Global Attractivity of a Periodic Delayed 𝑁-Species Model of Facultative Mutualism

Abstract

Two classes of periodic 𝑁-species Lotka-Volterra facultative mutualism systems with distributed delays are discussed. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin and the Lyapunov function method, some new sufficient conditions on the existence and global attractivity of positive periodic solutions are established.

1. Introduction

Mutualism is the interaction of two species of organisms that benefits both [1]. In general, mutualism may be either obligate or facultative. Obligate mutualist may survive only by association, and facultative mutualist, while benefiting from the presence of each other, may also survive in the absence of any of them [2]. As it is well known, in recent years the nonautonomous and periodic population dynamical systems are extensively studied. The basic and important studied questions for these systems are the persistence, permanence, and extinction of species, global stability of systems and the existence of positive periodic solutions, positive almost periodic solutions and strictly positive solutions, and so forth. Many important and influential results have been established and can be found in many articles and books. Particularly, the existence of positive periodic solutions for various type population dynamical systems has been extensively studied in [1–16] and the references cited therein.

In [7], the authors studied the following delayed two-species model of facultative mutualism: 𝑦̇₁(𝑡)=𝑦₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑦₁(𝑡)−𝑏₁(𝑡)𝑦₁𝑡−𝜏₁(𝑡)+𝑐₁(𝑡)𝑦₂𝑡−𝑝₁(𝑡),𝑦̇₂(𝑡)=𝑦₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦₂(𝑡)−𝑏₂(𝑡)𝑦₂𝑡−𝜏₁(𝑡)+𝑐₂(𝑡)𝑦₁𝑡−𝑝₁(𝑡). (1) By using the technique of coincidence degree and the Lyapunov functionals method, the sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions are obtained for system (1). In [2], the authors considered the following periodic delayed two-species model of facultative mutualism: 𝑦̇₁(𝑡)=𝑦₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑦₁(𝑡)+𝑏₁(𝑡)𝑦₁𝑡−𝜏₁(𝑡)+𝑐₁(𝑡)𝑦₂𝑡−𝑝₁(𝑡),𝑦̇₂(𝑡)=𝑦₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦₂(𝑡)+𝑏₂(𝑡)𝑦₂𝑡−𝜏₁(𝑡)+𝑐₂(𝑡)𝑦₁𝑡−𝑝₁(𝑡). (2) By means of the methods of coincidence degree and the Lyapunov functional, the sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions are established for system (2). In [12], the following 𝑛-species periodic Lotka-Volterra type competitive systems with feedback controls and finite and infinite distributed delays are discussed: 𝑦̇_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−𝑎_𝑖𝑖(𝑡)𝑦_𝑖(𝑡)−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗(𝑡)^𝜔₀𝐾_𝑖𝑗(𝑠)𝑦_𝑗(𝑡−𝑠)𝑑𝑠−𝛼_𝑖(𝑡)^𝜔₀𝐻_𝑖(𝑠)𝑢_𝑖(),𝑢̇𝑡−𝑠𝑑𝑠_𝑖(𝑡)=−𝜂_𝑖(𝑡)+𝑎_𝑖(𝑡)^𝜔₀𝐾_𝑖(𝑠)𝑦_𝑖(𝑡−𝑠)𝑑𝑠, (3) where 𝑖=1,2,…,𝑛. By using the technique of coincidence degree and the Lyapunov functionals method, the sufficient conditions for the existence and global stability of positive periodic solutions are obtained for system (3).

Motivated by the above works, in this paper, we investigate the following two classes of 𝑛 species periodic model of facultative mutualism with finite distributed delays: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−𝛼_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛𝑚_𝑗=1𝑙=1𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)𝑥_𝑗(𝑡+𝑠)𝑑𝑠𝑖=1,2,…𝑛,𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−𝛼_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑚_𝑙=1𝑎_𝑖𝑖𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑖𝑙(𝑠)𝑥_𝑖()+𝑡+𝑠𝑑𝑠_{𝑛𝑚𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)𝑥_𝑗(𝑡+𝑠)𝑑𝑠𝑖=1,2,…𝑛. (4)(5)

By using the technique of coincidence degree developed by Gaines and Mawhin in [17] and the Lyapunov functional method, we will establish some new sufficient conditions which guarantee that the system has at least one positive periodic solution and is globally attractive.

The organization of this paper is as follows. In the next section we will present some basic assumptions and main definitions and lemmas. In Section 3, conditions for the existence and global attractivity of positive periodic solution. In Section 4, two examples are given to illustrate that our main results are applicable. In the final section, we will discuss what we study in this paper and what we had in this paper.

2. Preliminaries

In systems (4) and (5), we have that 𝑥_𝑖(𝑡)(𝑖=1,2,…𝑛) represent the density of 𝑛 species 𝑥_𝑖(𝑖=1,2,…𝑛) at time 𝑡, respectively; 𝑟_𝑖(𝑡)(𝑖=1,2,…𝑛) represent the intrinsic growth rate of species 𝑥_𝑖(𝑖=1,2,…𝑛) at time 𝑡, respectively; 𝛼_𝑖(𝑡)(𝑖=1,2,…,𝑛) represent the intrapatch restriction density of species 𝑥_𝑖(𝑖=1,2,…𝑛) at time 𝑡, respectively; 𝑎_𝑖𝑗𝑙(𝑡)(𝑙=1,2,…,𝑚,𝑖≠𝑗,𝑖,𝑗=1,2,…,𝑛) represent the mutualism coefficients between 𝑛 species 𝑥_𝑖(𝑖=1,2,…𝑛) at time 𝑡, respectively, while 𝜏≥0 is a constant and 𝜏 may be +∞. System (4) involves positive feedback terms ^𝑚_𝑙=1𝑎_𝑖𝑖𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑖𝑙(𝑠)𝑥_𝑖(𝑡+𝑠)𝑑𝑠,𝑖=1,2,…,𝑛 (6) which can be regarded as the passive effect on the growth rate of a species, and system (5) involves negative feedback terms −^𝑚_𝑙=1𝑎_𝑖𝑖𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑖𝑙(𝑠)𝑥_𝑖(𝑡+𝑠)𝑑𝑠,𝑖=1,2,…,𝑛 (7) which are due to gestation. In this paper, we always assume the following:(H1)??𝑟_𝑖(𝑡) are 𝜔-periodic continuous functions with ∫^𝜔₀𝑟_𝑖(𝑡)𝑑𝑡>0(𝑖=1,2,…,𝑛); 𝛼_𝑖(𝑡)(𝑖=1,2,…,𝑛) and 𝑎_𝑖𝑗𝑙(𝑡)(𝑖,𝑗=1,2,…,𝑛;𝑙=1,2,…,𝑚) are positive 𝜔-periodic continuous functions; 𝑘_𝑖𝑗𝑙(𝑠)(𝑖,𝑗=1,2,…,𝑛;𝑙=1,2,…,𝑚) are nonnegative integrable functions on [−𝜏,0] satisfying ∫⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)𝑑𝑠=1.

From the viewpoint of mathematical biology, in this paper for systems (4) and (5) we only consider the solution with the following initial condition: 𝑥_𝑖(𝑡)=𝜙_𝑖(𝑠),∀𝑠∈[−𝜏,0],𝑖=1,2,…,𝑛, (8) where 𝜙_𝑖(𝑠)(𝑖=1,2,…,𝑛) are nonnegative continuous functions defined on [−𝜏,0] satisfying 𝜙_𝑖(0)>0(𝑖=1,2,…,𝑛).

In this paper, for any 𝜔-periodic continuous function 𝑓(𝑡) we denote the following: 𝑓^𝐿=min_{[]𝑡∈0,𝜔}𝑓(𝑡),𝑓^𝑀=max_{[]𝑡∈0,𝜔}𝑓(𝑡),1𝑓=𝜔^𝜔₀𝑓(𝑡)𝑑𝑡. (9)

In order to obtain the existence of positive 𝜔-periodic solutions of systems (4) and (5), we will use the continuation theorem developed by Gaines and Mawhin in [17]. For the reader’s convenience, we will introduce the continuation theorem in the following.

Let 𝑋 and 𝑍 be two normed vetor spaces. Let 𝐿∶Dom𝐿⊂𝑋→𝑍??be a linear operator and 𝑁∶𝑋→𝑍??be a continuous operator. The operator 𝐿 is called a Fredholm operator of index zero, if??dimKer𝐿=codimIm𝐿<∞ and Im𝐿 is a closed set in 𝑍. If 𝐿 is a Fredholm operator of index zero, then there exist continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 such that Im𝑃=Ker𝐿 and Im𝐿=Ker𝑄=Im(𝐼−𝑄). It follows that 𝐿∣Dom𝐿∩Ker𝑃∶Dom𝐿∩Ker𝑃→Im𝐿 is invertible and its inverse is denoted by 𝐾_𝑃 and denote by 𝐽∶Im𝑄→Ker𝐿 an isomorphism of Im𝑄 onto Ker?𝐿. Let Ω be a bounded open subset of 𝑋, we say that the operator 𝑁 is 𝐿-compact on Ω, where Ω denotes the closure of Ω in 𝑋, if 𝑄𝑁(Ω) is bounded and 𝐾_𝑃(𝐼−𝑄)𝑁∶Ω→𝑋 is compact.

Lemma 1 (see [17]). Let 𝐿 be a Fredholm operator of index zero and let 𝑁 be 𝐿-compact on Ω. If(a)for each 𝜆∈(0,1) and 𝑥∈𝜕Ω∩Dom𝐿, 𝐿𝑥≠𝜆𝑁𝑥;(b)for each 𝑥∈𝜕Ω∩Ker𝐿, 𝑄𝑁𝑥≠0;(c)deg{𝐽𝑄𝑁,Ω∩Ker𝐿,0}≠0,then the operator equation 𝐿𝑥=𝑁𝑥 has at least one solution lying in Dom𝐿∩Ω.

3. Main Results

Now, for the convenience of statements, we denote the function 𝑎_𝑖𝑗(𝑡)=^𝑚_𝑙=1𝑎_𝑖𝑗𝑙(𝑡),𝑖,𝑗=1,2,…,𝑛. (10)

The following theorem is about the existence and global attractivity of positive periodic solutions of system (4).

Theorem 2. Suppose that assumption (𝐇𝟏) holds and there exists a constant 𝜇_𝑖>0(𝑖=1,2,…,𝑛) such that min_{𝑡∈[0,𝜔]}𝜇_𝑖𝛼_𝑖(𝑡)−^𝑛𝑚_𝑗=1𝑙=1𝜇_𝑗⁰_−𝜏𝑎_𝑗𝑖𝑙(𝑡−𝑠)𝑘_𝑗𝑖𝑙(𝑠)𝑑𝑠=∶𝛿_𝑖>0,𝑖=1,2,…,𝑛, (11) and the algebraic equation 𝑟_𝑖−𝛼_𝑖𝑣_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗𝑣_𝑗=0,𝑖=1,2,…,𝑛 (12) has a unique positive solution. Then, system (4) has a positive 𝜔-periodic solution which is globally attractive.

Proof. We firstly consider the existence of positive periodic solutions of system (4). For system (4), we introduce new variables 𝑢_𝑖(𝑡)(𝑖=1,2,…,𝑛) such that 𝑥_𝑖(𝑡)=exp𝑢_𝑖(𝑡),𝑖=1,2,…,𝑛. (13) Then, system (4) is rewritten in the following form: 𝑢̇_𝑖(𝑡)=𝑟_𝑖(𝑡)−𝛼_𝑖(𝑡)exp𝑢_𝑖(𝑡+_𝑚𝑙=1)𝑎_𝑖𝑖𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑖𝑙(𝑠)exp𝑢_𝑖(+𝑡+𝑠)𝑑𝑠_{𝑛𝑚𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(𝑡+𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (14) In order to apply Lemma 1 to system (14), we introduce the normed vector spaces 𝑋 and 𝑍 as follows. Let 𝐶(𝑅,𝑅^𝑛) denote the space of all continuous function 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡))∶𝑅→𝑅^𝑛. We take 𝑋=𝑍=𝑢(𝑡)∈𝐶𝑅,𝑅^𝑛∶𝑢(𝑡)isan𝜔-periodicfunction (15) with norm ‖𝑢‖=^𝑛_𝑖=1max_{[]𝑡∈0,𝜔}𝑢_𝑖(𝑡). (16) It is obvious that 𝑋 and 𝑍 are the Banach spaces.
We define a linear operator 𝐿∶Dom𝐿⊂𝑋→𝑍 and a continuous operator 𝑁∶𝑋→𝑍 as follows: (𝑡)=𝑢̇(𝑡),(𝑡)=𝐿𝑢𝑁𝑢𝑁𝑢₁(𝑡),𝑁𝑢₂(𝑡),…,𝑁𝑢_𝑛(𝑡), (17) where 𝑁𝑢_𝑖(𝑡)=𝑟_𝑖(𝑡)−𝛼_𝑖(𝑡)exp𝑢_𝑖(𝑡+_𝑚𝑙=1)𝑎_𝑖𝑖𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑖𝑙(𝑠)exp𝑢_𝑖(+𝑡+𝑠)𝑑𝑠_{𝑛𝑚𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(𝑡+𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (18) Further, we define continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 as follows: (𝑡)=1𝑃𝑢𝜔^𝜔₀𝑢(𝑡)(𝑡)=1𝑑𝑡,𝑄𝑣𝜔^𝜔₀𝑣(𝑡)𝑑𝑡. (19) We easily see that??Im𝐿={𝑣∈𝑍∶∫^𝜔₀𝑣(𝑡)𝑑𝑡=0} and Ker𝐿=𝑅^𝑛. It is obvious that Im𝐿 is closed in 𝑍 and dimKer𝐿=𝑛. Since for any 𝑣∈𝑍 there are unique 𝑣₁∈𝑅^𝑛 and 𝑣₂∈Im𝐿 with 𝑣₁=1𝜔^𝜔₀𝑣(𝑡)𝑑𝑡,𝑣₂(𝑡)=𝑣(𝑡)−𝑣₁, (20) such that 𝑣(𝑡)=𝑣₁+𝑣₂(𝑡), we have co?dim?Im𝐿=𝑛. Therefore, 𝐿 is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to 𝐿) 𝐾_𝑝∶Im𝐿→Ker𝑃∩Dom𝐿 is given in the following form: 𝐾_𝑝𝑣(𝑡)=^𝑡₀𝑣(𝑠)1𝑑𝑠−𝜔^𝜔₀^𝑡₀𝑣(𝑠)𝑑𝑠𝑑𝑡. (21) For convenience, we denote 𝐹(𝑡)=(𝐹₁(𝑡),𝐹₂(𝑡),…,𝐹_𝑛(𝑡)) as follows: 𝐹_𝑖(𝑡)=𝑟_𝑖(𝑡)−𝛼_𝑖(𝑡)exp𝑢_𝑖(𝑡+_𝑚𝑙=1)𝑎_𝑖𝑖𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑖𝑙(𝑠)exp𝑢_𝑖(+𝑡+𝑠)𝑑𝑠_{𝑛𝑚𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(𝑡+𝑠)𝑑𝑠,𝑖=1,2,…𝑛. (22) Thus, we have (𝑡)=1𝑄𝑁𝑢𝜔^𝜔₀𝐹(𝑡)𝐾𝑑𝑡,_𝑝(𝐼−𝑄)𝑁𝑢(𝑡)=𝐾_𝑝𝐼𝑁𝑢(𝑡)−𝐾_𝑝(𝑡)=𝑄𝑁𝑢_𝑡0𝐹(𝑠)1𝑑𝑠−𝜔^𝜔₀^𝑡₀𝐹(𝑠)+1𝑑𝑠𝑑𝑡2−𝑡𝜔^𝜔₀𝐹(𝑠)𝑑𝑠. (23) From formulas (23), we easily see that 𝑄𝑁 and 𝐾_𝑝(𝐼−𝑄)𝑁 are continuous operators. Furthermore, it can be verified that 𝐾_𝑝(𝐼−𝑄)𝑁(Ω) is compact for any open bounded set Ω⊂𝑋 by using Arzela-Ascoli theorem and 𝑄𝑁(Ω) is bounded. Therefore, 𝑁 is 𝐿-compact on Ω for any open bounded subset Ω⊂𝑋.
Now, we reach the position to search for an appropriate open bounded subset Ω for the application of the continuation theorem (Lemma 1) to system (4).
Corresponding to the operator equation 𝐿𝑢(𝑡)=𝜆𝑁𝑢(𝑡) with parameter 𝜆∈(0,1), we have 𝑢̇_𝑖(𝑡)=𝜆𝐹_𝑖(𝑡),𝑖=1,2,…,𝑛, (24) where 𝐹_𝑖(𝑡)(𝑖=1,2,…,𝑛) is given in (22).
Assume that 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡))∈𝑋 is a solution of system (24) for some parameter 𝜆∈(0,1). By integrating system (24) with the interval [0,𝜔], we obtain the following: ^𝜔₀𝑟_𝑖(𝑡)−𝛼_𝑖(𝑡)exp𝑢_𝑖(𝑡+_𝑚𝑙=1)𝑎_𝑖𝑖𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑖𝑙(𝑠)exp𝑢_𝑖(+𝑡+𝑠)𝑑𝑠_{𝑛𝑚𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(𝑡+𝑠)𝑑𝑠𝑑𝑡=0,𝑖=1,2,…𝑛. (25) Consequently, ^𝜔₀𝛼_𝑖(𝑡)exp𝑢_𝑖(𝑡−_𝑚𝑙=1)𝑎_𝑖𝑖𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑖𝑙(𝑠)exp𝑢_𝑖(−𝑡+𝑠)𝑑𝑠_{𝑛𝑚𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(=𝑡+𝑠)𝑑𝑠𝑑𝑡𝑟_𝑖𝜔,𝑖=1,2,…𝑛. (26) From the continuity of 𝑢(𝑡)=(𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡)), there exist constants 𝜉_𝑖,𝜂_𝑖∈[0,𝜔](𝑖=1,2,…,𝑛) such that 𝑢_𝑖𝜉_𝑖=max_{[]𝑡∈0,𝜔}𝑢_𝑖(𝑡),𝑢_𝑖𝜂_𝑖=min_{[]𝑡∈0,𝜔}𝑢_𝑖(𝑡),𝑖=1,2,…,𝑛. (27) From (26) and (27), we obtain ^𝜔₀𝛼_𝑖(𝑡)exp𝑢_𝑖𝜉_𝑖𝑑𝑡≥𝑟_𝑖𝜔,𝑖=1,2,…,𝑛. (28) Therefore, we further have 𝑢_𝑖𝜉_𝑖≥ln𝑟_𝑖𝛼_𝑖,𝑖=1,2,…,𝑛. (29) For each 𝑖,𝑗=1,2,…,𝑛 and 𝑙=1,2,…,𝑚, we have ^𝜔₀𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(=𝑡+𝑠)𝑑𝑠𝑑𝑡_{0−𝜏𝜔0}𝑎_𝑖𝑗𝑙(𝑡)𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(=𝑡+𝑠)𝑑𝑡𝑑𝑠_{0−𝜏𝑠𝑠+𝜔}𝑎_𝑖𝑗𝑙(𝑣−𝑠)𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(𝑣=)𝑑𝑣𝑑𝑠_{0−𝜏𝜔0}𝑎_𝑖𝑗𝑙(𝑣−𝑠)𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(𝑣=)𝑑𝑣𝑑𝑠_{𝜔00−𝜏}𝑎_𝑖𝑗𝑙(𝑣−𝑠)𝑘_𝑖𝑗𝑙(𝑠)exp𝑢_𝑗(𝑣=)𝑑𝑠𝑑𝑣_{𝜔00−𝜏}𝑎_𝑖𝑗𝑙(𝑡−𝑠)𝑘_𝑖𝑗𝑙(𝑠)𝑑𝑠exp𝑢_𝑗(𝑡)𝑑𝑡. (30) Hence, from (26) we further obtain ^𝜔₀𝛼_𝑖(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_𝑖𝑖𝑙(𝑡−𝑠)𝑘_𝑖𝑖𝑙(𝑠)𝑑𝑠exp𝑢_𝑖(𝑡−_{𝑛𝑚𝑗≠𝑖𝑙=10−𝜏})𝑎_𝑖𝑗𝑙(𝑡−𝑠)𝑘_𝑖𝑗𝑙(𝑠)𝑑𝑠exp𝑢_𝑗(𝑡=)𝑑𝑡𝑟_𝑖𝜔,𝑖=1,2,…,𝑛. (31) Consequently, ^𝜔₀𝛼₁(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_11𝑙(𝑡−𝑠)𝑘_11𝑙(𝑠)𝑑𝑠exp𝑢₁(𝑡−_{𝑛𝑚𝑗≠1𝑙=10−𝜏})𝑎_1𝑗𝑙(𝑡−𝑠)𝑘_1𝑗𝑙(𝑠)𝑑𝑠exp𝑢_𝑗(𝑡+)𝑑𝑡_𝜔0𝛼₂(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_22𝑙(𝑡−𝑠)𝑘_22𝑙(𝑠)𝑢𝑑𝑠×exp₂(𝑡)−^𝑛𝑚_{𝑗≠2𝑙=1}⁰_−𝜏𝑎_2𝑗𝑙(𝑡−𝑠)𝑘_2𝑗𝑙(𝑠)𝑢𝑑𝑠×exp_𝑗(𝑡)𝑑𝑡+⋯+^𝜔₀𝛼_𝑛(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_𝑛𝑛𝑙(𝑡−𝑠)𝑘_𝑛𝑛𝑙(𝑠)𝑢𝑑𝑠×exp_𝑛(𝑡−_{𝑛𝑚𝑗≠𝑛𝑙=10−})_𝜏𝑎_𝑛𝑗𝑙(𝑡−𝑠)𝑘_𝑛𝑗𝑙(𝑠)𝑢𝑑𝑠×exp_𝑗(𝑡=)𝑑𝑡_𝜔0𝛼₁(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_11𝑙(𝑡−𝑠)𝑘_11𝑙(𝑠)+𝑑𝑠_{𝑛𝑗≠10−𝜏}𝑎_𝑗1𝑙(𝑡−𝑠)𝑘_𝑗1𝑙(𝑠)𝑢𝑑𝑠×exp₁(𝑡+)𝑑𝑡_𝜔0𝛼₂(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_22𝑙(𝑡−𝑠)𝑘_22𝑙(𝑠)+𝑑𝑠_{𝑛𝑗≠20−𝜏}𝑎_𝑗2𝑙(𝑡−𝑠)×𝑘_𝑗2𝑙(𝑠)𝑢𝑑𝑠×exp₂(𝑡)𝑑𝑡+⋯+^𝜔₀𝛼_𝑛(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_𝑛𝑛𝑙(𝑡−𝑠)𝑘_𝑛𝑛𝑙(𝑠)+𝑑𝑠_{𝑛𝑗≠𝑛0−𝜏}𝑎_𝑗𝑛𝑙(𝑡−𝑠)×𝑘_𝑗𝑛𝑙(𝑠)𝑢𝑑𝑠×exp_𝑛(𝑡=)𝑑𝑡_𝜔0𝛼₁(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_11𝑙(𝑡−𝑠)𝑘_11𝑙(𝑠)+^𝑛_𝑗≠1𝑎_𝑗1𝑙(𝑡−𝑠)×𝑘_𝑗1𝑙(𝑠)𝑢𝑑𝑠×exp₁(𝑡+)𝑑𝑡_𝜔0𝛼₂(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_22𝑙(𝑡−𝑠)𝑘_22𝑙(𝑠)+^𝑛_𝑗≠2𝑎_𝑗2𝑙(𝑡−𝑠)×𝑘_𝑗2𝑙(𝑠𝑢)𝑑𝑠×exp₂(𝑡)𝑑𝑡+⋯+^𝜔₀𝛼_𝑛(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_𝑛𝑛𝑙(𝑡−𝑠)𝑘_𝑛𝑛𝑙(𝑠)+^𝑛_𝑗≠𝑛𝑎_𝑗𝑛𝑙(𝑡−𝑠)×𝑘_𝑗𝑛𝑙(𝑠)𝑢𝑑𝑠×exp_𝑛(𝑡)𝑑𝑡=^𝑛_𝑖=1𝑟_𝑖𝜔. (32) From the assumptions of Theorem 2, we can obtain ^𝜔₀𝛼_𝑖(𝑡)−^𝑚_𝑙=1⁰_−𝜏𝑎_𝑖𝑖𝑙(𝑡−𝑠)𝑘_𝑖𝑖𝑙(𝑠)+^𝑛_𝑗≠𝑖𝑎_𝑗𝑖𝑙(𝑡−𝑠)×𝑘_𝑗𝑖𝑙(𝑠𝑢)𝑑𝑠×exp_𝑖(𝑡≤)𝑑𝑡_𝑛𝑖=1𝑟_𝑖𝜔,𝑖=1,2,…,𝑛. (33) Hence, 𝛿_𝑖^𝜔₀exp𝑢_𝑖(𝑡)𝑑𝑡≤^𝑛_𝑖=1𝑟_𝑖𝜔,𝑖=1,2,…,𝑛. (34) Consequently, exp𝑢^𝜔_0𝑖(𝑡∑)𝑑𝑡≤^𝑛𝑖=1𝑟_𝑖𝜔𝛿_𝑖,𝑖=1,2,…,𝑛. (35) From (35), we further obtain 𝑢_𝑖𝜂_𝑖∑≤ln^𝑛𝑖=1𝑟_𝑖𝛿_𝑖,𝑖=1,2,…,𝑛. (36) On the other hand, directly from system (14) we have 𝑢̇^𝜔_0𝑖(𝑡≤)𝑑𝑡_𝜔0𝑟_𝑖(𝑡)+𝛼_𝑖(𝑡)exp𝑢_𝑖(𝑡+_𝑚𝑙=1)𝑎_𝑖𝑖𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑖𝑙(𝑠)exp𝑢_𝑖(+𝑡+𝑠)𝑑𝑠_{𝑛𝑚𝑗≠1𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)×exp𝑢_𝑗(=𝑡+𝑠)𝑑𝑠𝑑𝑡_𝜔0𝑟_𝑖(𝑡+)𝑑𝑡_𝜔0𝛼_𝑖(𝑡)+^𝑚_𝑙=1⁰_−𝜏𝑎_𝑖𝑖𝑙(𝑡−𝑠)𝑘_𝑖𝑖𝑙(𝑠)𝑢𝑑𝑠×exp_𝑖(𝑡+)𝑑𝑡_{𝜔0𝑛𝑚𝑗≠𝑖𝑙=10−𝜏}𝑎_𝑖𝑗𝑙(𝑡−𝑠)𝑘_𝑖𝑗𝑙(𝑠)𝑢𝑑𝑠×exp_𝑗(𝑡≤)𝑑𝑡_𝜔0𝑟_𝑖(𝑡)𝑑𝑡+^𝜔₀𝛼_𝑖(𝑡)exp𝑢_𝑖(𝑡+)𝑑𝑡_𝑚𝑙=1𝑎^𝑀_𝑖𝑖𝑙^𝜔₀exp𝑢_𝑖(𝑡+)𝑑𝑡_{𝑛𝑚𝑗≠𝑖𝑙=1}𝑎^𝑀_𝑖𝑗𝑙^𝜔₀exp𝑢_𝑗(𝑡≤)𝑑𝑡𝑟_𝑖𝜔+^𝑛𝑚_𝑗=1𝑙=1𝑎^𝑀_𝑖𝑗𝑙∑^𝑛_𝑖=1𝑟_𝑖𝜔𝛿_𝑖+𝛼^𝑀_𝑖∑^𝑛_𝑖=1𝑟_𝑖𝜔𝛿_𝑖,𝑖=1,2,…,𝑛. (37) From (36) and (37), we have, for any 𝑡∈[0,𝜔], 𝑢_𝑖(𝑡)≤𝑢_𝑖𝜂_𝑖+^𝜔₀𝑢̇_𝑖(𝑡∑)𝑑𝑡≤ln^𝑛𝑖=1𝑟_𝑖𝛿_𝑖+𝑟_𝑖𝜔+^𝑛𝑚_𝑗=1𝑙=1𝑎^𝑀_𝑖𝑗𝑙∑^𝑛_𝑖=1𝑟_𝑖𝜔𝛿_𝑖+𝛼^𝑀_𝑖∑^𝑛_𝑖=1𝑟_𝑖𝜔𝛿_𝑖=∶𝑀_𝑖,𝑖=1,2,…,𝑛. (38)
Further, from (29) and (37), we have, for any 𝑡∈[0,𝜔], 𝑢_𝑖(𝑡)≥𝑢_𝑖𝜉_𝑖−^𝜔₀𝑢̇_𝑖(𝑡)𝑑𝑡≥ln𝑟_𝑖𝛼_𝑖−𝑟_𝑖𝜔−^𝑛𝑚_𝑗=1𝑙=1𝑎^𝑀_𝑖𝑗𝑙∑^𝑛_𝑖=1𝑟_𝑖𝜔𝛿_𝑖−𝛼^𝑀_𝑖∑^𝑛_𝑖=1𝑟_𝑖𝜔𝛿_𝑖=∶𝑁_𝑖,𝑖=1,2,…,𝑛. (39)
Therefore, from (38) and (39) we have max_{𝑡∈[0,𝜔]}𝑢_𝑖(𝑡)≤max𝑀_𝑖,𝑁_𝑖=∶𝐵_𝑖,𝑖=1,2,…,𝑛. (40) It can be seen that the constants 𝐵_𝑖(𝑖=1,2,…,𝑛) are independent of parameter 𝜆∈(0,1).
For any 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)∈𝑅^𝑛, from (18) we can obtain 𝑄𝑁𝑢=𝑄𝑁𝑢₁,𝑄𝑁𝑢₂,…,𝑄𝑁𝑢_𝑛, (41) where 𝑄𝑁𝑢=𝑟_𝑖−𝛼_𝑖−𝑎_𝑖𝑖exp𝑢_𝑖+^𝑛_𝑗≠𝑖𝑎_𝑖𝑗exp𝑢_𝑗,𝑖=1,2,…,𝑛. (42) We consider the following algebraic equation: 𝑟_𝑖−𝛼_𝑖−𝑎_𝑖𝑖𝑣_𝑖+^𝑛_𝑗≠𝑖𝑎_𝑖𝑗𝑣_𝑗=0,𝑖=1,2,…,𝑛. (43) From the assumption of Theorem 2, the equation has a unique positive solution 𝑣^∗=(𝑣^∗₁,𝑣^∗₂,…,𝑣^∗_𝑛). Hence, the equation 𝑄𝑁𝑢=0 has a unique solution 𝑢^∗=(𝑢^∗₁,𝑢^∗₂,…,𝑢^∗_𝑛)=(ln𝑣^∗₁,ln𝑣^∗₂,…,ln𝑣^∗_𝑛)∈𝑅^𝑛.
Choosing constant 𝐵>0 large enough such that |𝑢^∗₁|+|𝑢^∗₂|+⋯+|𝑢^∗_𝑛|<𝐵 and 𝐵>𝐵₁+𝐵₂+⋯+𝐵_𝑛, we define a bounded open set Ω⊂𝑋 as follows: Ω={𝑢∈𝑋∶‖𝑢‖<𝐵}. (44) It is clear that Ω satisfies conditions (a) and (b) of Lemma 1. On the other hand, by direct calculating we can obtain {(𝑓deg𝐽𝑄𝑁,Ω∩Ker𝐿,0,0,…,0)}=sgn^1𝑢1𝑓¹_𝑢2⋯𝑓¹_𝑢𝑛𝑓²_𝑢1𝑓²_𝑢2⋯𝑓²_𝑢𝑛𝑓⋯⋯⋯⋯_𝑛𝑢1𝑓^𝑛_𝑢2⋯𝑓^𝑛_𝑢𝑛, (45) where 𝑓^𝑖_𝑢𝑗=−𝛼_𝑖−𝑎_𝑖𝑗exp𝑢^∗_𝑗𝑓,𝑖=𝑗,_𝑖𝑢𝑗=𝑎_𝑖𝑗exp𝑢^∗_𝑗,𝑖≠𝑗,𝑖,𝑗=1,2,…,𝑛. (46)
From the assumption of Theorem 2, we have 𝑓¹_𝑢1𝑓¹_𝑢2⋯𝑓¹_𝑢𝑛𝑓²_𝑢1𝑓²_𝑢2⋯𝑓²_𝑢𝑛𝑓⋯⋯⋯⋯_𝑛𝑢1𝑓^𝑛_𝑢2⋯𝑓^𝑛_𝑢𝑛≠0. (47)
From this, we finally have deg{𝐽𝑄𝑁,Ω∩Ker𝐿,(0,0,…,0)}≠0. (48) This shows that Ω satisfies condition (c) of Lemma 1. Therefore, system (14) has a 𝜔-periodic solution 𝑢^∗(𝑡)=(𝑢^∗₁(𝑡),𝑢^∗₂(𝑡),…,𝑢^∗_𝑛(𝑡))∈Ω. Further, from (13), system (4) has a positive 𝜔-periodic solution 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡)).
Next, we will consider the global attractivity of positive periodic solutions 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡)) of system (4). Choose positive constants 𝑚_𝑖>0, 𝑀_𝑖>0 such that 𝑚_𝑖≤𝑥^∗_𝑖(𝑡)≤𝑀_𝑖,𝑖=1,2,…,𝑛. (49) From the assumption of Theorem 2, there exists constant 𝛽>0 such that for all 𝑡≥0 we have 𝛿_𝑖≥𝛽>0,𝑖=1,2,…,𝑛. (50) Let (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) be any solution of system (4), we define Lyapunov function as follows: 𝑉_𝑖(𝑡)=𝜇_𝑖ln𝑥^∗_𝑖(𝑡)−ln𝑥_𝑖(𝑡+_{𝑛𝑚𝑗=1𝑙=1})𝜇_𝑗⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)^𝑡_𝑡+𝑠𝑎_𝑖𝑗𝑙()×𝑥𝜃−𝑠_∗𝑗(𝜃)−𝑥_𝑗(𝜃)𝑑𝜃𝑑𝑠. (51) Calculating the upper right derivation of 𝑉_𝑖(𝑡) along system (4) for 𝑖=1,2,…,𝑛, we have 𝐷⁺𝑉_𝑖(𝑡)=sign𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡×)−𝜇_𝑖𝛼_𝑖(𝑡)𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡)+^𝑛𝑚_𝑗=1𝑙=1𝜇_𝑗𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)×𝑥^∗_𝑗(𝑡+𝜃)−𝑥_𝑗()+𝑡+𝜃𝑑𝑠_{𝑛𝑚𝑗=1𝑙=1}𝜇_𝑗⁰_−𝜏𝑎_𝑖𝑗𝑙(𝑡−𝑠)𝑘_𝑖𝑗𝑙(𝑠)𝑑𝑠𝑥^∗_𝑗(𝑡)−𝑥_𝑗(𝑡−_{𝑛𝑚𝑗=1𝑙=1})𝜇_𝑗𝑎_𝑖𝑗𝑙(𝑡)⁰_−𝜏𝑘_𝑖𝑗𝑙(𝑠)×𝑥^∗_𝑗(𝑡+𝜃)−𝑥_𝑗(𝑡+𝜃)𝑑𝑠≤−𝜇_𝑖𝛼_𝑖(𝑡)𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡+_{𝑛𝑚𝑗=1𝑙=1})𝜇_𝑗⁰_−𝜏𝑎_𝑖𝑗𝑙(𝑡−𝑠)𝑘_𝑖𝑗𝑙(𝑠)𝑑𝑠𝑥^∗_𝑗(𝑡)−𝑥_𝑗(𝑡). (52) Further, we define a Lyapunov function as follows: 𝑉(𝑡)=^𝑛_𝑖=1𝑉_𝑖(𝑡). (53) Calculating the upper right derivation of 𝑉(𝑡), from (52) we finally can obtain, for all 𝑡≥0, 𝐷⁺𝑉(𝑡)≤−^𝑛_𝑖=1𝛿_𝑖𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡). (54)
Integrating from 0 to 𝑡 on both sides of (54) and by (50) produces 𝑉(𝑡)+𝛽^𝑡₀^𝑛_𝑖=1𝑥^∗_𝑖(𝑠)−𝑥_𝑖(𝑠)𝑑𝑠≤𝑉(0),𝑡≥0, (55) then ^𝑡₀^𝑛_𝑖=1𝑥^∗_𝑖(𝑠)−𝑥_𝑖(𝑠𝑉(0))𝑑𝑠≤𝛽,𝑡≥0. (56) By the definition of 𝑉(𝑡) and (53), we have ^𝑛_𝑖=1𝜇_𝑖ln𝑥^∗_𝑖(𝑡)−ln𝑥_𝑖(𝑡)≤𝑉(𝑡)≤𝑉(0),𝑡≥0. (57) Therefore, for 𝑖=1,2,…,𝑛 we have 𝜇_𝑖ln𝑥^∗_𝑖(𝑡)−ln𝑥_𝑖(𝑡)≤𝑉(0),𝑡≥0, (58) which, together with (49), leads to 𝑚_𝑖−𝑉(0)exp𝜇_𝑖≤𝑥_𝑖(𝑡)≤𝑀_𝑖𝑉(0)exp𝜇_𝑖,𝑖=1,2,…,𝑛, (59) and, hence, ∑^𝑛_𝑖=1|𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡)|∈𝐿¹[0,+∞). From the boundedness of 𝑥^∗_𝑖(𝑡) and (58), it follows that 𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛) are bounded for 𝑡≥0. It is obvious that both 𝑥_𝑖(𝑡) and 𝑥^∗_𝑖(𝑡) satisfy the equations of system (4), then by system (4) and the boundedness of 𝑥_𝑖(𝑡) and 𝑥^∗_𝑖(𝑡), we know that the derivatives 𝑥̇_𝑖(𝑡) and 𝑥̇^∗_𝑖(𝑡) are bounded. Furthermore, we can obtain that 𝑥̇^∗_𝑖(𝑡)−𝑥̇_𝑖(𝑡)(𝑖=1,2,…,𝑛) and their derivatives remain bounded on [0,+∞). Therefore ∑^𝑛_𝑖=1|𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡)| is uniformly continuous on [0,+∞). Thus, from (56), we have lim^𝑛_{𝑡→+∞𝑖=1}𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡)=0. (60) Therefor, lim_𝑡→+∞𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡)=0,𝑖=1,2,…,𝑛. (61) This completes the proof of Theorem 2.

From the proof of Theorem 2, on the existence and global attractivity of positive periodic solutions of system (5), we have the following result.

Corollary 3. Suppose that assumption (𝐇𝟏) holds and there exists a constant 𝜌_𝑖>0(𝑖=1,2,…,𝑛) such that min_{𝑡∈[0,𝜔]}𝜌_𝑖𝛼_𝑖(𝑡)−^𝑛𝑚_𝑗=1𝑙=1𝜌_𝑗⁰_−𝜏𝑎_𝑗𝑖𝑙(𝑡−𝑠)𝑘_𝑗𝑖𝑙(𝑠)𝑑𝑠=∶𝜆_𝑖>0,𝑖=1,2,…,𝑛, (62) and the algebraic equation 𝑟_𝑖−𝛼_𝑖+𝑎_𝑖𝑖𝑣_𝑖+^𝑛_𝑗≠𝑖𝑎_𝑖𝑗𝑣_𝑗=0,𝑖=1,2,…,𝑛, (63) has a unique positive solution. Then, system (5) has a positive 𝜔-periodic solution which is globally attractive.

4. Two Examples

Example 4. First, we consider the following delayed system: 𝑥̇₁(𝑡)=𝑥₁(𝑡)2+cos(𝑡)−(6+cos(𝑡))𝑥₁(𝑡)+2+cos(𝑡)3⁰_−𝜏𝑘₁₁₁(𝑠)𝑥₁()+(𝑡)𝑡+𝑠𝑑𝑠2+cos3⁰_−𝜏𝑘₁₂₁(𝑠)𝑥₂()+(𝑡)𝑡+𝑠𝑑𝑠2+cos3⁰_−𝜏𝑘₁₃₁(𝑠)𝑥₃(),𝑥̇𝑡+𝑠𝑑𝑠₂(𝑡)=𝑥₂(𝑡)2+cos(𝑡)−(5+cos(𝑡))𝑥₂(𝑡)+2+cos(𝑡)7⁰_−𝜏𝑘₂₂₁(𝑠)𝑥₂()+(𝑡)𝑡+𝑠𝑑𝑠3+cos4⁰_−𝜏𝑘₂₁₁(𝑠)𝑥₁()+(𝑡)𝑡+𝑠𝑑𝑠3+cos4⁰_−𝜏𝑘₂₃₁(𝑠)𝑥₃(),𝑥̇𝑡+𝑠𝑑𝑠₃(𝑡)=𝑥₃(𝑡)2+cos(𝑡)−(6+cos(𝑡))𝑥₃(𝑡)+3+cos(𝑡)4⁰_−𝜏𝑘₃₃₁(𝑠)𝑥₃()+(𝑡)𝑡+𝑠𝑑𝑠4+cos5⁰_−𝜏𝑘₃₁₁(𝑠)𝑥₁()+(𝑡)𝑡+𝑠𝑑𝑠3+cos5⁰_−𝜏𝑘₃₂₁(𝑠)𝑥₂(𝑡+𝑠)𝑑𝑠. (64) Corresponding to system (4), 𝑛=3,??𝑚=1,??𝜔=2𝜋, by direct calculation, we can get 𝜎₁≈2,𝜎₂≈1.6,𝜎₃≈2.2, (65) and the following equations have unique positive solutions: 𝑎₁₁𝑣₁−𝑎₁₂𝑣₂−𝑎₁₃𝑣₃=𝑟₁,𝑎₂₂𝑣₁−𝑎₂₁𝑣₂−𝑎₂₃𝑣₃=𝑟₂,𝑎₃₃𝑣₁−𝑎₃₁𝑣₂−𝑎₃₂𝑣₃=𝑟₃, (66) where 𝑣₁≈0.1944,𝑣₂≈0.5625,𝑣₃≈0.8819. (67) It is clear that all the conditions of Theorem 2 hold. Hence, system (64) has a positive periodic solution which is globally attractive.

Example 5. Next, we consider the following delayed system: 𝑥̇₁(𝑡)=𝑥₁(𝑡(𝑡)−(𝑡))2+cos5+4cos𝑥20¹(𝑡)+3+2cos(𝑡)2⁰_−𝜏𝑘₁₁₁(𝑠)𝑥₁()+(𝑡)𝑡+𝑠𝑑𝑠3+2cos4⁰_−𝜏𝑘₁₂₁(𝑠)𝑥₂()+(𝑡)𝑡+𝑠𝑑𝑠2+cos2⁰_−𝜏𝑘₁₃₁(𝑠)𝑥₃(),𝑥̇𝑡+𝑠𝑑𝑠₂(𝑡)=𝑥₂(𝑡(𝑡)−(𝑡))2+cos41+40cos𝑥100²(𝑡)+(2+cos(𝑡))⁰_−𝜏𝑘₂₂₁(𝑠)𝑥₂()+(𝑡)𝑡+𝑠𝑑𝑠9+6cos8⁰_−𝜏𝑘₂₁₁(𝑠)𝑥₁()+(𝑡)𝑡+𝑠𝑑𝑠3+2cos4⁰_−𝜏𝑘₂₃₁(𝑠)𝑥₃(),𝑥̇𝑡+𝑠𝑑𝑠₃(𝑡)=𝑥₃(𝑡(𝑡)−(𝑡))2+cos5+4cos𝑥20³(𝑡)+(4+cos(𝑡))⁰_−𝜏𝑘₃₃₁(𝑠)𝑥₃()+(𝑡)𝑡+𝑠𝑑𝑠5+4cos8⁰_−𝜏𝑘₃₁₁(𝑠)𝑥₁()+(𝑡)𝑡+𝑠𝑑𝑠6+3cos8⁰_−𝜏𝑘₃₂₁(𝑠)𝑥₂(𝑡+𝑠)𝑑𝑠. (68) Corresponding to system (4), 𝑛=3,??𝑚=1,??𝜔=2𝜋, by direct calculation we can get 𝜎₁≈−5.2,𝜎₂≈−6.1,𝜎₃≈−7.2, (69) and the following equations have a unique positive solutions: 𝑎₁₁𝑣₁−𝑎₁₂𝑣₂−𝑎₁₃𝑣₃=𝑟₁,𝑎₂₂𝑣₁−𝑎₂₁𝑣₂−𝑎₂₃𝑣₃=𝑟₂,𝑎₃₃𝑣₁−𝑎₃₁𝑣₂−𝑎₃₂𝑣₃=𝑟₃, (70) where 𝑣₁=−0.1622,𝑣₂=0.7007,𝑣₃=1.2717. (71) Clearly, the conditions of Theorem 2 do not hold.
From Figure 1 we can see that system (68) has no globally attractive positive periodic solution.

Figure 1 

Numerical simulation for system (68). Here, we take the initial value 𝑥0=(𝑥10,𝑥20,𝑥30)=(1,1.5,2).

Remark 6. From these two examples, we can see that if the conditions of Theorem 2 hold, then the system has a globally attractive positive periodic solution. If the conditions of Theorem 2 do not hold, then the system has no globally attractive positive periodic solution.

5. Conclusions

Mawhin’s continuation theorem is a powerful tool for studying the existence of periodic solutions of periodic high-dimensional time-delayed problems. When dealing with time-delayed problem, it is very convenient and the result is relatively simple. The most critical thing in the using of the theorem is the calculation of topological degree, that is, the condition (c) of the theorem.

In this paper, motivated by [2, 7] of Liu et al. we propose two classes of periodic 𝑁-species Lotka-Volterra facultative mutualism systems with distributed delays. By applying the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin and the Lyapunov function method, we easily obtain sufficient conditions for the existence and global attractivity of positive periodic solutions of the system. From Theorem 2 and Corollary 3, we can see that the distributed time delays have effect on the existence and global attractivity of positive periodic solutions, and conditions (11) and (62) are very crucial to find the criteria for globally attractive positive periodic solutions. Further, the conditions (11) and (62) indicate that the undelayed intraspecific competition dominates the delayed intraspecific reproduction, and the intraspecific competition is more significant than the interspecific cooperation.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants nos. 11271312, 11261056, and 11261058), the China Postdoctoral Science Foundation (Grant no. 20110491750), and the Natural Science Foundation of Xinjiang (Grant no. 2011211B08).