# Global Attractivity and Extinction of a Discrete Competitive System with Infinite Delays and Single Feedback Control

Authors: Yalong Xue; Xiangdong Xie; Qifa Lin; Fengde Chen
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

A nonautonomous discrete two-species competition system with infinite delays and single feedback control is considered in this paper. Based on the discrete comparison theorem, a set of sufficient conditions which guarantee the permanence of the system is obtained. Then, 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, by choosing some suitable feedback control variable, one of two species will be driven to extinction.


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## Body

## 1. Introduction

Two or more species compete for the same limited food source or in some way inhibit each other’s growth. For example, competition may be for territory which is directly related to food resources. The importance of species competition in nature is obvious. Tradition two-species Lotka-Volterra competition system is as follows:(1)x˙1t=x1tr1-a11x1t-a12x2t,x˙2t=x2tr2-a21x1t-a22x2t.However, system (1) has a property which is considered as a disadvantage and that is the linearity of the above system. Ayala et al. [1] presented the following nonlinear competitive system with continuous time version: (2)x˙1t=x1tr1-x1t-a1x2t-c1x22t,x˙2t=x2tr2-x2t-a2x1t-c2x12t.Assume that each species needs some time to mature and the competition occurs after some time lag required for maturity of the species; Gopalsamy [2] discussed the following system with discrete delays: (3)x˙1t=x1tr1-x1t-a1x2t-τ2-c1x22t-τ2,x˙2t=x2tr2-x2t-a2x1t-τ1-c2x12t-τ1.Such systems are not well studied in the sense that most results are continuous time cases related (see [3, 4]). As we know, a discrete time system governed by difference equations is more approximate than the continuous ones when the populations have nonoverlapping generations or a short-life expectancy. Discrete time system can also provide efficient computation for numerical simulations (see [5–10]). Considering the biological parameters naturally being subject to almost periodic fluctuation in time, Tan and Liao [11] established the following nonautonomous discrete competition system: (4)x1k+1=x1kexp⁡r1k-x1k-a1kx2k-τ2-c1kx22k-τ2,x2k+1=x2kexp⁡r2k-x2k-a2kx1k-τ1-c2kx12k-τ1.In the real world, ecosystems are disturbed by unpredictable forces which can result in some changes of parameters. In order to accurately describe such a system, scholars introduced feedback control into ecosystems. Recently, the ecosystems with feedback controls have been extensively studied and obtained many interesting results (see [12–18]), noting that models in [10, 12–15] considered at least two feedback controls variables, which means that, for the different species, different control strategy is adopted, whereas, in the real world, the strategy adopted for one species may also affect the other species. For example, spraying pesticide not only can reduce the number of weeds but also have a negative impact on the growth of corps or beneficial animals [19]. Therefore, how to keep these negative effects caused by feedback controls to a minimum? One strategy is to reduce the number of feedback controls like [16–18]. Motivated by the above, in this paper, we study the following discrete competitive system with delays and single feedback control: (5)x1n+1=x1nexp⁡r1n-a11nx1n-a12n∑s=0+∞K12sx2n-s-b12n∑s=0+∞H12sx22n-s-c1n∑s=0+∞Q1sun-s,x2n+1=x2nexp⁡r2n-a22nx2n-a21n∑s=0+∞K21sx1n-s-b21n∑s=0+∞H21sx12n-s-c2n∑s=0+∞Q2sun-s,un+1=1-enun+d1n∑s=0+∞P1sx1n-s+d2n∑s=0+∞P2sx2n-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.( H 1 ): for any nonnegative bounded sequence {f(n)} defined on Z, we use the notations fl=infn∈Zf(n) and fu=supn∈Zf(n).( H 2 ): ri(n), aij(n), bij(n), ci(n), di(n)(i=1,2), and e(n) are bounded nonnegative sequences of real numbers defined on Z such that (6)0<ril≤rin≤riu,0<aijl≤aijn≤aiju,0<bijl≤bijn≤biju,0<cil≤cin≤ciu,0<dil≤din≤diu,0<el≤en≤eu<1.( H 3 ): Kij(s), Hij(s), Qi(s), and Pi(s)(i=1,2) are nonnegative bounded sequences such that (7)∑s=0+∞Kijs=1,∑s=0+∞Hijs=1,∑s=0+∞Qis=1,∑s=0+∞Pis=1;Θij=∑s=0+∞Kijss<+∞,Λij=∑s=0+∞Hijss<+∞,Δi=∑s=0+∞Qiss<+∞,Ωi=∑s=0+∞Piss<+∞.We consider the solution of system (5) with the following initial conditions: (8)xiθ=φiθ≥0,xi0=φi0>0,supn∈Z-⁡φin<+∞,uθ=ψθ≥0,u0=ψ0>0,supn∈Z-⁡ψn<+∞,where i=1,2 and s=···,-n,-n+1,···,-1,0. One can easily show that the solutions of (5) with initial condition (7) are defined and remain positive for all n∈N+.The remaining part of this paper is organized as follows. We introduce some useful lemmas in Section2 and then state and prove the main results in Sections 3, 4, and 5, respectively. Two examples together with their numeric simulations are presented to show the feasibility of the main results in Section 6.


## 2. Preliminaries

This section is concerned with some lemmas which will be used for our main results. Consider the following difference equation:(9)yn+1=ayn+b,where a, b are positive constants.Lemma 1 (see [20]).
Assume thata<1, and, for any initial value y(0), there exists a unique solution y(n) of (9), which can be expressed as follows: (10)yn=any0-y∗+y∗,where y∗=b/(1-a). Thus, for any solution y(n) of (10), we have (11)limn→+∞yn=y∗.Lemma 2 (see [20]).
Letn∈Nn0+={n0,n0+1,···,n0+l,···} and r≥0. For any fixed n, g(n,r) is nondecreasing function with respect to r, and, for n≥n0, the following inequalities hold: (12)yn+1≤gn,yn,un+1≥gn,un.If y(n0)≤u(n0), then y(n)≤u(n) for all n≥n0.Lemma 3 (see [5]).
Assume thatr(n)>0, x(n) satisfies x(n)>0, and (13)xn+1≤xnexp⁡rn1-axn,for n∈[n1,+∞), where a is a positive constant. Then (14)limsupn→+∞⁡xn≤1aruexp⁡ru-1.Lemma 4 (see [5]).
Assume thatr(n)>0, x(n) satisfies x(n)>0, and (15)xn+1≥xnexp⁡rn1-axn,for n∈[n1,+∞), limsupn→+∞x(n)≤x∗, and x(n1)>0, where a and x∗ are positive constants such that ax∗>1. Then (16)liminfn→+∞⁡xn≥1aexp⁡ru1-ax∗.Similarly, according to the proof of lemma 2.3 in [5], we have the following lemma.Lemma 5.
Letx:Z→R be a nonnegative bounded sequences, and let H:N→R be a nonnegative sequence such that ∑n=0+∞H(n)=1. For any fixed k∈N+, then (17)liminfn→+∞⁡xkn≤liminfn→+∞⁡∑s=-∞nHn-sxks≤limsupn→+∞⁡∑s=-∞nHn-sxks≤limsupn→+∞⁡xkn.If m≤liminfn→+∞x(n)≤limsupn→+∞x(n)≤M holds, then (18)mk≤liminfn→+∞⁡xkn≤limsupn→+∞⁡xkn≤Mk.Proof.
Letx∗=limsupn→+∞xk(n) and sup{xk(n):n∈Z}<M. Given ε>0, let N be an integer such that, for all n≥N, (19)xkn<x∗+ε2,∑s=n+∞Hs<ε2M.Therefore, for all n≥2N, (20)∑s=-∞nHn-sxks≤∑s=-∞NHn-sxks+∑s=NnHn-sxks<∑s=-∞NHn-sM+x∗+ε2∑s=NnHn-s=∑s=n-N+∞HsM+x∗+ε2∑s=0n-NHs<Mε2M+x∗+ε2=x∗+ε.Then (21)limsupn→+∞⁡∑s=-∞nHn-sxks≤x∗+ε.
Settingε→0, we have (22)limsupn→+∞⁡∑s=-∞nHn-sxks≤limsupn→+∞⁡xkn.
Letx∗=liminfn→+∞xk(n). If x∗=0, the result is trivial. If x∗>0, then, given ε∈(0,x∗), there exists an integer N such that, for all n≥N, (23)xkn>x∗-ε2,∑s=0NHs>1-ε.Therefore, for all n≥2N, (24)∑s=-∞nHn-sxks=∑s=-∞N-1Hn-sxks+∑s=NnHn-sxks≥∑s=NnHn-sxks=x∗-ε2∑s=0n-NHs>x∗-ε21-ε.Then (25)liminfn→+∞⁡∑s=-∞nHn-sxks≥x∗-ε21-ε.Setting ε→0, we have (26)liminfn→+∞⁡∑s=-∞nHn-sxks≥liminfn→+∞⁡xkn.
Iflimsupn→+∞x(n)≤M, Given ε>0, there exists an integer N such that, for all n>N, x(n)<M+ε, therefore, xk(n)<(M+ε)k, and then limsupn→+∞xk(n)≤(M+ε)k. Setting ε→0, we have limsupn→+∞xk(n)≤Mk.
Ifliminfn→+∞x(n)≥m and if m=0, the result is trivial. If m>0, given ε∈(0,m), there exists an integer N such that, for all n>N, x(n)>m-ε, therefore, xk(n)>(m-ε)k, and then liminfn→+∞xk(n)≥(m-ε)k. Setting ε→0, we have limsupn→+∞xk(n)≥mk.
This ends the proof of Lemma5.


## 3. Permanence

Concerned with the persistent property of system (5), we have the following result.Theorem 6.
Assume that(27)ril-aijuMj-bijuMj2-ciuB>0,i,j=1,2;i≠j,holds, and, then, for any positive solution (x1(n),x2(n),u(n)) of system (5), we have (28)mi≤liminfn→+∞⁡xin≤limsupn→+∞⁡xin≤Mi,A≤liminfn→+∞⁡un≤limsupn→+∞⁡un≤B,i=1,2,where (29)Di=aiiuril-aijuMj-bijuMj2-ciuB,mi=1Di·exp⁡aiiuDi-aiiuMi,Mi=1aiilexp⁡riu-1,A=∑i=12dilmieu,B=∑i=12diuMiel,i,j=1,2;i≠j.Proof.
From the first and second equations of system (5), we have (30)xin+1=xinexp⁡rin1-aiilriuxin,i=1,2.And so, from Lemma 3, for any solution (x1(n),x2(n),u(n)) of system (5), we can obtain (31)limsupn→+∞⁡xin≤1aiilexp⁡riu-1=defMi,i=1,2.According to Lemma 5 and the above inequality, for i,j=1,2(i≠j), one has (32)limsupn→+∞⁡∑s=0+∞Kjisxin-s=limsupn→+∞⁡∑s=-∞nKjin-sxis≤limsupn→+∞⁡xin≤Mi,limsupn→+∞⁡∑s=0+∞Pisxin-s=limsupn→+∞⁡∑s=-∞nPin-sxis≤limsupn→+∞⁡xin≤Mi,limsupn→+∞⁡∑s=0+∞Hjisxi2n-s=limsupn→+∞⁡∑s=-∞nHjin-sxi2s≤limsupn→+∞⁡xi2n≤Mi2.For any ε>0, there exists a positive integer N1 such that, for all n>N1, (33)xin≤Mi+ε,∑s=0+∞Kjisxin-s≤Mi+ε,∑s=0+∞Pisxin-s≤Mi+ε,∑s=0+∞Hjisxi2n-s≤Mi2+ε,i,j=1,2;i≠j.By the third equation of system (5) and (33), we have (34)un+1≤1-elun+∑i=12diuMi+ε.Hence, by applying Lemma 1 and Lemma 2 to (34), we obtain (35)limsupn→+∞⁡un≤∑i=12diuMi+εel.Setting ε→0, it follows that (36)limsupn→+∞un≤∑i=12diuMiel=defB.According to Lemma 5 and the above inequality, (37)limsupn→+∞∑s=0+∞Qisun-s=limsupn→+∞⁡∑s=-∞nQin-sus≤limsupn→+∞⁡un≤B.Condition (27) implies that, for enough small positive constant ε, the following inequalities hold: (38)ril-aijuMj+ε-bijuMj2+ε-ciuB+ε>0,i,j=1,2;i≠j.It follows from (37) that there exists a positive integer N2>N1 such that, for all n>N2, (39)un≤B+ε,∑s=0+∞Qisun-s≤B+ε.Thus, for all n>N2, from (33), (39), and the first two equations of system (5), we have (40)xin+1≥xinexp⁡ril-aijuMj+ε-bijuMj2+ε-ciuB+ε1-Diεxin,where Diε=aiiu/(ril-aiju(Mj+ε)-biju(Mj2+ε)-ciu(B+ε)) for i,j=1,2, i≠j. Noting the fact that exp(x-1)>x, for x>0, we have (41)exp⁡riu-1ril-aijuMj+ε-bijuMj2+ε-ciuB+ε>1,and then (42)Diε·Mi=exp⁡riu-1ril-aijuMj+ε-bijuMj2+ε-ciuB+ε·aiiuaiil>1.Hence, according to Lemma 4, (43)liminfn→+∞⁡xin≥1Diε·exp⁡ril-aijuMj+ε-bijuMj2+ε-ciuB+ε-aiiuMi.Setting ε→0, it follows that (44)liminfn→+∞⁡xin≥34·1Di·exp⁡ril-aijuMj-bijuMj2-ciuB-aiiuMi=defmi,where Di=aiiu/(ril-aijuMj-bijuMj2-ciuB) for i,j=1,2,i≠j.
According to Lemma5, from (44) we have that, for any ε1>0 small enough (without loss of generality, assume that ε1≤(1/2)min{m1,m2}), there exists an N3>N2, such that (45)∑s=0+∞Pisxin-s≥mi-ε1,∀n>N3,i=1,2.For n>N3, from (45) and the last equation of system (5), we have (46)un+1≥1-euun+∑i=12dilmi-ε1.Hence, by applying Lemmas 1 and 2 to (44), (47)liminfn→+∞⁡un≥∑i=12dilmi-ε1eu.Setting ε1→0, it follows that (48)liminfn→+∞⁡un≥34·∑i=12dilmieu=defA.This ends the proof Theorem 6.


## 4. Global Attractivity

Concerned with the stability property of system (5), we have the following result.Theorem 7.
Assume that there exist positive constantsα1, α2, and α3, such that (49)α1A11-α2a21u+2b21uM1-α3d1u>0,α2A22-α1a12u+2b12uM2-α3d2u>0,α3el-α1c1u-α2c2u>0hold; then, for any two positive solutions (x1(n),x2(n),u(n)) and (x1∗(n),x2∗(n),u∗(n)) of system (5), we have (50)limn→+∞⁡xin-xi∗n=0,limn→+∞⁡un-u∗n=0,i=1,2,where (51)Aii=min⁡aiil,2Mi-aiiui=1,2.Proof.
By (49), we can choose enough small positive constants δ and ε such that (52)α1A11ε-α2a21u+2b21uM1-α3d1u>δ,α2A22ε-α1a12u+2b12uM2-α3d2u>δ,α3el-α1c1u-α2c2u>δ,where (53)Aiiε=min⁡aiil,2Mi+ε-aiiui=1,2.
Let(x1(n),x2(n),u(n)) be any positive solution of system (5). For the above ε, from (31) and (36), there exists an enough large integer n∗>N1, such that (54)xin<Mi+ε,un<B+ε,∀n>n∗,i=1,2.
Now, let us define a Lyapunov functional(55)Vn=∑i=13αiVin,where αi, i=1,2,3 are positive constants and (56)V1n=ln⁡x1n-ln⁡x1∗n+∑s=0+∞K12s∑r=n-sn-1a12r+sx2r-x2∗r+∑s=0+∞H12s∑r=n-sn-1b12r+sx22r-x2∗2r+∑s=0+∞Q1s∑r=n-sn-1c1r+sur-u∗r,V2n=ln⁡x2n-ln⁡x2∗n+∑s=0+∞K21s∑r=n-sn-1a21r+sx1r-x1∗r+∑s=0+∞H21s∑r=n-sn-1b21r+sx12r-x1∗2r+∑s=0+∞Q2s∑r=n-sn-1c2r+sur-u∗r,V3n=un-u∗n+∑s=0+∞P1s∑r=n-sn-1d1r+sx1r-x1∗r+∑s=0+∞P2s∑r=n-sn-1d2r+sx2r-x2∗r.Then, from the definition of Vi(n), i=1,2,3, one can easily see that Vi(n)>0 for all n∈Z+. Also, for any fixed n∗∈Z+, (57)Vn∗≤α1ln⁡x1n∗-ln⁡x1∗n∗+a12usupr∈Z+,r≤n∗⁡x2r-x2∗r∑s=0+∞K12ss+b12usupr∈Z+,r≤n∗⁡x22r-x2∗2r∑s=0+∞H12ss+c1usupr∈Z+,r≤n∗⁡ur-u∗r∑s=0+∞Q1ss+α2ln⁡x2n∗-ln⁡x2∗n∗+a21usupr∈Z+,r≤n∗⁡x1r-x1∗r∑s=0+∞K21ss+b21usupr∈Z+,r≤n∗⁡x12r-x1∗2r∑s=0+∞H21ss+c2usupr∈Z+,r≤n∗⁡ur-u∗r∑s=0+∞Q2ss+α3un-u∗n+d1usupr∈Z+,r≤n∗⁡x1r-x1∗r∑s=0+∞P1ss+d2usupr∈Z+,r≤n∗x2r-x2∗r∑s=0+∞P2ss<+∞.Also, from the first equation of system (5) and using the Mean Value Theorem, for all n>n∗, (58)ΔV1n≤ln⁡x1n-ln⁡x1∗n-a11nx1n-x1∗n-ln⁡x1n-ln⁡x1∗n+∑s=0+∞K12sa12n+sx2n-x2∗n+∑s=0+∞H12sb12n+sx22n-x2∗2n+∑s=0+∞Q1sc1n+sun-u∗n≤-1φ1n-1φ1n-a11nx1n-x1∗n+a12ux2n-x2∗n+b12ux2n+x2∗nx2n-x2∗n+c1uun-u∗n≤-min⁡a11l,2M1+ε-a11ux1n-x1∗n+a12u+2b12uM2+εx2n-x2∗n+c1uun-u∗n.
Similarly to the analysis of (58), we can obtain (59)ΔV2n≤-min⁡a22l,2M2+ε-a22ux2n-x2∗n+a21u+2b21uM1+εx1n-x1∗n+c2uun-u∗n,ΔV3n≤-elun-u∗n+∑i=12diuxin-xi∗n,where φi(n) lies between xi(n) and xi∗(n), i=1,2.
From (58) and (59), we have (60)ΔVn≤-α1A11ε-α2a21u-2α2b21uM1+ε-α3d1ux1n-x1∗n-α2A22ε-α1a12u-2α1b12uM2+ε-α3d2ux2n-x2∗n-α3el-α1c1u-α2c2uun-u∗n≤-δ∑i=12xin-xi∗n+un-u∗n.Summating both sides of the above inequalities from n∗ to n, (61)∑p=n∗nVp+1-Vp≤-δ∑p=n∗n∑i=12xip-xi∗p+up-u∗p.Hence, (62)Vn+1+δ∑p=n∗n∑i=12xip-xi∗p+up-u∗p≤Vn∗<+∞.Then, we have (63)∑p=n∗n∑i=12xip-xi∗p+up-u∗p≤Vn∗δ.Therefore (64)∑p=n∗+∞∑i=12xip-xi∗p+up-u∗p≤+∞,which means that (65)limn→+∞⁡∑i=12xin-xi∗n+un-u∗n=0.Consequently (66)limn→+∞⁡un-u∗n=0,limn→+∞⁡xin-xi∗n=0,i=1,2.
This completes the proof of Theorem7.


## 5. Extinction

Concerned with the extinction property of system (5), we have the following results.Theorem 8.
Assume that(67)r1ur2l<min⁡a12lel+c1ld2ua22uel+c2ud2u,a11lel+c1ld1ua21u+b21uM1el+c2ud1u,hold, let x1(n),x2(n),u(n) be any positive solution of system (5), and then (68)limn→+∞x1n=0,where M1 is defined in Theorem 6.Theorem 9.
Assume that(69)r2ur1l<min⁡a21lel+c2ld1ua11uel+c1ud1u,a22lel+c2ld2ua12u+b12uM2el+c1ud2u,hold, let x1(n),x2(n),u(n) be any positive solution of system (5), and then (70)limn→+∞⁡x2n=0,where M2 is defined in Theorem 6.Proof of Theorem8.
By condition (67), we can choose positive constants β1 and β2 such that (71)r1ur2l<β2β1<min⁡a12lel+c1ld2ua22uel+c2ud2u,a11lel+c1ld1ua21u+b21uM1el+c2ud1u.Thus, there exists a positive constant η such that (72)β1r1u-β2r2l<-η<0,-β1a11l+β2a21u+b21uM1+-β1c1l+β2c2ud1uel<0,β2a22u-β1a12l+-β1c1l+β2c2ud2uel<0.There exists a constant β3 such that (-β1c1l+β2c2u)/el<β3. Thus, for enough small positive constant ε, we have (73)-β1a11l+β2a21u+b21uM1+ε+β3d1u<0,β2a22u-β1a12l+β3d2u<0,-β3el-β1c1l+β2c2u<0.
Consider the following Lyapunov functional:(74)V4n=x1β1nx2-β2nexp⁡-β1∑s=0+∞K12s∑r=n-sn-1a12r+sx2r+∑s=0+∞H12s∑r=n-sn-1b12r+sx22r+∑s=0+∞Q1s∑r=n-sn-1c1r+sur+β2∑s=0+∞K21s∑r=n-sn-1a21r+sx1r+∑s=0+∞H21s∑r=n-sn-1b21r+sx12r+∑s=0+∞Q2s∑r=n-sn-1c2r+sur+β3un+∑i=12∑s=0+∞Pis∑r=n-sn-1dir+sxir.
From (74), we obtain (75)V4n+1V4n=exp⁡β1r1n-β2r2n+β2∑s=0+∞H21sb21n+sx12n-β1∑s=0+∞H12sb12n+sx22n+-β1a11n+β2∑s=0+∞K21sa21n+s+β3∑s=0+∞P1sd1n+sx1n+β2a22n-β1∑s=0+∞K12sa12n+s+β3∑s=0+∞P2sd2n+sx2n+-β3en-β1∑s=0+∞Q1sc1n+s+β2∑s=0+∞Q2sc2n+sun≤exp⁡β1r1u-β2r2l+-β1a11l+β2a21u+b21uM1+ε+β3d1ux1n+β2a22u-β1a12l+β3d2ux2n+-β3el-β1c1l+β2c2uun.From (73) and (75), we can obtain (76)V4n+1≤V4nexp⁡-η.Therefore, (77)V4n≤V40exp⁡-nη.From (31) and (36) we know that there exists an M>0 such that (78)maxi=1,2xin,xi2nun<M,∀n∈Z,and so (79)V40<x1β10x2-β20expβ1Ma12u∑s=0+∞K12ss+b12u∑s=0+∞H12ss+c1u∑s=0+∞Q1ss+β2Ma21u∑s=0+∞K21ss+b21u∑s=0+∞H21ss+c2u∑s=0+∞Q2ss+β3M1+d1u∑s=0+∞P1ss+d2u∑s=0+∞P2ss=x1β10x2-β20exp⁡β1Ma12uΘ12+b12uΛ12+c1uΔ1+β2Ma21uΘ21+b21uΛ21+c2uΔ2+β3M1+∑i=12diuΩi<+∞.On the other hand, we also have (80)V4n≥x1β1nx2-β2nexp⁡-β1∑s=0+∞K12s∑r=n-sn-1a12r+sx2r+∑s=0+∞H12s∑r=n-sn-1b12r+sx22r+∑s=0+∞Q1s∑r=n-sn-1c1r+sur>x1β1nM-β2exp⁡-β1Ma12uΘ12+b12uΛ12+c1uΔ1.Combining inequalities (77), (79), and (80), (81)x1n≤λexp⁡-ηβ1n,where (82)λ=V401/β1Mβ2/β1exp⁡β1Ma12uΘ12+b12uΛ12+c1uΔ1<+∞.Hence we obtain that (83)limn→+∞⁡x1n=0.This ends the proof of Theorem 8.Proof of Theorem9.
Define the following Lyapunov functional:(84)V5n=x1-λ1nx2λ2nexp⁡λ1∑s=0+∞K12s∑r=n-sn-1a12r+sx2r+∑s=0+∞H12s∑r=n-sn-1b12r+sx22r+∑s=0+∞Q1s∑r=n-sn-1c1r+sur-λ2∑s=0+∞K21s∑r=n-sn-1a21r+sx1r+∑s=0+∞H21s∑r=n-sn-1b21r+sx12r+∑s=0+∞Q2s∑r=n-sn-1c2r+sur+λ3un+∑i=12∑s=0+∞Pis∑r=n-sn-1dir+sxir.Similarly to the analysis of the proof of Theorem 8, we have limn→+∞x2(n)=0.
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:(85)x2n+1=x2nexp⁡r2n-a22nx2n-c2n∑s=0+∞Q2sun-s,un+1=1-enun+d2n∑s=0+∞P2sx2n-s.(86)x1n+1=x1nexp⁡r1n-a11nx1n-c1n∑s=0+∞Q1sun-s,un+1=1-enun+d1n∑s=0+∞P1sx1n-s.Theorem 10.
Assume that (67) holds and also (87)d2u<A22elc2uholds; then, for any positive solution (x1(n),x2(n),u(n)) of system (5) and any positive solution (x2∗(n),u∗(n)) of system (85), we have (88)limn→+∞⁡x1n=0,limn→+∞⁡x2n-x2∗n=0,limn→+∞⁡un-u∗n=0,where A22 is defined in Theorem 7.Theorem 11.
Assume that (69) holds and also (89)d1u<A11elc1uholds; then, for any positive solution (x1(n),x2(n),u(n)) of system (5) and any positive solution (x1∗(n),u∗(n)) of system (86), we have (90)limn→+∞⁡x2n=0,limn→+∞⁡x1n-x1∗n=0,limn→+∞⁡un-u∗n=0,where A11 is defined in Theorem 7.Proof of Theorem10.
By condition (87), we can choose positive constants ω1 and ω2 such that (91)d2uA22<ω1ω2<elc2u.Thus, there exist enough small positive constants ε and γ such that (92)ω1A22ε-ω2d2u>γ,ω2el-ω1c2u>γ,where A22ε is defined in (53).
From (81), we have (93)x12n≤λ2exp⁡-2ηβ1n.By applying the Direct Comparison Test to (81) and (93), we obtain ∑n=1+∞x1(n) and ∑n=1+∞x12(n) are absolute convergence.
Now, we define a Lyapunov functional:(94)V6n=ω1ln⁡x2n-ln⁡x2∗n+∑s=0+∞K21s∑r=n-sn-1a21r+sx1r+∑s=0+∞H21s∑r=n-sn-1b21r+sx12r+∑s=0+∞Q2s∑r=n-sn-1c2r+sur-u∗r+ω2un-u∗n+∑s=0+∞P1s∑r=n-sn-1d1r+sx1r+∑s=0+∞P2s∑r=n-sn-1d2r+sx2r-x2∗rand one could easily see that V6(n)≥0 for all n∈Z+. Also, for any fixed n∗∈Z+, from (94) one could see that (95)V6n∗<ω1ln⁡x2n∗-ln⁡x2∗n∗+a21uM∑s=0+∞K21ss+b21uM∑s=0+∞H21ss+c2usupq∈Z+,q≤n∗⁡uq-u∗q∑s=0+∞Q2ss+ω2un∗-u∗n∗+d1uM∑s=0+∞P1ss+d2usupq∈Z+,q≤n∗⁡x2q-x2∗q∑s=0+∞P2ss<+∞.It follows from system (5) and (85) and the Mean Value Theorem that (96)ΔV6n≤-ω11φ2n-1φ2n-a22n-ω2d2ux2n-x2∗n-ω2el-ω1c2uun-u∗n+ω1a21u+ω2d1ux1n+ω1b21ux12n≤-ω1A22ε-ω2d2ux2n-x2∗n-ω2el-ω1c2uun-u∗n+ω1a21u+ω2d1ux1n+ω1b21ux12n≤-γx2n-x2∗n+un-u∗n+ω1a21u+ω2d1ux1n+ω1b21ux12n.Summating both sides of the above inequality from n∗ to n, we have (97)∑p=n∗nV6p+1-V6p<-γ∑p=n∗nx2p-x2∗p+up-u∗p+∑p=n∗nω1a21u+ω2d1ux1p+ω1b21ux12p.Hence (98)V6n+1+γ∑p=n∗nx2p-x2∗p+up-u∗p<V6n∗+∑p=n∗nω1a21u+ω2d1ux1p+ω1b21ux12p.Then, from (95) we have (99)∑p=n∗nx2p-x2∗p+up-u∗p<V6n∗+∑p=n∗nω1a21u+ω2d1ux1p+ω1b21ux12pγ<+∞.Therefore (100)∑p=n∗+∞x2p-x2∗p+up-u∗p<+∞,which means that (101)limn→+∞⁡x2n-x2∗n+un-u∗n=0.Consequently (102)limn→+∞⁡x2n-x2∗n=0,limn→+∞⁡un-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.


## 6. Numerical Simulations

In this section, we give an example to check the feasibility of our result.Example 12.
Consider the following system:(103)x1n+1=x1nexp⁡0.87+0.02sin⁡2n-x1n-0.025-0.001sin⁡3n∑s=0+∞e-1ee-sx2n-s-0.015+0.002sin⁡2n∑s=0+∞e2-1e2e-2sx22n-s-0.25+0.05cos⁡3n∑s=0+∞e3-1e3e-3sun-s.x2n+1=x2nexp⁡0.95+0.03cos⁡3n-x2n-0.02+0.001cos⁡3n∑s=0+∞e4-1e4e-4sx1n-s-0.027+0.002cos⁡2n∑s=0+∞e5-1e5e-5sx12n-s-0.35+0.05sin⁡3n∑s=0+∞e6-1e6e-6sun-sun+1=1-0.93+0.03cos⁡3nun+0.17+0.05cos⁡2n∑s=0+∞e7-1e7e-7sx1n-s+0.16+0.03sin⁡3n∑s=0+∞e8-1e8e-8sx2n-s,One could easily see that conditions (H1), (H2), and (H3) are satisfied. Also, by calculating, one has (104)A11=min⁡a11l,2M1-a11u=1,A22=min⁡a22l,2M2-a22u=1.Now, let us take α1=0.0144, α2=0.0141, and α3=0.0222, and then (105)α1A11-α2a21u+2b21uM1-α3d1u=0.01>0,α2A22-α1a12u+2b12uM2-α3d2u=0.01>0,α3el-α1c1u-α2c2u=0.01>0.Clearly, condition (49) is satisfied, and so from Theorem 7 we have limn→+∞(xi(n)-xi∗(n))=0 and limn→+∞(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.Figure 1
Dynamic behaviors of the solutions(x1(n),x2(n),u(n)) of system (103) with the initial conditions (x1(θ),x2(θ),u(θ))=(0.2,0.4,0.6),(0.45,0.5,0.55),(0.75,0.85,0.8) for θ=⋯,-n,-n+1,⋯,-1,0, respectively.Example 13.
Consider the following system:(106)x1n+1=x1nexp⁡0.4+0.2sin⁡2n-3x1n-1.4+0.1sin⁡3n∑s=0+∞e-1ee-sx2n-s-0.35+0.05sin⁡3n∑s=0+∞e2-1e2e-2sx22n-s-0.15+0.05cos⁡3n∑s=0+∞e3-1e3e-3sun-s,x2n+1=x2nexp⁡1.2+0.1cos⁡3n-x2n-2.8+0.2sin⁡3n∑s=0+∞e4-1e4e-4sx1n-s-0.17+0.03cos⁡2n∑s=0+∞e5-1e5e-5sx12n-s-0.15+0.05sin⁡3n∑s=0+∞e6-1e6e-6sun-s,un+1=1-0.7+0.1cos⁡2nun+1.1+0.1sin⁡2n∑s=0+∞e7-1e7e-7sx1n-s+0.9+0.3sin⁡3n∑s=0+∞e8-1e8e-8sx2n-s.One could easily see that conditions (H1), (H2), and (H3) are satisfied. Also, by calculating, one has (107)min⁡a12lel+c1ld2ua22uel+c2ud2u,a11lel+c1ld1ua21u+b21uM1el+c2ud1u=0.929,r1ur2l=0.5455,1.2=d2u<A22elc2u=1.449.Clearly, conditions (67) and (87) are satisfied, and so from Theorems 8 and 10 we know that 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
Dynamic behaviors of the solutions(x1(n),x2(n),u(n)) of system (106) with the initial conditions (x1(θ),x2(θ),u(θ))=(0.2,0.4,0.6),(0.35,0.8,0.7),(0.5,0.1.2,0.9) for θ=⋯,-n,-n+1,⋯,-1,0, respectively.


## 7. Discussion

During the past decade, many scholars investigated the dynamic behaviors of the feedback control ecosystem. However, by using a feedback control variable to control all the species, it is much difficult. In this paper, we focused our attention on the nonlinear competition system with single feedback control, and the dynamic behavior of the system is investigated. The study shows that the feedback control variable plays a crucial role in both of global attractivity and partial extinction. i.e., under some suitable conditions the two species can survive well, but under other conditions one of two species will be driven to extinction.



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*Source: 1893181-2018-06-25.xml*