# Global Positive Periodic Solutions of Generalizedn-Species Gilpin-Ayala Delayed Competition Systems with Impulses

Authors: Zhenguo Luo; Liping Luo; Jianhua Huang; Binxiang Dai
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

We consider the following generalizedn-species Lotka-Volterra type and Gilpin-Ayala type competition systems with multiple delays and impulses: xi′(t)=xi(t)[ai(t)-bi(t)xi(t)-∑j=1n‍cij(t)xjαij(t-ρij(t))-∑j=1n‍dij(t)xjβij(t-τij(t))-∑j=1n‍eij(t)∫-ηij0‍kij(s)xjγij(t+s)ds-∑j=1n‍fij(t)∫-θij0‍Kij(ξ)xiδij(t+ξ)xjσij(t+ξ)dξ],a.e, t>0, t≠tk; xi(tk+)-xi(tk-)=hikxi(tk), i=1,2,…,n, k∈Z+. By applying the Krasnoselskii fixed-point theorem in a cone of Banach space, we derive some verifiable necessary and sufficient conditions for the existence of positive periodic solutions of the previously mentioned. As applications, some special cases of the previous system are examined and some earlier results are extended and improved.


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## Body

## 1. Introduction

In the recent decades, the traditional Lotka-Volterra competition systems have been studied extensively. One of the models is the following competition system:(1)dxi(t)dt=xi(t)[bi(t)-∑j=1n‍aij(t)xj(t)],i=1,2,…,n.Many results concerned with the permanence, global asymptotic stability, and the existence of positive periodic solutions of system (1) are obtained; we refer to [1–10] and the reference therein. However, the Lotka-Volterra type models have often been severely criticized. One of the criticisms is that, in such a model, the per capita rate of change of the density of each species is a linear function of densities of the interacting species. In 1973, Ayala et al. [11] conducted experiments on fruit fly dynamics to test the validity of ten models of competitions. One of the models accounting best for the experimental results is given by
(2)x1′(t)=r1x1(t)[1-(x1(t)K1)θ1-α12x2(t)K2],x2′(t)=r2x2(t)[1-(x2(t)K2)θ2-α21x1(t)K1].In order to fit data in their experiments and to yield significantly more accurate results, Gilpin and Ayala [12] claimed that a slightly more complicated model was needed and proposed the following competition model:
(3)dxi(t)dt=rixi(t)[1-(xi(t)Ki)θi-∑j=1,j≠in‍aijxj(t)Kj],11111111111111111111111111111i=1,2,…,n,wherexiis the population density of the ith species,riis the intrinsic exponential growth rate of the ith species,Kiis the environmental carrying capacity of speciesiin the absence of competition,θiprovides a nonlinear measure of interspecific interference, andaijprovides a measure of interspecific interference. [13–15] obtained sufficient conditions which guarantee the global asymptotic stability of system (3). Chen [16] investigated the followingn-species nonautonomous Gilpin-Ayala competitive model:
(4)dxi(t)dt=xi(t)[bi(t)-∑j=1n‍aij(t)xjθij(t)],11111111111111111111111i=1,2,…,n.For eachr≤n, they established a series of criteria under whichrof the species of system (4) were permanent while the remainingn-rspecies were driven to extinction. In [17], Fan and Wang further studied the following delay Gilpin-Ayala type competition model:
(5)dxi(t)dt=xi(t)[bi(t)-∑j=1n‍aij(t)xjθij(t-τij(t))],111111111111111111111111111111i=1,2,…,n.They obtained a set of easily verifiable sufficient conditions for the existence of at least one positive periodic solution of the system (5) by applying the coincidence degree theory. Recently, in [18], Chen investigated the followingn-species Gilpin-Ayala type competition systems:
(6)yi′(t)=yi(t)[ri(t)-∑j=1n‍aij(t)xjαij(t)1111111111111-∑j=1n‍bij(t)xjβij(t-τij(t))1111111111111-∑j=1n‍∫-σij0‍cij(t,s)yjγij(t+s)ds],11111111111111111111111111111i=1,2,…,n.He established a series of criteria under whichrof the species in the system (6) were permanent while the remainingn-rspecies were driven to extinction. In [19], Xia et al. considered the following almost periodic nonlinearn-species competitive Lotka-Volterra model:
(7)yi′(t)=yi(t)[ri(t)-∑j=1N‍aij(t)xjαij(t)111111111111-∑j=1N‍bij(t)xjαij(t-τij(t))111111111111-∑j=1N‍cij(t)yiαii(t)yjαij(t)],11111111111111111111111i=1,2,…,n.By using comparison theorem and constructing suitable Lyapunov functional, they derived a set of sufficient conditions for the existence and global attractivity of a unique positive almost periodic solution of the previously mentioned model. Motivated by the previous ideas, in [20], Yan considered the following generalized periodicn-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form:
(8)yi′(t)=yi(t)[ri(t)-∑j=1n‍aij(t)xjαij(t)1111111111111-∑j=1n‍bij(t)xjβij(t-τij(t))1111111111111-∑j=1n‍cij(t)∫-σij0‍Kij(ξ)yiγij111111111111111111111111111×(t+ξ)yjδij(t+ξ)dξ∫-σij0‍],111111111111111111111111111111111111i=1,2,…,n.By using a fixed point theorem in cone and the proof by contradiction, he obtained a necessary and sufficient condition for the existence of positive periodic solutions (with strictly positive components) of the system (8).However, the ecological system is often deeply perturbed by human exploitation activities such as planting and harvesting, which makes it unsuitable to be considered continually. For having a more accurate description of such a system, we need to consider the impulsive differential equations. The theory of impulsive differential equations is not only richer than the corresponding theory of differential equations without impulses, but also represents a more natural framework for mathematical modeling of many real-world phenomena (see [21–23]). In recent years, some impulsive equations have been recently introduced in population dynamics in relation to population ecology; we refer the reader to [24–36] and the reference therein. However, to this day, only a little work has been done on the existence of positive periodic solutions to the generalized periodicn-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form and impulses. Motivated by this, in this paper, we mainly consider the followingn-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form and impulses:
(9)xi′(t)=xi(t)[×∫-θij0‍Kij(ξ)xiδij(t+ξ)xjσij(t+ξ)dξ]ai(t)-bi(t)xi(t)1111111111111-∑j=1n‍cij(t)xjαij(t-ρij(t))1111111111111-∑j=1n‍dij(t)xjβij(t-τij(t))-∑j=1n‍eij(t)1111111111111×∫-ηij0‍kij(s)xjγij(t+s)ds-∑j=1n‍fij(t)1111111111111×∫-θij0‍Kij(ξ)xiδij(t+ξ)xjσij(t+ξ)dξ],11111111111111111111111111111111a.e,t>0,t≠tk;xi(tk+)-xi(tk-)=hikxi(tk),i=1,2,…,n,k∈Z+,with initial conditions(10)xi(ϑ)=ϕi(ϑ),ϑ∈[-τ,0],ϕi(0)>0,ϕi∈C([-τ,0),[0,+∞)),i=1,2,…,n,wherexirepresents the density of the ith speciesXiat timet,ai(t)is the intrinsic growth rate of the ith speciesXiat this time,ai(t)may be negative whilea¯i=(1/ω)∫0ω‍ai(t)dt>0;cij(t),dij(t),eij(t)(i≠j)denote the competitive coefficient between the ith speciesXiand jth speciesXj, andai(t),bi(t),cij(t),dij(t),eij(t)are continuousω-periodic functions withbi(t)≥0,cij(t)≥0,dij(t)≥0,eij(t)≥0,fij(t)≥0,  ρij(t)≥0,τij(t)≥0,ηij≥0,θij≥0corresponding to the time delays withτ=maxi,j∈[1,n]{ρij(t),τij(t),ηij,θij}≥0;αij,βij,γij,δij,σijare positive constants. Assume thathik,i=1,2,…,n,k∈Z+={1,2,3,…}are constants and there exists an integerq>0such thattk+q=tk+ω,hi(k+q)=hik,i=1,2,…,n,where0<t1<t2<⋯<tq<ω.Throughout this paper, we make the following notation and assumptions:Cω={x∣x∈C(R,R),x(t+ω)=x(t)}, with the norm defined by|x|0=maxt∈[0,ω]|x(t)|;Cω1={x∣x∈C1(R,R),x(t+ω)=x(t)}, with the norm defined by∥x∥=maxt∈[0,ω]{|x|0,|x′|0};PC={x∣x:R→R+,lims→tx(s)=x(t),ift≠tk,limt→tk-x(t)=x(tk),limt→tk+x(t)exists,k∈Z+};PC1={x∣x:R→R+,x′∈PC};PCω={x∣x∈PC,x(t+ω)=x(t)}, with the norm defined by|x|0=maxt∈[0,ω]|x(t)|;PCω1={x∣x∈PC1,x(t+ω)=x(t)}, with the norm defined by∥x∥=maxt∈[0,ω]{|x|0,|x′|0}.The previously mentioned spaces are all Banach spaces. We also denote(11)g¯=1ω∫0ω‍g(t)dt,foranyg∈PCω,Δik=1+hik,i=1,2,…,n,k∈Z+,and make the following assumptions:(H1)  ai(t),bi(t),cij(t),dij(t),eij(t),fij(t),τij(t),ρij(t)∈PCω,i,j=1,2,…,n, andηij≥0,θij≥0.(H2)  {tk},k∈Z+satisfies0<t1<t2<⋯<tk<⋯andlimk→+∞tk=+∞.(H3)  {hik}is a real sequence withΔik=hik+1>0, and∏0<tk<t‍Δikis anω-periodic function.(H4)  kij∈C([-ηij,0],R+),Kij∈C([-θij,0],R+)are constants such that∫-ηij0‍kij(t)dt=1,∫-θij0‍Kij(t)dt=1.To conclude this section, we summarize in the following definition and lemma that will be needed in our arguments.Definition 1.
A functionxi:R→(0,+∞)is said to be a positive solution of the system (9) and (10), if the following conditions are satisfied: (a)
xi(t)is absolutely continuous on each(tk,tk+1).(b)
for eachk∈Z+,xi(tk+)andxi(tk-)exist, andxi(tk-)=xi(tk).(c)
xi(t)satisfies the first equation of the system (9) and (10) for almost everywhere (for short a.e.) in[0,∞]∖{tk}and satisfiesxi(tk+)=Δikxi(tk)fort=tk,k∈Z+.
Under the previously mentioned hypotheses (H1)–(H4), we consider the following nonimpulsive Lotka-Volterra competitive systems:
(12)dyi(t)dt=yi(t)[×∫-θij0‍Kij(ξ)yiδij(t+ξ)yjσij(t+ξ)dξ]ai(t)-Bi(t)yi(t)11111111111111-∑j=1n‍Cij(t)yiαij(t-ρij(t))11111111111111-∑j=1n‍Dij(t)yjβij(t-τij(t))-∑j=1n‍Eij(t)1111111111111×∫-ηij0‍kij(s)yjγij(t+s)ds-∑j=1n‍Fij(t)1111111111111×∫-θij0‍Kij(ξ)yiδij(t+ξ)yjσij(t+ξ)dξ],kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkii=1,2,…,n,
with initial conditions(13)yi(ζ)=φi(ζ),ζ∈[-τ,0],φi(0)>0,φi∈C([-τ,0),[0,+∞)),i=1,2,…,n,
where(14)Bi(t)=bi(t)∏0<tk<t‍(1+θik),Cij(t)=cij(t)∏0<tk<t-ρij(t)‍(1+θik)αij,Dij(t)=dij(t)∏0<tk<t-τij(t)‍(1+θik)βij,Eij(t)=eij(t)∏0<tk<t‍(1+θik)γij,Fij(t)=fij(t)∏0<tk<t‍(1+θik)δij+σij,i=j=1,2,…,n.
By a solutiony(t)=(y1(t),…,yn(t))Tof the system (12) and (13), it means an absolutely continuous functiony(t)=(y1(t),…,yn(t))Tdefined on[-τ,0]that satisfies (12) and (13).
The following lemma will be used in the proofs of our results. The proof of Lemma2 is similar to that of Theorem1in [24].Lemma 2.
Suppose that (H1)–(H4) hold. Then (i)
ifyi(t)(i=1,2,…,n)is a solution of (12) and (13) on[-τ,+∞), thenxi(t)=∏0<tk<t‍Δikyi(t)(i=1,2,…,n)is a solution of (9) and (10) on[-τ,+∞);(ii)
ifxi(t)(i=1,2,…,n)is a solution of (9) and (10) on[-τ,+∞), thenyi(t)=∏0<tk<t‍Δik-1xi(t)(i=1,2,…,n)is a solution of (12) and (13) on[-τ,+∞).Proof.
(i) It is easy to see thatxi(t)=∏0<tk<t‍Δikyi(t)(i=1,2,…,n)is absolutely continuous on every interval(tk,tk+1],t≠tk,k=1,2,…,(15)xi′(t)-xi(t)[ai(t)-bi(t)xi(t)-∑j=1n‍cij(t)xjαij(t-ρij(t))111111111111-∑j=1n‍dij(t)xjβij(t-τij(t))-∑j=1n‍eij(t)111111111111×∫-ηij0‍kij(v)xjγij(t+v)dv-∑j=1n‍fij(t)111111111111×∫-ξij0‍Kij(v)xiδij(t+v)xjσij11111111111111111×(t+v)dv[ai(t)-bi(t)xi(t)-∑j=1n‍cij(t)xjαij(t-ρij(t))]=∏0<tk<t‍Δikyi′(t)-∏0<tk<t‍Δikyi(t)×[ai(t)-bi(t)∏0<tk<t‍Δikyi(t)1111111-∑j=1n‍cij(t)∏0<tk<t-ρij(t)‍Δikαijyjαij(t-ρij(t))1111111-∑j=1n‍dij(t)∏0<tk<t-τij(t)‍Δikβijyjβij(t-τij(t))1111111-∑j=1n‍eij(t)∏0<tk<t‍Δikγij∫-ηij0‍kij(v)yjγij(t+v)dv1111111-∑j=1n‍fij(t)∏0<tk<t‍Δikδij+σij1111111×∫-ξij0‍Kij(v)yiδij(t+v)yjσij(t+v)dv[ai(t)-bi(t)∏0<tk<t‍Δikyi(t)]=∏0<tk<t‍Δik{×∫-ξij0‍Kij(v)yiδij(t+v)yjσij(t+v)dv[ai(t)-Bi(t)yi(t)-∑j=1n‍Cij(t)yiαij(t-ρij(t))]}yi′(t)-yi(t)1111111111111×[×∫-ξij0‍Kij(v)yiδij(t+v)yjσij(t+v)dv[ai(t)-Bi(t)yi(t)-∑j=1n‍Cij(t)yiαij(t-ρij(t))]ai(t)-Bi(t)yi(t)1111111111111111-∑j=1n‍Cij(t)yiαij(t-ρij(t))1111111111111111-∑j=1n‍Dij(t)yjβij(t-τij(t))-∑j=1n‍Eij(t)1111111111111111×∫-ηij0‍kij(u)yjγij(t+u)du-∑j=1n‍Fij(t)1111111111111111×∫-ξij0‍Kij(v)yiδij(t+v)yjσij1111111111111111111111×(t+v)dv[ai(t)-Bi(t)yi(t)-∑j=1n‍Cij(t)yiαij(t-ρij(t))]}=0.
On the other hand, for anyt=tk,k=1,2,…,(16)xi(tk+)=limt→tk+∏0<tj<t‍Δikyi(t)=∏0<tj≤tk‍Δikyi(tk),xi(tk)=∏0<tj<tk‍Δikyi(tk),i=1,2,…,n;
thus,(17)xi(tk+)=Δikxi(tk),i=1,2,…,n,k=1,2,….
It follows from (15)–(17) thatxi(t)(i=1,2,…,n)is a solution of the system (9) and (10). Similarly, ifyi(t)(i=1,2,…,n)is a solution of the system (12) and (13), we can prove thatxi(t)(i=1,2,…,n)is a solution of the system (9) and (10).
(ii) Sincexi(t)=∏0<tk<t‍Δikyi(t)is absolutely continuous on every interval(tk,tk+1],t≠tk,k=1,2,…,and in view of (16), it follows that for anyk=1,2,…,(18)yi(tk+)=∏0<tj≤tk‍Δik-1xi(tk+)=∏0<tj<tk‍Δik-1xi(tk)=yi(tk),yi(tk-)=∏0<tj<tk‍Δik-1xi(tk-)=∏0<tj≤tk-‍Δik-1xi(tk-)=yi(tk),
which implies thatyi(t)is continuous on[-τ,+∞). It is easy to prove thatyi(t)is absolutely continuous on[-τ,+∞). Similar to the proof of(i), we can check thatyi(t)=∏0<tk<t‍Δik-1xi(t)(i=1,2,…,n)are solutions of (8) on[-τ,+∞). Similarly, ifxi(t)(i=1,2,…,n)is a solution of the system (9) and (10), we can prove thatyi(t)(i=1,2,…,n)is a solution of the system (12) and (13). The proof of Lemma 2 is completed.In the following section, we only discuss the existence of a periodic solution for the system (12) and (13).The paper is organized as follows. In the next section, we give some definitions and lemmas. In Section3, we derive a necessary and sufficient condition ensuring at least one positive periodic solution of the system, by using the Krasnoselskii fixed-point theorem in the cone of Banach space. In Section 4, as applications, we consider some particular cases of the system which have been investigated extensively in the references mentioned previously.


## 2. Preliminaries

We will first make some preparations and list a few preliminary results. LetX={y=(y1(t),y2(t),…,yn(t))T∈C(R,Rn)∣yi(t+ω)=yi(t)}with the norm∥y∥=∑i=1n|yi|0,|yi|0=supt∈[0,ω]|yi(t)|.It is easy to verify that(X,∥·∥)is a Banach space.We define an operatorA:X→Xas follows:
(19)(Ay)(t)=((Ay)1(t),(Ay)2(t),…,(Ay)n(t))T,
where
(20)(Ay)i(t)=∫tt+ω‍{×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}Gi(t,v)yi(v)1111111111×[∑j=1n‍Bi(v)yi(v)11111111111111+∑j=1n‍Cij(v)yiαij(v-ρij(v))11111111111111+∑j=1n‍Dij(v)yjβij(v-τij(v))+∑j=1n‍Eij(v)1111111111111×∫-ηij0‍kij(s)yjγij(v+s)ds+∑j=1n‍Fij(v)1111111111111×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij1111111111111111111×(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}dv,i=1,2,…,n,Gi(t,v)=e-∫tv‍ai(u)du1-e-∫0ω‍ai(u)du=e∫vt+ω‍ai(u)due∫0ω‍ai(u)du-1,G(t,v)=diag[G1(t,v),G2(t,v),…,Gn(t,v)],1111111111111111111111111111t≤v≤t+ω.It is clear thatGi(t+ω,v+ω)=Gi(t,v),∂Gi(t,v)/∂t=ai(t)Gi(t,v),Gi(t,t)-Gi(t,t+ω)=1.In view of(H1), for1≤i≤n, we define
(21)αi:=min0≤t≤s≤ω|Gi(t,v)|=1e∫0ω‍ai(u)du-1,βi:=max0≤t≤s≤ω|Gi(t,v)|=e∫0ω‍ai(u)due∫0ω‍ai(u)du-1,α=min1≤i≤nαi,β=max1≤i≤nβi,σ=αβ∈(0,1).DefinePas a cone inXby
(22)P={y=(y1(t),y2(t),…,yn(t))T∈X:yi(t)11111≥σ|yi|0,t∈[0,ω]{y=(y1(t),y2(t),…,yn(t))T∈X:yi(t)}.We easily verify thatPis a cone inX. For convenience of expressions, we define an operatorT:P→Xby
(23)(Ty)i(t)=yi(t)[Bi(t)yi(t)+∑j=1n‍Cij(t)yiαij(t-ρij(t))111111111111111+∑j=1n‍Dij(t)yjβij(t-τij(t))+∑j=1n‍Eij(t)111111111111111×∫-ηij0‍kij(s)yjγij(t+s)ds+∑j=1n‍Fij(t)111111111111111×∫-θij0‍Kij(ξ)yiδij(t+ξ)yjσij(t+ξ)dξ[Bi(t)yi(t)+∑j=1n‍Cij(t)yiαij(t-ρij(t))],1111111111111111111111111111111111111i=1,2,…,n.The proof of the main result in this paper is based on an application of the Krasnoselskii fixed-point theorem in cones. Firstly, we need to introduce some definitions and lemmas.Definition 3.
LetXbe a real Banach space andPbe a closed, nonempty subset ofX.Pis said to be a cone if (1)
αx+βy∈Pfor allx,y∈P, andα,β>0,(2)
x,-x∈Pimplyx=0.Lemma 4 (see [37–39]).
LetPbe a cone in a real Banach spaceX. Assume thatΩr1andΩr2are open subsets ofXwith0∈Ωr1⊂Ωr1¯⊂Ωr2,whereΩri={x∈X:∥x∥<ri},i=1,2.LetT:P∩(Ωr2∖Ωr1¯)→Pbe a continuous and completely continuous operator satisfying(1)
∥Tx∥≤∥x∥, for anyx∈P∩∂Ωr1;(2)
The fact that there existsϕ∈P∖{0}such thaty≠Ty+λy, for anyy∈P∩∂Ωr2andλ>0.
Then,Thas a fixed point inP∩(Ωr2¯∖Ωr1).The same conclusion remains valid if (1) holds for for anyx∈P∩∂Ωr2and (2) holds for anyy∈P∩∂Ωr1andλ>0.Lemma 5.
Assume that (H1)–(H4) hold. Then the solutions of the system (12) and (13) are defined on[-τ,∞)and are positive.Proof.
By Lemma2, we only need to prove that the solutionsyi(t)(i=1,2,…,n)of (12) and (13) are defined on[-τ,∞)and are positive on[0,∞). From (12), we have that for anyφi∈C([-τ,0),R+)(i=1,2,3,…,n)andt>0(24)yi(t)=φi(0)exp{∫0t‍[×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij(v+ξ)dξ]ai(v)-Bi(v)yi(v)11111111111111111-∑j=1n‍Cij(v)yiαij(v-ρij(v))11111111111111111-∑j=1n‍Dij(v)yjβij(v-τij(v))-∑j=1n‍Eij(v)11111111111111111×∫-ηij0‍kij(s)yjγij(v+s)ds-∑j=1n‍Fij(v)11111111111111111×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij11111111111111111111111×(v+ξ)dξ∫-θij0‍]dv},111111111111111111111111111i=1,2,…,n.
Therefore,yi(t)(i=1,2,…,n)are defined on[-τ,∞)and are positive on[0,∞). The proof of Lemma 5 is complete.Lemma 6.
Assume that (H1)–(H4) hold. ThenA:P→Pis well defined.Proof.
In view of the definitions ofPandT, for anyy∈P, we have
(25)(Ay)i(t)=∫tt+ω‍Gi(t,v)(Ty)i(v)dv,(Ay)i(t+ω)=∫t+ωt+2ω‍Gi(t,v+ω)(Ty)i(v+ω)dv(Ay)i(t+ω)=∫tt+ω‍Gi(t,v)(Ty)i(v)dv=(Ay)i(t).
Therefore,(Ay)∈X. Furthermore, for anyy∈P, it follows from (20) that
(26)|(Ay)i|0≤βi∫0ω‍(Ty)i(v)dv.
On the other hand, for anyy∈P, we obtain
(27)(Ay)i(t)≥αi∫0ω‍(Ty)i(v)dv(Ay)i(t)≥αiβi|(Ay)i|0≥σ|(Ay)i|0.
Therefore,Ay∈P.The proof of Lemma 6 is complete.Lemma 7.
The operatorA:P→Pis continuous and completely continuous.Proof.
By using a standard argument one can show thatψis continuous onP. Now, we show thatAis completely continuous. Letrbe any positive constant andSr={y∈X:|yi|0≤r}a bounded set. For anyy∈Sr, by (20), we have
(28)|(Ay)i|0≤βi∫0ω‍(Ty)i(v)dv=βi∫0ω‍yi(v)[×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij(v+ξ)dξ]Bi(v)yi(v)111111111111111+∑j=1n‍Cij(v)yiαij(v-ρij(v))111111111111111+∑j=1n‍Dij(v)yjβij(v-τij(v))+∑j=1n‍Eij(v)111111111111111×∫-ηij0‍kij(s)yjγij(v+s)ds+∑j=1n‍Fij(v)111111111111111×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij1111111111111111111111×(v+ξ)dξ∫-θij0‍]dv≤ωβir[B¯ir+∑j=1n‍(C¯ijrαij+D¯ijrβij+E¯ijrγij1111111111111111111+F¯ijrδij+σij)[B¯ir+∑j=1n‍(C¯ijrαij+D¯ijrβij+E¯ijrγij]∶=Ri.
Therefore, for anyy∈Sr, we obtain
(29)∥Ay∥=∑i=1n‍|(Ay)i|0≤∑i=1n‍Ri∶=R,
which implies thatA(Sr)is a uniformly bounded set. On the other hand, in view of the definitions ofAandT, we have
(30)d[(Ay)i(t)]dt=Gi(t,t+ω)(Ty)i(t+ω)-Gi(t,t)(Ty)i(t)d[(Ay)i(t)]dt=+∫tt+ω‍dGi(t,v)dt(Ty)i(v)dvd[(Ay)i(t)]dt=-(Ty)i(t)+ai(t)∫tt+ω‍Gi(t,v)(Ty)i(v)dvd[(Ay)i(t)]dt=ai(t)(Ay)i(t)-(Ty)i(t).
Again, from (20), we obtain
(31)|d[(Ay)i(t)]dt|≤aiMRi+r|d[(Ay)i(t)]dt|=×[B¯ir+∑j=1n‍(C¯ijrαij+D¯ijrβij1111111111111111111111111+E¯ijrγij+F¯ijrδij+σij)[B¯ir+∑j=1n‍(C¯ijrαij+D¯ijrβij]|d[(Ay)i(t)]dt|:=Ri¯≤M:=maxi∈[1,n]{Ri¯},
which implies thatd[(Ay)i(t)]/dt, for anyy∈Sr, is also uniformly bounded. Hence,A(Sr)⊂Xis a family of uniformly bounded and equi-continuous functions. By the well-known Ascoli-Arzela theorem, we know that the operatorψis completely continuous. The proof of Lemma 7 is complete.Lemma 8.
Assume that (H1)–(H4) hold. The existence of positiveω-periodic solution of the system (12) and (13) is equivalent to that of nonzero fixed point ofAinP.Proof.
Assume thaty=(y1(t),y2(t),…,yn(t))T∈Xis a periodic solution of (12) and (13). Then, we have
(32)[yi(t)e-∫0t‍ai(u)du]=-e-∫0t‍ai(u)duyi(t)11111×[Bi(t)yi(t)+∑j=1n‍Cij(t)yiαij(t-ρij(t))111111111+∑j=1n‍Dij(t)yjβij(t-τij(t))+∑j=1n‍Eij(t)111111111×∫-ηij0‍kij(s)yjγij(t+s)ds+∑j=1n‍Fij(t)111111111×∫-θij0‍Kij(ξ)yiδij(t+ξ)yjσij111111111111111×(t+ξ)dξ[Bi(t)yi(t)+∑j=1n‍Cij(t)yiαij(t-ρij(t))],i=1,2,…,n.
Integrating the previous equation over[t,t+ω], we can have
(33)[yi(v)e-∫0v‍ai(u)du][yi(v)e-∫0v‍ai(u)du]|tt+ω=-∫tt+ω‍{×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}e-∫0v‍ai(u)duyi(v)111111111111×[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))111111111111111+∑j=1n‍Dij(v)yjβij(v-τij(v))+∑j=1n‍Eij(v)111111111111111×∫-ηij0‍kij(s)yjγij(v+s)ds+∑j=1n‍Fij(v)111111111111111×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij11111111111111111111×(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}dv,1111111111111111111111111i=1,2,…,n.
Therefore, we have(34)yi(t)e-∫0t‍ai(u)du[1-e-∫0ω‍ai(u)du]=∫tt+ω‍{×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}e-∫0v‍ai(u)duyi(v)1111111111×[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))1111111111111+∑j=1n‍Dij(v)yjβij(v-τij(v))+∑j=1n‍Eij(v)1111111111111×∫-ηij0‍kij(s)yjγij(v+s)ds+∑j=1n‍Fij(v)1111111111111×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij1111111111111111111×(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}dv,111111111111111111111111i=1,2,…,n,
which can be transformed into(35)yi(t)=∫tt+ω‍{×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}e-∫0v‍ai(u)due-∫0t‍ai(u)du[1-e-∫0ω‍ai(u)du]yi(v)111111111111×[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))111111111111111+∑j=1n‍Dij(v)yjβij(v-τij(v))+∑j=1n‍Eij(v)111111111111111×∫-ηij0‍kij(s)yjγij(v+s)ds+∑j=1n‍Fij(v)111111111111111×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij111111111111111111111×(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}dv=∫tt+ω‍{×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}Gi(t,v)yi(v)1111111111×[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))1111111111111+∑j=1n‍Dij(v)yjβij(v-τij(v))+∑j=1n‍Eij(v)1111111111111×∫-ηij0‍kij(s)yjγij(v+s)ds+∑j=1n‍Fij(v)1111111111111×∫-θij0‍Kij(ξ)yiδij(v+ξ)yjσij1111111111111111111×(v+ξ)dξ[Bi(v)yi(v)+∑j=1n‍Cij(v)yiαij(v-ρij(v))]}dv=∫tt+ω‍Gi(t,v)(Ty)i(v)dv,i=1,2,…,n.
Thus,yi(t)(i=1,2,…,n)is a periodic solution for system (12) and (13).
Ify=(y1(t),y2(t),…,yn(t))T∈X, andAy=((Ay)1,(Ay)2,…,(Ay)n)T=ywithy≠0, then for anyt=tkderivative the two sides of (20) aboutt,
(36)d[(Ay)i(t)]dt=Gi(t,t+ω)(Ty)i(t+ω)-Gi(t,t)(Ty)i(t)111111+∫tt+ω‍dGi(t,v)dt(Ty)i(v)dv=-(Ty)i(t)+ai(t)∫tt+ω‍Gi(t,v)(Ty)i(v)dv=ai(t)yi(t)-(Ty)i(t)=yi(t)[×∫-θij0‍Kij(ξ)yiδij(t+ξ)yjσij(t+ξ)dξ]ai(t)-Bi(t)yi(t)111111111111-∑j=1n‍Cij(t)yiαij(t-ρij(t))111111111111-∑j=1n‍Dij(t)yjβij(t-τij(t))-∑j=1n‍Eij(t)111111111111×∫-ηij0‍kij(s)yjγij(t+s)ds-∑j=1n‍Fij(t)111111111111×∫-θij0‍Kij(ξ)yiδij(t+ξ)yjσij(t+ξ)dξ]=dyi(t)dt.
Hence,y(t)=(y1(t),y2(t),…,yn(t))T∈Xis a positiveω-periodic solution of (12) and (13). Thus we complete the proof of Lemma 8.


## 3. Existence of Periodic Solution of the System

Now, we are at the position to study the existence of positive periodic solutions of system (9) and (10). We mainly apply the Krasnoselskii fixed-point theorem in the cone of Banach space under some conditions to prove the main Theorem 9.Theorem 9.
Assume (H1)–(H4). System (9) and (10) has at least one positiveω-periodic solution if and only if the condition
(37)l0=min1≤i≤n{B¯i+∑j=1n‍(C¯ij+D¯ij+E¯ij+F¯ij)}>0
holds.Proof (sufficiency).
Let(38)L0=maxi∈[1,n]{b¯i+∑j=1n‍(C¯ij+D¯ij+E¯ij+F¯ij)},
by condition (37), we know thatL0≥l0>0. Choose a constantL≥L0such that1/ωβiL<1. Letr=1/ωβiLand
(39)Ωr={y(t)=(y1(t),y2(t),…,yn(t))T∈X:|yi|0111111111<r,i=1,2,…,n{y(t)=(y1(t),y2(t),…,yn(t))T∈X:|yi|0}.
For anyy=y(t)∈P∩∂Ωr,σ|yi|0≤yi(t)≤|yi|0, from (20), we obtain
(40)|(Ay)i|0≤βi∫0ω‍(Ty)i(t)dt=βi∫0ω‍yi(t)[Bi(t)yi(t)+∑j=1n‍Cij(t)yiαij(t-ρij(t))111111111111111+∑j=1n‍Dij(t)yjβij(t-τij(t))+∑j=1n‍Eij(t)111111111111111×∫-ηij0‍kij(s)yjγij(t+s)ds+∑j=1n‍Fij(t)111111111111111×∫-θij0‍Kij(ξ)yiδij(t+ξ)yjσij(t+ξ)dξ∑j=1n‍]dt≤ωβi|yi|0[B¯ir+∑j=1n‍(C¯ijrαij+D¯ijrβij111111111111+E¯ijrγij+F¯ijrδij+σij)∑j=1n‍]≤ωβi|yi|0[B¯i+∑j=1n‍(C¯ij+D¯ij+E¯ij+F¯ij)]r≤ωβi|yi|0L0r≤|yi|0.
Hence, for anyy=y(t)∈P∩∂Ωr,σ|yi|0≤yi(t)≤|yi|0, we have
(41)∥Ay∥=∑j=1n‍|(Ay)i|0≤∑j=1n‍|yi|0=∥y∥,
which implies that condition (1) in Lemma 4 is satisfied.
On the other hand, we choose0<l≤l0such that1/ωσαil>1. LetR=1/ωσαil>1and supposeu=(u1,u2,…,un)T∈P/{0}. We show that, for anyy=y(t)∈P∩∂ΩRandλ>0,y≠ψy+λu. Otherwise, there existy0=y0(t)∈P∩∂ΩRandλ0>0, such thaty0=ψy0+λ0u. Letui0≠0(1≤i0≤n); sinceyi0(t)≥σ|yi0|0, it follows that
(42)yi0=(Ay)i0(t)+λ0ui0yi0=∫tt+ω‍Gi0(t,v)(Ty)i0(v)dv+λ0ui0yi0≥σαi|yi0|0∫0ω‍[∫-θij0‍Bi(t)yi(t)1111111111111111+∑j=1n‍Cij(t)yiαij(t-ρij(t))1111111111111111+∑j=1n‍Dij(t)yjβij(t-τij(t))+∑j=1n‍Eij(t)1111111111111111×∫-ηij0‍kij(s)yjγij(t+s)ds+∑j=1n‍Fij(t)1111111111111111×∫-θij0‍Kij(ξ)yiδij(t+ξ)yjσij1111111111111111111111×(t+ξ)dξ∫-θij0‍]dt+λ0ui0yi0≥ωσαi|yi0|0[B¯iR+∑j=1n‍(C¯ijRαij+D¯ijRβij11111111111111+E¯ijRγij+F¯ijRδij+σij)∑j=1n‍]yi0≥ωσαi|yi|0[B¯i+∑j=1n‍(C¯ij+D¯ij+E¯ij+F¯ij)]R+λ0ui0yi0≥ωσαil0R|yi|0+λ0ui0≥|yi0|0+λ0ui0>|yi0|0,
which is a contradiction. This proves that condition (2) in Lemma 4 is also satisfied. By Lemmas 4 and 8, system (12) and (13) has at least one positive omega-periodic solution. From Lemma 2, system (9) and (10) has at least one positiveω-periodic solution.
(Necessity). Suppose that (37) does not hold. Then, there exists at least ani0(1≤i0≤n)such that
(43)B¯i0=0,C¯i0j=D¯i0j=E¯i0j=F¯i0j=0,j∈[1,n].
If system (12) and (13) has a positiveω-periodic solutiony(t)=(y1(t),y2(t),…,yn(t))T, then we have
(44)dyi0(t)dt=ai0(t)yi0(t).
Integrating the previous equation over[t,t+ω],we can have
(45)0=lnyi0(t+ω)yi0(t)=∫0ω‍ai0(t)dt>0,
which is a contradiction. The proof of Theorem9 is complete.


## 4. Applications

In this section, we apply the results obtained in the previous section to somen-species competition systems which are mentioned in Section 1. If we consider the environmental or biological factors, the assumption of the periodic oscillation of the parameters and impulse functions seems realistic and reasonable in view of any seasonal phenomena which they might be subject to, for example, mating habits, availability of food, weather conditions, and so forth.Application 1. We consider the following three classes of periodicn-species competition systems with impulses:
(46)dxi(t)dt=xi(t)[bi(t)-∑j=1n‍aij(t)xj(t)],t∈R,11111111111111111111111t≠tk,i=1,2,…,n,Δxi(tk)=hikxi(tk),k∈Z+,i=1,2,…,n;(47)dxi(t)dt=xi(t)[bi(t)-∑j=1n‍aij(t)xjθij(t)],t∈R,111111111111111111111111t≠tk,i=1,2,…,n,Δxi(tk)=hikxi(tk),k∈Z+,i=1,2,…,n;(48)dxi(t)dt=xi(t)[bi(t)-∑j=1n‍aij(t)xjθij(t-τij(t))],t∈R,1111111111111111111111111111111t≠tk,i=1,2,…,n,Δxi(tk)=hikxi(tk),k∈Z+,i=1,2,…,n,which are special cases of system (9), wherebi(t),aij(t),τij(t),θij,hikare the same as in (H1)–(H4). We denote
(49)Aij(t)=aij(t)∏0<tk<t‍(1+hik),A*ij(t)=aij(t)∏0<tk<t-τij(t)‍(1+hik),from Theorem9, we have the following results.Corollary 10.
Assume that (H1)–(H4) hold. The systems (46) and (47) have at least one positiveω-periodic solution if and only if the following condition
(50)min1≤i≤n{A¯ij}>0
holds.Corollary 11.
Assume that (H1)–(H4) hold. The system (48) has at least one positiveω-periodic solution if and only if the following condition
(51)min1≤i≤n{A*¯ij}>0
holds.Application 2. We consider the followingn-species Gilpin-Ayala type competition systems with impulses:
(52)yi′(t)=yi(t)[ri(t)-∑j=1n‍aij(t)yjαij(t)1111111111111-∑j=1n‍bij(t)yjβij(t-τij(t))-∑j=1n‍cij(t)1111111111111×∫-σij0‍kij(s)yjγij(t+s)ds∑j=1n‍],111111111111111a.e,t>0,t≠tk,i=1,2,…,n,yi(tk+)-yi(tk-)=hikyi(tk),i=1,2,…,n,k∈Z+,which is a special case of system (9), whereri(t),aij(t),bij(t),cij(t),τij(t)∈C(R,R+)areω-periodic,kij∈C([-ηij,0],R+)are constants such that∫-ηij0‍kij(t)dt=1, andαij≥0,βij≥0,γij≥0,i=1,2,…,n. We denote
(53)Aij(t)=aij(t)∏0<tk<t‍(1+hik),Bij(t)=bij(t)∏0<tk<t-τij(t)‍(1+hik),Cij(t)=cij(t)∏0<tk<t-σij‍(1+hik),from Theorem9, we have the following result.Corollary 12.
Assume that (H1)–(H4) and the condition
(54)R2=min1≤i≤n{∑j=1n‍(A¯ij+B¯ij+C¯ij)}>0
hold, the system (52) has at least one positiveω-periodic solution.Application 3. We study the following periodic nonlinearn-species competitive Lotka-Volterra models with impulses:
(55)yi′(t)=yi(t)[ri(t)-∑j=1N‍aij(t)yjαij(t)1111111111111-∑j=1N‍bij(t)yjβij(t-τij(t))1111111111111-∑j=1N‍cij(t)yiγij(t)yjδij(t)],11111111111a.e,t>0,t≠tk,i=1,2,…,N,yi(tk+)-yi(tk-)=hikyi(tk),i=1,2,…,N,k∈Z+,which is a special case of system (9), whereri(t),aij(t),bij(t),cij(t),τij(t)∈C(R,R+)areω-periodic, andαij≥0,βij≥0,γij≥0,δij≥0. We denote
(56)Aij(t)=aij(t)∏0<tk<t‍(1+hik)αij,Bij(t)=bij(t)∏0<tk<t-τij(t)‍(1+hik)βij,Cij(t)=cij(t)∏0<tk<t‍(1+hik)γij+δij,from Theorem9, we have the following result.Corollary 13.
Assume that (H1)–(H4) and the condition
(57)R3=min1≤i≤N{∑j=1N‍(A¯ij+B¯ij+C¯ij)}>0
hold, the system (55) has at least one positiveω-periodic solution.Application 4. We study the following generalized periodicn-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form and impulses:
(58)yi′(t)=yi(t)[ri(t)-∑j=1n‍aij(t)yjαij(t)111111111111-∑j=1n‍bij(t)yjβij(t-τij(t))-∑j=1n‍cij(t)111111111111×∫-σij0‍Kij(ξ)yiγij(t+ξ)yjδij111111111111×(t+ξ)dξ∑j=1n‍],t≠tk,Δyi(tk)=θky(tk),i=1,2,…,n,k∈Z+,t=tk,which is a special case of system (9), whereri(t),aij(t),bij(t),cij(t),τij(t)∈C(R,R+)areω-periodic,K
i
j
∈
C
(
[
-
σ
i
j
,
0
]
,
R
+
)are constants such that∫-σij0‍Kij(t)dt=1, andαij≥0,βij≥0,γij≥0,δij≥0,i=1,2,…,n,1+Hik>0. We denote
(59)Aij(t)=aij(t)∏0<tk<t‍(1+hik)αij,Bij(t)=bij(t)∏0<tk<t-τij(t)‍(1+hik)βij,Cij(t)=cij(t)∏0<tk<t-σij(t)‍(1+hik)γij+δij,from Theorem9, we have the following result.Corollary 14.
Assume that (H1)–(H4) and the condition
(60)R4=min1≤i≤2{A¯i+∑i=1n‍(B¯ij+C¯ij)}>0
hold, the system (58) has at least one positiveω-periodic solution.Application 5. We study the following two-species competitive systems with impulses:
(61)y1′(t)=y1(t)[∑j=1n‍r1(t)-a1(t)y1(t)11111111111111-∑j=1n‍b1j(t)y1α1j(t-τj(t))11111111111111-∑j=1n‍c1j(t)y2β2j(t-ρj(t))],y2′(t)=y2(t)[∑j=1n‍r2(t)-a2(t)y2(t)11111111111111-∑j=1n‍b2j(t)y2α2j(t-ηj(t))11111111111111-∑j=1n‍c2j(t)y1β1j(t-σj(t))],t≠tk,Δyi(tk)=hikyi(tk),i=1,2,k∈Z+,t=tk,which is a special case of system (9), whereri(t),aij(t),bij(t),cij(t),τj(t),ρj(t),ηj(t),σj(t)∈C(R,R+)(i=1,2)areω-periodic. We denote
(62)Ai(t)=ai(t)∏0<tk<t‍(1+θik),Bij(t)=bij(t)∏0<tk<t-τj(t)‍(1+θik)αij(i=1,2),C1j(t)=C1j(t)∏0<tk<t-ρj(t)‍(1+θik)β2jC2j(t)C1j(t)=c2j(t)∏0<tk<t-σj(t)‍(1+θik)β1j,from Theorem9, we have the following result.Corollary 15.
Assume that (H1)–(H4) and the condition
(63)R4=min1≤i≤2{A¯i+∑i=1n‍(B¯ij+C¯ij)}>0
hold, the system (61) has at least one positiveω-periodic solution.Remark 16.
We apply the main results obtained in the previous section to study some examples which have some biological implications; from the previous corollaries, we see that, under the appropriate conditions, the impulsive perturbations do not affect the existence of periodic solution of systems. However, if the impulsive perturbations are unbounded, some properties of the solution of systems could be changed significantly, which will be our further work.



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*Source: 617824-2013-11-26.xml*