# Periodic Solutions forn-Species Lotka-Volterra Competitive Systems with Pure Delays

Authors: Ahmadjan Muhammadhaji; Rouzimaimaiti Mahemuti; Zhidong Teng
Journal: Chinese Journal of Mathematics
 (2015)

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## Abstract

We study a class of periodic generaln-species competitive Lotka-Volterra systems with pure delays. Based on the continuation theorem of the coincidence degree theory and Lyapunov functional, some new sufficient conditions on the existence and global attractivity of positive periodic solutions for the n-species competitive Lotka-Volterra systems are established. As an application, we also examine some special cases of the system, which have been studied extensively in the literature.


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## Body

## 1. Introduction

As we know well, in recent years the application of the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin to the existence of positive periodic solutions in population dynamical systems has been studied extensively [1–21]. For example, in [1–3, 12, 14–20], the authors studied existence of positive periodic solutions for population competition systems, in [4, 5, 8, 13, 21], the authors studied existence of positive periodic solutions for population cooperative systems, in [9], the authors studied existence of positive periodic solutions for population predator-prey system, and in [6, 11], the authors studied existence of positive periodic solutions for single species systems. The continuation theorem is a powerful tool to study the existence of periodic solutions of periodic high-dimensional time-delayed problems. When dealing with time-delayed problem, it is very convenient and the result is relatively simple [4, 5].Frequently, the environments of most natural populations undergo temporal variation, causing changes in the growth characteristics of these populations. One of the methods of incorporating temporal nonuniformity of the environments in models is to assume that the parameters are periodic with the same period of the time variable [14].However, in the real world, the growth rate of a natural species will not often respond immediately to changes in its own population or that of an interacting species but will rather do so after a time lag [22]. Research [23, 24] has shown that time delays have a great destabilizing influence on the species population. Usually, time delays are of two types: discrete delay and distributed time delay. For a competition system, competitors have both an instantaneous competition and a memory competition in the past. Therefore, we should introduce distributed delay into model foundation, which will have more resemblance to the real ecosystem. In fact, during the last decades, most of the authors study dynamics of population with delays [1–7, 10–21, 25–31], which is useful for the control of the population of mankind, animals, and environment.It is well known that the focus in theoretical models of population and community dynamics must be not only on how populations depend on their own population densities or the population densities of other organisms but also on how populations change in response to the physical environment [19]. To consider periodic environmental factor, it is reasonable to study Lotka-Volterra systems with periodic coefficients. One of the celebrated population dynamical systems is Lotka-Volterra competition system. Since the Lotka-Volterra competition system has been established and was accepted by many scientists, now it has became the most important means to explain the ecological phenomenon. Recently, a great deal of Lotka-Volterra competition system with delays have been proposed to study the existence of periodic solutions [1–3, 12, 14–21, 25–31] and many good results were obtained by using of the continuation theorem [1–3, 12, 14–21].In [1], the authors studied the following nonautonomous N-species Lotka-Volterra competitive systems with continuous time delays: (1)x˙=xtrit-∑j=1naijtxjt-τijt,i=1,2,…,n.By using the method of coincidence degree and Lyapunov functional, a set of easily verifiable sufficient conditions of the existence and global attractivity of positive periodic solutions are established. In [2], the authors considered the following nonautonomous N-species Lotka-Volterra competitive systems with distributed time delays, (2)u˙=utrit-aiituit-∑j=1,j≠inaijt∫-Tij0Kijsujt+sds,i=1,2,…,n,and the periodic state dependent delay Lotka-Volterra competition system, (3)u˙=utrit-aiituit-∑j=1,j≠inaijtujt-τjt,u1t,…,unt,i=1,2,…,n.By using the Mawhin’s continuation theorem, the sufficient conditions on the existence of positive periodic solutions are established. In [3], the authors considered the following nonautonomous delay N-species competitive systems with delays: (4)u˙=utFit,uit-τiit-∑j=1,j≠inaijtujt-τijt,t∈R,i=1,2,…,n.By means of the Mawhin’s continuation theorem and Lyapunov function method, the sufficient conditions for the existence and global attractivity of positive periodic solutions are established. In [21], the authors studied the following nonautonomous N-species Lotka-Volterra cooperative systems with continuous time delays and feedback controls: (5)x˙it=xitrit-∑l=1maiiltxit-τiilt+∑j≠in∑l=1maijltxjt-τijlt-dituit-eit·uit-εit,u˙it=-bituit+βitxit+γitxit-σit,i=1,2,…,n.The sufficient conditions for the existence of positive periodic solutions are established, based on the Mawhin’s continuation theorem.Motivated by the above works, in this paper, we investigate the followingn species periodic Lotka-Volterra type competitive systems with pure delays (6)x˙it=xitrit-∑l=1maiiltxit-τiilt-∑j≠in∑l=1maiilt∫-τijl0kijlsxjt+sds,i=1,2,…,n.By using the technique of coincidence degree developed by Gaines and Mawhin in [32], we will establish some new sufficient conditions, which guarantee that the system has at least one positive periodic solution. By means of the Lyapunov functionals we also will further establish the sufficient conditions on the global attractivity of the positive periodic solution.


## 2. Preliminaries

In system (6), we have that xi(t)(i=1,2,…,n) represent the density of n competitive species xi(i=1,2,…,n) at time t, respectively; ri(t)(i=1,2,…,n) represent the intrinsic growth rate of species xi(i=1,2,…,n) at time t, respectively; aiil(t)(i=1,2,…,n,l=1,2,…,m) represent the intrapatch restriction density of species xi(i=1,2,…,n) at time t, respectively; aijl(t)(l=1,2,…,m, i≠j,i,j=1,2,…,n) represent the competitive coefficients between n species xi(i=1,2,…,n) at time t, respectively. In this paper, we always assume the following.(H1) τijl>0 and τiil(t)(l=1,2,…,m,i=1,2,…,n), ri(t)(i=1,2,…,n) are continuous ω-periodic functions with τiil′(t)<1 and ∫0ωri(t)dt>0. aijl(t)(i,j=1,2,l=1,2,…,m), (i=1,2,…,n) are continuous positive ω-periodic functions. kijl(s)(i,j=1,2,…,n;l=1,2,…,m) are nonnegative integrable functions on [-τijl,0] satisfying ∫-τijl0kijl(s)ds=1.From the viewpoint of mathematical biology, in this paper for system (6) we only consider the solution with the following initial conditions: (7)xit=ϕit,∀t∈-γ,0,i=1,2,…,n,where ϕi(t)(i=1,2,…,n) are nonnegative continuous functions defined on [-γ,0] satisfying ϕi(0)>0(i=1,2,…,n) with γ=maxt∈[0,ω]{τijl(t),(i,j=1,2,l=1,2,…,m)}.In this paper, for anyω-periodic continuous function f(t), we denote (8)fL=mint∈0,ω⁡ft,fM=maxt∈0,ω⁡ft,f-=1ω∫0ωftdt.In order to obtain the existence of positiveω-periodic solutions of system (6), we will use the continuation theorem developed by Gaines and Mawhin in [32]. For the reader’s convenience, we will introduce the continuation theorem in the following.LetX and Z be two normed vector spaces. Let L:Dom⁡L⊂X→Z be a linear operator and N:X→Z a continuous operator. The operator L is called a Fredholm operator of index zero, if dim⁡Ker⁡L=codimIm⁡L<∞ and Im⁡L is a closed set in Z. If L is a Fredholm operator of index zero, then there exist continuous projectors P:X→X and Q:Z→Z such that Im⁡P=Ker⁡L and Im⁡L=Ker⁡Q=Im⁡(I-Q). It follows that L∣Dom⁡L∩Ker⁡P:Dom⁡L∩Ker⁡P→Im⁡L is invertible and its inverse is denoted by KP and denote by J:Im⁡Q→Ker⁡L an isomorphism of Im⁡Q onto KerL. Let Ω be a bounded open subset of X; we say that the operator N is L-compact on Ω¯, where Ω¯ denotes the closure of Ω in X, if QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is compact. Such definitions can be found in [4, 5].Lemma 1 (see [10]).
Supposeτ∈C1(R,R) with τ(t+ω)≡τ(t) and τ′(t)<1, ∀t∈[0,ω]. Then the function t-τ(t) has a unique inverse function μ(t) satisfying μ∈C(R,R), μ(u+ω)=μ(u)+ω, ∀u∈R.Lemma 2 (see [32]).
LetL be a Fredholm operator of index zero and let N be L-compact on Ω¯. If(a)
for eachλ∈(0,1) and x∈∂Ω∩Dom⁡L, Lx≠λNx,(b)
for eachx∈∂Ω∩Ker⁡L, QNx≠0,(c)
deg⁡{JQN,Ω∩Ker⁡L,0}≠0,then the operator equation Lx=Nx has at least one solution lying in Dom⁡L∩Ω¯.


## 3. Main Results

Now, for convenience of statements we denote the functions(9)aijt=∑l=1maijlt,i,j=1,2,…,n.The following theorem is about the existence of positive periodic solutions of system (6).Theorem 3.
Suppose that assumption (H1) holds and there exists a constant θi>0, i=1,2,…,n, such that (10)mint∈ω⁡r-iθi-∑j≠in∑l=1maijlMθj∑i=1nr-iAj≕Bi>0,where (11)mint∈ω⁡∑l=1mδiilt+∑j≠in∫-τ0ajilt-skjilsds≕Ai,δiilt=aiilφiilt1-τiil′φiilt,i=1,2,…,n,l=1,2,…,m,and the algebraic equation, (12)r-i-a-iivi-∑j≠ina-ijvj=0,i=1,2,…,n,has a unique positive solution. Then system (6) has at least one positive ω-periodic solution.Proof.
For system (6) we introduce new variables yi(t)(i=1,2,…,n) such that (13)xit=exp⁡yit,i=1,2,…,n.Then system (6) is rewritten in the following form: (14)y˙it=rit-∑l=1maiiltexp⁡yit-τiilt-∑j≠in∑l=1maijlt∫-τijl0kijlsexp⁡yjt+sds,i=1,2,…,n.
In order to apply Lemma2 to system (14), we introduce the normed vector spaces X and Z as follows. Let C(R,Rn) denote the space of all continuous function y(t)=(y1(t),y2(t),…,yn(t)):R→Rn. We take (15)X=Z=yt∈CR,Rn:ytis anω-periodic function,with norm (16)y=∑i=1nmaxt∈0,ω⁡yit.It is obvious that X and Z are the Banach spaces. We define a linear operator L:Dom⁡L⊂X→Z and a continuous operator N:X→Z as follows: (17)Lyt=y˙t,Nyt=Ny1t,Ny2t,…,Nynt,where (18)Nyit=rit-∑l=1maiiltexp⁡yit-τiilt-∑j≠in∑l=1maijlt∫-τijl0kijlsexp⁡yjt+sds.Further, we define continuous projectors P:X→X and Q:Z→Z as follows: (19)Pyt=1ω∫0ωytdt,Qvt=1ω∫0ωvtdt.We easily see Im⁡L={v∈Z:∫0ωv(t)dt=0} and Ker⁡L=Rn. It is obvious that Im⁡L is closed in Z and dimKer⁡L=n. Since for any v∈Z there are unique v1∈Rn and v2∈Im⁡L with (20)v1=1ω∫0ωvtdt,v2t=vt-v1,such that v(t)=v1+v2(t), we have codimIm⁡L=n. Therefore, L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) Kp:Im⁡L→Ker⁡P∩Dom⁡L is given in the following form: (21)Kpvt=∫0tvsds-1ω∫0ω∫0tvsdsdt.For convenience, we denote F(t)=(F1(t),F2(t),…,Fn(t)) as follows: (22)Fit=rit-∑l=1maiiltexp⁡yit-τiilt-∑j≠in∑l=1maijlt∫-τijl0kijlsexp⁡yjt+sds.Thus, we have (23)QNyt=1ω∫0ωFtdt,KpI-QNut=KpINut-KpQNut=∫0tFsds-1ω∫0ω∫0tFsdsdt+12-tω∫0ωFsds.From formulas (23), we easily see that QN and Kp(I-Q)N are continuous operators. Furthermore, it can be verified that Kp(I-Q)N(Ω¯)¯ is compact for any open bounded set Ω⊂X by using Arzela-Ascoli theorem and QN(Ω¯) is bounded. Therefore, N is L-compact on Ω¯ for any open bounded subset Ω⊂X.
Now, we reach the position to search for an appropriate open bounded subsetΩ for the application of the continuation theorem (Lemma 2) to system (14).
Corresponding to the operator equationLy(t)=λNy(t) with parameter λ∈(0,1), we have (24)y˙it=λFit,i=1,2,…,n,where Fi(t)(i=1,2,…,n) are given in (22).
Assume thaty(t)=(y1(t),y2(t),…,yn(t))∈X is a solution of system (24) for some parameter λ∈(0,1). By integrating system (24) over the interval [0,ω], we obtain (25)∫0ωrit-∑l=1maiiltexp⁡yit-τiilt-∑j≠in∑l=1maijlt∫-τijl0kijlsexp⁡yjt+sdsdt=0,i=1,2,…,n.Consequently, (26)∫0ω∑l=1maiiltexp⁡yit-τiiltdt+∫0ω∑j≠in∑l=1maijlt∫-τijl0kijlsexp⁡yjt+sdsdt=r-iω,i=1,2,…,n.Let siil(t)=t-τiil(t)(i=1,2,…,n,l=1,2,…,m); then from Lemma 1 and (H1) we get that function siil(t) has a unique ω periodic inverse function φiil(t); then for every i=1,2,…,n, l=1,2,…,m, we have (27)∫0ωaiiltexp⁡yit-τiiltdt=∫-τiil0ω-τiilωaiilφiilt1-τiil′φiiltexp⁡yitdt.One can see that (28)aiilφiilt1-τiil′φiilt≕δiilt,i=1,2,…,n,l=1,2,…,m,are ω periodic functions. Then for every i=1,2,…,n, l=1,2,…,m, we have (29)∫0ωaiiltexp⁡yit-τiiltdt=∫0ωδiiltexp⁡yitdt.For each i,j=1,2,…,n, and l=1,2,…,m, we have(30)∫0ωaijlt∫-τ0kijlsexp⁡yjt+sdsdt=∫-τ0∫0ωaijltkijlsexp⁡yjt+sdtds=∫-τ0∫ss+ωaijlv-skijlsexp⁡yjvdvds=∫-τ0∫0ωaijlv-skijlsexp⁡yjvdvds=∫0ω∫-τ0aijlv-skijlsexp⁡yjvdsdv=∫0ω∫-τ0aijlt-skijlsdsexp⁡yjtdt.From ((26)–(30)), we obtain(31)∑i=1n∫0ω∑l=1mδiiltexp⁡uit+∑j≠in∑l=1m∫-τ0aijlt-skijlsdsexp⁡ujtdt=∑i=1n∫0ω∑l=1mδiiltexp⁡uit+∑j≠in∑l=1m∫-τ0ajilt-skjilsdsexp⁡uitdt=∑i=1n∫0ω∑l=1mδiilt+∫-τ0∑j≠inajilt-skjilsdsexp⁡uitdt=∑i=1nr-iω.From the above equality we have (32)∫0ω∑l=1mδiilt+∫-τ0∑j≠inajilt-skjilsds·exp⁡uitdt≤∑i=1nr-iω,i=1,2,…,n.From (32) we can obtain (33)Ai∫0ωexp⁡yitdt≤∑i=1nr-iω,i=1,2,…,n,where (34)Ai=mint∈ω⁡∑l=1mδiilt+∑j≠in∫-τ0ajilt-skjilsds.Consequently, (35)∫0ωexp⁡yitdt≤∑i=1nr-iωAi,i=1,2,…,n.From the continuity of y(t)=(y1(t),y2(t),…,yn(t)), there exist constants ξi,ηi∈[0,ω](i=1,2,…,n) such that (36)yiξi=maxt∈0,ω⁡yit,yiηi=mint∈0,ω⁡yit,i=1,2,…,n.From (35) and (36), we further obtain (37)yiηi≤ln⁡∑i=1nr-iAi,i=1,2,…,n.From ((26)–(30)), we can obtain (38)∫0ω∑l=1mδiiltexp⁡yitdt+∑j≠in∫0ω∑l=1m∫-τijl0aijlt-skijlsdsexp⁡yjtdt=r-iω,i=1,2,…,n.From (38) we can obtain (39)∑l=1mσiilM∫0ωexp⁡yitdt≥r-iω-∑j≠in∑l=1maijlM∫0ωexp⁡yjtdt,i=1,2,…,n.From the assumptions of Theorem 3 and (35), (38), and (39) we further obtain (40)yiξi≥ln⁡Bi∑l=1mσiilM,i=1,2,…,n,where (41)Bi=mint∈ω⁡r-i-∑j≠in∑l=1maijlM∑i=1nr-iAj.On the other hand, directly from system (14) we have (42)∫0ωy˙itdt≤∫0ωritdt+∫0ω∑l=1maiiltexp⁡yit-τiilt+∑j≠in∑l=1maijlt∫-τijl0kijlsexp⁡yjt+sdsdt≤∫0ωritdt+∑l=1mδiilM∫0ωexp⁡yitdt+∑l=1m∑j≠inaijlM∫0ωexp⁡yjtdt≤∫0ωritdt+∑l=1mδiilM∑i=1nr-iωAi+∑l=1m∑j≠inaijlM∑i=1nr-iωAj≤r-iω+Ci,i=1,2,…,n,where (43)Ci=∑l=1mδiilM∑i=1nr-iωAi+∑j≠inaijlM∑i=1nr-iωAj,i=1,2,…,n.From (37), (40), and (42), we have for any t∈[0,ω](44)yit≤yiηi+∫0ωy˙itdt≤ln⁡∑i=1nr-iAi+r-iω+Ci≕Mi,i=1,2,…,n,(45)yit≥yiξi-∫0ωy˙itdt≥ln⁡Bi∑l=1mσiilM-r-iω-Ci≕Ni,i=1,2,…,n.Therefore, from (44) and (45), we have (46)maxt∈0,ω⁡yit≤max⁡Mi,Ni≕Bi,i=1,2,…,n.It can be seen that the constants Bi(i=1,2,…,n) are independent of parameter λ∈(0,1). For any y=(y1,y2,…,yn)∈Rn, from (18) we can obtain (47)QNy=QNy1,QNy2,…,QNyn,QNyi=r-i-a-iiexp⁡yi-∑j≠ina-ijexp⁡yj,i=1,2,…,n.We consider the following algebraic equation: (48)r-i-a-iivi-∑j≠ina-ijvj=0i=1,2,…,n.From the assumption of Theorem 3, the equation has a unique positive solution v∗=(v1∗,v2∗,…,vn∗). Hence, the equation QNy=0 has a unique solution y∗=(y1∗,y2∗,…,yn∗)=(lnv1∗,lnv2∗,…,lnvn∗)∈Rn.
Choosing constantB>0 large enough such that y1∗+y2∗+⋯+yn∗<B and B>B1+B2+⋯+Bn, we define a bounded open set Ω⊂X as follows: (49)Ω=y∈X:y<B.It is clear that Ω satisfies conditions (a) and (b) of Lemma 2. On the other hand, by directly calculating we can obtain (50)deg⁡JQN,Ω∩Ker⁡L,0,0,…,0=sgn⁡fy11fy21⋯fyn1fy12fy22⋯fyn2⋯⋯⋯⋯fy1nfy2n⋯fynn,where (51)fyji=-a-ijexp⁡yj∗,i=j,fyji=-a-ijexp⁡yj∗,i≠j,i,j=1,2,…,n.
From the assumption of Theorem3, we have (52)fy11fy21⋯fyn1fy12fy22⋯fyn2⋯⋯⋯⋯fy1nfy2n⋯fynn≠0.From this, we finally have (53)deg⁡JQN,Ω∩Ker⁡L,0,0,…,0≠0.This shows that Ω satisfies condition (c) of Lemma 2. Therefore, system (14) has a ω-periodic solution y∗(t)=(y1∗(t),y2∗(t),…,yn∗(t))∈Ω-. Finally, we have system (6) which has a positive ω-periodic solution. This completes the proof.Theorem 4.
Suppose that the conditions of Theorem3 hold and τiil(t)≡0(i,j=1,2,l=1,2,…,m); further, there exists a constant πi>0(i=1,2,…,n) such that (54)mint∈0,ω⁡∑l=1mπiaiilt-∑j≠inπj∫-τijl0aijlt-skijlsds≕δi>0,i=1,2,…,n;then system (6) has a positive ω-periodic solution which is globally attractive.Proof.
From Theorem3, we can obtain that system (6) has a positive periodic solution.
Letx∗(t)=(x1∗(t),x2∗(t),…,xn∗(t)) be a positive periodic solution of system (6). Choose positive constants mi>0, Mi>0 such that (55)mi≤xi∗t≤Mi,i=1,2,…,n.From the assumptions of Theorem 4, there exist constant β>0 such that for all t≥0 we have (56)δi≥β>0,i=1,2,…,n.Let (x1(t),x2(t),…,xn(t)) be any solution of system (6); we define Lyapunov functional as follows: (57)Vit=πiln⁡xi∗t-ln⁡xit+∑j≠in∑l=1mπj∫-τijl0kijls∫t+staijlθ-sxj∗θ-xjθdθds.Calculating the upper right derivation of Vi(t) along system (6) for i=1,2,…,n, we have (58)D+Vit=sign⁡xi∗t-xit-∑l=1mπiaiiltxi∗t-xit+∑j≠in∑l=1mπjaijlt·∫-τ0kijlsxj∗t+θ-xjt+θds+∑j≠in∑l=1mπj∫-τ0aijlt-skijlsdsxj∗t-xjt-∑j≠in∑l=1mπjaijlt·∫-τ0kijlsxj∗t+θ-xjt+θds≤-∑l=1mπiaiiltxi∗t-xit+∑j≠in∑l=1mπj∫-τ0aijlt-skijlsdsxj∗t-xjt.Further, we define a Lyapunov function as follows:(59)Vt=∑i=1nVit.Calculating the upper right derivation of V(t), from (58) we finally can obtain, for all t≥0, (60)D+Vt≤-∑i=1nδixi∗t-xit.Integrating both sides of (60) from 0 to t and by (56), we derive(61)Vt+β∫0t∑i=1nxi∗s-xisds≤V0,t≥0;then (62)∫0t∑i=1nxi∗s-xisds≤V0β,t≥0.By the definition of V(t) and (59) we have (63)∑i=1nμiln⁡xi∗t-ln⁡xit≤Vt≤V0,t≥0.Therefore, for i=1,2,…,n we have (64)μiln⁡xi∗t-ln⁡xit≤V0,t≥0,which, together with (55), lead to (65)miexp⁡-V0μi≤xit≤Miexp⁡V0μi,i=1,2,…,n,and hence ∑i=1nxi∗t-xit∈L1[0,+∞). From the boundedness of xi∗(t) and (64), it follows that xi(t)(i=1,2,…,n) are bounded for t≥0. It is obvious that both xi(t) and xi∗(t) satisfy the equations of system (9); then by system (9) and the boundedness of xi(t) and xi∗(t) we know that the derivatives x˙i(t) and x˙i∗(t) are bounded. Furthermore, we can obtain that x˙i∗(t)-x˙i(t)(i=1,2,…,n) and their derivatives remain bounded on [0,+∞). Therefore ∑i=1nxi∗t-xit is uniformly continuous on [0,+∞). Thus from (62), we have (66)limt→+∞⁡∑i=1nxi∗t-xit=0.Therefore, (67)limt→+∞⁡xi∗t-xit=0,i=1,2,…,n.This completes the proof of Theorem 4.Corollary 5.
Suppose that the conditions of Theorem4 hold; then system (6) is permanent.Proof.
From the global attractivity of bounded positive solutions, we can get that the permanence of system (6).


## 4. Applications

In this section, to illustrate the generality of our result, we apply Theorem3 to some particular Lotka-Volterra type competition systems with pure delays. Consider the following periodic n-species competition systems: (68)x˙it=xitrit-∑l=1m∑j=1naijltxjt-τijlt,i=1,2,…n.(69)x˙it=xit·rit-∑j=1n∑l=1maijlt∫-τijl0kijlsxjt+sds,i=1,2,…n.(70)x˙it=xitrit-∑l=1m∑j=1naijltxjt-τijl,i=1,2,…,n.For system ((68)–(70)), we assume the following:(H2) τijl(t)(l=1,2,…,m,i,j=1,2,…,n), ri(t)(i=1,2,…,n) are continuous ω-periodic functions with τijl′(t)<1 and ∫0ωri(t)dt>0. aijl(t)(i,j=1,2,l=1,2,…,m) are continuous, positive ω-periodic functions.(H3) ri(t)(i=1,2,…,n) are ω-periodic continuous functions with ∫0ωri(t)dt>0; aijl(t)(i,j=1,2,…,n;l=1,2,…,m) are positive ω-periodic continuous functions; kijl(s)(i,j=1,2,…,n;l=1,2,…,m) are nonnegative integrable functions on [-τijl,0](i,j=1,2,…,n;l=1,2,…,m) satisfying ∫-τijl0kijl(s)ds=1.(H4) τijl>0(l=1,2,…,m,i,j=1,2,…,n) are constants, ri(t)(i=1,2,…,n) are continuous ω-periodic functions with ∫0ωri(t)dt>0. aijl(t)(i,j=1,2,l=1,2,…,m) are continuous, positive ω-periodic functions. Thus from Theorem 3 we have the following.Corollary 6.
Suppose that assumption (H2) holds and there exists a constant ζi>0, i=1,2,…,n, such that (71)mint∈0,ω⁡r-iζi-∑j≠in∑l=1mδijlMζj∑i=1nr-iEj≕Di>0,i=1,2,…,n,where (72)mint∈0,ω⁡∑l=1mδiilt+∑j≠inδjilt≕Ei,i=1,2,…,n,δijlt=aijlφijlt1-τijl′φijlt,i,j=1,2,…,n,l=1,2,…,m,and the algebraic equation, (73)r-i-a-iivi-∑j≠ina-ijvj=0,i=1,2,…,n,has a unique positive solution. Then system (68) has at least one positive ω-periodic solution.Corollary 7.
Suppose that assumption (H3) holds and there exists a constant βi>0, i=1,2,…,n, such that (74)mint∈0,ω⁡r-iβi-∑j≠in∑l=1maijlMβj∑i=1nr-iKj≕Gi>0,i=1,2,…,n,where (75)mint∈0,ω⁡∑l=1m∫-τ0aiilt-skiils+∑j≠inajilt-skjilsds≕Ki,i=1,2,…,n,and the algebraic equation, (76)r-i-a-iivi-∑j≠ina-ijvj=0,i=1,2,…,n,has a unique positive solution. Then system (69) has at least one positive ω-periodic solution.Corollary 8.
Suppose that assumption (H4) holds and there exists a constant γi>0, i=1,2,…,n, such that (77)mint∈0,ω⁡r-iγi-∑j≠in∑l=1maijlMγj∑i=1nr-iLj≕Hi>0,i=1,2,…,n,where (78)mint∈0,ω⁡∑l=1maiilt+τiil+∑j≠inajilt+τjil≕Li,i=1,2,…,n,and the algebraic equation, (79)r-i-a-iivi-∑j≠ina-ijvj=0,i=1,2,…,n,has a unique positive solution. Then system (70) has at least one positive ω-periodic solution.


## 5. One Example

In this section, we will give one example to illustrate the results obtained in this paper. From the example, we will see that if the conditions of Theorem3 hold, then the system has a positive periodic solution. If the conditions of Theorem 4 hold, then the system has a positive periodic solution and which is globally attractive.Example. We consider the following periodic three-species competition systems:(80)x˙1t=x1t1.1+cos⁡t-6+sin⁡tx1t-τ111t-22+11sin⁡t20∫-τ1210k121sx2t+sds-22+11sin⁡t20∫-τ1310k131sx3t+sds,x˙2t=x2t1.1+sin⁡t-6+sin⁡tx2t-τ221t-22+11sin⁡t20∫-τ2110k211sx1t+sds-2+sin⁡t2∫-τ2310k231sx3t+sds,x˙3t=x3t1.1+cos⁡t-6+cos⁡tx3t-τ331t-27+9cos⁡t20∫-τ3110k311sx1t+sds-18+9cos⁡t20∫-τ3210k321sx2t+sds,where τ111(t)=τ221(t)=τ331(t)=0, n=3, m=1, ω=2π; by direct calculation we can get (81)B1=0.5,B2=0.65,B3=1.25,and the following system of equations has a unique positive solution (82)2v1-0.5v2-0.5v3=1.9802,2v2-0.5v1-0.4v3=1.9608,3v3-0.75v1-0.5v2=1.9417,and get (83)v1=1.7554,v2=1.6929,v3=1.3682.It is clear that all the conditions of Theorems 3 and 4 hold.From the Figure1. we can see, system (80) is permanent and has a positive periodic solution which is globally attractive.Figure 1
Dynamic behaviors of system (80).
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