# Permanence and Positive Periodic Solutions of a Discrete Delay Competitive System

Authors: Wenjie Qin; Zhijun Liu
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

A discrete time non-autonomous two-species competitive system with delays is proposed, which involves the influence of many generations on the density of species population. Sufficient conditions for permanence of the system are given. When the system is periodic, by using the continuous theorem of
coincidence degree theory and constructing a suitable Lyapunov discrete function, sufficient conditions which guarantee the existence and global attractivity of positive periodic solutions are obtained. As an application, examples and their numerical simulations are presented to illustrate the feasibility of our main results.


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## Body

## 1. Introduction

In recent years, the application of theories of functional differential equations in mathematical ecology has developed rapidly. Various delayed models have been proposed in the study of population dynamics, ecology, and epidemic. In fact, more realistic population dynamics should take into account the effect of delay. Also, delay differential equations may exhibit much more complicated dynamic behaviors than ordinary differential equations since a delay could cause a stable equilibrium to become unstable and cause the population to fluctuate (see [1]). One of the famous models for dynamics of population is the delay Lotka-Volterra competitive system. Owing to its theoretical and practical significance, various delay competitive systems have been studied extensively (see [2–8]). Although much progress has been seen for Lotka-Volterra competitive systems, such systems are not well studied in the sense that most results are continuous time versions related. Many authors [9–11] have argued that the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have non-overlapping generations. Discrete time models can also provide efficient computational models of continuous models for numerical simulations. Therefore, the dynamic behaviors of population models governed by difference equations have been studied by many authors, see [12–18] and the references cited therein. Noting that some studies of the dynamics of natural populations indicate that the density-dependent population regulation probably takes place over many generations [19, 20], many authors have discussed the influence of many past generations on the density of species population and discussed the dynamic behaviors of competitive, predator-prey, and cooperative systems (see [21–24]).Motivated by the above work [19–24], in this paper we will investigate the following discrete time non-autonomous two-species competitive system with delays: (1.1)x1(k+1)=x1(k)exp[r1(k)-∑l=0ma1l(k)x1(k-l)-∑l=0mc2l(k)x2(k-l)1+x2(k-l)],x2(k+1)=x2(k)exp[r2(k)-∑l=0ma2l(k)x2(k-l)-∑l=0mc1l(k)x1(k-l)1+x1(k-l)],
with the initial conditions (1.2)xi(-l)≥0,xi(0)>0,l=0,1,…,m,i=1,2,
where xi(k) represents the density of population xi at the kth generation, ri(k) is the intrinsic growth rate of population xi at the kth generation, ail(k) measures the intraspecific influence of the (k-l)th generation of population xi on the density of its own population, and cjl(k) stands for the interspecific influence of the (k-l)th generation of population xj on population xi, i,j=1,2 and i≠j. The coefficients {ri(k)}, {ail(k)}, and {cil(k)}(i=1,2) are bounded nonnegative sequences. The exponential form of the equations in system (1.1) ensures that any forward trajectory {(x1(k),x2(k))⊤} of system (1.1) with initial conditions (1.2) remains positive for all k∈{0,1,2,…}. For the investigations of some continuous versions of (1.1) we refer to [8, 25, 26] and the references cited therein.The principle aim of this paper is to study the dynamic behaviors of system (1.1), such as permanence, existence, and global attractivity of positive periodic solutions. To the best of our knowledge, no work has been done for the discrete non-autonomous difference system (1.1). The paper is organized as follows. In Section 2, we obtain sufficient conditions which guarantee the permanence of system (1.1). In Section 3, a good understanding of the existence and global attractivity of positive periodic solutions of system (1.1) is gained by using the method of coincidence degree theory and a Lyapunov discrete function. Some illustrative examples are given to demonstrate the feasibility of the obtained results in Section 4. To do this, we need to give the following notations and Definitions 1.1 and 1.2.For the simplicity and convenience of exposition, throughout this paper we letZ,Z+,R+, and R2 denote the sets of all integers, nonnegative integers, nonnegative real numbers and two-dimensional Euclidian vector space, respectively. Meanwhile, we denote that b*=supk∈Z+b(k), b*=infk∈Z+b(k) for any bounded sequence b(k).Definition 1.1.
System  (1.1) is said to be permanent if there exist positive constants mi and Mi such that
(1.3)mi≤liminfk→+∞xi(k)≤lim supk→+∞xi(k)≤Mi,i=1,2
for any positive solution {(x1(k),x2(k))⊤} of system (1.1).Definition 1.2.
A positive periodic solution{(x̃1(k),x̃2(k))⊤} of system (1.1) is said to be globally attractive if each other solution {(x1(k),x2(k))⊤} of system (1.1) satisfies
(1.4)limk→+∞|x1(k)-x̃1(k)|=0,limk→+∞|x2(k)-x̃2(k)|=0.


## 2. Permanence

In this section, we will establish sufficient conditions for the permanence of system (1.1). To do this, we first give two lemmas which will be useful for establishing our main result in this section.Lemma 2.1 (see [27, Lemma 1]).
Assume that{x(k)} satisfies x(k)>0 and
(2.1)x(k+1)≤x(k)exp[r(k)(1-αx(k))]
for k∈[k1,+∞), where α is a positive constant and k1∈Z+. Then
(2.2)lim supk→+∞x(k)≤1αr*exp(r*-1).Lemma 2.2 (see [27, Lemma 2]).
Assume that{x(k)} satisfies x(k)>0 and
(2.3)x(k+1)≥x(k)exp[r(k)(1-αx(k))]
for k∈[k2,+∞), lim supk→+∞x(k)≤M, and x(k2)>0, where α is a constant such that αM>1 and k2∈Z+. Then
(2.4)liminfn→+∞x(k)≥1αexp[r*(1-αM)].Before stating Theorem2.3, for the sake of convenience, we set (2.5)Mi=defexp(ri*-1)∑l=0mail*exp(-ri*l),Δi=deflimɛ→0Δiɛ,Δiɛ=def∑l=0mail*Ĝiri*-∑l=0mcjl*,Ĝi=defexp{[(Mi+ɛ)∑l=0mail*+∑l=0mcjl*-ri*]l},
where i,j=1,2, i≠j, ɛ>0 is a sufficiently small constant.We are now in a position to state our main result of this section on the permanence of system (1.1).Theorem 2.3.
If the following assumptions:(2.6)min{r1*-∑l=0mc2l*,r2*-∑l=0mc1l*}>0,min{M1Δ1,M2Δ2}>1
hold, then system (1.1) is permanent.Proof.
Clearly, any solution{(x1(k),x2(k))⊤} of system (1.1) satisfies x1(k)>0, x2(k)>0. The following two steps are considered.Step 1.
According to Definition1.1, we will prove that any positive solution of system (1.1) satisfies lim supk→+∞xi(k)≤Mi for i=1,2.
It follows from the first equation of system (1.1) that
(2.7)x1(k+1)≤x1(k)exp[r1(k)].
We will make a convention that∏i=k1k2b(i)=1 if k1>k2 for any bounded sequence b(i). For k≥m, l=0,1,…,m, we can obtain
(2.8)∏i=k-lk-1x1(i+1)≤∏i=k-lk-1x1(i)exp[r1(i)],
that is,
(2.9)x1(k)≤x1(k-l)exp[∑i=k-lk-1r1(i)],
in other words,
(2.10)x1(k-l)≥x1(k)exp[-∑i=k-lk-1r1(i)].
Consequently, we have(2.11)x1(k+1)≤x1(k)exp{r1(k)-∑l=0m[a1l(k)x1(k)exp[-∑i=k-lk-1r1(i)]]}≤x1(k)exp{r1(k)-∑l=0m[a1l*exp(-r1*l)]x1(k)}≤x1(k)exp{r1(k)[1-∑l=0ma1l*exp(-r1*l)r1*x1(k)]}.
By Lemma2.1, we can derive that
(2.12)lim supk→+∞x1(k)≤exp(r1*-1)∑l=0ma1l*exp(-r1*l)=M1.
Similar to the above argument, we can verify that(2.13)lim supk→+∞x2(k)≤exp(r2*-1)∑l=0ma2l*exp(-r2*l)=M2.Step 2.
By a similar procedure to Step1, we will prove that any positive solution of system (1.1) satisfies lim infk→+∞xi(k)≥mi, where
(2.14)mi=defexp[(ri*-∑l=0mcjl*)(1-ΔiMi)]Δi,i,j=1,2,i≠j.
For any sufficiently smallɛ>0, according to (2.5), there exists a positive integer k0 such that x1(k)≤M1+ɛ for all k≥k0. Thus, for k≥k0+m, it follows from the first equation of system (1.1) that
(2.15)x1(k+1)≥x1(k)exp[r1(k)-(M1+ɛ)∑l=0ma1l(k)-∑l=0mc2l(k)].
Therefore, for alll=0,1,…,m and k≥k0+m, it follows that
(2.16)∏i=k-lk-1x1(i+1)≥∏i=k-lk-1{x1(i)exp[r1(i)-(M1+ɛ)∑l=0ma1l(i)-∑l=0mc2l(i)]},
that is,
(2.17)x1(k-l)≤x1(k)exp{∑i=k-lk-1[(M1+ɛ)∑l=0ma1l(i)+∑l=0mc2l(i)-r1(i)]}=x1(k)G1,
where
(2.18)G1=defexp{∑i=k-lk-1[(M1+ɛ)∑l=0ma1l(i)+∑l=0mc2l(i)-r1(i)]}.
Combining (2.5) and (2.17) with the first equation of system (1.1) leads to
(2.19)x1(k+1)≥x1(k)exp[r1*-∑l=0mc2l*-∑l=0ma1l*x1(k-l)]≥x1(k)exp[r1*-∑l=0mc2l*-∑l=0ma1l*G1x1(k)]≥x1(k)exp{(r1*-∑l=0mc2l*)[1-∑l=0ma1l*Ĝ1r1*-∑l=0mc2l*x1(k)]}=x1(k)exp{(r1*-∑l=0mc2l*)[1-Δ1ɛx1(k)]}.
And hence, by applying Lemma2.2 and letting ɛ→0, it follows from (2.5)-(2.6) and (2.19) that
(2.20)liminfk→+∞x1(k)≥exp[(r1*-∑l=0mc2l*)(1-Δ1M1)]Δ1=m1.
Analogously, from the second equation of system (1.1), we can verify that
(2.21)liminfk→+∞x2(k)≥exp[(r2*-∑l=0mc1l*)(1-Δ2M2)]Δ2=m2.
The proof of Theorem2.3 is completed by combining Steps 1 and 2.


## 3. Existence and Global Attractivity of Positive Periodic Solutions

In this section, we will give two main results. We first derive sufficient conditions for the existence of positive periodic solutions of system (1.1). We further assume that ri,ail,cil:Z→R+ are positive ω-periodic for system (1.1), that is, (3.1)ri(k+ω)=ri(k),ail(k+ω)=ail(k),cil(k+ω)=cil(k),l=0,1,2,…,m,i=1,2
for any k∈Z, where ω, a fixed positive integer, denotes the prescribed common period of the parameters in system (1.1).In order to obtain sufficient conditions for the existence of positive periodic solutions of system (1.1), we will use the method of coincidence degree. For convenience, we will summarize in the following a few concepts and results from [28] that will be useful in this section.LetX and Y be two Banach spaces. Consider an operator equation (3.2)Lx=λNx,λ∈(0,1),
where L:DomL∩X→Y is a linear operator and λ is a parameter. Let P and Q denote two projectors such that (3.3)P:X∩DomL→KerL,Q:Y→YImL.Denote thatJ:ImQ→KerL is an isomorphism of ImQ onto KerL. Recall that a linear mapping L:DomL∩X→Y with KerL=L-1(0) and ImL=L(DomL) will be called a Fredholm mapping if the following two conditions hold:(i)
KerL has a finite dimension;(ii)
ImL is closed and has a finite codimension.Recall also that the codimension ofImL is the dimension of Y/ImL, that is, the dimension of the cokernel coker L of L.WhenL is a Fredholm mapping, its index is the integer IndL=dimKerL-codimImL.We will say that a mappingN is L-compact on Ω if the mapping QN:Ω¯→Y is continuous. QN(Ω¯) is bounded and Kp(I-Q)N:Ω¯→X is compact, that is, it is continuous and Kp(I-Q)N(Ω¯) is relatively compact, where Kp:ImL-DomL∩KerP is an inverse of the restriction Lp of L to DomL∩KerP, so that LKp=I and Kp=I-P.Lemma 3.1 (see [28, Continuation Theorem]).
LetX and Y be two Banach spaces and let L be a Fredholm mapping of index zero. Assume that N:Ω¯→Y is L-compact on Ω¯ with Ω open bounded in X. Furthermore assume that(a)
for eachλ∈(0,1), x∈∂Ω∩DomL, Lx≠λNx,(b)
QNx≠0 for each x∈∂Ω∩KerL,(c)
deg{JQNx,Ω∩KerL,0}≠0.Then the equation Lx=Nx has at least one solution in DomL∩Ω¯.In what follows, we will use the following notations:(3.4)Iω={0,1,…,ω-1},f¯=1ω∑k=0ω-1f(k),fL=mink∈Iωf(k),fU=maxk∈Iωf(k),
where {f(k)} is an ω-periodic sequence of real numbers defined for k∈Z.Lemma 3.2 (see [29, Lemma 3.2]).
Letf:Z→R be ω-periodic, that is, f(k+ω)=f(k), then for any fixed k1,k2∈Iω and any k∈Z, one has
(3.5)f(k)≤f(k1)+∑s=0ω-1|f(s+1)-f(s)|,f(k)≥f(k2)-∑s=0ω-1|f(s+1)-f(s)|.Denote that(3.6)l2={x={x(k)}:x(k)∈R2,k∈Z}.
For a=(a1,a2)⊤∈R2, define |a|=max{a1,a2}. Let lω⊂l2 denote the subspace of all ω-periodic sequences equipped with the usual supremum norm ∥·∥, that is, (3.7)∥x∥=maxk∈Iω|x(k)|forx={x(k):k∈Z}∈lω.
Then it follows that lω is a finite dimensional Banach space.Let(3.8)l0ω={x={x(k)∈lω}:∑k=0ω-1x(k)=0},lcω={x={x(k)∈lω}:x(k)=h∈R2,k∈Z}.
Then it follows that l0ω and lcω are both closed linear subspaces of lω and (3.9)lω=l0ω⊕lcω,dimlcω=2.We are now in a position to state one of the main results of this section on the existence of positive periodic solutions of system (1.1).Theorem 3.3.
Assume that(3.10)r¯1>∑l=0mc¯2l,r¯2>∑l=0mc¯1l.
Then system (1.1) has at least one positive ω-periodic solution.Proof.
We first make the change of variables(3.11)x1(k)=exp{y1(k)},x2(k)=exp{y2(k)}.
By substituting (3.11) into system (1.1), we can get
(3.12)y1(k+1)-y1(k)=r1(k)-∑l=0ma1l(k)exp{y1(k-l)}-∑l=0mc2l(k)exp{y2(k-l)}1+exp{y2(k-l)},y2(k+1)-y2(k)=r2(k)-∑l=0ma2l(k)exp{y2(k-l)}-∑l=0mc1l(k)exp{y1(k-l)}1+exp{y1(k-l)}.
It is easy to see that if system (3.12) has one ω-periodic solution, then system (1.1) has one positive ω-periodic solution. Therefore, to complete the proof, it is only to show that system (3.12) has at least one ω-periodic solution.
SetX=Y=lω. Denote by L:X→X the difference operator given by Ly={(Ly)(k)} with
(3.13)(Ly)(k)=y(k+1)-y(k)fory∈X,k∈Z,
and N:X→X as follows:
(3.14)N[y1y2]=[r1(k)-∑l=0ma1l(k)exp{y1(k-l)}-∑l=0mc2l(k)exp{y2(k-l)}1+exp{y2(k-l)}r2(k)-∑l=0ma2l(k)exp{y2(k-l)}-∑l=0mc1l(k)exp{y1(k-l)}1+exp{y1(k-l)}]
for any (y1,y2)⊤∈X and k∈Z. It is easy to see that L is a bounded linear operator and
(3.15)KerL=lcω,ImL=l0ω,dimKerL=2=codimImL,
then we get that L is a Fredholm mapping of index zero.
Define(3.16)P[y1y2]=Q[y1y2]=[1ω∑s=0ω-1y1(s)1ω∑s=0ω-1y2(s)],[y1y2]∈X=Y.
It is not difficult to show that P and Q are continuous projectors such that
(3.17)ImP=KerL,KerQ=ImL=Im(I-Q).
Furthermore, the inverse (toL) Kp:ImL→DomL∩KerP exists and is given by
(3.18)Kp(y)=∑s=0k-1y(s)-1ω∑s=0ω-1(ω-s)y(s).
ThenQN:X→Y and Kp(I-Q)N:X→X are given by
(3.19)QNy=1ω∑s=0ω-1Ny(s),Kp(I-Q)Ny=∑s=0k-1Ny(s)-1ω∑s=0ω-1(ω-s)Ny(s)-(kω-1+ω2ω)∑s=0ω-1Ny(s).
In order to apply Lemma3.1, we need to search for an appropriately open, bounded subset Ω.
Corresponding to the operator equationLy=λNy, λ∈(0,1), we have
(3.20)y1(k+1)-y1(k)=λ[r1(k)-∑l=0ma1l(k)exp{y1(k-l)}-∑l=0mc2l(k)exp{y2(k-l)}1+exp{y2(k-l)}],y2(k+1)-y2(k)=λ[r2(k)-∑l=0ma2l(k)exp{y2(k-l)}-∑l=0mc1l(k)exp{y1(k-l)}1+exp{y1(k-l)}].
Suppose that y={y(k)}={(y1(k),y2(k))⊤}∈X is a solution of (3.20) for a certain λ∈(0,1). Summing both sides of (3.20) from 0 to ω-1 with respect to k, we can derive
(3.21)r¯1ω=∑k=0ω-1∑l=0ma1l(k)exp{y1(k-l)}+∑k=0ω-1∑l=0mc2l(k)exp{y2(k-l)}1+exp{y2(k-l)},r¯2ω=∑k=0ω-1∑l=0ma2l(k)exp{y2(k-l)}+∑k=0ω-1∑l=0mc1l(k)exp{y1(k-l)}1+exp{y1(k-l)}.
Since y={y(k)}∈X, there exist ξi∈Iω such that
(3.22)yi(ξi)=mink∈Iω{yi(k)},i=1,2.
It follows from (3.21) that
(3.23)r¯iω≥∑k=0ω-1∑l=0mail(k)exp{yi(ξi)}+∑k=0ω-1∑l=0mcjl(k)exp{yj(ξi)}1+exp{yj(ξi)}>∑k=0ω-1∑l=0mail(k)exp{yi(ξi)}=ω∑l=0ma¯ilexp{yi(ξi)},i,j=1,2,i≠j,
which implies
(3.24)yi(ξi)<lnr¯i∑l=0ma¯il=lnr¯iAi,i=1,2,
where Ai=def∑l=0ma¯il, besides, from (3.20) and (3.21)
(3.25)∑k=0ω-1|yi(k+1)-yi(k)|≤λ{∑k=0ω-1[ri(k)+∑l=0mail(k)exp{yi(k-l)}+∑l=0mcjl(k)exp{yj(k-l)}1+exp{yj(k-l)}]}≤2r¯iω,i,j=1,2,i≠j.
By (3.24), (3.25), and Lemma 3.2, we have
(3.26)yi(k)≤yi(ξi)+∑s=0ω-1|yi(s+1)-yi(s)|≤lnr¯iAi+2r¯iω.
On the other hand, there also existηi∈Iω such that
(3.27)yi(ηi)=maxk∈Iω{yi(k)},i=1,2.
In view of (3.21), we can obtain
(3.28)r¯iω≤∑k=0ω-1∑l=0mail(k)exp{yi(ηi)}+∑k=0ω-1∑l=0mcjl(k)=[∑l=0ma¯ilexp{yi(ηi)}+∑l=0mc¯jl]ω,i,j=1,2,i≠j.
Therefore,
(3.29)r¯i≤∑l=0ma¯ilexp{yi(ηi)}+∑l=0mc¯jl,i,j=1,2,i≠j.
Then
(3.30)exp{yi(ηi)}>r¯i-∑l=0mc¯jl∑l=0ma¯il=defBi,i,j=1,2,i≠j.
That is,
(3.31)yi(ηi)>lnBi,i=1,2.
By (3.25), (3.31), and Lemma 3.2, we have
(3.32)yi(k)≥yi(ηi)-∑s=0ω-1|yi(s+1)-yi(s)|≥lnBi-2r¯iω,i=1,2.
Inequalities (3.26) and (3.32) imply
(3.33)|yi(k)|≤max{|lnr¯iAi+2r¯iω|,|lnBi-2r¯iω|}=defHi,i=1,2.
Obviously,Bi,Ai, and Hi in (3.33) are independent of λ, respectively. Denote H=H1+H2+h0, where h0 is taken sufficiently large such that any solution {(ỹ1,ỹ2)⊤} of the system of algebraic equations
(3.34)∑l=0ma¯1lexp(y1)+∑l=0mc¯2lexp(y2)1+exp(y2)=r¯1,∑l=0ma¯2lexp(y2)+∑l=0mc¯1lexp(y1)1+exp(y1)=r¯2
satisfies ∥(ỹ1,ỹ2)⊤∥=max{|ỹ1|,|ỹ2|}<h0 (If system (3.34) has at least one solution). Let Ω=def{y:(y1,y2)∈X∣∥y∥<H}, thus condition (a) in Lemma 3.1 holds. When y∈∂Ω∩KerL=∂Ω∩R2, y={(y1,y2)⊤}, (y1,y2)⊤ is a constant vector in R2 with ∥y∥=H. If system (3.34) has at least one solution, then
(3.35)QN[y1y2]=[r¯1-∑l=0ma¯1lexp(y1)-∑l=0mc¯2lexp(y2)1+exp(y2)r¯2-∑l=0ma¯2lexp(y2)-∑l=0mc¯1lexp(y1)1+exp(y1)]≠[00].
If system (3.34) does not have one solution, then it is obvious that
(3.36)QN[y1y2]≠[00].
This implies that condition (b) in Lemma 3.1 is satisfied.
Now we prove that condition (c) in Lemma3.1 holds. Define Φ:DomL×[0,1]→X as follows:
(3.37)Φ(y1,y2,μ)=[r¯1-∑l=0ma¯1lexp(y1)r¯2-∑l=0ma¯2lexp(y2)]+μ[-∑l=0mc¯2lexp(y2)1+exp(y2)-∑l=0mc¯1lexp(y1)1+exp(y1)],
where μ is a parameter with μ∈[0,1]. When y={(y1,y2)⊤}∈∂Ω∩KerL=∂Ω∩R2, (y1,y2)⊤ is a constant vector in R2 with ∥y∥=H. We will show that {(y1,y2)⊤}∈∂Ω∩KerL, Φ(y1,y2,μ)≠0. If the conclusion is not true, then there is a constant vector (y1,y2)⊤∈R2 with ∥y∥=H satisfying Φ(y1,y2,μ)=0, that is,
(3.38)r¯1-∑l=0ma¯1lexp(y1)-μ∑l=0mc¯2lexp(y2)1+exp(y2)=0,r¯2-∑l=0ma¯2lexp(y2)-μ∑l=0mc¯1lexp(y1)1+exp(y1)=0.
A similar argument to the above shows that ∥y∥<H, which is a contradiction. Using the property of topological degree and taking J=I:ImQ→KerL, (y1,y2)⊤→(y1,y2)⊤, we have
(3.39)deg{JQN(y1,y2)⊤,Ω∩KerL,(0,0)⊤}=deg{Φ(y1,y2,1)⊤,Ω∩KerL,(0,0)⊤}=deg{Φ(y1,y2,0)⊤,Ω∩KerL,(0,0)⊤}=deg{(r¯1-∑l=0ma¯1lexp(y1),r¯2-∑l=0ma¯2lexp(y2))⊤,Ω∩KerL,(0,0)⊤}.
Obviously, the following equations:
(3.40)r¯1-∑l=0ma¯1lexp(u1)=0,r¯2-∑l=0ma¯2lexp(u2)=0
have the unique solution (ũ1,ũ2)⊤∈R2. Therefore, we have
(3.41)deg{JQN(y1,y2)⊤,Ω∩KerL,(0,0)⊤}=sign{∑l=0ma¯1lexp(ũ1)∑l=0ma¯2lexp(ũ2)}=1≠0.
Finally, we will show thatN is L-compact on Ω¯. For any y∈Ω¯, we have
(3.42)∥QNy∥=∥1ω∑s=0ω-1Ny(x)∥≤max{r1U+∑l=0ma1lUexp(H1)+∑l=0mc2lU,r2U+∑l=0ma2lUexp(H2)+∑l=0mc1lU}=defE.
Hence, QN(Ω¯) is bounded. Obviously, QNy:Ω¯→Y is continuous.
It is easy to see that(3.43)∥Kp(I-Q)Ny∥≤∑s=0ω-1∥Ny(s)∥+1ω∑s=0ω-1(ω-s)∥Ny(s)∥+1+3ω2ω∑s=0ω-1∥Ny(s)∥≤1+7ω2E.
For any y∈Ω¯, k1,k2∈Iω, without loss of generality, let k2>k1, then we have
(3.44)|Kp(I-Q)Ny(k2)-Kp(I-Q)Ny(k1)|=|∑s=k1k2-1Ny(s)-k2-k1ω∑s=0ω-1Ny(s)|≤∑s=k1k2-1|Ny(s)|+k2-k1ω∑s=0ω-1|Ny(s)|≤2E|k2-k1|.
Thus, the set{Kp(I-Q)Ny∣y∈Ω¯} is equicontinuous and uniformly bounded. By using the Arzela-Ascoli theorem, we see that Kp(I-Q)N:Ω¯→X is compact. Consequently, N is L-compact.
By now, we know thatΩ verifies all the requirements in Lemma 3.1 and then system (3.12) has at least one ω-periodic solution. By the medium of (3.11), we derive that system (1.1) has at least one ω-periodic solution. This completes the proof of Theorem 3.3.Next, by constructing a suitable Lyapunov-like discrete function, we further investigate the global attractivity of positive periodic solutions of system (1.1).Theorem 3.4.
In addition to (3.10), assume further that there exists a constant η>0 such that
(3.45)min{ai0*,2Mi-ai0*}-maiM-(m+1)ciM≥η,i=1,2,
where Mi  (i=1,2) are defined in  (2.5) and
(3.46)aiM=max{ail*:l=1,2,3,…,m},ciM=max{cil*:l=0,1,2,…,m},i=1,2.
Then the positive periodic solution of system (1.1) is globally attractive.Proof.
Let{(x̃1(k),x̃2(k))⊤} be a positive periodic solution of system (1.1). To finish the proof of Theorem 3.4, we will consider the following two steps.Step 1.
LetV11(k)=|lnx1(k)-lnx̃1(k)|, then it follows from the first equation of system (1.1) that
(3.47)V11(k+1)=|lnx1(k+1)-lnx̃1(k+1)|=|[lnx1(k)+r1(k)-∑l=0ma1l(k)x1(k-l)-∑l=0mc2l(k)x2(k-l)1+x2(k-l)]-[lnx̃1(k)+r1(k)-∑l=0ma1l(k)x̃1(k-l)-∑l=0mc2l(k)x̃2(k-l)1+x̃2(k-l)]|≤|lnx1(k)-lnx̃1(k)-a10(k)[x1(k)-x̃1(k)]|+∑l=1ma1l(k)|x1(k-l)-x̃1(k-l)|+∑l=0mc2l(k)|x2(k-l)-x̃2(k-l)|.
By the mean value theorem, we have(3.48)x1(k)-x̃1(k)=exp[lnx1(k)]-exp[lnx̃1(k)]=ξ1(k)lnx1(k)x̃1(k),
that is,
(3.49)lnx1(k)x̃1(k)=1ξ1(k)[x1(k)-x̃1(k)],
where ξ1(k) lies between x1(k) and x̃1(k). Then we have
(3.50)|lnx1(k)x̃1(k)-a10(k)[x1(k)-x̃1(k)]|=|lnx1(k)x̃1(k)|-|lnx1(k)x̃1(k)|+|lnx1(k)x̃1(k)-a10(k)[x1(k)-x̃1(k)]|=|lnx1(k)x̃1(k)|-1ξ1(k)|x1(k)-x̃1(k)|+|1ξ1(k)[x1(k)-x̃1(k)]-a10(k)[x1(k)-x̃1(k)]|=|lnx1(k)x̃1(k)|-1ξ1(k)|x1(k)-x̃1(k)|+|1ξ1(k)-a10(k)|×|x1(k)-x̃1(k)|=|lnx1(k)x̃1(k)|-[1ξ1(k)-|1ξ1(k)-a10(k)|]×|x1(k)-x̃1(k)|.
And hence it follows from (3.47) and (3.50) that
(3.51)ΔV11(k)=V11(k+1)-V11(k)≤-[1ξ1(k)-|1ξ1(k)-a10(k)|]×|x1(k)-x̃1(k)|+∑l=1ma1l(k)|x1(k-l)-x̃1(k-l)|+∑l=0mc2l(k)|x2(k-l)-x̃2(k-l)|.Step 2.
Let(3.52)V12(k)=∑l=1m∑s=k-lk-1a1l(s+l)|x1(s)-x̃1(s)|+∑l=0m∑s=k-lk-1c2l(s+l)|x2(s)-x̃2(s)|.
For the sake of convenience, we will make a convention that∏i=k1k2b(i)=1 if k1>k2 for any bounded sequence b(i). By a simple calculation, it derives that
(3.53)ΔV12(k)=V12(k+1)-V12(k)=∑l=1m∑s=k-l+1ka1l(s+l)|x1(s)-x̃1(s)|+∑l=0m∑s=k-l+1kc2l(s+l)|x2(s)-x̃2(s)|-∑l=1m∑s=k-lk-1a1l(s+l)|x1(s)-x̃1(s)|-∑l=0m∑s=k-lk-1c2l(s+l)|x2(s)-x̃2(s)|=∑l=1ma1l(k+l)|x1(k)-x̃1(k)|-∑l=1ma1l(k)|x1(k-l)-x̃1(k-l)|+∑l=0mc2l(k+l)|x2(k)-x̃2(k)|-∑l=0mc2l(k)|x2(k-l)-x̃2(k-l)|.
Now, we are in a position to defineV1(k) by
(3.54)V1(k)=V11(k)+V12(k).
Therefore, it follows from (3.51) and (3.53) that
(3.55)ΔV1(k)=ΔV11(k)+ΔV12(k)≤-[1ξ1(k)-|1ξ1(k)-a10(k)|-∑l=1ma1l(k+l)]×|x1(k)-x̃1(k)|+∑l=0mc2l(k+l)|x2(k)-x̃2(k)|.
By a similar argument, we can defineV2(k) by
(3.56)V2(k)=V21(k)+V22(k),
where
(3.57)V21(k)=|lnx2(k)-lnx̃2(k)|,V22(k)=∑l=1m∑s=k-lk-1a2l(s+l)|x2(s)-x̃2(s)|+∑l=0m∑s=k-lk-1c1l(s+l)|x1(s)-x̃1(s)|.
Then it is easy to derive that
(3.58)ΔV2(k)=ΔV21(k)+ΔV22(k)≤-[1ξ2(k)-|1ξ2(k)-a20(k)|-∑l=1ma2l(k+l)]×|x2(k)-x̃2(k)|+∑l=0mc1l(k+l)|x1(k)-x̃1(k)|,
where ξ2(k) is between x2(k) and x̃2(k).
Now we can define a Lyapunov-like discrete functionV(k) by
(3.59)V(k)=V1(k)+V2(k).
It is easy to see thatV(k)≥0 for all k∈Z and V(k0+m)<+∞. For the arbitrariness of ɛ and by (3.45), we can choose a small enough ɛ>0 such that
(3.60)min{ai0*,2Mi+ɛ-ai0*}-maiM-(m+1)ciM≥η,i=1,2.
Therefore, it follows from (3.55)–(3.60) that
(3.61)ΔV(k)=ΔV1(k)+ΔV2(k)≤-∑i=12{1ξi(k)-|1ξi(k)-ai0(k)|-∑l=1mail(k+l)-∑l=0mcil(k+l)}×|xi(k)-x̃i(k)|≤-∑i=12{min{ai0*,2Mi+ɛ-ai0*}-∑l=1mail*-∑l=0mcil*}×|xi(k)-x̃i(k)|≤-∑i=12{min{ai0*,2Mi+ɛ-ai0*}-maiM-(m+1)ciM}×|xi(k)-x̃i(k)|≤-η∑i=12|xi(k)-x̃i(k)|fork≥k0+m.
Thus, by (3.61) we obtain
(3.62)V(k+1)+η∑p=k0+mk∑i=12|xi(p)-x̃i(p)|≤V(k0+m)foranyk≥k0+m,∑k=k0+m+∞∑i=12|xi(k)-x̃i(k)|≤V(k0+m)η<+∞,
from which we conclude that ∑i=12|xi(k)-x̃i(k)|=0 when k→+∞, that is,
(3.63)limk→+∞|x1(k)-x̃1(k)|=0,limk→+∞|x2(k)-x̃2(k)|=0.
According to Definition1.2, this result implies that the positive periodic solution {(x̃1(k),x̃2(k))⊤} is globally attractive. This completes the proof of Theorem 3.4.


## 4. Example and Numerical Simulation

In this paper, a discrete time non-autonomous two-species competitive system with delays is investigated. By using difference inequality technique, continuous theorem of coincidence degree theory, and Lyapunov discrete function, sufficient conditions for the permanence of system (1.1) and the existence and global attractivity of positive periodic solutions of system (1.1) are obtained, respectively.To substantiate our analytical results, we construct the following example:(4.1)x1(k+1)=x1(k)exp[r1(k)-a10(k)x1(k)-a11(k)x1(k-1)-c20(k)x2(k)1+x2(k)-c21(k)x2(k-1)1+x2(k-1)],x2(k+1)=x2(k)exp[r2(k)-a20(k)x2(k)-a21(k)x2(k-1)-c10(k)x1(k)1+x1(k)-c11(k)x1(k-1)1+x1(k-1)].We first verify the sufficient conditions for permanence of system (4.1) and choose the coefficients (4.2)r1(k)=0.85+0.04sink,r2(k)=0.87+0.01sink,a10(k)=a11(k)=1.83+0.02sink,a20(k)=a21(k)=1.20+0.03sink,c10(k)=c11(k)=0.05+0.01sink,c20(k)=c21(k)=0.04+0.01sink
and the initial conditions (4.3)x1(-1)=0.2,x1(0)=0.018,x2(-1)=0.3,x2(0)=0.017.
It is easy to see that system (4.1) satisfies assumption (2.6) of Theorem 2.3, and hence system (4.1) is permanent (see Figure 1).Permanence of system (4.1) with x1(-1)=0.2, x1(0)=0.018, x2(-1)=0.3, and x2(0)=0.017. (a) Time series of x1. (b) Time series of x2.
(a)(b)In the following, we will consider the existence and global attractivity of positive periodic solutions of system (4.1). We assume that (4.4)r1(k)=0.0695+0.0055cosπk,r2(k)=0.0665-0.0005cosπk,a10(k)=1.0080+0.0010cosπk,a20(k)=1.0425-0.0005cosπk,a11(k)=0.0080-0.0010cosπk,a21(k)=0.0425-0.0005cosπk,c10(k)=0.00035+0.00015cosπk,c20(k)=0.00035-0.00005cosπk,c11(k)=0.00045+0.00015cosπk,c21(k)=0.00045-0.00005cosπk.
It is easy to verify that assumptions (3.10) of Theorem 3.3 are satisfied.  Figure 2 shows that system (4.1) has a 2-periodic solution {(x1(k),x2(k))⊤}, which implies that the two species x1 and x2 can coexist. Furthermore, a calculation can show that the assumptions (3.34) of Theorem 3.4 are satisfied, so {(x1(k),x2(k))⊤} is globally attractive, that is, any positive solution {(x1*(k),x2*(k))⊤} of system (4.1) tends to {(x1(k),x2(k))⊤} (see  Figure 3). From  Figure 3(a), we see that x1* with x1*(-1)=0.03 and x1*(0)=0.0662 will tend to x1 with x1(-1)=0.02 and x1(0)=0.0668. Similarly, from  Figure 3(b), we see that x2* with x2*(-1)=0.04 and x2*(0)=0.0606 will tend to x2 with x2(-1)=0.05 and x2(0)=0.0607.Dynamic behaviors of system (4.1) with x1(-1)=0.02, x1(0)=0.0660, x2(-1)=0.03, and x2(0)=0.0608. (a) Time series of x1. (b) Time series of x2. (c) Phase portrait of x1 and x2 with k over [0,120]. (d) Phase portrait of x1 and x2 with k over [110,120].
(a)(b)(c)(d)Time series ofx1*,x1 and x2*,x2. (a) Time series of x1*,x1 with x1*(-1)=0.03, x1*(0)=0.0662, x1(-1)=0.02, and x1(0)=0.0668. (b) Time series of x2*,x2 with x2*(-1)=0.04, x2*(0)=0.0606, x2(-1)=0.05, and x2(0)=0.0607.
(a)(b)



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*Source: 381750-2010-05-06.xml*