IJDEInternational Journal of Differential Equations1687-96511687-9643Hindawi Publishing Corporation87169310.1155/2011/871693871693Research ArticleExistence of Positive Periodic Solutions for a Class of n-Species Competition Systems with ImpulsesGuoPeilianLiuYanshengEzzinbiKhalilDepartment of Mathematics, Shandong Normal University, Jinan 250014Chinasdnu.edu.cn20111392011201120052011110720112011Copyright © 2011 Peilian Guo and Yansheng Liu.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By using the fixed point theorem on cone, some sufficient conditions are obtained on the existence of positive periodic solutions for a class of n-species competition systems with impulses. Meanwhile, we point out that the conclusion of (Yan, 2009) is incorrect.

1. Introduction

In recent years, the problem of periodic solutions of the ecological species competition systems has always been one of the active areas of research and has attracted much attention. For instance, the traditional Lotka-Volterra competition system is a rudimentary model on mathematical ecology which can be expressed as follows: ẋi(t)=xi(t)[ri(t)-j=1naij(t)xj],i=1,2,,n. Owing to its theoretical and practical significance, the systems have been studied extensively by many researchers. And many excellent results which concerned with persistence, extinction, global attractivity of periodic solutions, or almost periodic solutions have been obtained.

However, the Lotka-Volterra competition systems ignore many important factors, such as the age structure of a population or the effect of toxic substances. So, more complicated competition systems are needed. In 1973, Ayala and Gilpin proposed several competition systems. One of the systems is the following competition system: Ṅi(t)=riNi[1-(NiKi)θi-j=1,jinαijNjKj],i=1,2,,n, where Ni is the population density of the ith species; ri is the intrinsic exponential growth rate of the ith species; Ki is the environmental carrying capacity of species i in the absence of competition; θij provides a nonlinear measure of interspecific interference, αij provides a nonlinear measure of interspecific interference.

On the other hand, in the study of species competition systems, the effect of some impulsive factors has been neglected, which exists widely in the real world. For example, the harvesting or stocking occur at fixed time, natural disaster such as fire or flood happen unexpectedly, and some species usually migrate seasonally. Consequently, such processes experience short-time rapid change which can be described by impulses. Therefore, it is important to study the existence of the periodic solutions of competitive systems with impulse perturbation (see  and the references therein).

For example, by using the method of coincidence degree, Wang  considered the existence of periodic solutions for the following n-species Gilpin-Ayala impulsive competition system: ẋi(t)=xi(t)[ri(t)-j=1naij(t)xjαij(t)-j=1nbij(t)xjαij(t-τij(t))-j=1ncij(t)xiαii(t)xjαij(t)],ttk;Δxi(tk)=xi(tk+)-xi(tk-)=pkixi(tk),k=1,2,3,, where the constant pki satisfied -1<pki<0,  i=1,2,,n. What is more,  also obtained several results for the persistence and global attractivity of the periodic solution of the model.

In , Yan applied the Krasnoselskii fixed point theorem to investigate the following n-species competition system: ẏi(t)=yi(t)[ri(t)-j=1naij(t)yjαij(t)-j=1nbij(t)yjβij(t-τij(t))-j=1ncij(t)-σij0Kij(ξ)yiγij(t+ξ)yjδij(t+ξ)dξ],i=1,2,,n, where the constants αij,βij,γij1,  i,j=1,2,,n. He obtained a necessary and sufficient condition for the existence of periodic solutions of system (1.4). Unfortunately, its last conclusion is wrong. Please see the remark in Section 3 of this paper.

Motivated by [1, 2], in this paper, we investigate the following impulsive n-species competition system: ẏi(t)=yi(t)[ri(t)-j=1naij(t)yjαij(t)-j=1nbij(t)yjβij(t-τij(t))-j=1ncij(t)-σij0Kij(ξ)yiγij(t+ξ)yjδij(t+ξ)dξ],  ttk,  i=1,2,,n,Δyi(tk)=yi(tk+)-yi(tk-)=Iik(y1(tk),  y2(tk),,yn(tk)),k=1,2,,m, where yi(t) is the population density of the ith species at time t; ri(t) is the intrinsic exponential growth rate of the ith species at time t; τij(t) is the time delay; σij is a positive constant; aij(t),bij(t),cij(t)  (ij) measure the amount of competition between the species Yi and Yj; αij,  βij,  δij  (ij) provide a nonlinear measure of interspecific interference; yi(tk+) (yi(tk-)) is the left (right) limits of yi(t) at t=tk,  i,j=1,2,,n,  k=1,2,,m.

The main features of the present paper are as follows. The Gilpin-Ayala species competition system (1.5) has impulsive effects. As is known to us, there were few papers to study such system. Finally, we point out that the conclusion of  is incorrect.

For an ω-periodic function u(t)C(,), let u¯=1/ω0ωu(t)dt. Throughout this paper, assume the following conditions hold.

ri,  aij,  bij,  cij,  τij are continuous ω-periodic functions, and r¯i>0,  aij(t),  bij(t),  cij(t)0,  i,j=1,2,,n, and there exists i0    (1i0n) such that min1jn(a¯i0j+b¯i0j)>0.

KijC([-σij,0],),  Kij0, σij is a positive constant, and -σij0Kij(t)dt=1,  i,j=1,2,,n.

IikC(n,[0,+)), and for 0<t1<t2<<tm<ω, there exists an positive integer l>0 such that tk+lm=tk+lω,Iik(y1,y2,,yn)=Ii(k+lm)(y1,y2,,yn), where (y1,y2,,yn)n,  i=1,2,,n,  k=1,2,,m.

αij,βij,δij>0,  γij0 are constants, i,j=1,2,,n.

In order to prove our main result, now we state the fixed point theorem of cone expansion and compression.

Lemma 1.1 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let E be a Banach space, and let P be a cone in E. Assume that Ω1,  Ω2 are open subsets of E with 0Ω1,  Ω¯1Ω2. Let A:P(Ω¯2Ω1)P be a completely continuous operator such that one of the following two conditions is satisfied:

Ax for xPΩ1; Axœ for xPΩ2,

Ax for xPΩ2; Axœ for xPΩ1,

Then, A has a fixed point in P(Ω¯2Ω1).

The organization of this paper is as follows. In the next section, we introduce some lemmas and notations. In Section 3, the main result will be stated and proved on the existence of periodic solutions of system (1.5).

2. Preliminaries

Let PC(,n)={y(t)=(y1(t),y2(t),yn(t)):nyi(t) be continuous at ttk, left continuous at t=tk, and the right limit yi(tk+) exists for i=1,2,,n,  k=1,2,,m}. Evidently, E={y(t)=(y1(t),y2(t),  yn(t))PC(R,Rn)yi(t)=yi(t+ω),  i=1,2,,n}   is a Banach space with the norm y=i=1n|yi|0, where |yi|0=maxt[0,ω]|yi(t)|.

Define an operator T:EE by (Ty)(t)=((Ty)1(t),(Ty)1(t),,(Ty)n(t)), where (Ty)i(t)=tt+ωGi(t,s)yi(s)[j=1naij(s)yjαij(s)+j=1nbij(s)yjβij(s-τij(s))+j=1ncij(s)-σij0Kij(ξ)yiγij(s+ξ)yjδij(s+ξ)dξ]ds+k=1mGi(t,tk+qkm)Iik(y1(tk),y2(tk),  ,yn(tk)), where Gi(t,s)=exp(-tsri(ν)dν)/(1-exp(-ωr¯i)),  tst+ω,  i=1,2,,n, and tk+qkm=tk+qkω[t,t+ω], where qk is a positive integer, k=1,2,,m.

It is obvious that the functions Gi(t,s),  i=1,2,,n have the following properties.

Gi(t,s)>0 for (t,s)2, and Gi(t,s)=Gi(t+ω,s+ω).

AGi(t,s)B for (t,s)2, where A=min1in{exp(-ωr¯i)/(1-exp(-ωr¯i))}, B=min1in{exp(ωr¯i)/(1-exp(-ωr¯i))}.

Now, we choose a set defined by P={y(t)=(y1(t),y2(t),  yn(t))Eyi(t)σ|yi|0,  t[0,ω],  i=1,2,,n}, where σ=A/B. Clearly, P is a cone in E.

For the sake of convenience, we define an operator F:PE by (Fy)(t)=((Fy)1(t),(Fy)2(t),,(Fy)n(t)), where (Fy)i(t)=yi(t)[j=1naij(t)yjαij(t)+j=1nbij(t)yjβij(t-τij(t))+j=1ncij(t)-σij0Kij(ξ)yiγij(t+ξ)yjδij(t+ξ)dξ],i=1,2,,n.

Lemma 2.1.

The operator T:PP is completely continuous.

Proof.

First, it is easy to see T:PC(,n)PC(,n). Next, since (Ty)i(t+ω)=t+ωt+2ωGi(t+ω,s)(Fy)i(s)ds+k=1mGi(t+ω,tk+qkm+ω)Iik(y1(tk),y2(tk),  ,yn(tk))=tt+ωGi(t+ω,υ+ω)(Fy)i(υ+ω)dυ+k=1mGi(t,tk+qkm)Iik(y1(tk),y2(tk),,yn(tk))=tt+ωGi(t,υ)(Fy)i(υ)dυ+k=1mGi(t,tk+qkm)Iik(y1(tk),y2(tk),,yn(tk))=(Ty)i(t), we have TyE.

Observe that AGi(t,s)B,  i=1,2,,n, for all s[t,t+ω]. Hence, we obtain that, for yP, |(Ty)i|0B0ω(Fy)i(s)ds+Bk=1mIik(y1(tk),y2(tk),,yn(tk)),(Ty)i(t)A0ω(Fy)i(s)ds+Ak=1mIik(y1(tk),y2(tk),,yn(tk))AB|(Ty)i|0=σ|(Ty)i|0. Thus, TyP, that is, T(P)P.

Obviously, the operator T is continuous. Next, we show that T is compact. Let SE be a bounded subset; that is, there exists d>0 such that |yi|0d,  i=1,2,,n for all yS. From the continuity of F,Ik,  k=1,2,,m, we have, for all yS, |(Ty)i|0B0ω(Fy)i(s)ds+Bk=1mIik(y1(tk),  y2(tk),,yn(tk))Bωdj=1n(a¯ijdαij+b¯ijdβij+c¯ijdγij+δij)+BmEi=:Di, where Ei=maxyS|Iik(y1(tk),y2(tk),,yn(tk)|,  i=1,2,,n.

Therefore, Ty=i=1n|(Ty)i|0i=1nDi=:D, which implies that T(S) is uniformly bounded.

On the other hand, noticing that ddt(Ty)i(t)=ri(t)(Ty)i(t)+Gi(t,t+ω)(Fy)i(t+ω)-Gi(t,t)(Fy)i(t)=ri(t)(Ty)i(t)+[Gi(t,t+ω)-Gi(t,t)](Fy)i(t)=ri(t)(Ty)i(t)+(Fy)i(t),ttk,  k=1,2,,m. This guarantees that, for each yS, we have |ddt(Ty)i(t)|riMDi+dj=1n(aijMdαij+bijMdβij+cijMdγij+δij)=:DĩD̃=i=1nDĩ, where riM=maxt[0,ω]ri(t),  aijM=maxt[0,ω]aij(t), bijM=maxt[0,ω]bij(t), cijM=maxt[0,ω]cij(t), i=1,2,,n.

Consequently, T(S) is equicontinuous on Jk,  k=0,1,2,,m, where J0=[0,t1),  J1=[t1,t2),,Jm-1=[tm-1,tm),  Jm=[tm,ω). By the Ascoli-Arzela theorem, the function T:PP is completely continuous from P to P.

Lemma 2.2.

The system (1.5) has a positive ω-periodic solution in P if and only if T has a fixed point in P.

Proof.

For yP satisfying Ty=y, that is, (Ty)i(t)=yi(t),  t[0,ω],  i=1,2,,n, it follows from (2.2) and (2.4) that ẏi(t)=ddt(Ty)i(t)=ri(t)(Ty)i(t)+Gi(t,t+ω)(Fy)i(t+ω)-G(t,t)(Fy)i(t)=ri(t)(Ty)i(t)+[G(t,t+ω)-G(t,t)](Fy)i(t)=ri(t)(Ty)i(t)+(Fy)i(t),ttk,  k=1,2,,m. And for t=tk,  k=1,2,,m, Δyi(tk)=yi(tk+)-yi(tk-)=Iik(y1(tk),y2(tk),,yn(tk)),i=1,2,,n, which implies that y(t) is a positive ω-periodic solution of (1.5).

Conversely, assume that yP is an ω-periodic solution of system (1.5). Then, the system (1.5) can be transformed into (ẏi(t)+r(t)yi(t))e0tr(ν)dν=(Fy)i(t)e0tr(ν)dν, that is, (yi(t)e0tr(ν)dν)=(Fy)i(t)e0tr(ν)dν.

So, integrating the above equality from t to t+ω and noticing that yi(t)=yi(t+ω), we have (Ty)i(t)=tt+ωGi(t,s)yi(s)[j=1naij(s)yjαij(s)+j=1nbij(s)yjβij(s-τij(s))+j=1ncij(s)-σij0Kij(ξ)yiγij(s+ξ)yjδij(s+ξ)dξ]ds+k=1mGi(t,tk+qkm)Iik(y1(tk),y2(tk),,yn(tk)), where     Gi(t,s)=exp(-tsri(ν)dν)/(1-exp(-ωr¯i)),  tst+ω,  i=1,2,,n, and tk+qkm=tk+qkω[t,t+ω], where qk is a positive integer, k=1,2,,m, that is, Ty=y.

Therefore, yP is a fixed point of the operator T. The proof of the Lemma is complete.

3. Main Results

Theorem 3.1.

Suppose (H1)–(H4) hold, and lim|v|0Iik(v)/|v|=0,  k=1,2,,m,  i=1,2,,n, where v=(v1,v2,,vn),  |v|=min1in|vi|.   Then system (1.5) has at least one positive ω-periodic solution.

Proof.

Let M0=max1in{j=1n(a¯ij+b¯ij+c¯ij)}>0. Choose M1M0 and ε=1/(BωM1+Bm)>0. Then, there exists δ>0 such that, for 0<x<δ and 0<|v|<δ, we have xαij<ε,xβij<ε,xγij+δij<ε,Iik(v)<ε|v|,k=1,2,,m,  i,j=1,2,,n.

Choose r<δ. Let Ω1={y(t)=(y1(t),y2(t),,yn(t))E|yi|0<r,  i=1,2,,n}.

Now, we prove that Tyy,yPΩ1.

Suppose (3.2) does not hold. Then, there exists some yPΩ1 such that Tyy. Since yPΩ1, we have σ|yi|0yi(t)|yi|0 for t[0,ω], i=1,2,,n. From (2.2) and (2.4), it follows that (Tyi)(t)Btt+ω(Fy)i(s)ds+Bk=1mIik(y1(tk),y2(tk),,yn(tk))Bω|yi|0(i=1na¯ijrαij+b¯ijrβij+c¯ijrγij+δij)+Bmε|yi|0(BM0ωε+Bmε)|yi|0<|yi|0, which implies |(Ty)i|0<|yi|0, a contradiction. Hence, Tyy for yPΩ1.

On the other hand, let m0=min1jn{a¯i0jσαij+b¯i0jσβij}, on the account of (H1), we know m0>0. Choose 0<m1m0 and M=1/Aσωm1>0. Then, there exists R1>0 such that, for x>R1, we know that xαij>M,xβij>M,i,j=1,2,,n.

Choose R>max{R1,r}. Let Ω2={y(t)=(y1(t),y2(t),yn(t))E|yi|0<R,  i=1,2,,n}. Then, Ω2={y(t)=(y1(t),y2(t),yn(t))E there exist some integers j0(1j0n) such that |yj0|0=R;|yi|0R for ij0}.

Next we show that Tyy,yPΩ2.

In fact, if there exists some yPΩ2 such that Tyy and since yPΩ2, we have σ|yi|0yi(t)|yi|0,  i=1,2,,n for t[0,ω], and there exists some j0 such that |yj0|0=R. Therefore, this together with (H1) guarantees that, for i0  (1i0n), yi0(t)(Ty)i0(t)Att+ω(Fy)i0(s)dsAσ|yi0|0tt+ω(ai0j0(s)yj0αi0j0(s)+bi0j0(s)yj0βi0j0(s-τ(s)))dsAσ|yi0|0tt+ω(ai0j0(s)σαi0j0|yj0|0αi0j0+bi0j0(s)σβi0j0|yj0|0βi0j0)dsAσ|yi0|0ωM(a¯i0j0σαi0j0+b¯i0j0σβi0j0)AσωMm0|yi0|0>|yi0|0, which is a contradiction. Thus, (3.5) is satisfied.

From all the above, the condition (i) of Lemma 1.1 is satisfied. So the operator T has a fixed point in P(Ω¯2Ω1). That is, system (1.5) has at least one positive periodic solution.

Remark 3.2.

For any R>0, if we let Ω={y(t)=(y1(t),y2(t),,yn(t))E|yi|0<R,  i=1,2,,n}, then Ω={y(t)=(y1(t),y2(t),,yn(t))E    j0(1j0n),    |yj0|0=R,  |yi|0R,  ij0}. However, in the proof of Theorem  1.1 of , it is regarded mistakenly as Ω={y(t)=(y1(t),y2(t),,yn(t))E|yi|0=R,  i=1,2,,n}. Therefore, the proof of its sufficiency is not correct. So the result of  is incorrect.

Acknowledgments

This research was supported by the Natural Science Foundation of Shandong Province (ZR2009AM006), the Key Project of Chinese Ministry of Education (no. 209072), and the Science and Technology Development Funds of Shandong Education Committee (J08LI10).

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