# Application of Mawhin's Coincidence Degree and Matrix Spectral Theory to a Delayed System

Authors: Yong-Hui Xia; Xiang Gu; Patricia J. Y. Wong; Syed Abbas
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to a predator-prey model withM-predators and N-preys. The method is different from that used in the previous work. Some new sufficient conditions are obtained for the existence and global asymptotic stability of the periodic solution. The existence and stability conditions are given in terms of spectral radius of explicit matrices which are much different from the conditions given by the algebraic inequalities. Finally, an example is given to show the feasibility of our results.


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## Body

## 1. Introduction and Motivation

### 1.1. History and Motivations

Mawhin's coincidence degree theory has been applied extensively to study the existence of periodic solutions for nonlinear differential systems (e.g. see [1–16] and references therein). The most important step of applying Mawhin’s degree theory to nonlinear differential equations is to obtain the priori bounds of unknown solutions to the operator equation Lx=λNx. However, different estimation techniques for the priori bounds of unknown solutions to the equation Lx=λNx may lead to different results. Most of papers obtained the priori bounds by employing the inequalities:
(1.1)x(t)≤x(ξ)+∫0ω‍|x˙(t)|dt,x(t)≥x(η)-∫0ω‍|x˙(t)|dt,x(ξ)=mint∈[0,ω]x(t),x(η)=maxt∈[0,ω]x(t).
These inequalities lead to a relatively strong condition given in terms of algebraic inequality or classic norms (see e.g., [3–16]). Different from standard consideration, in this paper, we employ matrix spectral theory to obtain the priori bounds, not the above inequalities. So in this paper, the existence and stability of periodic solution for a multispecies predator-prey model is studied by jointly employing Mawhin's coincidence degree and matrix spectral theory.


### 1.2. Model Formulation

One of classical Lotka-Vlterra system is predator-prey models which have been investigated extensively by mathematicians and ecologist. Many good results have been obtained for stability, bifurcations, chaos, uniform persistence, periodic solution, almost periodic solutions. It has been observed that most of works focus on either two or three species model. There are few paper considering the multispecies model. To model the dynamic behavior of multispecie predator-prey system, Yang and Rui [17] proposed a predator-prey model with M-predators and N-preys of the form:
(1.2)x˙i(t)=xi(t)[bi(t)-∑k=1N‍aik(t)xk(t)-∑l=1M‍cil(t)yl(t)],i=1,2,…,N,y˙j(t)=yj(t)[-rj(t)+∑k=1N‍djk(t)xk(t)-∑l=1M‍ejl(t)yl(t)],j=1,2,…,M,
where xi(t) denotes the density of prey species Xi at time t, yj(t) denotes the density of predator species Yj at time t. The coefficients bi(t),rj(t),aik(t),cil(t),djk(t), and ejl(t),(i,k=1,…,N;j,l=1,…,M) are nonnegative continuous periodic functions defined on t∈(-∞,+∞). The coefficient bi is the intrinsic growth rate of prey species Xi, rj is the death rate of the predator species Yj, aik measures the amount of competition between the prey species Xi and Xk(k≠i,i,k=1,…,N), ejl measures the amount of competition between the predator species Yj and Yk(k≠j,j,k=1,…,M), and the constant k~ij=Δdij/cij denotes the coefficient in conversing prey species Xi into new individual of predator species Yj(i=1,…,N;j=1,…,M). By using the differential inequality, Zhao and Chen [18] improved the results of Yang and Rui [17]. Recently, Xia et al. [19] obtained some sufficient conditions for the existence and global attractivity of a unique almost periodic solution of the system (1.2).It is more natural to consider the delay model because most of the species start interacting after reaching a maturity period. Hence many scholars think that the delayed models are more realistic and appropriate to be studied than ordinary model. Delayed system is important also because sometimes time delays may lead to oscillation, bifurcation, chaos, instability which may be harmful to a system. Inspired by the above argument, Wen [20] considered a periodic delayed multispecies predator-prey system as follows:
(1.3)x˙i(t)=xi(t)[bi(t)-aii(t)xi(t)-∑k=1,k≠iN‍aik(t)xk(t-τik)-∑l=1M‍cil(t)yl(t-ηil)],y˙j(t)=yj(t)[-rj(t)+∑k=1N‍djk(t)xk(t-δjk)-ejj(t)yj(t)-∑l=1,l≠jM‍ejl(t)yl(t-ξjl)],
where bi(t), rj(t), aik(t), cil(t), djk(t), ejl(t), ejl(t)(i,k=1,2,…,N;j,l=1,2,…,M) are assumed to be continuous ω-periodic functions and the delays τik, δjk, ηil, ξjl are assumed to be positive constants. The system (1.3) is supplemented with the initial condition:
(1.4)xi(θ)=ϕi(θ),yj(θ)=ψi(θ),θ∈[-τ,0],ϕi(0)>0,ψi(0)>0,
where
(1.5)τ=max{max1≤i,k≤nτik,max1≤i≤N,1≤l≤Mηil,max1≤j,l≤Mξjl,max1≤j≤M,1≤k≤Nδjk}>0.
It is easy to see that for such given initial conditions, the corresponding solution of the system (1.3) remains positive for all t≥0. The purpose of this paper is to obtain some new and interesting criteria for the existence and global asymptotic stability of periodic solution of the system (1.3).


### 1.3. Comparison with Previous Work

To obtain the periodic solutions of the system (1.3), the method used in [20] is based on employing the differential inequality and Brower fixed point theorem. Different from consideration taken by [20], our method is based on combining matrix spectral theory with Mawhin's degree theory. In our method, we study the global asymptotic stability by combining matrix's spectral theory with Lyapunov functional method. The existence and stability conditions are given in terms of spectral radius of explicit matrices. These conditions are much different from the sufficient conditions obtained in [20].


### 1.4. Outline of This Work

The structure of this paper is as follows. In Section2, some new and interesting sufficient conditions for the existence of periodic solution of system (1.3) are obtained. Section 3 is devoted to examining the stability of the periodic solution obtained in the previous section. In Section 4, some corollaries are presented to show the effectiveness of our results. Finally, an example is given to show the feasibility of our results.


## 1.1. History and Motivations

Mawhin's coincidence degree theory has been applied extensively to study the existence of periodic solutions for nonlinear differential systems (e.g. see [1–16] and references therein). The most important step of applying Mawhin’s degree theory to nonlinear differential equations is to obtain the priori bounds of unknown solutions to the operator equation Lx=λNx. However, different estimation techniques for the priori bounds of unknown solutions to the equation Lx=λNx may lead to different results. Most of papers obtained the priori bounds by employing the inequalities:
(1.1)x(t)≤x(ξ)+∫0ω‍|x˙(t)|dt,x(t)≥x(η)-∫0ω‍|x˙(t)|dt,x(ξ)=mint∈[0,ω]x(t),x(η)=maxt∈[0,ω]x(t).
These inequalities lead to a relatively strong condition given in terms of algebraic inequality or classic norms (see e.g., [3–16]). Different from standard consideration, in this paper, we employ matrix spectral theory to obtain the priori bounds, not the above inequalities. So in this paper, the existence and stability of periodic solution for a multispecies predator-prey model is studied by jointly employing Mawhin's coincidence degree and matrix spectral theory.


## 1.2. Model Formulation

One of classical Lotka-Vlterra system is predator-prey models which have been investigated extensively by mathematicians and ecologist. Many good results have been obtained for stability, bifurcations, chaos, uniform persistence, periodic solution, almost periodic solutions. It has been observed that most of works focus on either two or three species model. There are few paper considering the multispecies model. To model the dynamic behavior of multispecie predator-prey system, Yang and Rui [17] proposed a predator-prey model with M-predators and N-preys of the form:
(1.2)x˙i(t)=xi(t)[bi(t)-∑k=1N‍aik(t)xk(t)-∑l=1M‍cil(t)yl(t)],i=1,2,…,N,y˙j(t)=yj(t)[-rj(t)+∑k=1N‍djk(t)xk(t)-∑l=1M‍ejl(t)yl(t)],j=1,2,…,M,
where xi(t) denotes the density of prey species Xi at time t, yj(t) denotes the density of predator species Yj at time t. The coefficients bi(t),rj(t),aik(t),cil(t),djk(t), and ejl(t),(i,k=1,…,N;j,l=1,…,M) are nonnegative continuous periodic functions defined on t∈(-∞,+∞). The coefficient bi is the intrinsic growth rate of prey species Xi, rj is the death rate of the predator species Yj, aik measures the amount of competition between the prey species Xi and Xk(k≠i,i,k=1,…,N), ejl measures the amount of competition between the predator species Yj and Yk(k≠j,j,k=1,…,M), and the constant k~ij=Δdij/cij denotes the coefficient in conversing prey species Xi into new individual of predator species Yj(i=1,…,N;j=1,…,M). By using the differential inequality, Zhao and Chen [18] improved the results of Yang and Rui [17]. Recently, Xia et al. [19] obtained some sufficient conditions for the existence and global attractivity of a unique almost periodic solution of the system (1.2).It is more natural to consider the delay model because most of the species start interacting after reaching a maturity period. Hence many scholars think that the delayed models are more realistic and appropriate to be studied than ordinary model. Delayed system is important also because sometimes time delays may lead to oscillation, bifurcation, chaos, instability which may be harmful to a system. Inspired by the above argument, Wen [20] considered a periodic delayed multispecies predator-prey system as follows:
(1.3)x˙i(t)=xi(t)[bi(t)-aii(t)xi(t)-∑k=1,k≠iN‍aik(t)xk(t-τik)-∑l=1M‍cil(t)yl(t-ηil)],y˙j(t)=yj(t)[-rj(t)+∑k=1N‍djk(t)xk(t-δjk)-ejj(t)yj(t)-∑l=1,l≠jM‍ejl(t)yl(t-ξjl)],
where bi(t), rj(t), aik(t), cil(t), djk(t), ejl(t), ejl(t)(i,k=1,2,…,N;j,l=1,2,…,M) are assumed to be continuous ω-periodic functions and the delays τik, δjk, ηil, ξjl are assumed to be positive constants. The system (1.3) is supplemented with the initial condition:
(1.4)xi(θ)=ϕi(θ),yj(θ)=ψi(θ),θ∈[-τ,0],ϕi(0)>0,ψi(0)>0,
where
(1.5)τ=max{max1≤i,k≤nτik,max1≤i≤N,1≤l≤Mηil,max1≤j,l≤Mξjl,max1≤j≤M,1≤k≤Nδjk}>0.
It is easy to see that for such given initial conditions, the corresponding solution of the system (1.3) remains positive for all t≥0. The purpose of this paper is to obtain some new and interesting criteria for the existence and global asymptotic stability of periodic solution of the system (1.3).


## 1.3. Comparison with Previous Work

To obtain the periodic solutions of the system (1.3), the method used in [20] is based on employing the differential inequality and Brower fixed point theorem. Different from consideration taken by [20], our method is based on combining matrix spectral theory with Mawhin's degree theory. In our method, we study the global asymptotic stability by combining matrix's spectral theory with Lyapunov functional method. The existence and stability conditions are given in terms of spectral radius of explicit matrices. These conditions are much different from the sufficient conditions obtained in [20].


## 1.4. Outline of This Work

The structure of this paper is as follows. In Section2, some new and interesting sufficient conditions for the existence of periodic solution of system (1.3) are obtained. Section 3 is devoted to examining the stability of the periodic solution obtained in the previous section. In Section 4, some corollaries are presented to show the effectiveness of our results. Finally, an example is given to show the feasibility of our results.


## 2. Existence of Periodic Solutions

In this section, we will obtain some sufficient conditions for the existence of periodic solution of the system (1.3).


### 2.1. Preliminaries on the Matrix Theory and Degree Theory

For convenience, we introduce some notations, definitions, and lemmas. Throughout this paper, we use the following notations.(i)
We always usei,k=1,…,N;j,l=1,…,M, unless otherwise stated.(ii)
Iff(t) is a continuous ω-periodic function defined on R, then we denote
(2.1)f_=mint∈[0,ω]|f(t)|,f¯=maxt∈[0,ω]|f(t)|,m(f)=1ω∫0ω‍f(t)dt.We use x=(x1,…,xn)T∈ℝn to denote a column vector, 𝒟=(dij)n×n is an n×n matrix, 𝒟T denotes the transpose of 𝒟, and En is the identity matrix of size n. A matrix or vector 𝒟>0 (resp., 𝒟≥0) means that all entries of 𝒟 are positive (resp., nonnegative). For matrices or vectors 𝒟 and E, 𝒟>E (resp., 𝒟≥E) means that 𝒟-E>0 (resp., 𝒟-E≥0). We denote the spectral radius of the matrix 𝒟 by ρ(𝒟).Ifv=(v1,v2,…,vn)T∈ℝn, then we have a choice of vector norms in ℝn, for instance ∥v∥1, ∥v∥2, and ∥v∥∞ are the commonly used norms, where
(2.2)∥v∥1=∑j=1n‍|vi|,∥v∥2={∑j=1n‍|vi|2}1/2,∥v∥∞=max1≤i≤n|vi|.
We recall the following norms of matrices induced by respective vector norms. For instance if 𝒜=(aij)n×n, the norm of the matrix ∥𝒜∥ induced by a vector norm ∥·∥ is defined by
(2.3)∥𝒜∥p=supv∈ℝn,v≠0∥𝒜v∥p∥v∥p=sup∥v∥p=1∥𝒜v∥p=sup∥v∥p≤1∥𝒜v∥p.
In particular one can show that ∥𝒜∥1=max1≤j≤n∑i=1n|aij|  (column  norm), ∥𝒜∥2=[λmax(𝒜T𝒜)]1/2=[max.eigenvalueof(𝒜T𝒜)]1/2and∥𝒜∥∞=max1≤i≤n∑j=1n|aij| (row  norm).Definition 2.1 (see [1, 21]).
LetX,Z be normed real Banach spaces, let L:DomL⊂X→Z be a linear mapping, and N:X→Z be a continuous mapping. The mapping L is called a Fredholm mapping of index zero, if dimKer L=codimImL<+∞ and Im L is closed in Z. If L is a Fredholm mapping of index zero and there exist continuous projectors P:X→X and Q:Z→Z such that Im P=KerL, KerQ=ImL=Im(I-Q), then L∣domL∩KerP:(I-P)X→ImL is invertible. We denote the inverse of that map by KP. If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω¯ if QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J:ImQ→KerL.Definition 2.2 (see [1, 22]).
LetΩ⊂ℝn be open and bounded, f∈C1(Ω,ℝn)∩C(Ω¯,ℝn) and y∈ℝn/f(∂Ω∪Nf), that is, y is a regular value of f. Here, Nf={x∈Ω:Jf(x)=0}, the critical set of f and Jf is the Jacobian of f at x. Then the degree deg{f,Ω,y} is defined by
(2.4)deg{f,Ω,y}=∑x∈f-1(y)‍sgnJf(x),
with the agreement that ∑‍ϕ=0. For more details about Degree Theory, the reader may consult Deimling [22].Lemma 2.3 (Continuation Theorem [1]).
LetΩ⊂X be an open and bounded set and L be a Fredholm mapping of index zero and N be L-compact on Ω¯ (i.e., QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is compact). Assume(i)
for eachλ∈(0,1), x∈∂Ω∩DomL, Lx≠λNx;(ii)
for eachx∈∂Ω∩KerL, QNx≠0 and deg{JQN,Ω∩KerL,0}≠0. Then Lx=Nx has at least one solution in Ω¯∩DomL.Definition 2.4 (see [23, 24]).
A realn×n matrix 𝒜=(aij) is said to be an M-matrix if aij≤0, i,j=1,2,…,n, i≠j, and 𝒜-1≥0.Lemma 2.5 (see [23, 24]).
Let𝒜≥0 be an n×n matrix and ρ(𝒜)<1, then (En-𝒜)-1≥0, where En denotes the identity matrix of size n.Now we introduce some function spaces and their norms, which will be valid throughout this paper. Denote(2.5)X={U(t)=(u(t),v(t))T∈C1(ℝ,ℝN+M)∣U(t+ω)=U(t)∀t∈ℝ},Z={U(t)=(u(t),v(t))T∈C(ℝ,ℝN+M)∣U(t+ω)=U(t)∀t∈ℝ}.
The norms are given by
(2.6)|Un(t)|0=maxt∈[0,ω]|Un(t)|,|Un(t)|1=|Un(t)|0+|U˙n(t)|0,i=1,2,…,N+M,∥Un(t)∥0=max1≤n≤N+M{|Un(t)|0},∥Un(t)∥1=max1≤n≤N+M{|Un(t)|1}.
Obviously, X and Z, respectively, endowed with the norms ∥·∥1 and ∥·∥0 are Banach spaces.


### 2.2. Result on the Existence of Periodic Solutions

Theorem 2.6.
Assume that the following conditions hold:(H1):
the system of algebraic equations:(2.7)m(bi)-m(aii)ui-∑k=1,k≠iN‍m(aik)uk-∑l=1M‍m(cil)wl=0,i=1,2,…,N,m(-rj)+∑k=1N‍m(djk)uk-m(ejj)wj-∑l=1,l≠jM‍m(ejl)wl=0,j=1,2,…,M,has finite solution(u1*,…,uN*,w1*,…,wM*)T∈ℝ+N+M with u*>0,w*>0;(H2):
ρ(𝒦)<1,        where 𝒦=(𝒫N×N𝒬N×MℳM×N𝒩M×M)(N+M)×(N+M),
(2.8)𝒫N×N=(pik)N×N,pik={0,i=k,a¯ika_kk-1,i≠k.𝒬N×M=(qil)N×M,qil=c¯ile_ll-1,ℳM×N=(mjk)M×N,mjk=d¯jka_kk-1,𝒩M×M=(njl)M×M,njl={0,j=l,e¯jle_ll-1,j≠l.
Then system (1.3) has at least one positive ω-periodic solution.Proof.
Note that every solution(2.9)U(t)=(u(t),v(t))T=(u1(t),…,uN(t),v1(t),…,vM(t))T∈X
of the system (1.3) with the initial condition is positive. By using the following changes of variables:
(2.10)ui(t)=lnxi(t),vj(t)=lnyj(t),i=1,2,…,N,j=1,2,…,M,
the system (1.3) can be rewritten as
(2.11)u˙i(t)=bi(t)-aii(t)eui(t)-∑k=1,k≠iN‍aik(t)euk(t-τik)-∑l=1M‍cil(t)evl(t-ηil),i=1,2,…,N,v˙j(t)=-rj(t)+∑k=1N‍djk(t)euk(t-δjk)-ejj(t)evj(t)-∑l=1,l≠jM‍ejl(t)evl(t-ξjl),j=1,2,…,M.
Obviously, system (1.3) has at least one ω-periodic solution which is equivalent to the system (2.11) having at least one ω-periodic solution. To prove Theorem 2.6, our main tasks are to construct the operators (i.e., L, N, P, and Q) appearing in Lemma 2.3 and to find an appropriate open set Ω satisfying conditions (i), (ii) in Lemma 2.3.
For anyU(t)∈X, in view of the periodicity, it is easy to check that
(2.12)Δi(U,t)=bi(t)-aii(t)eui(t)-∑k=1,k≠iN‍aik(t)euk(t-τik)-∑l=1M‍cil(t)evl(t-ηil)∈Z,Δj(U,t)=-rj(t)+∑k=1N‍djk(t)euk(t-δjk)-ejj(t)evj(t)-∑l=1,l≠jM‍ejl(t)evl(t-ξjl)∈Z.
Now, we define the operators L,N as follows:
(2.13)L:DomL⊂X→Z,L(u(t),v(t))=(dui(t)dt,dvj(t)dt)∈Z,N:X→ZisdefinedbyNU=(Δi(U,t)Δj(U,t)).
Define, respectively, the projectors P:X→X and Q:Z→Z by
(2.14)PU=1ω∫0ω‍U(t)dt,U∈X,QU=1ω∫0ω‍U(t)dt,U∈Z.
It is obvious that the domain of L in X is actually the whole space, and
(2.15)KerL={x(t)∈X∣Lx(t)=0,i.e.x˙(t)=0}=ℝN+M,ImL={z(t)∈Z∣∫0ω‍z(t)dt=0}isclosedinZ.
Moreover, P,Q are continuous operators such that
(2.16)ImP=ℝN=KerL,ImL=KerQ=Im(I-Q),dim KerL=codim ImL=N+M<+∞.
It follows that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) KP:ImL→DomL∩KerP exists, which is given by
(2.17)KP(y)=∫0t‍y(s)ds-1ω∫0ω‍∫0t‍y(s)dsdt.
Then QN:X→Z and KP(I-Q)N:X→X are defined by
(2.18)QNU=(1ω∫0ω‍Δi(U,t)dt1ω∫0ω‍Δj(U,t)dt),KP(I-Q)Nx=∫0t‍NU(s)ds-1ω∫0ω‍∫0t‍NU(s)dsdt-(tω-12)∫0ω‍NU(s)ds.
Clearly, QN and KP(I-Q)N are continuous. By using the generalized Arzela-Ascoli theorem, it is not difficult to prove that (KP(I-Q)N)(Ω¯) is relatively compact in the space (X,∥·∥1). The proof of this step is complete.
Then, in order to apply condition (i) of Lemma2.3, we need to search for an appropriate open bounded subset Ω, denoted by
(2.19)Ω=Un(t)∈X∣|Un(t)|1=|Un(t)|0+|U˙n(t)|0<hn.
Specifically, our aim is to find an appropriate hn. Corresponding to the operator equation Lx=λNx for each λ∈(0,1), we have
(2.20)u˙i(t)=λ[bi(t)-aii(t)eui(t)-∑k=1,k≠iN‍aik(t)euk(t-τik)-∑l=1M‍cil(t)evl(t-ηil)],v˙j(t)=λ[-rj(t)+∑k=1N‍djk(t)euk(t-δjk)-ejj(t)evj(t)-∑l=1,l≠jM‍ejl(t)evl(t-ξjl)].
Since U(t)∈X, each Un(t),n=1,2,…,N+M, as components of U(t), is continuously differentiable and ω-periodic. In view of continuity and periodicity, there exists ti∈[0,ω] such that ui(ti)=maxt∈[0,ω]|ui(t)|, i=1,2,…,N, and there also exists tN+j∈[0,ω] such that vj(tN+j)=maxt∈[0,ω]|vj(t)|, j=1,2,…,M. Accordingly, u˙i(ti)=0, v˙j(tN+j)=0, and we get
(2.21)bi(ti)-aii(ti)eui(ti)-∑k=1,k≠iN‍aik(ti)euk(ti-τik)-∑l=1M‍cil(ti)evl(tN+j-ηil)=0,-rj(tN+j)+∑k=1N‍djk(tN+j)euk(ti-δjk)-ejj(tN+j)evj(tN+j)-∑l=1,l≠jM‍ejl(tN+j)evl(tN+j-ξjl)=0.
That is,
(2.22)aii(ti)eui(ti)=bi(ti)-∑k=1,k≠iN‍aik(ti)euk(ti-τik)-∑l=1M‍cil(ti)evl(tN+j-ηil),ejj(tN+j)evj(tN+j)=-rj(tN+j)+∑k=1N‍djk(tN+j)euk(ti-δjk)-∑l=1,l≠jM‍ejl(tN+j)evl(tN+j-ξjl).
Note that uk(tk)=maxt∈[0,ω]|uk(t)| and vl(tN+l)=maxt∈[0,ω]|vl(t)|, which implies
(2.23)uk(ti)≤uk(tk),uk(ti-τik)≤uk(tk),uk(ti-δjk)≤uk(tk);vl(tN+j)≤vl(tN+l),vl(tN+j-ηil)≤vl(tN+l),vl(tN+j-ξjl)≤vl(tN+l).
It follows that
(2.24)a_iieui(ti)≤|aii(ti)eui(ti)|=|bi(ti)-∑k=1,k≠iN‍aik(ti)euk(ti-τik)-∑l=1M‍cil(ti)evl(tN+j-ηil)|≤b¯i+∑k=1,k≠iN‍a¯ikeuk(ti-τik)+∑l=1M‍c¯ilevl(tN+j-ηil)≤b¯i+∑k=1,k≠iN‍a¯ikeuk(tk)+∑l=1M‍c¯ilevl(tN+l),e_jjevj(tN+j)≤|ejj(tN+j)evj(tN+j)|=|-rj(tN+j)+∑k=1N‍djk(tN+j)euk(ti-δjk)-∑l=1,l≠jM‍ejl(tN+j)evl(tN+j-ξjl)|≤r¯j+∑k=1N‍d¯jkeuk(ti-δjk)+∑l=1,l≠jM‍e¯jlevl(tN+j-ξjl)≤r¯j+∑k=1N‍d¯jkeuk(tk)+∑l=1,l≠jM‍e¯jlevl(tN+l).
Let
(2.25)a_iieui(ti)=zi(ti),e_jjevj(tN+j)=z~j(tN+j).
Using (2.25), the inequalities (2.24) become
(2.26)zi(ti)≤b¯i+∑k=1,k≠iN‍a¯ika_kk-1zk(tk)+∑l=1M‍c¯ile_ll-1z~l(tN+l),z~j(tN+j)≤r¯j+∑k=1N‍d¯jka_kk-1zk(tk)+∑l=1,l≠jM‍e¯jle_ll-1z~l(tN+l),
or
(2.27)zi(ti)-∑k=1,k≠iN‍a¯ika_kk-1zk(tk)-∑l=1M‍c¯ile_ll-1z~l(tN+l)≤b¯i,z~j(tN+j)-∑k=1N‍d¯jka_kk-1zk(tk)-∑l=1,l≠jM‍e¯jle_ll-1z~l(tN+l)≤r¯j,
which implies
(2.28)(EN×N-𝒫N×N-𝒬N×M-ℳM×NEM×M-𝒩M×M)(N+M)×(N+M)×(z1(t1),…,zN(tN),z~1(tN+1),…,z~M(tN+M))≤(b¯1,…,b¯N,r¯1,…,r¯M),
where
(2.29)𝒫N×N=(pik)N×N,pik={0,i=k,a¯ika_kk-1,i≠k,𝒬N×M=(qil)N×M,qil=c¯ile_ll-1,ℳM×N=(mjk)M×N,mjk=d¯jka_kk-1,𝒩M×M=(njl)M×M,njl={0,j=l,e¯jle_ll-1,j≠l.
Set D=(b¯1,…,b¯N,r¯1,…,r¯M)T. It follows from (2.28) and (H2) that
(2.30)(E-𝒦)(z1(t1),…,zN(tN),z~1(tN+1),…,z~M(tN+M))T≤D.
In view of ρ(𝒦)<1 and Lemma 2.5, we get (EN+M-𝒦)-1≥0. Let
(2.31)H=(h~1,…,h~N,h~N+1,…,h~N+M)T:=(E-𝒦)-1D≥0.
Using (2.30) and (2.31), we get
(2.32)(z1(t1),…,zN(tN),z~1(tN+1),…,z~M(tN+M))T≤H,
or
(2.33)zi(ti)≤h~i,i=1,2,…,N,z~j(tN+j)≤h~N+j,j=1,2,…,M.
Then
(2.34)ui(ti)≤lnh~ia_ii,vj(tN+j)≤lnh~N+je_jj,
which implies
(2.35)|un(t)|0=maxt∈[0,ω]|un(t)|=maxt∈[0,ω]{ui(ti),vj(tN+j)}=max{lnh~ia_ii,lnh~N+je_jj}.
On the other hand, it follows from (2.31) that
(2.36)(E-𝒦)H=D,orH=𝒦H+D,
that is
(2.37)h~i=∑k=1,k≠iN‍pikh~k+∑l=1M‍qilh~N+l+b¯i,h~N+j=∑k=1N‍mjkh~k+∑l=1,l≠jM‍njlh~N+l+r¯j.
Estimating (2.20), by using (2.25), (2.33), and (2.37), we have
(2.38)|u˙i(t)|0=λ|bi(t)-aii(t)eui(t)-∑k=1,k≠iN‍aik(t)euk(t-τik)-∑l=1M‍cil(t)evl(t-ηil)|0≤b¯i+a¯ii|eui(t)|0+∑k=1,k≠iN‍a¯ik|euk(t-τik)|0+∑l=1M‍c¯il|evl(t-ηil)|0=b¯i+a¯iieui(ti)+∑k=1,k≠iN‍a¯ikeuk(tk)+∑l=1M‍c¯ilevl(tN+l)=b¯i+a¯iia_ii-1zi(ti)+∑k=1,k≠iN‍a¯ika_kk-1zk(tk)+∑l=1M‍c¯ile_ll-1z~l(tN+l)=b¯i+zi(ti)+∑k=1,k≠iN‍a¯ika_kk-1zk(tk)+∑l=1M‍c¯ile_ll-1z~l(tN+l)≤b¯i+h~i+∑k=1,k≠iN‍pikh~k+∑l=1M‍qilh~N+l=h~i+h~i=2h~i,|v˙j(t)|0=λ|-rj(t)+∑k=1N‍djk(t)euk(t-δjk)-ejj(t)evj(t)-∑l=1,l≠jM‍ejl(t)evl(t-ξjl)|0≤r¯j+∑k=1N‍d¯jk|euk(t-δjk)|0+∑l=1,l≠jM‍e¯jl|evl(t-ξjl)|0+e¯jj|evj(t)|0=r¯j+∑k=1N‍d¯jkeuk(tk)+∑l=1,l≠jM‍e¯jlevl(tN+l)+e¯jjevj(tN+j)=r¯j+∑k=1N‍d¯jka_kk-1zk(tk)+∑l=1,l≠jM‍e¯jle_ll-1z~l(tN+l)+e¯jje_jj-1zj(tN+j)=r¯j+∑k=1N‍d¯jka_kk-1zk(tk)+∑l=1,l≠jM‍e¯jle_ll-1z~l(tN+l)+z~j(tN+j)≤r¯j+∑k=1N‍mjkh~k+∑l=1,l≠jM‍njlh~N+l+h~N+j=h~N+j+h~N+j=2h~N+j.
The above relations imply
(2.39)|u˙n(t)|0=maxt∈[0,ω]|u˙n(t)|=maxt∈[0,ω]{u˙i(t),v˙j(t)}=max{2h~i,2h~N+j}.
Further, it follows form the definition of norm that
(2.40)|un(t)|1=|un(t)|0+|u˙n(t)|0=max{lnh~ia_ii,lnh~N+je_jj}+max{2h~i,2h~N+j}.
Let us set the following:
(2.41)hn=max{lnh~ia_ii,lnh~N+je_jj}+max{2h~i,2h~N+j}+d,
where d is any positive constant.
Then for any solution ofLx=λNx, we have |un(t)|1=|un(t)|0+|u˙n(t)|0<hn for all n=1,2,…,N+M. Obviously, hn are independent of λ and the choice of U(t). Consequently, by taking this hn, the open subset Ω satisfies that Ω∩DomL, that is, the open subset Ω satisfies the assumption (i) of Lemma 2.3.
Now in the last step of the proof, we need to verify that for the given open bounded setΩ obtained in Step 2, the assumption (ii) of Lemma 2.3 also holds. That is, for each U∈∂Ω∩KerL, QNU≠0 and deg{JQN,Ω∩KerL,0}≠0.
TakeU∈∂Ω∩KerL. Then, in view of KerL=ℝN+M, U is a constant vector in ℝN+M, denoted by U=(u1,…,uN,v1,…,vM)T and with the property
(2.42)|Un|=|Un|0=|Un|1=hn.
By operating U by QN gives
(2.43)(QNU)n=(m(bi)-m(aii)eui-∑k=1,k≠iN‍m(aik)euk-∑l=1M‍m(cil)evlm(-rj)+∑k=1N‍m(djk)euk-m(ejj)evj-∑l=1,l≠jM‍m(ejl)evl).
It is easy to obtain that (QNU)n and deg{JQN,Ω∩KerL,0}≠0, where deg(·) is the Brouwer degree and J is the identity mapping since ImQ=KerL. We have shown that the open subset Ω⊂X satisfies all the assumptions of Lemma 2.3. Hence, by Lemma 2.3, the system (2.11) has at least one positive ω-periodic solution in DomL∩Ω¯. By (2.10), the system (1.3) has at least one positive ω-periodic solution. This completes the proof of Theorem 2.6.


## 2.1. Preliminaries on the Matrix Theory and Degree Theory

For convenience, we introduce some notations, definitions, and lemmas. Throughout this paper, we use the following notations.(i)
We always usei,k=1,…,N;j,l=1,…,M, unless otherwise stated.(ii)
Iff(t) is a continuous ω-periodic function defined on R, then we denote
(2.1)f_=mint∈[0,ω]|f(t)|,f¯=maxt∈[0,ω]|f(t)|,m(f)=1ω∫0ω‍f(t)dt.We use x=(x1,…,xn)T∈ℝn to denote a column vector, 𝒟=(dij)n×n is an n×n matrix, 𝒟T denotes the transpose of 𝒟, and En is the identity matrix of size n. A matrix or vector 𝒟>0 (resp., 𝒟≥0) means that all entries of 𝒟 are positive (resp., nonnegative). For matrices or vectors 𝒟 and E, 𝒟>E (resp., 𝒟≥E) means that 𝒟-E>0 (resp., 𝒟-E≥0). We denote the spectral radius of the matrix 𝒟 by ρ(𝒟).Ifv=(v1,v2,…,vn)T∈ℝn, then we have a choice of vector norms in ℝn, for instance ∥v∥1, ∥v∥2, and ∥v∥∞ are the commonly used norms, where
(2.2)∥v∥1=∑j=1n‍|vi|,∥v∥2={∑j=1n‍|vi|2}1/2,∥v∥∞=max1≤i≤n|vi|.
We recall the following norms of matrices induced by respective vector norms. For instance if 𝒜=(aij)n×n, the norm of the matrix ∥𝒜∥ induced by a vector norm ∥·∥ is defined by
(2.3)∥𝒜∥p=supv∈ℝn,v≠0∥𝒜v∥p∥v∥p=sup∥v∥p=1∥𝒜v∥p=sup∥v∥p≤1∥𝒜v∥p.
In particular one can show that ∥𝒜∥1=max1≤j≤n∑i=1n|aij|  (column  norm), ∥𝒜∥2=[λmax(𝒜T𝒜)]1/2=[max.eigenvalueof(𝒜T𝒜)]1/2and∥𝒜∥∞=max1≤i≤n∑j=1n|aij| (row  norm).Definition 2.1 (see [1, 21]).
LetX,Z be normed real Banach spaces, let L:DomL⊂X→Z be a linear mapping, and N:X→Z be a continuous mapping. The mapping L is called a Fredholm mapping of index zero, if dimKer L=codimImL<+∞ and Im L is closed in Z. If L is a Fredholm mapping of index zero and there exist continuous projectors P:X→X and Q:Z→Z such that Im P=KerL, KerQ=ImL=Im(I-Q), then L∣domL∩KerP:(I-P)X→ImL is invertible. We denote the inverse of that map by KP. If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω¯ if QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is compact. Since ImQ is isomorphic to KerL, there exists an isomorphism J:ImQ→KerL.Definition 2.2 (see [1, 22]).
LetΩ⊂ℝn be open and bounded, f∈C1(Ω,ℝn)∩C(Ω¯,ℝn) and y∈ℝn/f(∂Ω∪Nf), that is, y is a regular value of f. Here, Nf={x∈Ω:Jf(x)=0}, the critical set of f and Jf is the Jacobian of f at x. Then the degree deg{f,Ω,y} is defined by
(2.4)deg{f,Ω,y}=∑x∈f-1(y)‍sgnJf(x),
with the agreement that ∑‍ϕ=0. For more details about Degree Theory, the reader may consult Deimling [22].Lemma 2.3 (Continuation Theorem [1]).
LetΩ⊂X be an open and bounded set and L be a Fredholm mapping of index zero and N be L-compact on Ω¯ (i.e., QN(Ω¯) is bounded and KP(I-Q)N:Ω¯→X is compact). Assume(i)
for eachλ∈(0,1), x∈∂Ω∩DomL, Lx≠λNx;(ii)
for eachx∈∂Ω∩KerL, QNx≠0 and deg{JQN,Ω∩KerL,0}≠0. Then Lx=Nx has at least one solution in Ω¯∩DomL.Definition 2.4 (see [23, 24]).
A realn×n matrix 𝒜=(aij) is said to be an M-matrix if aij≤0, i,j=1,2,…,n, i≠j, and 𝒜-1≥0.Lemma 2.5 (see [23, 24]).
Let𝒜≥0 be an n×n matrix and ρ(𝒜)<1, then (En-𝒜)-1≥0, where En denotes the identity matrix of size n.Now we introduce some function spaces and their norms, which will be valid throughout this paper. Denote(2.5)X={U(t)=(u(t),v(t))T∈C1(ℝ,ℝN+M)∣U(t+ω)=U(t)∀t∈ℝ},Z={U(t)=(u(t),v(t))T∈C(ℝ,ℝN+M)∣U(t+ω)=U(t)∀t∈ℝ}.
The norms are given by
(2.6)|Un(t)|0=maxt∈[0,ω]|Un(t)|,|Un(t)|1=|Un(t)|0+|U˙n(t)|0,i=1,2,…,N+M,∥Un(t)∥0=max1≤n≤N+M{|Un(t)|0},∥Un(t)∥1=max1≤n≤N+M{|Un(t)|1}.
Obviously, X and Z, respectively, endowed with the norms ∥·∥1 and ∥·∥0 are Banach spaces.


## 2.2. Result on the Existence of Periodic Solutions

Theorem 2.6.
Assume that the following conditions hold:(H1):
the system of algebraic equations:(2.7)m(bi)-m(aii)ui-∑k=1,k≠iN‍m(aik)uk-∑l=1M‍m(cil)wl=0,i=1,2,…,N,m(-rj)+∑k=1N‍m(djk)uk-m(ejj)wj-∑l=1,l≠jM‍m(ejl)wl=0,j=1,2,…,M,has finite solution(u1*,…,uN*,w1*,…,wM*)T∈ℝ+N+M with u*>0,w*>0;(H2):
ρ(𝒦)<1,        where 𝒦=(𝒫N×N𝒬N×MℳM×N𝒩M×M)(N+M)×(N+M),
(2.8)𝒫N×N=(pik)N×N,pik={0,i=k,a¯ika_kk-1,i≠k.𝒬N×M=(qil)N×M,qil=c¯ile_ll-1,ℳM×N=(mjk)M×N,mjk=d¯jka_kk-1,𝒩M×M=(njl)M×M,njl={0,j=l,e¯jle_ll-1,j≠l.
Then system (1.3) has at least one positive ω-periodic solution.Proof.
Note that every solution(2.9)U(t)=(u(t),v(t))T=(u1(t),…,uN(t),v1(t),…,vM(t))T∈X
of the system (1.3) with the initial condition is positive. By using the following changes of variables:
(2.10)ui(t)=lnxi(t),vj(t)=lnyj(t),i=1,2,…,N,j=1,2,…,M,
the system (1.3) can be rewritten as
(2.11)u˙i(t)=bi(t)-aii(t)eui(t)-∑k=1,k≠iN‍aik(t)euk(t-τik)-∑l=1M‍cil(t)evl(t-ηil),i=1,2,…,N,v˙j(t)=-rj(t)+∑k=1N‍djk(t)euk(t-δjk)-ejj(t)evj(t)-∑l=1,l≠jM‍ejl(t)evl(t-ξjl),j=1,2,…,M.
Obviously, system (1.3) has at least one ω-periodic solution which is equivalent to the system (2.11) having at least one ω-periodic solution. To prove Theorem 2.6, our main tasks are to construct the operators (i.e., L, N, P, and Q) appearing in Lemma 2.3 and to find an appropriate open set Ω satisfying conditions (i), (ii) in Lemma 2.3.
For anyU(t)∈X, in view of the periodicity, it is easy to check that
(2.12)Δi(U,t)=bi(t)-aii(t)eui(t)-∑k=1,k≠iN‍aik(t)euk(t-τik)-∑l=1M‍cil(t)evl(t-ηil)∈Z,Δj(U,t)=-rj(t)+∑k=1N‍djk(t)euk(t-δjk)-ejj(t)evj(t)-∑l=1,l≠jM‍ejl(t)evl(t-ξjl)∈Z.
Now, we define the operators L,N as follows:
(2.13)L:DomL⊂X→Z,L(u(t),v(t))=(dui(t)dt,dvj(t)dt)∈Z,N:X→ZisdefinedbyNU=(Δi(U,t)Δj(U,t)).
Define, respectively, the projectors P:X→X and Q:Z→Z by
(2.14)PU=1ω∫0ω‍U(t)dt,U∈X,QU=1ω∫0ω‍U(t)dt,U∈Z.
It is obvious that the domain of L in X is actually the whole space, and
(2.15)KerL={x(t)∈X∣Lx(t)=0,i.e.x˙(t)=0}=ℝN+M,ImL={z(t)∈Z∣∫0ω‍z(t)dt=0}isclosedinZ.
Moreover, P,Q are continuous operators such that
(2.16)ImP=ℝN=KerL,ImL=KerQ=Im(I-Q),dim KerL=codim ImL=N+M<+∞.
It follows that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to L) KP:ImL→DomL∩KerP exists, which is given by
(2.17)KP(y)=∫0t‍y(s)ds-1ω∫0ω‍∫0t‍y(s)dsdt.
Then QN:X→Z and KP(I-Q)N:X→X are defined by
(2.18)QNU=(1ω∫0ω‍Δi(U,t)dt1ω∫0ω‍Δj(U,t)dt),KP(I-Q)Nx=∫0t‍NU(s)ds-1ω∫0ω‍∫0t‍NU(s)dsdt-(tω-12)∫0ω‍NU(s)ds.
Clearly, QN and KP(I-Q)N are continuous. By using the generalized Arzela-Ascoli theorem, it is not difficult to prove that (KP(I-Q)N)(Ω¯) is relatively compact in the space (X,∥·∥1). The proof of this step is complete.
Then, in order to apply condition (i) of Lemma2.3, we need to search for an appropriate open bounded subset Ω, denoted by
(2.19)Ω=Un(t)∈X∣|Un(t)|1=|Un(t)|0+|U˙n(t)|0<hn.
Specifically, our aim is to find an appropriate hn. Corresponding to the operator equation Lx=λNx for each λ∈(0,1), we have
(2.20)u˙i(t)=λ[bi(t)-aii(t)eui(t)-∑k=1,k≠iN‍aik(t)euk(t-τik)-∑l=1M‍cil(t)evl(t-ηil)],v˙j(t)=λ[-rj(t)+∑k=1N‍djk(t)euk(t-δjk)-ejj(t)evj(t)-∑l=1,l≠jM‍ejl(t)evl(t-ξjl)].
Since U(t)∈X, each Un(t),n=1,2,…,N+M, as components of U(t), is continuously differentiable and ω-periodic. In view of continuity and periodicity, there exists ti∈[0,ω] such that ui(ti)=maxt∈[0,ω]|ui(t)|, i=1,2,…,N, and there also exists tN+j∈[0,ω] such that vj(tN+j)=maxt∈[0,ω]|vj(t)|, j=1,2,…,M. Accordingly, u˙i(ti)=0, v˙j(tN+j)=0, and we get
(2.21)bi(ti)-aii(ti)eui(ti)-∑k=1,k≠iN‍aik(ti)euk(ti-τik)-∑l=1M‍cil(ti)evl(tN+j-ηil)=0,-rj(tN+j)+∑k=1N‍djk(tN+j)euk(ti-δjk)-ejj(tN+j)evj(tN+j)-∑l=1,l≠jM‍ejl(tN+j)evl(tN+j-ξjl)=0.
That is,
(2.22)aii(ti)eui(ti)=bi(ti)-∑k=1,k≠iN‍aik(ti)euk(ti-τik)-∑l=1M‍cil(ti)evl(tN+j-ηil),ejj(tN+j)evj(tN+j)=-rj(tN+j)+∑k=1N‍djk(tN+j)euk(ti-δjk)-∑l=1,l≠jM‍ejl(tN+j)evl(tN+j-ξjl).
Note that uk(tk)=maxt∈[0,ω]|uk(t)| and vl(tN+l)=maxt∈[0,ω]|vl(t)|, which implies
(2.23)uk(ti)≤uk(tk),uk(ti-τik)≤uk(tk),uk(ti-δjk)≤uk(tk);vl(tN+j)≤vl(tN+l),vl(tN+j-ηil)≤vl(tN+l),vl(tN+j-ξjl)≤vl(tN+l).
It follows that
(2.24)a_iieui(ti)≤|aii(ti)eui(ti)|=|bi(ti)-∑k=1,k≠iN‍aik(ti)euk(ti-τik)-∑l=1M‍cil(ti)evl(tN+j-ηil)|≤b¯i+∑k=1,k≠iN‍a¯ikeuk(ti-τik)+∑l=1M‍c¯ilevl(tN+j-ηil)≤b¯i+∑k=1,k≠iN‍a¯ikeuk(tk)+∑l=1M‍c¯ilevl(tN+l),e_jjevj(tN+j)≤|ejj(tN+j)evj(tN+j)|=|-rj(tN+j)+∑k=1N‍djk(tN+j)euk(ti-δjk)-∑l=1,l≠jM‍ejl(tN+j)evl(tN+j-ξjl)|≤r¯j+∑k=1N‍d¯jkeuk(ti-δjk)+∑l=1,l≠jM‍e¯jlevl(tN+j-ξjl)≤r¯j+∑k=1N‍d¯jkeuk(tk)+∑l=1,l≠jM‍e¯jlevl(tN+l).
Let
(2.25)a_iieui(ti)=zi(ti),e_jjevj(tN+j)=z~j(tN+j).
Using (2.25), the inequalities (2.24) become
(2.26)zi(ti)≤b¯i+∑k=1,k≠iN‍a¯ika_kk-1zk(tk)+∑l=1M‍c¯ile_ll-1z~l(tN+l),z~j(tN+j)≤r¯j+∑k=1N‍d¯jka_kk-1zk(tk)+∑l=1,l≠jM‍e¯jle_ll-1z~l(tN+l),
or
(2.27)zi(ti)-∑k=1,k≠iN‍a¯ika_kk-1zk(tk)-∑l=1M‍c¯ile_ll-1z~l(tN+l)≤b¯i,z~j(tN+j)-∑k=1N‍d¯jka_kk-1zk(tk)-∑l=1,l≠jM‍e¯jle_ll-1z~l(tN+l)≤r¯j,
which implies
(2.28)(EN×N-𝒫N×N-𝒬N×M-ℳM×NEM×M-𝒩M×M)(N+M)×(N+M)×(z1(t1),…,zN(tN),z~1(tN+1),…,z~M(tN+M))≤(b¯1,…,b¯N,r¯1,…,r¯M),
where
(2.29)𝒫N×N=(pik)N×N,pik={0,i=k,a¯ika_kk-1,i≠k,𝒬N×M=(qil)N×M,qil=c¯ile_ll-1,ℳM×N=(mjk)M×N,mjk=d¯jka_kk-1,𝒩M×M=(njl)M×M,njl={0,j=l,e¯jle_ll-1,j≠l.
Set D=(b¯1,…,b¯N,r¯1,…,r¯M)T. It follows from (2.28) and (H2) that
(2.30)(E-𝒦)(z1(t1),…,zN(tN),z~1(tN+1),…,z~M(tN+M))T≤D.
In view of ρ(𝒦)<1 and Lemma 2.5, we get (EN+M-𝒦)-1≥0. Let
(2.31)H=(h~1,…,h~N,h~N+1,…,h~N+M)T:=(E-𝒦)-1D≥0.
Using (2.30) and (2.31), we get
(2.32)(z1(t1),…,zN(tN),z~1(tN+1),…,z~M(tN+M))T≤H,
or
(2.33)zi(ti)≤h~i,i=1,2,…,N,z~j(tN+j)≤h~N+j,j=1,2,…,M.
Then
(2.34)ui(ti)≤lnh~ia_ii,vj(tN+j)≤lnh~N+je_jj,
which implies
(2.35)|un(t)|0=maxt∈[0,ω]|un(t)|=maxt∈[0,ω]{ui(ti),vj(tN+j)}=max{lnh~ia_ii,lnh~N+je_jj}.
On the other hand, it follows from (2.31) that
(2.36)(E-𝒦)H=D,orH=𝒦H+D,
that is
(2.37)h~i=∑k=1,k≠iN‍pikh~k+∑l=1M‍qilh~N+l+b¯i,h~N+j=∑k=1N‍mjkh~k+∑l=1,l≠jM‍njlh~N+l+r¯j.
Estimating (2.20), by using (2.25), (2.33), and (2.37), we have
(2.38)|u˙i(t)|0=λ|bi(t)-aii(t)eui(t)-∑k=1,k≠iN‍aik(t)euk(t-τik)-∑l=1M‍cil(t)evl(t-ηil)|0≤b¯i+a¯ii|eui(t)|0+∑k=1,k≠iN‍a¯ik|euk(t-τik)|0+∑l=1M‍c¯il|evl(t-ηil)|0=b¯i+a¯iieui(ti)+∑k=1,k≠iN‍a¯ikeuk(tk)+∑l=1M‍c¯ilevl(tN+l)=b¯i+a¯iia_ii-1zi(ti)+∑k=1,k≠iN‍a¯ika_kk-1zk(tk)+∑l=1M‍c¯ile_ll-1z~l(tN+l)=b¯i+zi(ti)+∑k=1,k≠iN‍a¯ika_kk-1zk(tk)+∑l=1M‍c¯ile_ll-1z~l(tN+l)≤b¯i+h~i+∑k=1,k≠iN‍pikh~k+∑l=1M‍qilh~N+l=h~i+h~i=2h~i,|v˙j(t)|0=λ|-rj(t)+∑k=1N‍djk(t)euk(t-δjk)-ejj(t)evj(t)-∑l=1,l≠jM‍ejl(t)evl(t-ξjl)|0≤r¯j+∑k=1N‍d¯jk|euk(t-δjk)|0+∑l=1,l≠jM‍e¯jl|evl(t-ξjl)|0+e¯jj|evj(t)|0=r¯j+∑k=1N‍d¯jkeuk(tk)+∑l=1,l≠jM‍e¯jlevl(tN+l)+e¯jjevj(tN+j)=r¯j+∑k=1N‍d¯jka_kk-1zk(tk)+∑l=1,l≠jM‍e¯jle_ll-1z~l(tN+l)+e¯jje_jj-1zj(tN+j)=r¯j+∑k=1N‍d¯jka_kk-1zk(tk)+∑l=1,l≠jM‍e¯jle_ll-1z~l(tN+l)+z~j(tN+j)≤r¯j+∑k=1N‍mjkh~k+∑l=1,l≠jM‍njlh~N+l+h~N+j=h~N+j+h~N+j=2h~N+j.
The above relations imply
(2.39)|u˙n(t)|0=maxt∈[0,ω]|u˙n(t)|=maxt∈[0,ω]{u˙i(t),v˙j(t)}=max{2h~i,2h~N+j}.
Further, it follows form the definition of norm that
(2.40)|un(t)|1=|un(t)|0+|u˙n(t)|0=max{lnh~ia_ii,lnh~N+je_jj}+max{2h~i,2h~N+j}.
Let us set the following:
(2.41)hn=max{lnh~ia_ii,lnh~N+je_jj}+max{2h~i,2h~N+j}+d,
where d is any positive constant.
Then for any solution ofLx=λNx, we have |un(t)|1=|un(t)|0+|u˙n(t)|0<hn for all n=1,2,…,N+M. Obviously, hn are independent of λ and the choice of U(t). Consequently, by taking this hn, the open subset Ω satisfies that Ω∩DomL, that is, the open subset Ω satisfies the assumption (i) of Lemma 2.3.
Now in the last step of the proof, we need to verify that for the given open bounded setΩ obtained in Step 2, the assumption (ii) of Lemma 2.3 also holds. That is, for each U∈∂Ω∩KerL, QNU≠0 and deg{JQN,Ω∩KerL,0}≠0.
TakeU∈∂Ω∩KerL. Then, in view of KerL=ℝN+M, U is a constant vector in ℝN+M, denoted by U=(u1,…,uN,v1,…,vM)T and with the property
(2.42)|Un|=|Un|0=|Un|1=hn.
By operating U by QN gives
(2.43)(QNU)n=(m(bi)-m(aii)eui-∑k=1,k≠iN‍m(aik)euk-∑l=1M‍m(cil)evlm(-rj)+∑k=1N‍m(djk)euk-m(ejj)evj-∑l=1,l≠jM‍m(ejl)evl).
It is easy to obtain that (QNU)n and deg{JQN,Ω∩KerL,0}≠0, where deg(·) is the Brouwer degree and J is the identity mapping since ImQ=KerL. We have shown that the open subset Ω⊂X satisfies all the assumptions of Lemma 2.3. Hence, by Lemma 2.3, the system (2.11) has at least one positive ω-periodic solution in DomL∩Ω¯. By (2.10), the system (1.3) has at least one positive ω-periodic solution. This completes the proof of Theorem 2.6.


## 3. Globally Asymptotic Stability

Under the assumption of Theorem2.6, we know that system (1.3) has at least one positive ω-periodic solution, denoted by X*(t)=(x1*(t),…,xN*(t),y1*(t),…,yM*(t))T. The aim of this section is to derive a set of sufficient conditions which guarantee the existence and global asymptotic stability of the positive ω-periodic solution X*(t).Before the formal analysis, we recall some facts which will be used in the proof.Lemma 3.1 (see [25]).
Letf be a nonnegative function defined on [0,+∞] such that f is integrable on [0,+∞] and is uniformly continuous on [0,+∞]. Then limt→+∞f(t)=0.Lemma 3.2 (see [23, 24]).
Let𝒦=(Γij)n×n be a matrix with nonpositive off-diagonal elements. 𝒦 is an M-matrix if and only if there exists a positive diagonal matrix ξ=diag(ξ1,ξ2,…,ξn) such that
(3.1)ξia_ii>∑j≠i‍ξja¯ij,i=1,2,…,n.Theorem 3.3.
Assume that all the assumptions in Theorem2.6 hold. Then system (1.3) has a unique positive ω-periodic solution X*(t) which is globally asymptotically stable.Proof.
LetX(t)=(x(t),y(t))T=(x1(t),…,xN(t),y1(t),…,yM(t))T be any positive solution of system (1.3). It is easy to see that ρ(𝒦T)=ρ(𝒦)<1. Thus, in view of Lemma 2.5 and Definition 2.4, (E-𝒦T) is an M-matrix, where E denotes an identity matrix of size N+M. Therefore, by Lemma 3.2, there exists a diagonal matrix L=diag(α1,…,αN,β1,…,βM) with positive diagonal elements such that the product (E-𝒦T)L is strictly diagonally dominant with positive diagonal entries, namely,
(3.2)αia_ii>∑k=1,k≠iN‍αka¯ki+∑l=1M‍βld¯li,i=1,…,N,βje_jj>∑k=1N‍αkc¯kj+∑l=1,l≠jM‍βle¯lj,j=1,…,M.
Now, we define a Lyapunov function V(t) as follows:
(3.3)V(t)=∑i=1N‍αi[|lnxi(t)-lnxi*(t)|+∑K=1,K≠iN‍∫t-τiktaik(s+τik)|xk(t)-xk*(t)|ds+∑l=1M‍∫t-ηiltcil(s+ηil)|yl(t)-yl*(t)|ds∑K=1,K≠iN‍∫t-τiktaik(s+τik)|xk(t)-xk*(t)|]+∑j=1M‍βj[∑l=1,l≠jM‍|lnyj(t)-lnyj*(t)|+∑K=1N‍∫t-δjktdjk(s+δjk)|xk(t)-xk*(t)|ds+∑l=1,l≠jM‍∫t-ξjltejl(s+ξjl)|yl(t)-yl*(t)|ds∑l=1,l≠jM‍],t≥t0.
Calculating the upper right derivative of V(t) and using (3.2), we get
(3.4)D+V(t)≤∑i=1N‍αi[∑l=1,l≠jM‍-aii(t)|xi(t)-xi*(t)|+∑k=1,k≠iN‍aik(t+τik)|xk(t)-xk*(t)|+∑l=1M‍cil(t+ηil)|yl(t)-yl*(t)|∑l=1,l≠jM‍]+∑j=1M‍βj[∑l=1,l≠jM‍-ejj(t)|yj(t)-yj*(t)|+∑k=1N‍djk(t+δjk)|xk(t)-xk*(t)|+∑l=1,l≠jM‍ejl(t+ξjl)|yl(t)-yl*(t)|∑l=1,l≠jM‍]≤-∑i=1N‍αi[∑l=1,l≠jM‍-a_ii|xi(t)-xi*(t)|+∑k=1,k≠iN‍a¯ik|xk(t)-xk*(t)|+∑l=1M‍c¯il|yl(t)-yl*(t)|∑l=1,l≠jM‍]+∑j=1M‍βj[∑l=1,l≠jM‍-e_jj|yj(t)-yj*(t)|+∑k=1N‍d¯jk|xk(t)-xk*(t)|+∑l=1,l≠jM‍e¯jl|yl(t)-yl*(t)|∑l=1,l≠jM‍]=-∑i=1N‍(αia_ii-∑k=1,k≠iN‍αka¯ki-∑l=1M‍βld¯li)|xi(t)-xi*(t)|-∑j=1M‍(βje_jj-∑k=1N‍αkc¯kj-∑l=1,l≠jM‍βle¯lj)|yj(t)-yj*(t)|=-c{∑i=1N‍|xi(t)-xi*(t)|+∑j=1M‍|yj(t)-yj*(t)|},
where
(3.5)c=min{αia_ii-∑k=1,k≠iN‍αka¯ki-∑l=1M‍βld¯li,βje_jj-∑k=1N‍αkc¯kj-∑l=1,l≠jM‍βle¯lj}>0.
It follows from (3.4) that D+V(t)≤0. Obviously, the zero solution of (1.3) is Lyapunov stable. On the other hand, integrating (3.4) over [t0,t] leads to
(3.6)V(t)-V(t0)⩽-c∫t0t‍[∑i=1N‍|xi(s)-xi*(s)|+∑j=1M‍|yj(s)-yj*(s)|]ds,t⩾0,
or
(3.7)V(t)+c∫t0t‍[∑i=1N‍|xi(s)-xi*(s)|+∑j=1M‍|yj(s)-yj*(s)|]ds⩽V(t0)<+∞,t⩾t0.
Noting that V(t)⩾0, it follows that
(3.8)∫t0t‍[∑i=1N‍|xi(s)-xi*(s)|+∑j=1M‍|yj(s)-yj*(s)|]ds⩽V(t0)c<+∞,t⩾t0.
Therefore, by Lemma 3.1, it is not difficult to conclude that
(3.9)limt→+∞|Xi(t)-Xi*(t)|=0.
Theorem 3.3 follows.


## 4. Corollaries and Remarks

In order to illustrate some features of our main results, we will present some corollaries and remarks in this section. From the proofs of Theorems2.6 and 3.3, one can easily deduce the following corollary.Corollary 4.1.
In addition to(H1), further suppose that E-𝒦 or E-𝒦T is an M-matrix. Then system (1.3) has a unique positive ω-periodic solution which is globally asymptotically stable.Now recall that for a given matrix𝒦, its spectral radius ρ(𝒦) is equal to the minimum of all matrix norms of 𝒦, that is, for any matrix norm ∥·∥,ρ(𝒦)≤∥𝒦∥. Therefore, we have the following corollary.Corollary 4.2.
In addition to(H1), if one further supposes that there exist positive constants ξi,i=1,2,…,n,ηj,j=1,2,…,m such that one of the following inequalities holds.  (1)
max{max1≤k≤n{a_kk-1ξk-1[∑i=1,i≠knξia¯ik+∑l=1mηjd¯jk]},max1≤l≤m{e_ll-1ηl-1[∑i=1nξic¯li+∑j=1,j≠lmηje¯jl]}} < 1, or equivalently, for all k=1,…,n,l=1,2,…,n,
(4.1)ξka_kk>∑i=1,i≠kn‍ξia¯ik+∑l=1m‍ηjd¯jk,ηle_ll>∑i=1n‍ξic¯li+∑j=1,j≠lm‍ηje¯jl.(2)
∑i=1n+m‍∑j=1n+m‍(ξi-1ξjkij)2<1, where 𝒦=(kij)(n+m)×(n+m) has been defined in Theorem 2.6.(3)
max{max1≤i≤n{a_ii-1ξi-1[∑k=1,k≠inξka¯ki+∑l=1mηld¯li]},max1≤j≤m{e_jj-1ηj-1[∑k=1nξkc¯kj+∑l=1,l≠jmηle¯lj]}} < 1, or equivalently, for all i=1,…,n,j=1,2,…,n,
(4.2)ξiaii>∑k=1,k≠in‍ξk|aki|+∑l=1m‍ηl|dli|,ηjejj>∑k=1n‍ξk|ckj|+∑l=1,l≠jm‍ηl|elj|. Then system (1.3) has a unique positive ω-periodic solution which is globally asymptotically stable.Proof.
For any matrix norm∥·∥ and any nonsingular matrix S, ∥𝒦∥S=∥S-1𝒦S∥ also defines a matrix norm. Let us denote D=diag(ξ1,ξ2,…,ξn), then the conditions (1.2) and (1.3) correspond to the column norms and Frobenius norm of matrix D𝒦D-1, respectively. Condition (2.10) corresponds to the row norms of D𝒦TD-1 and note that ρ(D𝒦TD-1)=ρ(D𝒦D-1). Now Corollary 4.2 follows immediately.


## 5. Example

In this section, an example and its simulations are presented to illustrate the feasibility and effectiveness of our results.Example 5.1.
Consider the following periodic predator-prey model with2-predators and 1-prey:
(5.1)x˙1(t)=x1(t)[7+sint-x1(t)-14x2(t)-110y1(t-1)],x˙2(t)=x2(t)[7+cost-14x1(t)-x2(t)-14y1(t)],y˙1(t)=y1(t)[-120(1+cost)+32x1(t-1)+14x2(t)-12y1(t)].
Simple computation leads to
(5.2)𝒦=(0a22-1a12e11-1c11a11-1a210e11-1c21a11-1d11a22-1d120)=(01415140032140).
By using mathematica, we see that ρ(𝒦)=0.633982<1. Thus, the system (5.1) has a periodic solution which is globally asymptotically stable. Figure 1 shows the asymptotic behavior of the periodic solution.Figure 1
Asymptotic behavior of system (5.1) with initial values (x1(0),x2(0),y1(0))=(1,1,1),(0.3,0.3,0.3),(7,7,7), respectively, t∈[0,25].Remark 5.2.
In this example, one can observe that though the spectralρ(𝒦)<1, the matrix norms of the matrix 𝒦 are all bigger than 1. For instance, the column norm: is(5.3)∥𝒦∥1=max1≤j≤3{ajj-1∑i=1,i≠j3‍aij}=0+32+14>1.



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*Source: 940287-2012-10-02.xml*