Existence of Positive Periodic Solutions for a Class of n-Species Competition Systems with Impulses

Abstract

By using the fixed point theorem on cone, some sufficient conditions are obtained on the existence of positive periodic solutions for a class of 𝑛-species competition systems with impulses. Meanwhile, we point out that the conclusion of (Yan, 2009) is incorrect.

1. Introduction

In recent years, the problem of periodic solutions of the ecological species competition systems has always been one of the active areas of research and has attracted much attention. For instance, the traditional Lotka-Volterra competition system is a rudimentary model on mathematical ecology which can be expressed as follows: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗,𝑖=1,2,…,𝑛. (1.1) Owing to its theoretical and practical significance, the systems have been studied extensively by many researchers. And many excellent results which concerned with persistence, extinction, global attractivity of periodic solutions, or almost periodic solutions have been obtained.

However, the Lotka-Volterra competition systems ignore many important factors, such as the age structure of a population or the effect of toxic substances. So, more complicated competition systems are needed. In 1973, Ayala and Gilpin proposed several competition systems. One of the systems is the following competition system: 𝑁̇_𝑖(𝑡)=𝑟_𝑖𝑁_𝑖1−𝑁_𝑖𝐾_𝑖^𝜃𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝛼_𝑖𝑗𝑁_𝑗𝐾_𝑗,𝑖=1,2,…,𝑛, (1.2) where 𝑁_𝑖 is the population density of the 𝑖th species; 𝑟_𝑖 is the intrinsic exponential growth rate of the 𝑖th species; 𝐾_𝑖 is the environmental carrying capacity of species 𝑖 in the absence of competition; 𝜃_𝑖𝑗 provides a nonlinear measure of interspecific interference, 𝛼_𝑖𝑗 provides a nonlinear measure of interspecific interference.

On the other hand, in the study of species competition systems, the effect of some impulsive factors has been neglected, which exists widely in the real world. For example, the harvesting or stocking occur at fixed time, natural disaster such as fire or flood happen unexpectedly, and some species usually migrate seasonally. Consequently, such processes experience short-time rapid change which can be described by impulses. Therefore, it is important to study the existence of the periodic solutions of competitive systems with impulse perturbation (see [1–7] and the references therein).

For example, by using the method of coincidence degree, Wang [1] considered the existence of periodic solutions for the following 𝑛-species Gilpin-Ayala impulsive competition system: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝛼𝑖𝑗_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑥^𝛼𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥^𝛼𝑖𝑖_𝑖(𝑡)𝑥^𝛼𝑖𝑗_𝑗(𝑡),𝑡≠𝑡_𝑘;Δ𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝑝^𝑖_𝑘𝑥_𝑖𝑡_𝑘,𝑘=1,2,3,…, (1.3) where the constant 𝑝^𝑖_𝑘 satisfied −1<𝑝^𝑖_𝑘<0,𝑖=1,2,…,𝑛. What is more, [1] also obtained several results for the persistence and global attractivity of the periodic solution of the model.

In [2], Yan applied the Krasnoselskii fixed point theorem to investigate the following n-species competition system: 𝑦̇_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑦^𝛼𝑖𝑗_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛾𝑖𝑗_𝑖(𝑡+𝜉)𝑦^𝛿𝑖𝑗_𝑗(𝑡+𝜉)𝑑𝜉,𝑖=1,2,…,𝑛, (1.4) where the constants 𝛼_𝑖𝑗,𝛽_𝑖𝑗,𝛾_𝑖𝑗≥1,𝑖,𝑗=1,2,…,𝑛. He obtained a necessary and sufficient condition for the existence of periodic solutions of system (1.4). Unfortunately, its last conclusion is wrong. Please see the remark in Section 3 of this paper.

Motivated by [1, 2], in this paper, we investigate the following impulsive 𝑛-species competition system: 𝑦̇_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑦^𝛼𝑖𝑗_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛾𝑖𝑗_𝑖(𝑡+𝜉)𝑦^𝛿𝑖𝑗_𝑗(𝑡+𝜉)𝑑𝜉,𝑡≠𝑡_𝑘,𝑖=1,2,…,𝑛,Δ𝑦_𝑖𝑡_𝑘=𝑦_𝑖𝑡⁺_𝑘−𝑦_𝑖𝑡⁻_𝑘=𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘,𝑘=1,2,…,𝑚, (1.5) where 𝑦_𝑖(𝑡) is the population density of the 𝑖th species at time 𝑡; 𝑟_𝑖(𝑡) is the intrinsic exponential growth rate of the 𝑖th species at time 𝑡; 𝜏_𝑖𝑗(𝑡) is the time delay; 𝜎_𝑖𝑗 is a positive constant; 𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),𝑐_𝑖𝑗(𝑡)(𝑖≠𝑗) measure the amount of competition between the species 𝑌_𝑖 and 𝑌_𝑗; 𝛼_𝑖𝑗,𝛽_𝑖𝑗,𝛿_𝑖𝑗(𝑖≠𝑗) provide a nonlinear measure of interspecific interference; 𝑦_𝑖(𝑡⁺_𝑘) (𝑦_𝑖(𝑡⁻_𝑘)) is the left (right) limits of 𝑦_𝑖(𝑡) at 𝑡=𝑡_𝑘,𝑖,𝑗=1,2,…,𝑛,𝑘=1,2,…,𝑚.

The main features of the present paper are as follows. The Gilpin-Ayala species competition system (1.5) has impulsive effects. As is known to us, there were few papers to study such system. Finally, we point out that the conclusion of [2] is incorrect.

For an 𝜔-periodic function 𝑢(𝑡)∈𝐶(ℝ,ℝ), let 𝑢=1/𝜔∫^𝜔₀𝑢(𝑡)𝑑𝑡. Throughout this paper, assume the following conditions hold.(H1)𝑟_𝑖,𝑎_𝑖𝑗,𝑏_𝑖𝑗,𝑐_𝑖𝑗,𝜏_𝑖𝑗 are continuous 𝜔-periodic functions, and 𝑟_𝑖>0,𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),𝑐_𝑖𝑗(𝑡)≥0,𝑖,𝑗=1,2,…,𝑛, and there exists 𝑖₀(1≤𝑖₀≤𝑛) such that min_{1≤𝑗≤𝑛}(𝑎_𝑖0𝑗+𝑏_𝑖0𝑗)>0.(H2)𝐾_𝑖𝑗∈𝐶([−𝜎_𝑖𝑗,0],ℝ),𝐾_𝑖𝑗≥0, 𝜎_𝑖𝑗 is a positive constant, and ∫⁰_{−𝜎𝑖𝑗}𝐾_𝑖𝑗(𝑡)𝑑𝑡=1,𝑖,𝑗=1,2,…,𝑛.(H3)𝐼_𝑖𝑘∈𝐶(ℝ^𝑛,[0,+∞)), and for 0<𝑡₁<𝑡₂<⋯<𝑡_𝑚<𝜔, there exists an positive integer 𝑙>0 such that 𝑡_{𝑘+𝑙𝑚}=𝑡_𝑘+𝑙𝜔,𝐼_𝑖𝑘(𝑦₁,𝑦₂,…,𝑦_𝑛)=𝐼_{𝑖(𝑘+𝑙𝑚)}(𝑦₁,𝑦₂,…,𝑦_𝑛), where (𝑦₁,𝑦₂,…,𝑦_𝑛)∈ℝ^𝑛,𝑖=1,2,…,𝑛,𝑘=1,2,…,𝑚.(H4)𝛼_𝑖𝑗,𝛽_𝑖𝑗,𝛿_𝑖𝑗>0,𝛾_𝑖𝑗≥0 are constants, 𝑖,𝑗=1,2,…,𝑛.

In order to prove our main result, now we state the fixed point theorem of cone expansion and compression.

Lemma 1.1 (see [4]). Let 𝐸 be a Banach space, and let 𝑃 be a cone in 𝐸. Assume that Ω₁,Ω₂ are open subsets of 𝐸 with 0∈Ω₁,Ω₁⊆Ω₂. Let 𝐴∶𝑃∩(Ω₂⧵Ω₁)→𝑃 be a completely continuous operator such that one of the following two conditions is satisfied:(i)𝐴𝑥_╱≧ for 𝑥∈𝑃∩𝜕Ω₁; 𝐴𝑥_╱≦ for 𝑥∈𝑃∩𝜕Ω₂,(ii)𝐴𝑥_╱≧ for 𝑥∈𝑃∩𝜕Ω₂; 𝐴𝑥_╱≦ for 𝑥∈𝑃∩𝜕Ω₁,Then, 𝐴 has a fixed point in 𝑃∩(Ω₂⧵Ω₁).

The organization of this paper is as follows. In the next section, we introduce some lemmas and notations. In Section 3, the main result will be stated and proved on the existence of periodic solutions of system (1.5).

2. Preliminaries

Let 𝑃𝐶(ℝ,ℝ^𝑛)={𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),⋯𝑦_𝑛(𝑡))∶ℝ→ℝ^𝑛∣𝑦_𝑖(𝑡) be continuous at 𝑡≠𝑡_𝑘, left continuous at 𝑡=𝑡_𝑘, and the right limit 𝑦_𝑖(𝑡⁺_𝑘) exists for 𝑖=1,2,…,𝑛,𝑘=1,2,…,𝑚}. Evidently, 𝐸=𝑦(𝑡)=𝑦₁(𝑡),𝑦₂(𝑡),…𝑦_𝑛(𝑡)∈𝑃𝐶ℝ,ℝ^𝑛∣𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡+𝜔),𝑖=1,2,…,𝑛 (2.1) is a Banach space with the norm ‖𝑦‖=∑^𝑛_𝑖=1|𝑦_𝑖|₀, where |𝑦_𝑖|₀=max_{𝑡∈[0,𝜔]}|𝑦_𝑖(𝑡)|.

Define an operator 𝑇∶𝐸→𝐸 by (𝑇𝑦)(𝑡)=((𝑇𝑦)₁(𝑡),(𝑇𝑦)₁(𝑡),…,(𝑇𝑦)_𝑛(𝑡)), where 𝑇𝑦_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑦_𝑖(𝑠)^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑦^𝛼𝑖𝑗_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)𝑦^𝛽𝑖𝑗_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛾𝑖𝑗_𝑖(𝑠+𝜉)𝑦^𝛿𝑖𝑗_𝑗()+𝑠+𝜉𝑑𝜉𝑑𝑠_𝑚𝑘=1𝐺_𝑖𝑡,𝑡_{𝑘+𝑞𝑘𝑚}𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘, (2.2) where 𝐺_𝑖(𝑡,𝑠)=exp(−∫^𝑠_𝑡𝑟_𝑖(𝜈)𝑑𝜈)/(1−exp(−𝜔𝑟_𝑖)),𝑡≤𝑠≤𝑡+𝜔,𝑖=1,2,…,𝑛, and 𝑡_{𝑘+𝑞𝑘𝑚}=𝑡_𝑘+𝑞_𝑘𝜔∈[𝑡,𝑡+𝜔], where 𝑞_𝑘 is a positive integer, 𝑘=1,2,…,𝑚.

It is obvious that the functions 𝐺_𝑖(𝑡,𝑠),𝑖=1,2,…,𝑛 have the following properties.(i)𝐺_𝑖(𝑡,𝑠)>0 for (𝑡,𝑠)∈ℝ², and 𝐺_𝑖(𝑡,𝑠)=𝐺_𝑖(𝑡+𝜔,𝑠+𝜔).(ii)𝐴≤𝐺_𝑖(𝑡,𝑠)≤𝐵 for (𝑡,𝑠)∈ℝ², where 𝐴=min_{1≤𝑖≤𝑛}{exp(−𝜔𝑟_𝑖)/(1−exp(−𝜔𝑟_𝑖))}, 𝐵=min_{1≤𝑖≤𝑛}{exp(𝜔𝑟_𝑖)/(1−exp(−𝜔𝑟_𝑖))}.

Now, we choose a set defined by 𝑃=𝑦(𝑡)=𝑦₁(𝑡),𝑦₂(𝑡),…𝑦_𝑛(𝑡)∈𝐸∣𝑦_𝑖(𝑡)≥𝜎𝑦_𝑖0,𝑡∈[0,𝜔],𝑖=1,2,…,𝑛, (2.3) where 𝜎=𝐴/𝐵. Clearly, 𝑃 is a cone in 𝐸.

For the sake of convenience, we define an operator 𝐹∶𝑃→𝐸 by (𝐹𝑦)(𝑡)=((𝐹𝑦)₁(𝑡),(𝐹𝑦)₂(𝑡),…,(𝐹𝑦)_𝑛(𝑡)), where 𝐹𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡)^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑦^𝛼𝑖𝑗_𝑗(𝑡)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛾𝑖𝑗_𝑖(𝑡+𝜉)𝑦^𝛿𝑖𝑗_𝑗(𝑡+𝜉)𝑑𝜉,𝑖=1,2,…,𝑛. (2.4)

Lemma 2.1. The operator 𝑇∶𝑃→𝑃 is completely continuous.

Proof. First, it is easy to see 𝑇∶𝑃𝐶(ℝ,ℝ^𝑛)→𝑃𝐶(ℝ,ℝ^𝑛). Next, since 𝑇𝑦_𝑖()=𝑡+𝜔_{𝑡+2𝜔𝑡+𝜔}𝐺_𝑖(𝑡+𝜔,𝑠)𝐹𝑦_𝑖(𝑠)𝑑𝑠+^𝑚_𝑘=1𝐺_𝑖𝑡+𝜔,𝑡_{𝑘+𝑞𝑘𝑚}+𝜔𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡+𝜔,𝜐+𝜔)𝐹𝑦_𝑖(𝜐+𝜔)𝑑𝜐+^𝑚_𝑘=1𝐺_𝑖𝑡,𝑡_{𝑘+𝑞𝑘𝑚}𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝜐)𝐹𝑦_𝑖(𝜐)𝑑𝜐+^𝑚_𝑘=1𝐺_𝑖𝑡,𝑡_{𝑘+𝑞𝑘𝑚}𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘=𝑇𝑦_𝑖(𝑡), (2.5) we have 𝑇𝑦∈𝐸.
Observe that 𝐴≤𝐺_𝑖(𝑡,𝑠)≤𝐵,𝑖=1,2,…,𝑛, for all 𝑠∈[𝑡,𝑡+𝜔]. Hence, we obtain that, for 𝑦∈𝑃, 𝑇𝑦_𝑖0≤𝐵^𝜔₀𝐹𝑦_𝑖(𝑠)𝑑𝑠+𝐵^𝑚_𝑘=1𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘,𝑇𝑦_𝑖(𝑡)≥𝐴^𝜔₀𝐹𝑦_𝑖(𝑠)𝑑𝑠+𝐴^𝑚_𝑘=1𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘≥𝐴𝐵𝑇𝑦_𝑖0=𝜎𝑇𝑦_𝑖0. (2.6) Thus, 𝑇𝑦∈𝑃, that is, 𝑇(𝑃)⊂𝑃.
Obviously, the operator 𝑇 is continuous. Next, we show that 𝑇 is compact. Let 𝑆⊂𝐸 be a bounded subset; that is, there exists 𝑑>0 such that |𝑦_𝑖|₀≤𝑑,𝑖=1,2,…,𝑛 for all 𝑦∈𝑆. From the continuity of 𝐹,𝐼_𝑘,𝑘=1,2,…,𝑚, we have, for all 𝑦∈𝑆, 𝑇𝑦_𝑖0≤𝐵^𝜔₀𝐹𝑦_𝑖(𝑠)𝑑𝑠+𝐵^𝑚_𝑘=1𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘≤𝐵𝜔𝑑^𝑛_𝑗=1𝑎_𝑖𝑗𝑑^𝛼𝑖𝑗+𝑏_𝑖𝑗𝑑^𝛽𝑖𝑗+𝑐_𝑖𝑗𝑑^{𝛾𝑖𝑗+𝛿𝑖𝑗}+𝐵𝑚𝐸_𝑖=∶𝐷_𝑖, (2.7) where 𝐸_𝑖=max_𝑦∈𝑆|𝐼_𝑖𝑘(𝑦₁(𝑡_𝑘),𝑦₂(𝑡_𝑘),…,𝑦_𝑛(𝑡_𝑘)|,𝑖=1,2,…,𝑛.
Therefore, ‖𝑇𝑦‖=∑^𝑛_𝑖=1|(𝑇𝑦)_𝑖|₀≤∑^𝑛_𝑖=1𝐷_𝑖=∶𝐷, which implies that 𝑇(𝑆) is uniformly bounded.
On the other hand, noticing that 𝑑𝑑𝑡𝑇𝑦_𝑖(𝑡)=𝑟_𝑖(𝑡)𝑇𝑦_𝑖(𝑡)+𝐺_𝑖(𝑡,𝑡+𝜔)𝐹𝑦_𝑖(𝑡+𝜔)−𝐺_𝑖(𝑡,𝑡)𝐹𝑦_𝑖(𝑡)=𝑟_𝑖(𝑡)𝑇𝑦_𝑖(𝑡)+𝐺_𝑖(𝑡,𝑡+𝜔)−𝐺_𝑖(𝑡,𝑡)𝐹𝑦_𝑖(𝑡)=𝑟_𝑖(𝑡)𝑇𝑦_𝑖(𝑡)+𝐹𝑦_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑘=1,2,…,𝑚. (2.8) This guarantees that, for each 𝑦∈𝑆, we have 𝑑𝑑𝑡𝑇𝑦_𝑖(𝑡)≤𝑟^𝑀_𝑖𝐷_𝑖+𝑑^𝑛_𝑗=1𝑎^𝑀_𝑖𝑗𝑑^𝛼𝑖𝑗+𝑏^𝑀_𝑖𝑗𝑑^𝛽𝑖𝑗+𝑐^𝑀_𝑖𝑗𝑑^{𝛾𝑖𝑗+𝛿𝑖𝑗}=∶𝐷_𝑖≤𝐷=^𝑛_𝑖=1𝐷_𝑖, (2.9) where 𝑟^𝑀_𝑖=max_{𝑡∈[0,𝜔]}𝑟_𝑖(𝑡),𝑎^𝑀_𝑖𝑗=max_{𝑡∈[0,𝜔]}𝑎_𝑖𝑗(𝑡), 𝑏^𝑀_𝑖𝑗=max_{𝑡∈[0,𝜔]}𝑏_𝑖𝑗(𝑡), 𝑐^𝑀_𝑖𝑗=max_{𝑡∈[0,𝜔]}𝑐_𝑖𝑗(𝑡), 𝑖=1,2,…,𝑛.
Consequently, 𝑇(𝑆) is equicontinuous on 𝐽_𝑘,𝑘=0,1,2,…,𝑚, where 𝐽₀=[0,𝑡₁),𝐽₁=[𝑡₁,𝑡₂),…,𝐽_𝑚−1=[𝑡_𝑚−1,𝑡_𝑚),𝐽_𝑚=[𝑡_𝑚,𝜔). By the Ascoli-Arzela theorem, the function 𝑇∶𝑃→𝑃 is completely continuous from 𝑃 to 𝑃.

Lemma 2.2. The system (1.5) has a positive 𝜔-periodic solution in 𝑃 if and only if 𝑇 has a fixed point in 𝑃.

Proof. For 𝑦∈𝑃 satisfying 𝑇𝑦=𝑦, that is, (𝑇𝑦)_𝑖(𝑡)=𝑦_𝑖(𝑡),𝑡∈[0,𝜔],𝑖=1,2,…,𝑛, it follows from (2.2) and (2.4) that 𝑦̇_𝑖(𝑡)=𝑑𝑑𝑡𝑇𝑦_𝑖(𝑡)=𝑟_𝑖(𝑡)𝑇𝑦_𝑖(𝑡)+𝐺_𝑖(𝑡,𝑡+𝜔)𝐹𝑦_𝑖(𝑡+𝜔)−𝐺(𝑡,𝑡)𝐹𝑦_𝑖(𝑡)=𝑟_𝑖(𝑡)𝑇𝑦_𝑖(𝑡)+[𝐺(𝑡,𝑡+𝜔)−𝐺(𝑡,𝑡)]𝐹𝑦_𝑖(𝑡)=𝑟_𝑖(𝑡)𝑇𝑦_𝑖(𝑡)+𝐹𝑦_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑘=1,2,…,𝑚. (2.10) And for 𝑡=𝑡_𝑘,𝑘=1,2,…,𝑚, Δ𝑦_𝑖𝑡_𝑘=𝑦_𝑖𝑡⁺_𝑘−𝑦_𝑖𝑡⁻_𝑘=𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘,𝑖=1,2,…,𝑛, (2.11) which implies that 𝑦(𝑡) is a positive 𝜔-periodic solution of (1.5).
Conversely, assume that 𝑦∈𝑃 is an 𝜔-periodic solution of system (1.5). Then, the system (1.5) can be transformed into 𝑦̇_𝑖(𝑡)+𝑟(𝑡)𝑦_𝑖(𝑡)𝑒^{∫𝑡0𝑟(𝜈)𝑑𝜈}=𝐹𝑦_𝑖(𝑡)𝑒^{∫𝑡0𝑟(𝜈)𝑑𝜈}, (2.12) that is, 𝑦_𝑖(𝑡)𝑒^{∫𝑡0𝑟(𝜈)𝑑𝜈′}=𝐹𝑦_𝑖(𝑡)𝑒^{∫𝑡0𝑟(𝜈)𝑑𝜈}. (2.13)
So, integrating the above equality from 𝑡 to 𝑡+𝜔 and noticing that 𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡+𝜔), we have 𝑇𝑦_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝑦_𝑖(𝑠)^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑦^𝛼𝑖𝑗_𝑗(𝑠)+^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)𝑦^𝛽𝑖𝑗_𝑗𝑠−𝜏_𝑖𝑗(𝑠)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝑠)⁰_{−𝜎𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛾𝑖𝑗_𝑖(𝑠+𝜉)𝑦^𝛿𝑖𝑗_𝑗()+𝑠+𝜉𝑑𝜉𝑑𝑠_𝑚𝑘=1𝐺_𝑖𝑡,𝑡_{𝑘+𝑞𝑘𝑚}𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘, (2.14) where 𝐺_𝑖(𝑡,𝑠)=exp(−∫^𝑠_𝑡𝑟_𝑖(𝜈)𝑑𝜈)/(1−exp(−𝜔𝑟_𝑖)),𝑡≤𝑠≤𝑡+𝜔,𝑖=1,2,…,𝑛, and 𝑡_{𝑘+𝑞𝑘𝑚}=𝑡_𝑘+𝑞_𝑘𝜔∈[𝑡,𝑡+𝜔], where 𝑞_𝑘 is a positive integer, 𝑘=1,2,…,𝑚, that is, 𝑇𝑦=𝑦.
Therefore, 𝑦∈𝑃 is a fixed point of the operator 𝑇. The proof of the Lemma is complete.

3. Main Results

Theorem 3.1. Suppose (H1)–(H4) hold, and lim_|𝑣|→0𝐼_𝑖𝑘(𝑣)/|𝑣|=0,𝑘=1,2,…,𝑚,𝑖=1,2,…,𝑛, where 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛),|𝑣|=min_{1≤𝑖≤𝑛}|𝑣_𝑖|. Then system (1.5) has at least one positive 𝜔-periodic solution.

Proof. Let 𝑀₀=max_{1≤𝑖≤𝑛}{∑^𝑛_𝑗=1(𝑎_𝑖𝑗+𝑏_𝑖𝑗+𝑐_𝑖𝑗)}>0. Choose 𝑀₁≥𝑀₀ and 𝜀=1/(𝐵𝜔𝑀₁+𝐵𝑚)>0. Then, there exists 𝛿>0 such that, for 0<𝑥<𝛿 and 0<|𝑣|<𝛿, we have 𝑥^𝛼𝑖𝑗<𝜀,𝑥^𝛽𝑖𝑗<𝜀,𝑥^{𝛾𝑖𝑗+𝛿𝑖𝑗}𝐼<𝜀,_𝑖𝑘(𝑣)<𝜀|𝑣|,𝑘=1,2,…,𝑚,𝑖,𝑗=1,2,…,𝑛. (3.1)
Choose 𝑟<𝛿. Let Ω₁={𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))∈𝐸∣|𝑦_𝑖|₀<𝑟,𝑖=1,2,…,𝑛}.
Now, we prove that 𝑇𝑦_╱≧𝑦,∀𝑦∈𝑃∩𝜕Ω₁. (3.2)
Suppose (3.2) does not hold. Then, there exists some 𝑦∈𝑃∩𝜕Ω₁ such that 𝑇𝑦≥𝑦. Since 𝑦∈𝑃∩𝜕Ω₁, we have 𝜎|𝑦_𝑖|₀≤𝑦_𝑖(𝑡)≤|𝑦_𝑖|₀ for 𝑡∈[0,𝜔], 𝑖=1,2,…,𝑛. From (2.2) and (2.4), it follows that 𝑇𝑦_𝑖(𝑡)≤𝐵_𝑡^𝑡+𝜔𝐹𝑦_𝑖(𝑠)𝑑𝑠+𝐵^𝑚_𝑘=1𝐼_𝑖𝑘𝑦₁𝑡_𝑘,𝑦₂𝑡_𝑘,…,𝑦_𝑛𝑡_𝑘≤𝐵𝜔𝑦_𝑖0^𝑛_𝑖=1𝑎_𝑖𝑗𝑟^𝛼𝑖𝑗+𝑏_𝑖𝑗𝑟^𝛽𝑖𝑗+𝑐_𝑖𝑗𝑟^{𝛾𝑖𝑗+𝛿𝑖𝑗}+𝐵𝑚𝜀𝑦_𝑖0≤𝐵𝑀₀𝜔𝜀+𝐵𝑚𝜀𝑦_𝑖0<𝑦_𝑖0, (3.3) which implies |(𝑇𝑦)_𝑖|₀<|𝑦_𝑖|₀, a contradiction. Hence, 𝑇𝑦_╱≧𝑦 for 𝑦∈𝑃∩𝜕Ω₁.
On the other hand, let 𝑚₀=min_{1≤𝑗≤𝑛}{𝑎_𝑖0𝑗𝜎^𝛼𝑖𝑗+𝑏_𝑖0𝑗𝜎^𝛽𝑖𝑗}, on the account of (H1), we know 𝑚₀>0. Choose 0<𝑚₁≤𝑚₀ and 𝑀=1/𝐴𝜎𝜔𝑚₁>0. Then, there exists 𝑅₁>0 such that, for 𝑥>𝑅₁, we know that 𝑥^𝛼𝑖𝑗>𝑀,𝑥^𝛽𝑖𝑗>𝑀,𝑖,𝑗=1,2,…,𝑛. (3.4)
Choose 𝑅>𝑚𝑎𝑥{𝑅₁,𝑟}. Let Ω₂={𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…𝑦_𝑛(𝑡))∈𝐸∣|𝑦_𝑖|₀<𝑅,𝑖=1,2,…,𝑛}. Then, 𝜕Ω₂={𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…𝑦_𝑛(𝑡))∈𝐸∣ there exist some integers 𝑗₀(1≤𝑗₀≤𝑛) such that |𝑦_𝑗0|₀=𝑅;|𝑦_𝑖|₀≤𝑅 for 𝑖≠𝑗₀}.
Next we show that 𝑇𝑦_╱≦𝑦,∀𝑦∈𝑃∩𝜕Ω₂. (3.5)
In fact, if there exists some 𝑦∈𝑃∩𝜕Ω₂ such that 𝑇𝑦≤𝑦 and since 𝑦∈𝑃∩𝜕Ω₂, we have 𝜎|𝑦_𝑖|₀≤𝑦_𝑖(𝑡)≤|𝑦_𝑖|₀,𝑖=1,2,…,𝑛 for 𝑡∈[0,𝜔], and there exists some 𝑗₀ such that |𝑦_𝑗0|₀=𝑅. Therefore, this together with (H1) guarantees that, for 𝑖₀(1≤𝑖₀≤𝑛), 𝑦_𝑖0(𝑡)≥𝑇𝑦_𝑖0(𝑡)≥𝐴_𝑡^𝑡+𝜔𝐹𝑦_𝑖0(𝑠)𝑦𝑑𝑠≥𝐴𝜎_𝑖00𝑡^𝑡+𝜔𝑎_𝑖0𝑗0(𝑠)𝑦^{𝛼𝑖0𝑗0}_𝑗0(𝑠)+𝑏_𝑖0𝑗0(𝑠)𝑦^{𝛽𝑖0𝑗0}_𝑗0((𝑠𝑦𝑠−𝜏))𝑑𝑠≥𝐴𝜎_𝑖00𝑡^𝑡+𝜔𝑎_𝑖0𝑗0(𝑠)𝜎^{𝛼𝑖0𝑗0}𝑦_𝑗0^{𝛼𝑖0𝑗0}₀+𝑏_𝑖0𝑗0(𝑠)𝜎^{𝛽𝑖0𝑗0}𝑦_𝑗0^{𝛽𝑖0𝑗0}₀𝑦𝑑𝑠≥𝐴𝜎_𝑖00𝜔𝑀𝑎_𝑖0𝑗0𝜎^{𝛼𝑖0𝑗0}+𝑏_𝑖0𝑗0𝜎^{𝛽𝑖0𝑗0}≥𝐴𝜎𝜔𝑀𝑚₀𝑦_𝑖00>𝑦_𝑖00, (3.6) which is a contradiction. Thus, (3.5) is satisfied.
From all the above, the condition (i) of Lemma 1.1 is satisfied. So the operator 𝑇 has a fixed point in 𝑃∩(Ω₂⧵Ω₁). That is, system (1.5) has at least one positive periodic solution.

Remark 3.2. For any 𝑅>0, if we let Ω={𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))∈𝐸∣|𝑦_𝑖|₀<𝑅,𝑖=1,2,…,𝑛}, then 𝜕Ω={𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))∈𝐸∣∃𝑗₀(1≤𝑗₀≤𝑛),|𝑦_𝑗0|₀=𝑅,|𝑦_𝑖|₀≤𝑅,𝑖≠𝑗₀}. However, in the proof of Theorem  1.1 of [2], it is regarded mistakenly as 𝜕Ω={𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))∈𝐸∣|𝑦_𝑖|₀=𝑅,𝑖=1,2,…,𝑛}. Therefore, the proof of its sufficiency is not correct. So the result of [2] is incorrect.

Acknowledgments

This research was supported by the Natural Science Foundation of Shandong Province (ZR2009AM006), the Key Project of Chinese Ministry of Education (no. 209072), and the Science and Technology Development Funds of Shandong Education Committee (J08LI10).