# Global Positive Periodic Solutions of Generalizedn-Species Competition Systems with Multiple Delays and Impulses

Authors: Zhenguo Luo; Liping Luo
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

By applying the fixed point theorem in a cone of Banach space, we obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of two kinds of generalizedn-species competition systems with multiple delays and impulses as follows:
xi′(t)=xi(t)[ai(t)-bi(t)xi(t)-∑j=1ncij(t)xi(t-τij(t))-∑j=1ndij(t)xj(t-γij(t))-∑j=1neij(t)∫-σij0‍fij(s)xj(t+s)ds],a.e.,t>0,t≠tk,k∈Z+,i=1,2,…,n;xi(tk+)-xi(tk-)=θikxi(tk),i=1,2,…,n,k∈Z+;andxi′(t)=xi(t)[ai(t)-bi(t)xi(t)+∑j=1ncij(t)xi(t-τij(t))-∑j=1ndij(t)xj(t-γij(t))-∑j=1neij(t)∫-σij0‍fij(s)xj(t+s)ds],a.e.,t>0,t≠tk,k∈Z+,i=1,2,…,n;xi(tk+)-xi(tk-)=θikxi(tk),i=1,2,…,n,k∈Z+. It improves and generalizes a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.


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## Body

## 1. Introduction

LetR=(-∞,+∞), R+=[0,+∞), R-=(-∞,0], Rn={(x1,…,xn)T:xi≥0,1≤i≤n}, J⊂R, Z+={1,2,3,…}, respectively, and let ω>0 be a constant and 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}; gM=maxt∈[0,ω]{g(t)},gL=mint∈[0,ω]{g(t)}, for any g(t)∈PCω.The theory of impulsive delay differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of nonimpulsive delay differential equations. Many evolution processes in nature are characterized by the fact that at certain moments of time they experience abrupt change of state. That was the reason for the development of the theory of impulsive differential equations and impulsive delay differential equations, see the monographs [1–4]. In recent years, the existence theory of positive periodic solutions of delay differential equations with impulsive effects or without impulsive effects has been an object of active research, which is referred to as [5–12]. However, only a little work has been done on the existence of positive periodic solutions to the high-dimension impulsive differential equations based on the theory in cones. Motivated by this, in this paper, we mainly consider the following two classes of n-species Lotka-Volterra competitive systems with multiple delays and impulses:
(1)xi′(t)=xi(t)[ai(t)-bi(t)xi(t)-∑j=1ncij(t)xi(t-τij(t))+=xi(t)-∑j=1n‍dij(t)xj(t-γij(t))=xi(t)+-∑j=1n‍eij(t)∫-σij0‍fij(s)xj(t+s)ds],++,,′a.e.,t>0,t≠tk,k∈Z+,i=1,2,…,n,xi(tk+)-xi(tk-)=θikxi(tk),i=1,2,…,n;k∈Z+;(2)xi′(t)=xi(t)[ai(t)-bi(t)xi(t)+∑j=1ncij(t)xi(t-τij(t))=xi(t)+-∑j=1n‍dij(t)xj(t-γij(t))=xi(t)+-∑j=1n‍eij(t)∫-σij0‍fij(s)xj(t+s)ds],++,,′a.e.,t>0,t≠tk,k∈Z+,i=1,2,…,n,xi(tk+)-xi(tk-)=θikxi(tk),i=1,2,…,n;k∈Z+,
where ai(t),bi(t),cij(t),dij(t),eij(t),τij(t),γij(t)∈PCω, i,j=1,2,…,n, with ai¯=(1/ω)∫0ω‍ai(t)dt>0. Moreover, fij∈PC,σij>0 is a constant, and ∫-σij0‍fij(s)ds=1. Furthermore, Δxi(tk)=xi(tk+)-xi(tk), i=1,2,…,n (here x(tk+) represents the right limit of x(t) at the point tk); θk=(θ1k,θ2k,…,θnk)∈C(R+n,R-n); that is, xi(t) changes decreasingly suddenly at times tk. We assume that there exists an integer p>0 such that tk+p=tk+ω, θi(k+p)=θik, i=1,2,…,n, where 0<t1<t2<⋯<tp<ω.References [13, 14], G. Seifert investigated the following periodic single-species population growth models with periodic delay:
(3)y′(t)=y(t)[r(t)-a(t)y(t)-b(t)y(t-τ(t))].
They had assumed that the net birth r(t), the self-inhibition rate a(t), and the delay τ(t) are continuously differentiable ω-periodic functions, and r(t)>0, a(t)>0, b(t)≥0, τ(t)≥0, for t∈R. The negative feedback term -b(t)y(t-τ(t)) in the average growth rate of species has a negative time delay (the sign of the time delay term is negative), which can be regarded as the deleterious effect of time delay on a species growth rate. They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (3). But the discussion of global attractivity is only confined to the special case when the periodic delay is constant.In [15], Freedman and Wu proposed the following periodic single-species population growth models with periodic delay:
(4)y′(t)=y(t)[r(t)-a(t)y(t)+b(t)y(t-τ(t))].
They had assumed that the net birth r(t), the self-inhibition rate a(t), and the delay τ(t) are continuously differentiable ω-periodic functions, and r(t)>0, a(t)>0, b(t)≥0, τ(t)≥0 for t∈R. The positive feedback term b(t)y(t-τ(t)) in the average growth rate of species has a positive time delay (the sign of the time delay term is positive), which is a delay due to gestation. They had established sufficient conditions which guarantee that system (4) has a positive periodic solution which is globally asymptotically stable.References [16, 17] have studied the following two-species competitive system without delay:
(5)y1′(t)=y1(t)[r1(t)-a1(t)y1(t)-c1(t)y2(t)],y2′(t)=y2(t)[r2(t)-a2(t)y2(t)-c2(t)y1(t)].
They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (5) by using differential inequalities and topological degree, respectively. In fact, in many practical situations the time delay occurs so often. A more realistic model should include some of the past states of the system. Motivated by the previous ideas, Liu et al. [18] considered two corresponding periodic Lotka-Volterra competitive systems involving multiple delays:
(6)y1′(t)=y1(t)[(t-ρj(t))∑j=1mc1j(t)y2(t-ρj(t))r1(t)-a1(t)y1(t)+∑i=1nb1i(t)y1(t-τi(t))=y1(t)+′-∑j=1mc1j(t)y2(t-ρj(t))],y2′(t)=y2(t)[r2(t)-a2(t)y2(t)+∑j=1mb2j(t)y2(t-ηj(t))=y2(t)+-∑i=1nc2i(t)y1(t-σi(t))∑j=1mb2j(t)y2(t-ηj(t))],y1′(t)=y1(t)[∑j=1m‍c1j(t)y2(t-ρj(t))r1(t)-a1(t)y1(t)-∑i=1n‍b1i(t)y1(t-τi(t))+=y1(t)-∑j=1m‍c1j(t)y2(t-ρj(t))],y2′(t)=y2(t)[r2(t)-a2(t)y2(t)-∑j=1m‍b2j(t)y2(t-ηj(t))=y2(t)+-∑i=1n‍c2i(t)y1(t-σi(t))∑j=1m‍b2j(t)y2(t-ηj(t))],
where b1i(t),b2j(t)∈C(R,[0,+∞)), a1(t),a2(t),c1j(t),c2i(t)∈C(R,[0,+∞)),τi(t),ρj(t),ηj(t),σi(t)∈C1(R,[0,+∞))(i=1,2,…,n;j=1,2,…,m) are ω-periodic functions. Here, the intrinsic growth rates rk(t)∈C(R,R) are ω-periodic functions with ∫0ω‍rk(t)dt>0, k=1,2. They had derived the same criteria for the existence and globally asymptotic stability of positive periodic solutions of the previous two competitive systems by using Gaines and Mawhin's continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional. However, to this day, no scholars had done works on the existence of positive periodic solution of (1) and (2). One could easily see that systems (3)–(6) are all special cases of system (1) and (2).Throughout the paper, we make the following assumptions:(H1)
ai(t),bi(t),cij(t),dij(t),eij(t),τij(t),γij(t)∈PCω, i,j=1,2,…,n, with ai¯=(1/ω)∫0ω‍ai(t)dt>0, τ=max1≤i≤nmax1≤j≤n{τij(t),γij(t)};(H2)
{tk},k∈Z+ satisfies 0<t1<t2<⋯<tk<⋯ and limk→+∞tk=+∞;(H3)
{θik} is a real sequence with Δik=θik+1>0, i=1,2,…,n, k∈Z+;(H4)
∏0<tk<t‍Δik is a ω-periodic function.To conclude this section, we summarize in the following a few definitions and lemmas that will be needed in our arguments.Definition 1 (see [1]).
A functionxi:R→(0,+∞) is said to be a positive solution of (1) and (2), if the following conditions are satisfied:(a)
xi(t) is absolutely continuous on each (tk,tk+1);(b)
for eachk∈Z+, xi(tk+) and xi(tk-) exist and xi(tk-)=xi(tk);(c)
xi(t) satisfies the first equation of (1) and (2) for almost everywhere (for short a.e.) in [0,∞]∖{tk} and satisfies xi(tk+)=Δikxi(tk) for t=tk, k∈Z+={1,2,…}.
Under the previous hypotheses(H1)–(H4), we consider the following two classes of nonimpulsive Lotka-Volterra competitive systems:
(7)dyi(t)dt=yi(t)[ai(t)-Bi(t)yi(t)-∑j=1n‍Cij(t)yi(t-τij(t))=yi(t)++-∑j=1n‍Dij(t)yj(t-γij(t))-∑j=1n‍Eij(t)=yi(t)++×∫-σij0fij(s)yj(t+s)ds∑j=1nCij],a.e.,t≥0,(8)dyi(t)dt=yi(t)[ai(t)-Bi(t)yi(t)+∑j=1nCij(t)yi(t-τij(t))=yi(t)+++-∑j=1n‍Dij(t)yj(t-γij(t))-∑j=1nEij(t)=yi(t)+++×∫-σij0fij(s)yj(t+s)ds∑j=1nCij],a.e.,t≥0,
where
(9)Bi(t)=bi(t)∏0<tk<tΔik,Cij(t)=cij(t)∏0<tk<t-τij(t)Δik,Dij(t)=dij(t)∏0<tk<t-γij(t)‍Δik,Eij(t)=eij(t)∏0<tk<t-σij‍Δik.
By a solution y(t)=(y1(t),…,yn(t))T of (7) and (8), it means an absolutely continuous function y(t)=(y1(t),…,yn(t))T defined on [-τ,0] that satisfies (7) and (8).The following lemmas will be used in the proofs of our results. The proof of the first lemma is similar to that of Theorem 1 in [5].Lemma 2.
Suppose that(H1)–(H4) hold. Then(1)
ifyi(t)(i=1,2,…,n) are solutions of (7) and (8) on [-τ,+∞), then xi(t)=∏0<tk<t‍Δikyi(t)(i=1,2,…,n) are solutions of (1) and (2) on [-τ,+∞);(2)
ifxi(t)(i=1,2,…,n) are solutions of (1) and (2) on [-τ,+∞), then yi(t)=∏0<tk<t‍Δik-1xi(t)(i=1,2,…,n) are solutions of (7) and (8) on [-τ,+∞).Proof.
(1) It is easy to see thatxi(t)=∏0<tk<t‍Δikyi(t)(i=1,2,…,n) are absolutely continuous on every interval (tk,tk+1],t≠tk,k=1,2,…,
(10)xi′(t)-xi(t)[ai(t)-bi(t)xi(t)-∑j=1ncij(t)xi(t-τij(t))xi′(t)-xi(t)+-∑j=1n‍dij(t)xj(t-γij(t))-∑j=1neij(t)xi′(t)-xi(t)+×∫-σij0fij(s)xj(t+s)ds∑j=1ncij(t)xi(t-τij(t))]+=∏0<tk<t‍Δikyi′(t)-∏0<tk<t‍Δikyi(t)++×[ai(t)-bi(t)∏0<tk<tΔikyi(t)-∑j=1ncij(t)++++×∏0<tk<t-τij(t)Δikyi(t-τij(t))-∑j=1n‍dij(t)++++×∏0<tk<t-γij(t)‍Δikyj(t-γij(t))++++-∑j=1n∏0<tk<t-σijΔikeij(t)∫-σij0‍fij(s)yj(t+s)ds]+=∏0<tk<t‍Δik{[ai(t)-bi(t)∏0<tk<tΔikyi(t)yi′(t)-yi(t)+=∏0<tk<t‍Δik+×[ai(t)-bi(t)∏0<tk<tΔikyi(t)++=∏0<tk<t‍Δik++-∑j=1n‍cij(t)∏0<tk<t-τij(t)‍Δikyi++=∏0<tk<t‍Δik++×(t-τij(t))-∑j=1ndij(t)++=∏0<tk<t‍Δik++×∏0<tk<t-γij(t)Δikyj(t-γij(t))++=∏0<tk<t‍Δik++-∑j=1n∏0<tk<t-σijΔikeij(t)∫-σij0‍fij(s)yj++=∏0<tk<t‍Δik++×(t+s)ds∏0<tk<tΔikyi]}+=∏0<tk<t‍Δik{[ai(t)-Bi(t)yi(t)-∑j=1n‍Cij(t)yi(t-τij(t))yi′(t)-yi(t)+=∏0<tk<t‍Δik+×[ai(t)-Bi(t)yi(t)-∑j=1n‍Cij(t)yi++=∏0<tk<t‍Δik++×(t-τij(t))-∑j=1nDij(t)yj(t-γij(t))++=∏0<tk<t‍Δik++-∑j=1nEij(t)∫-σij0fij(s)yj(t+s)ds]}+=0.
On the other hand, for any t=tk, k=1,2,…,
(11)xi(tk+)=limt→tk+∏0<tj<tΔikyi(t)=∏0<tj≤tkΔikyi(tk),xi(tk)=∏0<tj<tkΔikyi(tk),i=1,2,…,n;
thus
(12)xi(tk+)=Δikxi(tk),i=1,2,…,n,k=1,2,….
It follows from (10)–(12) that xi(t)(i=1,2,…,n) are solutions of (1). Similarly, if yi(t)(i=1,2,…,n) are solutions of (8), we can prove that xi(t)(i=1,2,…,n) are solutions of (2).
(2) Sincexi(t)=∏0<tk<t‍Δikyi(t) is absolutely continuous on every interval (tk,tk+1],t≠tk,k=1,2,…, and in view of (12), it follows that for any k=1,2,…,
(13)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 that yi(t) is continuous on [-τ,+∞). It is easy to prove that yi(t) is absolutely continuous on [-τ,+∞). Similar to the proof of (1), we can check that yi(t)=∏0<tk<t‍Δik-1xi(t)(i=1,2,…,n) are solutions of (7) on [-τ,+∞). Similarly, if xi(t), i=1,2,…,n, is a solution of (2), we can prove that yi(t)(i=1,2,…,n) are solutions of (8). The proof of Lemma 2 is completed.In the following section, we only discuss the existence of a periodic solution for (7) and (8).Definition 3 (see [19]).
LetX be a real Banach space and E a closed, nonempty subset of X. E is said to be a cone if(1)
αx+βy∈E for all x,y∈E, and α,β>0,(2)
x,-x∈E imply x=0.Lemma 4 (see Krasnoselskii [20], Deimling [21], and Guo and Lakshmikantham [22]).
LetE be a cone in a real Banach space X. Assume that Ωr1 and Ωr2 are open subsets of X with 0∈Ωr1⊂Ωr1¯⊂Ωr2, where Ωri={x∈X:∥x∥<ri},i=1,2. Let T:E∩(Ωr2∖Ωr1¯)→E be a continuous and completely continuous operator satisfying(1)
∥Tx∥≤∥x∥, for any x∈E∩∂Ωr1;(2)
there existsϕ∈E∖{0} such that y≠Ty+λy, for any y∈E∩∂Ωr2 and λ>0.
ThenT has fixed points in E∩(Ωr2¯∖Ωr1). The same conclusion remains valid if (1) holds for any x∈E∩∂Ωr2  and (2) holds for any y∈E∩∂Ωr1 and λ>0.The paper is organized as follows. In Section2, firstly, we give some definitions and lemmas. Secondly, we derive a necessary and sufficient condition for at least one positive periodic solution of (1) which is established by using the fixed-point theorem in the cone of Banach space under some conditions. In the following section, we also get a necessary and sufficient condition for at least one positive periodic solution of (2) that is also established by applying the fixed-point theorem in the cone of Banach space under some conditions. Finally, as applications, we study some particular cases of system (1) and (2) which have been investigated extensively in the references mentioned earlier.


## 2. Existence of Periodic Solution of (1)

We establish the existence of positive periodic solutions of (1) by applying Lemma 4. We will first make some preparations and list later a few preliminary results. Let X={y=(y1(t),y2(t),…,yn(t))T∈C(R,Rn)∣y(t+ω)=y(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 operatorψ:X→X as follows:
(14)(ψy)(t)=((ψy)1(t),(ψy)2(t),…,(ψy)n(t))T,
where
(15)(ψy)i(t)+=∫tt+ω{[Bi(ξ)yi(ξ)+∑j=1nCij(ξ)yi(ξ-τij(ξ))Gi(t,ξ)yi(ξ)++++"+×[Bi(ξ)yi(ξ)+∑j=1nCij(ξ)yi(ξ-τij(ξ))++++++++∑j=1n‍Dij(ξ)yj(ξ-γij(ξ))++++++++∑j=1n‍Eij(ξ)∫-σij0‍fij(s)yj(ξ+s)ds]}dξ,+++++++++++++++++++++i=1,2,…,n,
where
(16)Gi(t,ξ)=e-∫tξ‍ai(u)du1-e-∫0ω‍ai(u)du=e∫ξt+ω‍ai(u)due∫0ω‍ai(u)du-1,G(t,ξ)=diag[G1(t,ξ),G2(t,ξ),…,Gn(t,ξ)],t≤ξ≤t+ω.
It is clear that Gi(t+ω,ξ+ω)=Gi(t,ξ), ∂Gi(t,ξ)/∂t=ai(t)Gi(t,ξ), Gi(t,t)-Gi(t,t+ω)=1. In view of (H1), we also define for 1≤i≤n(17)αi:=min0≤t≤ξ≤ω|Gi(t,ξ)|=1e∫0ω‍ai(u)du-1,βi:=max0≤t≤ξ≤ω|Gi(t,ξ)|=e∫0ωai(u)due∫0ωai(u)du-1,α=min1≤i≤nαi,β=max1≤i≤nβi,σ=αβ∈(0,1).
Define E to be a cone in X by
(18)E={y=(y1(t),y2(t),…,yn(t))T∈X:yi(t)+++≥σ|yi|0,t∈[0,ω](y1(t),y2(t),…,yn(t))T}.
We easily verify that E is a cone in X. For convenience of expressions, we define an operator φ:E→X by
(19)(φy)i(t)=yi(t)[∑j=1n‍Bi(t)yi(t)+yi+(t+)+∑j=1nCij(t)yi(ξ-τij(t))+yi+(t+)+∑j=1nDij(t)yj(t-γij(t))+yi+(t+)+∑j=1n‍Eij(t)∫-σij0fij(s)yj(t+s)ds],+!+++++++++++++++i=1,2,…,n.
The proof of the main result in this paper is based on an application of Krasnoselskii fixed-point theorem in cones. To make use of fixed point theorem in the cone, firstly, we need to introduce some definitions and lemmas.Lemma 5.
Assume that(H1)–(H4) hold. Then the solutions of (1) are defined on [-τ,∞) and are positive.Proof.
By Lemma2, we only need to prove that the solutions yi(t)(i=1,2,…,n) of (7) are defined on [-τ,∞) and are positive on [0,∞). From (7), we have that for any ui∈C([-τ,0),R+), i=1,2,3,…,n, and t>0,
(20)yi(t)=ui(0)exp{∫0t[∑j=1n‍ai(ξ)-Bi(ξ)yi(ξ)++++++++++-∑j=1n‍Cij(ξ)yi(ξ-τij(ξ))++++++++++-∑j=1n‍Dij(ξ)yj(ξ-γij(ξ))++++++++++-∑j=1n‍Eij(ξ)∫-σij0‍fij(s)yj++++++++++×(ξ+s)ds∑j=1n‍]dξ},++++++++++++i=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. Then ψ:E→E is well defined.Proof.
In view of the definitions ofE and F, for any y∈E, we have
(21)(ψy)i(t)=∫tt+ωGi(t,ξ)(φy)i(ξ)dξ,(ψy)i(t+ω)=∫t+ωt+2ωGi(t,ξ+ω)(φy)i(ξ+ω)dξ=∫tt+ωGi(t,ξ)(φy)i(ξ)dξ=(ψy)i(t).
Therefore, (ψy)∈X. Furthermore, for any y∈E, it follows from (15) that
(22)|(ψy)i|0≤βi∫0ω(φy)i(ξ)dξ.
On the other hand, we obtain
(23)(ψy)i(t)≥αi∫0ω(φy)i(ξ)dξ≥αiβi|(ψy)i|0≥σ|(ψy)i|0.
Therefore, ψy∈E. The proof of Lemma 6 is complete.Lemma 7.
The operatorψ:E→E is continuous and completely continuous.Proof.
By using a standard argument one can show thatψ is continuous on E. Now, we show that ψ is completely continuous. Let r be any positive constant and Sr={y∈X:|yi|0≤r} a bounded set. For any y∈Sr, by (15) we have
(24)|(ψy)i|0+≤βi∫0ω(φy)i(ξ)dξ+=βi∫0ωyi(t)[Bi(t)yi(t)+∑j=1nCij(t)yi(ξ-τij(t))+=βi∫0ωyi(t)++∑j=1nDij(t)yj(t-γij(t))+=βi∫0ωyi(t)++∑j=1neij(t)∫-σij0Fij(s)yj(t+s)ds]dξ+≤ωβir2[BiM+∑j=1n(CijM+DijM+EijM)]≔Ri.
Therefore, for any y∈Sr, we obtain
(25)∥ψy∥=∑i=1n|(ψy)i|0≤∑i=1nRi≔R,
which implies that ψ(Sr) is a uniformly bounded set. On the other hand, in view of the definitions of ψ and φ, we have
(26)d[(ψy)i(t)]dt=Gi(t,t+ω)(φy)i(t+ω)-Gi(t,t)(φy)i(t)++∫tt+ωdGi(t,ξ)dt(φy)i(ξ)dξ=-(φy)i(t)+ai(t)∫tt+ωGi(t,ξ)(φy)i(ξ)dξ=ai(t)(ψy)i(t)-(φy)i(t).
Again, from (15), we obtain
(27)|d[(ψy)i(t)]dt|≤aiMRi+r2[BiM+∑j=1n(CijM+DijM+EijM)]≔Ri¯≤M≔maxi∈[1,n]{Ri¯},
which implies that d[(ψy)i(t)]/dt, for any y∈Sr, is also uniformly bounded. Hence, ψ(Sr)⊂X is a family of uniformly bounded and equicontinuous functions. By the well-known Ascoli-Arzela theorem, we know that the operator ψ is completely continuous. The proof of Lemma 7 is complete.Lemma 8.
Assume that(H1)–(H4) hold. The existence of positive ω-periodic solution of (7) is equivalent to that of nonzero fixed point of ψ in E.Proof.
Assume thaty=(y1(t),y2(t),…,yn(t))T∈X is a periodic solution of (7). Then, we have
(28)[yi(t)e-∫0t‍ai(u)du]′+=-e-∫0tai(u)duyi(t)[Bi(t)yi(t)+∑j=1nCij(t)yi(t-τij(t))+=-e-∫0t‍ai(u)duyi(t)++∑j=1nDij(t)yj(t-γij(t))+=-e-∫0t‍ai(u)duyi(t)++∑j=1nEij(t)∫-σij0fij(s)yj(t+s)ds],++++++++++++++++++++++i=1,2,…,n.
Integrating the previous equation over [t,t+ω], we can have
(29)[yi(ξ)e-∫0ξai(u)du]|tt+ω+=-∫tt+ω{∑j=1nCije-∫0ξai(u)duyi(ξ)+++++++×[Bi(ξ)yi(ξ)+∑j=1nCij(ξ)yi(ξ-τij(ξ))++++++++++∑j=1n‍Dij(ξ)yj(ξ-γij(ξ))++++++++++∑j=1nEij(ξ)∫-σij0fij(s)yj(ξ+s)ds]}dξ,+++++++++++++++++++++++i=1,2,…,n.
Therefore, we have
(30)yi(t)e-∫0tai(u)du[1-e-∫0ωai(u)du]+=∫tt+ω{∑j=1nCije-∫0ξ‍ai(u)duyi(ξ)++++++×[Bi(ξ)yi(ξ)+∑j=1nCij(ξ)yi(ξ-τij(ξ))+++++++i+∑j=1nDij(ξ)yj(ξ-γij(ξ))+++++++i+∑j=1nEij(ξ)∫-σij0fij(s)yj(ξ+s)ds]}dξ,++++++++++++++++++++ii=1,2,…,n,
which can be transformed into
(31)yi(t)=∫tt+ω{e-∫0ξai(u)due-∫0tai(u)du[1-e-∫0ω‍ai(u)du]yi(ξ)+++++×[Bi(ξ)yi(ξ)+∑j=1nCij(ξ)yi(ξ-τij(ξ))++++++++∑j=1nDij(ξ)yj(ξ-γij(ξ))++++++++∑j=1nEij(ξ)∫-σij0fij(s)yj(ξ+s)ds]e-∫0ξai(u)due-∫0tai(u)du[1-e-∫0ω‍ai(u)du]}dξ=∫tt+ω‍{∑j=1nCijGi(t,ξ)yi(ξ)+++++×[Bi(ξ)yi(ξ)+∑j=1nCij(ξ)yi(ξ-τij(ξ))++++++++∑j=1nDij(ξ)yj(ξ-γij(ξ))++++++++∑j=1nEij(ξ)∫-σij0fij(s)yj(ξ+s)ds]}dξ=∫tt+ω‍Gi(t,ξ)(φy)i(ξ)dξ,i=1,2,…,n.
Thus, yi(t)(i=1,2,…,n) is a periodic solution for (15).
Ify=(y1(t),y2(t),…,yn(t))T∈X and ψy=((ψy)1,(ψy)2,…,(ψy)n)T=y with y≠0, then for any t=tk derivative the two sides of (15) about t,
(32)d[(ψy)i(t)]dt=Gi(t,t+ω)(φy)i(t+ω)-Gi(t,t)(φy)i(t)++∫tt+ωdGi(t,ξ)dt(φy)i(ξ)dξ=-(φy)i(t)+ai(t)∫tt+ωGi(t,ξ)(φy)i(ξ)dξ=ai(t)yi(t)-(φy)i(t)=yi(t)[∑j=1n‍ai(t)-Bi(t)yi(t)+++++-∑j=1nCij(t)yi(ξ-τij(t))+++++-∑j=1nDij(t)yj(t-γij(t))+++++-∑j=1nEij(t)∫-σij0fij(s)yj(t+s)ds]=dyi(t)dt.
Hence, y(t)=(y1(t),y2(t),…,yn(t))T∈X is a positive ω-periodic solution of (7). Thus we complete the proof of Lemma 8.Our main result of this paper is as follows.Theorem 9.
Assume(H1)–(H4). Then condition
(33)m0=min1≤i≤n{B¯i+∑j=1n‍(C¯ij+D¯ij+E¯ij)}>0
is necessary and sufficient for system (1) to have at least one positive ω-periodic solution.Proof.
(Sufficiency) Let(34)M0=maxi∈[1,n]{B¯i+∑j=1n‍(C¯ij+D¯ij+E¯ij)},
by condition (33), we know that M0≥m0>0. Choose a constant M≥M0 such that 1/ωβiM<1. Let r1=1/ωβiM and
(35)Ωr1={y(t)=(y1(t),y2(t),…,yn(t))T∈X:|yi|0++++′<r1,i=1,2,…,n(y1(t),y2(t),…,yn(t))T}.
For any y=y(t)∈E⋂‍∂Ωr1, σ|yi|0≤yi(t)≤|yi|0, from (15), we obtain
(36)|(ψy)i|0+≤βi∫0ω(φy)i(ξ)dξ+=βi∫0ωyi(t)[∑j=1nCijBi(t)yi(t)++++=+++++∑j=1nCij(t)yi(ξ-τij(t))+++++=++++∑j=1nDij(t)yj(t-γij(t))+++++=++++∑j=1nEij(t)∫-σij0fij(s)yj(t+s)ds]dξ+≤ωβi|yi|0[B¯i+∑j=1n(C¯ij+D¯ij+E¯ij)]r1+≤ωβi|yi|0M0r1+≤|yi|0.
Hence, for any y=y(t)∈E⋂∂Ωr1, σ|yi|0≤yi(t)≤|yi|0, we have
(37)∥ψy∥=∑j=1n|(ψy)i|0≤∑j=1n|yi|0=∥y∥,
which implies that condition (1) in Lemma 4 is satisfied.
On the other hand, we choose0<m≤m0 such that 1/ωσαim>1. Let r2=1/ωσαim>1 and suppose that u=(u1,u2,…,un)T∈E/{0}. We show that for any y=y(t)∈E⋂∂Ωr2 and λ>0, y≠ψy+λu. Otherwise, there exist y0=y0(t)∈E⋂∂Ωr2 and λ0>0, such that y0=ψy0+λ0u. Let ui0≠0(1≤i0≤n), since yi0(t)≥σ|yi0|0, it follows that
(38)yi0=(ψy)i0(t)+λ0ui0=∫tt+ωGi0(t,ξ)(φy)i0(ξ)dξ+λ0ui0≥ωσαi|yi0|0∫0ω[∑j=1n‍Bi(t)yi(t)+++++++++++∑j=1nCij(t)yi(ξ-τij(t))+++++++++++∑j=1nDij(t)yj(t-γij(t))+++++++++++∑j=1nEij(t)∫-σij0fij(s)yj++++++++++×(t+s)ds∑j=1n‍]dt+λ0ui0≥ωσαi|yi0|0[B¯i+∑j=1n(C¯ij+D¯ij+E¯ij)]r2+λ0ui0≥ωσαi|yi0|0m0r2+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 (7) has at least one positive ω-periodic solution. From Lemma 2, system (1) has at least one positive ω-periodic solution.
(Necessity) Suppose that (33) does not hold. Then there exists at least an i0(1≤i0≤n) such that
(39)B¯i0=0,C¯i0j=D¯i0j=E¯i0j=0,j∈[1,n].
If system (7) has a positive ω-periodic solution y(t)=(y1(t),y2(t),…,yn(t))T, then we have
(40)dyi0(t)dt=ai0(t)yi0(t).
Integrating the previos equation over [t,t+ω], we can have
(41)0=lnyi0(t+ω)yi0(t)=∫0ωai0(t)dt>0,
which is a contradiction. The proof of Theorem 9 is complete.


## 3. Existence of Periodic Solution of (2)

Now, we are at the position to study the existence of positive periodic solutions of (2). By carrying out similar arguments as in Section 2, it is not difficult to derive sufficient criteria for the existence of positive periodic solutions of (2). For simplicity, we prefer to list later the corresponding criteria for (2) without proof since the proofs are very similar to those in Section 2.For(t,ξ)∈R2, 1≤i≤n, we define
(42)Gi(t,ξ)=e-∫tξai(u)du1-e-∫0ωai(u)du=e∫ξt+ωai(u)due∫0ωai(u)du-1,G(t,ξ)=diag[G1(t,ξ),G2(t,ξ),…,Gn(t,ξ)],t≤ξ≤t+ω.
It is clear that Gi(t+ω,ξ+ω)=Gi(t,s), ∂Gi(t,ξ)/∂t=ai(t)Gi(t,ξ), Gi(t,t)-Gi(t,t+ω)=1. In view of (H1), we also define for 1≤i≤n(43)αi≔min0≤t≤ξ≤ω|Gi(t,ξ)|=1e∫0ωai(u)du-1,βi:=max0≤t≤ξ≤ω|Gi(t,ξ)|=e∫0ωai(u)due∫0ωai(u)du-1,α=min1≤i≤nαi,β=max1≤i≤nβi,σ=αβ∈(0,1).
Let X={y=(y1(t),y2(t),…,yn(t))T∈PC(R,Rn)∣y(t+ω)=y(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. Define P to be a cone in X by
(44)P={y=(y1(t),y2(t),…,yn(t))T∈X:yi(t)++++≥δ∥yi∥0,t∈[0,ω](y1(t),y2(t),…,yn(t))T}.
We easily verify that P is a cone in X. We define an operator A:P→X as follows:
(45)(Ay)(t)=((Ay)1(t),(Ay)2(t),…,(Ay)n(t))T,
where
(46)(Ay)i(t)=∫tt+ωGi(t,ξ)(ϕy)i(ξ)ds,
where
(47)(∅y)i(t)=yi(t)[Bi(t)yi(t)-∑j=1nCij(t)yi(ξ-τij(t))++++++∑j=1nDij(t)yj(t-γij(t))++++++∑j=1nEij(t)∫-σij0fij(s)yj(t+s)ds],++++++++++++++5+i=1,2,…,n.
The proof of the following lemmas and theorems is similar to those in the Section 2, so we all omit the details here.Lemma 10.
Assume that(H1)–(H4) hold. Then the solutions of (2) are defined on [-τ,∞) and are positive.Lemma 11.
Assume that(H1)–(H4) hold. Then A:E→E is well defined.Lemma 12.
The operatorA:E→E is continuous and completely continuous.Lemma 13.
Assume that(H1)–(H4) hold. The existence of positive ω-periodic solution of (8) is equivalent to that of nonzero fixed point of A in P.Theorem 14.
Assume(H1)–(H4). Moreover, if the condition
(48)l0=min1≤i≤n{B¯i+∑j=1n(-C¯ij+D¯ij+E¯ij)}>0
holds, then the system (2) has at least one positive ω-periodic solution.


## 4. Applications

In this section, as some applications of our main results, we will consider some special cases of systems (1) and (2), which have been investigated extensively in the literature.Application 1. We consider the following periodic single-species population growth models with periodic delay and impulse:
(49)y′(t)=y(t)[r(t)-a(t)y(t)-b(t)y(t-τ(t))],t∈R,t≠tk,Δy(tk)=θky(tk),k∈Z+,
which is a special case of system (1), and where a(t),b(t),τ(t)∈C(R,R+) are ω-periodic. Thus from Theorem 9 we have the following.Theorem 15.
Assume that(H1)–(H4) hold. Then condition:
(50)R1=A¯+B¯>0
is necessary and sufficient for system (49) to have at least one positive ω-periodic solution, where
(51)A(t)=a(t)∏0<tk<t(1+θk),B(t)=b(t)∏0<tk<t-τ(t)(1+θk).Application 2. We consider the following periodic single-species population growth models with periodic delay and impulse:
(52)y′(t)=y(t)[r(t)-a(t)y(t)+b(t)y(t-τ(t))],t∈R,t≠tk,Δy(tk)=θky(tk),k∈Z+,
which is a special case of system (2), and where a(t),b(t),τ(t)∈C(R,R+) are ω-periodic. Thus from Theorem 14 we have the following.Theorem 16.
Assume that(H1)–(H4) hold. Moreover, if the condition
(53)R2=A¯-B¯>0
holds, the system (52) has at least one positive ω-periodic solution, where
(54)A(t)=a(t)∏0<tk<t(1+θk),B(t)=b(t)∏0<tk<t-τ(t)(1+θk).Application 3. We study the following two-species competitive system with impulses:
(55)y1′(t)=y1(t)[r1(t)-a1(t)y1(t)-c1(t)y2(t)],y2′(t)=y2(t)[r2(t)-a2(t)y2(t)-c2(t)y1(t)],t≠tk,Δyi(tk)=θky(tk),i=1,2,k∈Z+,t=tk,
which is a special case of system (1), and where ai(t),ci(t)∈C(R,R+) are ω-periodic. Thus from Theorem 9 we have the following.Theorem 17.
Assume that(H1)–(H4) hold. Then condition
(56)R3=min1≤i≤2{A¯i+C¯i}>0
is necessary and sufficient for system (55) to have at least one positive ω-periodic solution, where
(57)Ai(t)=ai(t)∏0<tk<t(1+θk),Ci(t)=ci(t)∏0<tk<t(1+θk),i=1,2.Application 4. We study the following two-species competitive system with impulses:
(58)y1′(t)=y1(t)[∑j=1nb1jr1(t)-a1(t)y1(t)+++++-∑j=1nb1j(t)y1(t-τj(t))+++++-∑j=1nc1j(t)y2(t-ρj(t))],y2′(t)=y2(t)[∑j=1n‍r2(t)-a2(t)y2(t)+++++-∑j=1nb2j(t)y2(t-ηj(t))+++++-∑j=1nc2j(t)y1(t-σj(t))],+++++++++++++++t≠tk,Δyi(tk)=θky(tk),i=1,2,k∈Z+,t=tk,
which is a special case of system (1), and where ai(t),bij(t),cij(t)∈C(R,R+), i=1,2, are ω-periodic. Thus from Theorem 9 we have the following.Theorem 18.
Assume that(H1)–(H4) hold. Then condition
(59)R4=min1≤i≤2{A¯i+∑i=1n(B¯ij+C¯ij)}>0
is necessary and sufficient for system (58) to have at least one positive ω-periodic solution, where
(60)Ai(t)=ai(t)∏0<tk<t(1+θk),B1j(t)=b1j(t)∏0<tk<t-τj(t)(1+θk),B2j(t)=b2j(t)∏0<tk<t-ηj(t)(1+θk),C1j(t)=C1j(t)∏0<tk<t-ρj(t)(1+θk),C2j(t)=c2j(t)∏0<tk<t-σj(t)(1+θk),i=1,2.Application 5. We study the following two-species competitive system with impulses:
(61)y1′(t)=y1(t)[∑j=1n‍r1(t)-a1(t)y1(t)++++++∑j=1nb1j(t)y1(t-τj(t))+++++-∑j=1nc1j(t)y2(t-ρj(t))],y2′(t)=y2(t)[∑j=1n‍r2(t)-a2(t)y2(t)++++++∑j=1nb2j(t)y2(t-ηj(t))+++++-∑j=1nc2j(t)y1(t-σj(t))],+++++++++++++++t≠tk,Δyi(tk)=θky(tk),i=1,2,k∈Z+,t=tk,
which is a special case of system (2), and where ai(t),bij(t),cij(t)∈C(R,R+), i=1,2, are ω-periodic. Thus from Theorem 14 we have the following.Theorem 19.
Assume that(H1)–(H4) hold. Moreover, if the condition
(62)R5=min1≤i≤2{A¯i+∑i=1n(-B¯ij+C¯ij)}>0
holds, the system (61) has at least one positive ω-periodic solution, where
(63)Ai(t)=ai(t)∏0<tk<t(1+θk),B1j(t)=b1j(t)∏0<tk<t-τj(t)(1+θk),B2j(t)=b2j(t)∏0<tk<t-ηj(t)(1+θk),C1j(t)=C1j(t)∏0<tk<t-ρj(t)(1+θk),C2j(t)=c2j(t)∏0<tk<t-σj(t)(1+θk),i=1,2.Application 6. We investigate the following n-species competitive systems with impulses:
(64)xi′(t)=xi(t)[∑j=1ncijai(t)-bi(t)xi(t)+++++-∑j=1ncij(t)xj(t-τij(t))-∑j=1ndij(t)+++++×∫-σij0Kij(s)xj(t+s)ds∑j=1ncij],t≠tk,xi(tk+)-xi(tk-)=θikxi(tk),i=1,2,…,n,k∈Z+,
which is a special case of system (1), and where ai(t),bi(t),cij(t),dij(t),τij(t)∈C(R,R+), i=1,2,…,n, are ω-periodic. Moreover, Kij∈PC,σij>0 is a constant and ∫-σij0‍Kij(s)ds=1. Thus from Theorem 9 we have the following.Theorem 20.
Assume that(H1)–(H4) hold. Then condition
(65)R6=min1≤i≤2{B¯i+∑i=1n(C¯ij+D¯ij)}>0
is necessary and sufficient for system (64) to have at least one positive ω-periodic solution, where
(66)Bi(t)=bi(t)∏0<tk<t(1+θik),Cij(t)=Cij(t)∏0<tk<t-τij(t)(1+θik),Dij(t)=dij(t)∏0<tk<t-σij(1+θik).Remark 21.
We apply the main results obtained in the previous section to study some examples which have some biological implications. A very basic and important ecological problem associated with the study of population is that of the existence of positive periodic solutions which play the role played by the equilibrium of the autonomous models, meaning that the species is in an equilibrium state. From Theorems15–20, we see that under the appropriate conditions, the impulsive perturbations do not affect the existence of periodic solution of systems.



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