55)V10=x1α0x2−β0expη2u20−η1u10−α∑s=0∞K12s∑q=−s−1b1q+sx2q1+M2−α∑s=0∞H1s∑q=−s−1c1q+su1q+β∑s=0∞K21s∑q=−s−1b2q+sx1q1+m1+β∑s=0∞H2s∑q=−s−1c2q+su2q+η2∑s=0∞L2s∑q=−s−1d2q+sx2q−η1∑s=0∞L1s∑q=−s−1d1q+sx1q<x1α0x2−β0expη2M+βb2uM∑s=0∞K21ss1+m1+βc2uM∑s=0∞H2ss+η2d2uM∑s=0∞L2ss=x1α0x2−β0expMη2+βb2uMΘ211+m1+βc2uMΛ2+η2d2uMϒ2<+∞.
On the contrary, we also have(56)V1k≥x1αkx2−βkexp−η1u1k−α∑s=0∞K12s∑q=k−sk−1b1q+sx2q1+M2−α∑s=0∞H1s∑q=k−sk−1c1q+su1q−η1∑s=0∞L1s∑q=k−sk−1d1q+sx1q≥x1αkM−βexp−η1M−αb1uM∑s=0∞K12ss1+M2−αc1uM∑s=0∞H1ss−η1d1uM∑s=0∞L1ss=x1αkM−βexp−η1M+d1uMϒ1−αb1uMΘ121+M2+c1uMΛ1.
Combining inequalities (53), (55), and (56), we have(57)x1k≤Δexp−λ1αk,where(58)Δ=V101/αMβ/αexpη1αM+d1uMϒ1+b1uMΘ121+M2+c1uMΛ1<+∞.
Hence, we obtain that(59)limk⟶+∞x1k=0.
Similar to the corresponding proof of Theorem1 by Chen et al. [7], we can easily obtain that limk⟶+∞u1k=0. This ends the proof of Theorem 2.Proof of Theorem 3.
ConditionH2 is equivalent to(60)c1ue1l<r1lb2l1+M1r2ud1u−a1ud1u,c2le2u>r2ur1lb1ud2l−a2ld2l.
FromH3 and (60), there exist positive constants α,β,δ1,andδ2 and enough small positive ε such that(61)r1lr2u>βα,c1ue1l<δ1α<βb2l−1+M1+εαa1u1+M1+εαd1u<r1lb2l1+M1+εr2ud1u−a1ud1u,c2le2u>δ2β>αb1u−βa2lβd2l>r2ur1lb1ud2l−a2ld2l,1+M1+εa1ue1l+c1ud1ub2le1l<βα<r1l−γuM1+εM2+ε2r2u,b1ue2ua2le2u+c2ld2l<βα<r1l−γuM1+εM2+ε2r2u.That is,(62)αc1u−δ1e1l<0,δ2e2u−βc2l<0,αa1u−βb2l1+M1+ε+δ1d1u<0,αb1u−βa2l−δ2d2l<0,−αr1l+βr2u+αγuM1+εM2+ε2=−ξ1<0.
Letx1k,x2k,u1k,u2kT be a positive solution of system (5). For above ε, from Theorem 1, there exists an enough large T2, such that(63)xik<Mi+ε,uik<Ui+ε,t≥T2,i=1,2.
Define the following Lyapunov functional as follows:(64)V2k=x1−αkx2βkexpδ1u1k−δ2u2k+α∑s=0∞K12s∑q=k−sk−1b1q+sx2q+α∑s=0∞H1s∑q=k−sk−1c1q+su1q−β∑s=0∞K21s∑q=k−sk−1b2q+sx1q1+M1−β∑s=0∞H2s∑q=k−sk−1c2q+su2q−δ2∑s=0∞L2s∑q=k−sk−1d2q+sx2q+δ1∑s=0∞L1s∑q=k−sk−1d1q+sx1q.
Similar to the analysis of (51)–(58), we have(65)limk⟶+∞x2k=0,limk⟶+∞u2k=0.
This ends the proof of Theorem3.
From Theorems2 and 3, 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 species.
Consider the following discrete equations:(66)x2k+1=x2kexpr2k−a2kx2k−c2k∑s=0∞H2su2k−s,u2k+1=u2k1−e2k+d2k∑s=0∞L2sx2k−s.(67)x1k+1=x1kexpr1k−a1kx1k−c1k∑s=0∞H1su1k−s,u1k+1=u1k1−e1k+d1k∑s=0∞L1sx1k−s.Theorem 4.
Assume that (H1) holds; also,(68)c2u<A22e2ld2u,holds; then, for any positive solution x1k,x2k,u1k,u2k of system (5) and any positive solution x2∗k,u2∗k of system (66), we have(69)limk⟶+∞x1k=0,limk⟶+∞u1k=0,limk⟶+∞x2k−x2∗k=0,limk⟶+∞u2k−u2∗k=0,where A22=mina2l,2/M2−a2u.Theorem 5.
Assume that (H2) and (H3) hold; also,(70)c1u<A11e1ld1u,holds; then, for any positive solution x1k,x2k,u1k,u2k of system (5) and any positive solution x1∗k,u1∗k of system (67), we have(71)limk⟶+∞x1k−x1∗k=0,limk⟶+∞u1k−u1∗k=0,limk⟶+∞x2k=0,limk⟶+∞u2k=0,where A11=mina1l,2/M1−a1u.Proof of Theorem 4.
By conditions (68), we can choose positive constants ρ1andρ2 such that(72)e2lc2u>ρ1ρ2>d2uA22.
Thus, there exists enough small positive constantω and ε such that(73)ρ1A22ε−ρ2d2u>ω,ρ2e2l−ρ1c2u>ω,where A22ε=mina2l,2/M2+ε−a2u.
Now, we define a Lyapunov functional as follows:(74)G1k=ρ1lnx2k−lnx2∗k+∑s=0∞K21s∑q=n−sn−1b2q+sx1q+∑s=0∞H2s∑q=n−sn−1c2q+su2q−u2∗q+ρ2u2k−u2∗k+∑s=0∞L2s∑q=n−sn−1d2q+sx2q−x2∗q.
One could easily see thatG1k≥0 for all k∈Z+. Also, for any fixed k∗∈Z+, from (54), one could see that(75)G1k∗=ρ1lnx2k∗−lnx2∗k∗+ρ1Mb2u∑s=0∞K21ss+ρ1c2usupq∈Z+,q≤k∗u2q−u2∗q∑s=0∞H2ss+ρ2u2k∗−u2∗k∗+ρ2d2usupq∈Z+,q≤k∗x2q−x2∗q∑s=0∞L2ss<+∞.
It follows from the second equation of system (5) and the mean value theorem that(76)△G1k≤ρ1−1φ2k−1φ2k−a2kx2k−x2∗k+b2ux1k+c2uu2k−u2∗k+ρ2−e2lu2k−u2∗k+d2ux2k−x2∗k.≤−ρ1mina2l,2M2+ε−a2u−ρ2d2ux2k−x2∗k−ρ2e2l−ρ1c2uu2k−u2∗k+ρ1b2ux1k=−ρ1A22ε−ρ2d2ux2k−x2∗k−ρ2e2l−ρ1c2uu2k−u2∗k+ρ1b2ux1k≤−ωx2k−x2∗k+u2k−u2∗k+ρx1k,where ρ=ρ1b2u.
Summating both sides of the above inequality fromk∗ to k, we have(77)∑p=k∗kG1p+1−G1p≤−ω∑p=k∗kx2p−x2∗p+u2p−u2∗p+ρ∑p=k∗kx1p.
Hence,(78)G1k+1+ω∑p=k∗kx2p−x2∗p+u2p−u2∗p≤G1k∗+ρ∑p=k∗kx1p.
Then, from (75) and (57), we have(79)∑p=k∗kx2p−x2∗p+u2p−u2∗p≤G1k∗+ρ∑p=k∗kx1pω<+∞.
Therefore,(80)∑p=k∗∞x2p−x2∗p+u2p−u2∗p<+∞,which means that(81)limk⟶+∞x2k−x2∗k+u2k−u2∗k=0.
Consequently,(82)limk⟶+∞x2k−x2∗k=0,limk⟶+∞u2k−u2∗k=0.
This completes the proof of Theorem4.Proof of Theorem 5.
The proof of Theorem5 is similar to that of Theorem 4, and we omit the detail here.


## 5. Examples

In this section, we shall give three examples to illustrate the feasibility of main results.Example 1.
Consider the following equations:(83)x1k+1=x1kexp1−5.2+0.2sinkx1k−0.5∑s=0∞e−1/ee−sx2k−s1+∑s=0∞e−1/ee−sx2k−s−3x1kx22k−3∑s=0∞e2−1e2e−2su1k−s,x2k+1=x2kexp0.95−0.05cosk−0.2∑s=0∞e−1/ee−sx1k−s1+∑s=0∞e−1/ee−sx1k−s−2.5+0.5sinkx2k−0.8∑s=0∞e2−1e2e−2su2k−s,u1k+1=u1k1−0.5+0.25∑s=0∞e3−1e3e−3sx1k−s,u2k+1=u2k1−0.8+0.2∑s=0∞e3−1e3e−3sx2k−s.
One could easily see that conditions (C1) and (C2) are satisfied. Also, by calculating, one has(84)M1=1a1lexpr1u−1=0.2,M2=1a2lexpr2u−1=0.5,U1=d1ue1lM1=0.1,U2=d2ue2lM2=18,r1l−b1uM2−c1uU1=0.45>0,r2l−b2uM1−c2uU2=0.4>0,γu=3<r1l−b1uM2−c1uU1M1M22=22.5.
Clearly, condition (18) is satisfied, and so from Theorem 1, we have(85)mi≤lim infk⟶+∞xik<lim supk⟶+∞xik≤Mi,Bi≤lim infk⟶+∞uik<lim supk⟶+∞uik≤Ui,where x1k,x2k,u1k,u2k is any positive solution of system (83).
Figure1 shows the dynamic behaviors of system (83), which strongly supports the above assertions.Figure 1
Numeric simulations of the solutionsx1k,x2k,u1k,u2k of system (83), with the initial conditions x1s,x2s,u1s,u2s = 0.15,0.23,0.31,0.56, 0.32,0.4,0.43,0.7, 0.5,0.67,0.6,0.84, s=…,−n,−n+1,…,−1,0, respectively.Example 2.
Consider the following equations:(86)x1k+1=x1kexp1−3.2+0.2sinkx1k−5∑s=0∞e−1/ee−sx2k−s1+∑s=0∞e−1/ee−sx2k−s−0.00005xkx22k−0.3∑s=0∞e2−1e2e−2su1k−s,x2k+1=x2kexp0.95−0.05cosk−1.5∑s=0∞e−1/ee−sx1k−s1+∑s=0∞e−1/ee−sx1k−s−0.4x2k−0.1∑s=0∞e2−1e2e−2su2k−s,u1k+1=u1k1−0.9+5∑s=0∞e3−1e3e−3sx1k−s,u2k+1=u2k1−0.6+2∑s=0∞e3−1e3e−3sx2k−s.
One could easily see that conditions (C1) and (C2) are satisfied. Also, by calculating, one has(87)r1ur2l=109≈1.11,M2=1a2lexpr2u−1=2.5,a1le1u+c1ld1lb2ue1u≈3.11,11+M2b1le2la2ue2l+c2ud2u≈2.72,A22=mina2l,2M2−a2u=0.4,0.1=c2u<A22e2ld2u=0.12.
Clearly, condition (H1) and (68) are satisfied, and so from Theorem 2 and 4 we know that species x1 will be driven to extinction, while species x2 is global attractive.
Figure2 shows the dynamic behaviors of system (86), which strongly support our results.Figure 2
Numeric simulations of the solutionsx1k,x2k,u1k,u2k of system (86), with the initial conditions x1s,x2s,u1s,u2s = 0.3,0.2,2.7,0.4, 0.7,1.2,2,1.6, 1.1,0.7,1.5,2.3, s=…,−n,−n+1,…,−1,0, respectively.Example 3.
Consider the following equations:(88)x1k+1=x1kexp1−2.5+0.5sinkx1k−5∑s=0∞e−1/ee−sx2k−s1+∑s=0∞e−1/ee−sx2k−s−0.00005xkx22k−0.1∑s=0∞e2−1e2e−2su1k−s,x2k+1=x2kexp0.65−0.05cosk−5∑s=0∞e−1/ee−sx1k−s1+∑s=0∞e−1/ee−sx1k−s−3x2k−0.3∑s=0∞e2−1e2e−2su2k−s,u1k+1=u1k1−0.9+5∑s=0∞e3−1e3e−3sx1k−s,u2k+1=u2k1−0.9+2∑s=0∞e3−1e3e−3sx2k−s.
One could easily see that conditions (C1) and (C2) are satisfied. Also, by calculating, one has(89)M1=1a1lexpr1u−1=0.5,M2=1a2lexpr2u−1≈0.25,1+M1a1ue1l+c1ud1ub2le1l≈1.07,b1ue2ua2le2u+c2ld2l≈1.36,r1lr2u≈1.43>max1.07,1.36,1M1M22r1l−1+M1r2ua1ue1l+c1ud1ub2le1l=8,1M1M22r1l−r2ub1ue2ua2le2u+c2ld2l=1.6,A11=mina1l,2M1−a1u=1,0.1=c1u<A11e1ld1u=0.18.
Clearly, conditions (H2), (H3), and (70) are satisfied, and so from Theorem 3 and 5, we know that species x2 will be driven to extinction, while species x1 is global attractive.
Figure3 shows the dynamics behaviors of system (88), which strongly supports our results.Figure 3
Numeric simulations of the solutionsx1k,x2k,u1k,u2k of system (88), with the initial conditions x1s,x2s,u1s,u2s = 0.2,0.3,1.2,0.1, 0.3,0.5,1.5,0.4, 0.4,0.7,0.8,0.6, s=…,−n,−n+1,…,−1,0, respectively.


## 6. Discussion

(1)
From the conditions of Theorems2–5, we can easily find that the feedback control variables and toxic substances play a crucial role on the extinction of system (5). We find that, by choosing suitable feedback control variable, one of the species will be extinct or permanent, that is, feedback control variable, which represents the biological control or some harvesting procedures, is an unstable factor of the system. From the conditions of Theorem 2, we also find that, despite the second species could produce toxicity, if the toxic rate is very low such that inequality H3 holds; then, the second species is still driven to extinction; in other words, that lower rate of toxic production has no influence on the extinction property of system (5).
(2)
From the conditions of Theorem1, we find that the feedback control variables and toxic substances play an important role on the permanence of the system; only the rate of toxic production and feedback control variables are small enough such that inequality (18) holds, and the toxic substances and feedback controls have no effect on the permanence of the system.
(3)
Yu [7] obtained a set of sufficient conditions that guarantees the extinction of system (3), as direct corollaries of Theorems 2–5; one could also obtain other conditions for the extinction of system (3), which supplements and complements the results of Yu [7].
(4)
Now, concerned with the extinction property of system (2), with some minor revisions to the proof of Theorems 2 and 3, we could obtain the following results.Corollary 1.
Assume that(90)r1ur2l<mina1lb2u,b1l1+M2a2uholds, and let x1k,x2k be any positive solution of system (2), then(91)limk⟶+∞x1k=0.Corollary 2.
Assume that(92)r1lr2u>max1+M1a1ub2l,b1ua2l,(93)γu<min1M1M22r1l−1+M1r2ua1ub2l,r1l−r2ub1ua2l,hold; let x1k,x2k be any positive solution of system (2), then(94)limn⟶+∞x2k=0.It is interesting to investigate the stability property of the rest species under the assumption that one of the species in system (2) is driven to extinction. As direct corollaries of Theorems 4 and 5, by simple computation, we haveCorollary 3.
Assume that (90) holds; also,(95)a2ua2lexpr2u−1<2holds; then, for any positive solution x1k,x2k of system (2), we have(96)limn⟶+∞x1k=0,limn⟶+∞x2k−x2∗k=0,where x2∗k is any positive solution of the system x2k+1=x2kexpr2k−a2kx2k.Corollary 4.
Assume that (92) and (93) hold; also,(97)a1ua1lexpr1u−1<2holds; then, for any positive solution x1k,x2k of system (2), we have(98)limn⟶+∞x2k=0,limn⟶+∞x1k−x1∗k=0,where x1∗k is any positive solution of the system x1k+1=x1kexpr1k−a1kx1k.Obviously, Theorems 2.1, 2.5, 3.1, and 3.2 obtained by Yue [3] are corollaries 1–4 of Theorems 2–5, so corollaries 1–4 extend the results of Yue [3].



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