# Extinction in Two-Species Nonlinear Discrete Competitive System

Authors: Liqiong Pu; Xiangdong Xie; Fengde Chen; Zhanshuai Miao
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

We propose a nonlinear discrete system of two species with the effect of toxic substances. By constructing a suitable Lyapunov-type function, we obtain the sufficient conditions which guarantee that one of the components will be driven to extinction while the other will be globally attractive with any positive solution of a discrete equation. Two examples together with their numerical simulations illustrate the feasibility of our main results. The results not only improve but also complement some known results.


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## Body

## 1. Introduction

LetZ denote the set of all nonnegative integers. For any bounded sequence {f(n)}, set fu=supn∈Z⁡f(n) and fl=infn∈Z⁡f(n).In the real world, there are many types of interactions between two species. Competitive relations are among the most common ecological interactions. As we all know, the competitive system has been established and was accepted by many scientists and now it became the most important means to explain the ecological phenomenon. During the last decade, the study of the dynamic behaviors of competitive system with toxic substance or feedback control have been discussed by many authors; see, for example, [1–16]. However, most of the studies are based on the traditional Lotka-Volterra competitive system [5, 7, 8, 17–20]; seldom did scholars consider the nonlinear case [1–4, 8, 9, 11, 12, 15, 16, 21–25].In [1], Li and Chen studied the extinction property of the following two species competitive system: (1)x˙1t=x1tr1t-a1tx1t-b1tx2t-c1tx1tx2t,x˙2t=x2tr2t-a2tx1t-b2tx2t-c2tx1tx2t,where ri(t), ai(t), bi(t), ci(t), i=1,2, are assumed to be continuous and bounded above and below by positive constants and x1(t), x2(t) are population density of species x1 and x2 at time t, respectively.In fact, when the size of the population is relatively small, the discrete time models governed by difference equations are more appropriate than the continuous ones. Therefore, Li and Chen [2] and Guo et al. [3] studied the following discrete Lotka-Volterra competition system: (2)x1n+1=x1nexp⁡r1n-a1nx1n-b1nx2n-c1nx1nx2n,x2n+1=x2nexp⁡r2n-a2nx1n-b2nx2n-c2nx1nx2n,where {ri(n)},{ai(n)},{bi(n)}, and {ci(n)},i=1,2, are bounded nonnegative sequences defined on n∈Z. In [2], Li and Chen showed that if the coefficients of system (2) satisfy H0r2ur1l<min⁡a2la1u,b2lb1u,c2lc1u,species x2 will be driven to extinction. In [3], Guo et al. introduced the average growth rate and showed that if the coefficients of system (2) satisfy the following inequality: H0′Mr2mr1<min⁡a2la1u,b2lb1u,c2lc1u,then the same conclusion holds. Obviously, condition H0′ is weaker than that of H0.Since conditionsH0 and H0′ are all sufficient conditions, one of the interesting problems is whether the results still hold under the weaker condition. Now let us consider the following example.Example 1.
Consider the following system:(3)x1n+1=x1nexp⁡0.6+0.3sin⁡n-1+0.3sin⁡nx1n-1+0.5sin⁡nx2n-1.1+sin⁡nx1nx2n,x2n+1=x2nexp⁡0.6+0.3sin⁡n-2+0.6sin⁡nx1n-2+sin⁡nx2n-2.2+2sin⁡nx1nx2n.In this case (4)r1n=0.6+0.3sin⁡n,a1n=1+0.3sin⁡n,b1n=1+0.5sin⁡n,c1n=1.1+sin⁡n,r2n=0.6+0.3sin⁡n,a2n=2+0.6sin⁡n,b2n=2+sin⁡n,c2n=2.2+2sin⁡n.By simple computation, one can see that (5)a2la1u=1.41.3,b2lb1u=11.5,c2lc1u=0.22.1,(6)r2ur1l=0.90.3>0.22.1=min⁡a2la1u,b2lb1u,c2lc1u,(7)Mr2mr1=1>0.22.1=min⁡a2la1u,b2lb1u,c2lc1u,and (6) and (7) show that neither H0 nor H0′ holds; hence one could not draw any conclusion about the dynamic behaviors of the system. However, Figure 1 shows species x2 will be driven to extinction in this case. This motivates us to revisit the extinction property of system (2).Figure 1
Dynamic behaviors of system (3) with initial values (x1(0),x2(0))=(0.14,0.19), (0.27,0.69), and (0.8,0.39), respectively.On the other hand, Gilpin and Ayala [21] conducted experiment on fruit fly dynamics to test the validity of 10 models of competitions. One of the models accounting best for the experimental results is given by (8)x˙1t=r1x1t1-x1tK1θ1-α12x2tK2,x˙2t=r2x2t1-x2tK2θ2-α21x1tK1.Fan and Wang [22] studied the dynamic behaviors of the following nonautonomous n-species Gilpin-Ayala competitive system: (9)x˙it=xitbit-∑j=1naijtxjt-τijtθij,i=1,2,…,n,where bi(t), τij(t), and aij(t), i,j=1,2,…,n, are continuous for 0≤t<+∞andθij are positive constants.Chen et al. [23] studied a discrete n-species Gilpin-Ayala competitive system (10)xik+1=xikexp⁡bik-∑j=1naijkxjkθij,i=1,2,…,n,where bi(k), aij(k), i,j=1,2,…,n, are all positive sequences bounded above and below by positive constants. θij are positive constants.Recently, stimulated by the works of [1, 21, 22], Chen et al. [24] proposed the following, a nonautonomous nonlinear competition system: (11)x˙1t=x1tr1t-a1tx1α1t-b1tx2α2t-c1tx1α1tx2α2t,x˙2t=x2tr2t-a2tx1α1t-b2tx2α2t-c2tx1α1tx2α2t.Chen et al. showed that if the coefficients of system (10) satisfy (12)limsupt→+∞⁡r2tr1t<liminft→+∞⁡a2ta1t,b2tb1t,c2tc1t,the second species will be driven to extinction while the first one will stabilize at a certain solution of the system (13)x˙1t=x1tr1t-a1tx1α1t.Stimulated by the works of [1–3, 21–24], we propose the following a nonlinear discrete two species competition system: (14)x1n+1=x1nexp⁡r1n-a1nx1α1n-b1nx2α2n-c1nx1α1nx2α2n,x2n+1=x2nexp⁡r2n-a2nx1α1n-b2nx2α2n-c2nx1α1nx2α2n.We introduce the following assumptions:H1
{ri(n)} are bounded sequence defined on Z; {ai(n)}, {bi(n)}, and {ci(n)}, i=1,2, are bounded nonnegative sequences defined on Z; αi, i=1,2, are positive constants.H2
There exists positive integerω such that for each i=1,2(15)liminfn→∞⁡∑s=nn+ω-1ris>0.From the point of view of biology, we assume thatxi(0)>0, i=1,2; then system (14) has a positive solution (x1(n),x2(n)) passing through (x1(0),x2(0)).The aim of this paper is, by developing the analysis technique of Li and Chen [2], Chen et al. [4, 24], and Xu et al. [6], to study the extinction property of system (14).The organization of this paper is as follows. In Section2, sufficient conditions for the permanence of system (14) are obtained. In Section 3, we study the extinction of species x2. In Section 4, we study the global stability of species x1 when species x2 is eventual extinction. Examples are presented in Section 5 to show the feasibility of our main results.


## 2. Permanence

Lemma 2 (see [26]).
Assume that{x(n)} satisfy x(n)>0 and (16)xn+1≤xnexp⁡an-bnxn,n∈N,where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants. Then (17)limsupn→∞⁡xn≤expau-1bl.Lemma 3 (see [26]).
Assume that{x(n)} satisfies (18)xn+1≥xnexp⁡an-bnxn,n≥N0,limsupn→∞⁡x(n)≤x∗, and x(N0)>0, where a(n) and b(n) are nonnegative sequences bounded above and below by positive constants and N0∈N. Then (19)liminfn→∞⁡xn≥min⁡albuexp⁡al-bux∗,albu.Lemma 4.
Assume thatr1l>0; every positive solution (x1(n),x2(n)) of system (14) satisfies (20)limsupn→∞⁡x1n≤M1,where M1=(exp⁡{α1r1u-1}/α1a1l)1/α1.Proof.
By the first equation of system (14), we have (21)x1n≤x1nexp⁡r1n-a1nx1α1n.Suppose y1(n)=x1α1(n); then x1(n)=[y1(n)]1/α1.
From (21), we have (22)y1n+11/α1≤y1n1/α1exp⁡r1n-a1ny1n.That is, (23)y1n+1≤y1nexpα1r1n-α1a1ny1n.Applying Lemma 2 such that (24)limsupt→+∞⁡y1n≤exp⁡α1r1u-1α1a1l,hence (25)limsupt→+∞⁡x1n≤exp⁡α1r1u-1α1a1l1/α1=defM1.Lemma 5.
Assume thatr2l>0; every positive solution (x1(n),x2(n)) of system (14) satisfies(26)limsupn→∞⁡x2n≤M2,where M2=(exp⁡α2r2u-1/α2b2l)1/α2.Proof.
The proof of Lemma5 is similar to that of Lemma 4, so we omit the detail here.Lemma 6.
Assume thatH3
E1>0holds; every positive solution (x1(n),x2(n)) of system (14) satisfies (27)liminfn→∞⁡x1n≥m1,where m1=(E1/E2)1/α1exp⁡{E1-E2M1α1}, E1=r1l-b1uM2α2, and E2=a1u+c1uM2α2.Proof.
In view of (26), for each ε>0, there exists a N1>0 such that (28)x2n≤M2+ε,∀n≥N1.By the first equation of system (14), we have (29)x1n+1≥x1nexp⁡r1n-a1nx1α1n-b1nM2+εα2-c1nx1α1nM2+εα2=x1nexp⁡r1n-b1nM2+εα2-a1n+c1nM2+εα2x1α1n.Suppose y1(n)=x1α1(n); then x1(n)=[y1(n)]1/α1.
From (28), we have (30)y1n+11/α1≥y1n1/α1exp⁡r1n-b1nM2+εα2-a1n+c1nM2+εα2y1n.That is, (31)y1n+1≥y1nexp⁡α1r1n-α1b1nM2+εα2-α1a1n+α1c1nM2+εα2y1n≥y1nexp⁡α1r1l-α1b1uM2+εα2-α1a1u+α1c1uM2+εα2y1n=defy1nexp⁡α1E1ε-α1E2εy1n,where E1ε=r1l-b1u(M2+ε)α2andE2ε=a1u+c1u(M2+ε)α2.
Applying Lemma3 such that (32)liminfn→+∞⁡y1n≥min⁡E1E2exp⁡α1E1-α1E2M1α1,E1E2,where E1=r1l-b1uM2α2 and E2=a1u+c1uM2α2.
Hence(33)liminfn→+∞⁡x1n≥min⁡E1E21/α1exp⁡E1-E2M1α1,E1E21/α1.Note that (34)M1=exp⁡α1r1u-1α1a1l1/α1≥α1r1uα1a1l1/α1=r1ua1l1/α1.Thus (35)r1l-a1uM1α1≤0.And so(36)E1-E2M1α1≤0.Hence(37)liminfn→+∞⁡x1n≥E1E21/α1exp⁡E1-E2M1α1=defm1.Lemma 7.
Assume thatH4
F1>0holds; every positive solution (x1(n),x2(n)) of system (14) satisfies (38)liminfn→∞⁡x2n≥m2,where m2=(F1/F2)1/α2exp⁡{F1-F2M2α2}, F1=r2l-a2uM1α1, and F2=b2u+c2uM1α1.Proof.
The proof of Lemma7 is similar to that of Lemma 6, so we omit the detail here.Lemma 8.
Assume thatH1–H4 hold; then system (14) is permanent. That is, for every solution (x1(n),x2(n)) of system (14), one has (39)mi≤liminfn→+∞⁡xin≤limsupn→+∞⁡xin≤Mi,i=1,2.


## 3. Extinction

Theorem 9.
Assume thatH1,H2 hold; assume further that H5
limsupn→∞⁡(∑s=nn+ω-1r2s/∑s=nn+ω-1r1s)<liminfn→∞⁡{a2n/a1n,b2n/b1n,c2n/c1n∣n∈Z}.
Let(x1(n),x2(n)) be any positive solution of system (14); then x2(n)→0 as n→+∞.Proof.
By Lemma2 we know that there exists n1>0 such that (40)x1n≤2M1,n≥n1.By H2, there exist positive constants η0 and n2>n1 such that (41)∑s=nn+ω-1ris≥η0,n≥n2.By H5, we can choose positive constants α, β, and ε such that (42)limsupn→∞⁡∑s=nn+ω-1r2s∑s=nn+ω-1r1s<αβ-ε<αβ<liminfn→∞⁡a2na1n,b2nb1n,c2nc1n.Thus, there exists a n3>n2>0, such that for all n≥n3(43)∑s=nn+ω-1βr2s-αr1s<-εβ∑s=nn+ω-1r1s<-εβη0,(44)αa1n-βa2n<0,αb1n-βb2n<0,αc1n-βc2n<0.Let V(n)=x1-α(n)x2β(n); then (45)Vn+1Vn=x1n+1x1n-αx2n+1x2nβ=exp⁡-αr1n+αa1nx1α1n+αb1nx2α2n+αc1nx1α1nx2α2n+βr2n-βa2nx1α1n-βb2nx2α2n-βc2nx1α1nx2α2n=exp⁡βr2n-αr1n+αa1n-βa2nx1α1n+αb1n-βb2nx2α2n+αc1n-βc2nx1α1nx2α2n.From (43) and (44), it follows that (46)Vn+1≤Vnexp⁡βr2n-αr1n,n≥n3.For any n≥n3, we choose an integer m≥0 such that n∈[n3+mω,n3+(m+1)ω). Integrating (46) from n3 to n-1, from (43), we have (47)Vn≤Vn3exp⁡∑s=n3n-1βr2s-αr1s=Vn3exp⁡∑s=n3n3+mω-1+∑s=n3+mω-1n-1βr2s-αr1s≤Vn3exp⁡-εβη0m+A1<Vn3exp⁡-εβη0n-n3ω-1+A1=Vn3exp⁡-εβη0nω+A1∗,where A1∗=εβη0n3/ω+εβη0+A1 and A1=supn∈Z⁡βr2n-αr1nω.
(47) implies that (48)x2n<x1-αn3x2βn32M1αexp⁡A1∗1/βexp⁡-εη0nω,for all n≥n3. Hence, x2(n)→0 exponentially as n→+∞.


## 4. Global Stability

In Section3, we prove that species x2 will be driven to extinction if the conditions H1, H2, and H5 hold. Now we investigate the stability property of species x1 under the same conditions.Before we state the main result of this section, we first introduce some lemmas.Lemma 10.
Assume thatH1, H2, H5, and r1l>0 hold; let (x1(n),x2(n)) be any positive solution of system (14); then (49)m1′≤liminfn→∞⁡x1n≤limsupn→∞⁡x1n≤M1,where m1′=(r1l/a1u)1/α1exp{r1l-a1uM1α1}.Proof.
Under the assumption conditionsH1, H2, and H5, it follows from Theorem 9 that (50)limn→∞⁡x2n=0.From Lemma 4, we have (51)limsupn→∞⁡x1n≤M1.By Lemma 10, it is enough to show that (52)liminfn→∞⁡x1n≥m1′.In view of (50) and (51), for each ε>0, there exists an integer n0∈Z such that (53)x1n≤M1+ε,x2n≤ε,∀n≥n0.We consider the following two cases.
Case  1. We assume that there exists an l0≥n0 such that x1(l0+1)≤x1(l0). Note that for n≥l0(54)x1n+1=x1nexp⁡r1n-a1nx1α1n-b1nx2α2n-c1nx1α1nx2α2n≥x1nexp⁡r1l-a1ux1α1n-b1uεα2-c1uM1+εα1εα2.In particular, with n=l0, we obtain (55)r1l-a1ux1α1l0-b1uεα2-c1uM1+εα1εα2≤0,which implies that (56)x1l0≥r1l-b1u+c1uM1+εα1εα2a1u1/α1.From (54) and (56), it follows that (57)x1l0+1≥r1l-b1u+c1uM1+εα1εα2a1u1/α1exp⁡r1l-a1ux1α1l0-b1uεα2-c1uM1+εα1εα2≥r1l-b1u+c1uM1+εα1εα2a1u1/α1exp⁡r1l-a1uM1+εα1-b1uεα2-c1uM1+εα1εα2.Let (58)x1ε=r1l-b1u+c1uM1+εα1εα2a1u1/α1exp⁡r1l-a1uM1+εα1-b1uεα2-c1uM1+εα1εα2.Note that (59)M1=exp⁡α1r1u-1α1a1l1/α1≥α1r1uα1a1l1/α1≥r1ua1l1/α1,and thus r1l-a1uM1α1≤0; also, for arbitrary ε, (60)r1l-a1uM1+εα1-b1uεα2-c1uM1+εα1εα2≤0,or (61)r1l-b1u+c1uM1+εα1εα2a1u1/α1≥x1ε.We claim that (62)x1n≥x1ε,∀n≥l0.By way of contradiction, assume that there exists a p0>l0 such that x1(p0)<x1ε. Then p0≥l0+2. Let p~0≥l0+2 be the smallest integer such that x1(p~0)<x1ε. Then x1(p~0-1)>x1(p~0). The above argument produces that x1(p~0)≥x1ε, a contradiction. This proves the claim.
Case  2. We assume that x1(n+1)>x1(n) for n≥n0; then limn→∞⁡x1(n)=x_1. We claim that (63)x_1≥r1l-b1u+c1uM1+εα1εα2a1u1/α1.By way of contradiction, assume that (64)x_1<r1l-b1u+c1uM1+εα1εα2a1u1/α1.Taking limit in the first equation in system (14) gives (65)limn→∞⁡r1n-a1nx1α1n-b1nx2α2n-c1nx1α1nx2α2n=0,which is a contradiction since (66)limn→∞⁡r1n-a1nx1α1n-b1nx2α2n-c1nx1α1nx2α2n≥r1l-a1ux_1α1-b1uεα2-c1uM1+εα1εα2>0.This proves the claim; then we have (67)liminfn→∞⁡x1n=limn→∞⁡x1n=x_1≥x1ε.Combining Case  1 and Case  2, we see that (68)liminfn→∞⁡x1n≥x1ε.Setting ε→0, note that (69)limε→0⁡x1ε=r1la1u1/α1exp⁡r1l-a1uM1α1=defm1′.Now, we can easily see that (52) holds. This completes the proof of Lemma 10.We consider a discrete equation(70)xn+1=xnexp⁡r1n-a1nxα1n,n∈N,where {r1(n)} and {a1(n)} are bounded nonnegative sequences; similarly to the proof of Lemma 10, we can obtain the following lemma.Lemma 11.
For any positive solution{x(n)} of (70), one has (71)m≤liminfn→∞⁡xn≤limsupn→∞⁡xn≤M,where M=(exp{α1r1u-1}/α1a1l)1/α1 and m=(r1l/a1u)1/α1exp{r1l-a1uMα1}.
Now, we state the main result of this section.Theorem 12.
Assume thatH1, H2, H5, and r1l>0 hold; assume further that H6a1ua1lexp⁡α1r1u-1<2;then for any positive solution (x1(n),x2(n)) of system (14) and any positive solution {x(n)} of system (70), one has (72)limn→∞⁡x1n-xn=0,limn→∞⁡x2n=0.Proof.
SinceH1, H2, and H5 hold, it follows from Theorem 12 that (73)limn→∞⁡x2n=0.To prove limn→∞⁡x1n-xn=0, let (74)x1n=xnexp⁡yn.It follows from the first equation of system (14) and (74) that (75)yn+1=yn-a1nxα1nexp⁡α1yn-1-b1nx2α2n-c1nx1α1nx2α2n.Using the Mean Value Theorem, we get (76)exp⁡α1yn-1=α1exp⁡θnyn,θn∈0,α1yn.Then the first equation of system (14) is equivalent to (77)yn+1=1-α1a1nxα1nexp⁡θnyn-b1n+c1nx1α1nx2α2n,where θ(n)∈(0,α1y(n)).
To complete the proof, it suffices to show that(78)limn→∞⁡yn=0.We first assume that (79)λ=max⁡1-α1a1uM1α1,1-α1a1lmα1<1,and then we can choose positive constant ε>0 small enough such that (80)λε=max⁡1-α1a1uM1+εα1,1-α1a1lm-εα1<1.For above ε, according to Lemmas 10 and 11 and (73), there exists an integer n0∈Z such that (81)m-ε≤xn,x1n≤M1+ε,x2n≤ε,forn≥n0.It follows from (81) that (82)b1n+c1nx1α1n≤b1u+c1uM1+εα1=defMε,forn≥n0.Note that θ(n)∈(0,α1y(n)) implies that α1xα1nexp⁡θn lies between α1xα1(n) and α1x1α1(n). From (77) and (80)–(82), we get (83)yn+1≤max⁡1-α1a1uM1+εα1,1-α1a1lm-εα1yn+b1u+c1uM1+εα1εα2=λεyn+Mεεα2,forn≥n0.This implies that (84)yn≤λεn-n0yn0+1-λεn-n01-λεMεεα2,forn≥n0.Since λε<1 and ε is arbitrary small, we obtain limn→∞⁡y(n)=0; it means that (78) holds when λ<1.
Note that(85)1-α1a1uM1α1≤1-α1a1lmα1<1,and thus, λ<1 is equivalent to (86)1-α1a1uM1α1>-1, or (87)α1a1uM1α1=a1ua1lexp⁡α1r1u-1<2.
Now, we can conclude that (78) is satisfied as H6 holds, and so limn→∞⁡x1n-xn=0. This completes the proof of Theorem 12.As a direct corollary of Theorems9 and 12, for system (2), we have the following result.Corollary 13.
Assume thatH1, H2, H5, and r1l>0 hold; assume further that (88)a1ua1lexpr1u-1<2;then for any positive solution (x1(n),x2(n)) of system (2) and any positive solution {x(n)} of (89)x1n+1=x1nexp⁡r1n-a1nx1n,one has(90)limn→∞⁡x1n-xn=0,limn→∞⁡x2n=0.


## 5. Examples

The following examples show the feasibility of our main results.Example 14.
Now let us consider Example1; in this case, one can easily check that (91)liminfn→∞⁡∑s=nn+ω-1ris>0,limsupn→∞⁡∑s=nn+ω-1r2s∑s=nn+ω-1r1s=1,a2na1n=b2nb1n=c2nc1n=2;hence (92)limsupn→∞⁡∑s=nn+ω-1r2s∑s=nn+ω-1r1s<2=liminfn→∞⁡a2na1n,b2nb1n,c2nc1n.Also (93)a1ua1lexp⁡α1r1u-1=1.30.7exp⁡0.9-1≈1.680<2.Equations (91)–(93) show that all the conditions of Corollary 13 hold; then species x2 will be driven to extinction while species x1 will be globally attractive with any positive solution of the following discrete equation: (94)x1n+1=x1nexp0.6+0.3sin⁡n-1+0.3sin⁡nx1n.Example 15.
Consider the following system:(95)x1n+1=x1nexp⁡0.6-1.5+0.2sin⁡nx12n-1+0.5sin⁡nx21/2n-1.1+sin⁡nx12nx21/2n,x2n+1=x2nexp⁡0.3-1.4+0.5sin⁡nx12n-1.3+0.5sin⁡nx21/2n-1.2x12nx21/2n.In this case, corresponding to system (2), r1(n)=0.6, r2(n)=0.3, a1(n)=1.5+0.2sin⁡n, b1(n)=1+0.5sin⁡n, c1(n)=1.1+sin⁡n, a2(n)=1.4+0.5sin⁡n, b2(n)=1.3+0.5sin⁡(n), c2(n)=1.2, α1=2, and α2=1/2. By simple computation, one can see that (96)liminfn→∞⁡∑s=nn+ω-1ris>0,limsupn→∞⁡∑s=nn+ω-1r2s∑s=nn+ω-1r1s=12,(97)a2na1n≥0.91.7,b2nb1n≥0.81.5,c2nc1n≥1.22.1,(98)limsupn→∞⁡∑s=nn+ω-1r2s∑s=nn+ω-1r1s<0.91.7≤liminfn→∞⁡a2na1n,b2nb1n,c2nc1n,(99)a1ua1lexp⁡α1r1u-1≤1.71.3exp⁡2×0.6-1≈1.5972<2.Equations (97) and (99) show that all the conditions of Theorem 12 hold; thus species x2 is driven to extinction while species x1 is asymptotic to any positive solution of (100)xn+1=xnexp⁡0.6-1.5+0.2sin⁡nx2n.Figure 2 shows the dynamic behaviors of system (95).Figure 2
Dynamic behaviors of system (95) with initial values (x1(0),x2(0))=(0.14,0.19), (0.27,0.69), and (0.8,0.39), respectively.


## 6. Conclusion

In this paper, we consider a nonlinear discrete two species competition system with the effect of toxic substances. In Theorem9, by constructing a suitable Lyapunov-type function, we obtain a set of sufficient conditions which ensure species x2 will be driven to extinction. Our results improve and generalize Theorem  2.1 of [2] and Theorem  1.1 of [3].



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