Positive Periodic Solutions of Cooperative Systems with Delays and Feedback Controls

Abstract

This paper studies a class of periodic n species cooperative Lotka-Volterra systems with continuous time delays and feedback controls. Based on the continuation theorem of the coincidence degree theory developed by Gaines and Mawhin, some new sufficient conditions on the existence of positive periodic solutions are established.

1. Introduction

Mathematical ecological system has become one of the most important topics in the study of modern applied mathematics. Its dynamical behavior includes persistence, permanence and extinction of species, global stability of systems, the existence of positive periodic solutions, positive almost periodic solutions, and strictly positive solutions. The existence of positive periodic solutions has already become one of the most interesting subjects for scholars. In the recent years, the application of fixed-point theorems to the existence of positive periodic solutions in mathematical ecology has been studied extensively, for example, Brouwer’s fixed point theorem [1–4], Schauder’s fixed-point theorem [5–8], Krasnoselskii’s fixed-point theorem [9–14], Horn’s fixed-point theorem [15, 16], and Mawhins continuation theorem [17–36], and so forth. In particular, Mawhins continuation theorem is a powerful tool for studying the existence of periodic solutions of periodic high-dimensional time-delayed problems. When dealing with a time-delayed problem, it is very convenient and the result is relatively simple [30]. Recently, a considerable number of mathematical models with delays have been proposed in the study of population dynamics. One of the most celebrated models for population is the Lotka-Volterra system. Subsequently, a lot of the literature related to the study of the existence of positive periodic solutions for various Lotka-Volterra-type population dynamical systems with delays by using the method of continuation theorem was published and extensive research results were obtained [17–21, 24–34].

On the other hand, in some situations, people may wish to change the position of the existing periodic solution but to keep its stability. This is of significance in the control of ecology balance. One of the methods for the realization of it is to alter the system structurally by introducing some feedback control variables so as to get a population stabilizing at another periodic solution. The realization of the feedback control mechanism might be implemented by means of some biological control scheme or by harvesting procedure [21]. In fact, during the last decade, the existence of positive periodic solutions for the population dynamics with feedback control has been studied extensively [8, 14, 17–21, 24, 29]. To the best of our knowledge, studies on the existence of positive periodic solutions for cooperative systems with delays and feedback controls are fairly rare.

In [21], the authors studied the following neutral Lotka-Volterra system with feedback controls: 𝑦̇_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗𝑦̇_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖𝑡−𝜎_𝑖(𝑡)𝑢̇_𝑖(𝑡)=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑦_𝑖(𝑡)+𝛾_𝑖(𝑡)𝑦_𝑖(𝑡)𝑡−𝛿_𝑖(𝑡),𝑖=1,2,…,𝑛. (1) By using Mawhin’s continuation theorem, the sufficient conditions on the existence of positive periodic solutions are established. In [24], the authors considered the following delay differential system with feedback control: 𝑑𝑥𝑑𝑡=𝐹𝑡,𝑥𝑡−𝜏₁(𝑡),…,𝑥𝑡−𝜏_𝑛(𝑡),𝑢(𝑡−𝛿(𝑡))𝑑𝑢(𝑡)(𝑡)𝑥((𝑡.𝑑𝑡=−𝜂+𝑎𝑡−𝜎)) (2) A set of natural and easily verifiable sufficient conditions of the existence of positive periodic solutions are established, by using Mawhin’s continuation theorem. In [29], the authors considered the following single-species periodic logistic systems with feedback regulation and infinite distributed delay: 𝑁̇(𝑡)(𝑡)𝑁(𝑡)×1=𝑟1−𝐾(𝑡)^∞₀𝐻(𝑠)𝑁()(𝑡)𝑢(𝑡)𝑢̇(𝑡)(𝑡)𝑢(𝑡)(𝑡)𝑡−𝑠𝑑𝑠−𝑐=−𝑎+𝑏^∞₀𝐻(𝑠)𝑁²(𝑡−𝑠)𝑑𝑠. (3) The sufficient conditions for the existence of positive periodic solutions are established, based on Mawhin’s continuation theorem.

Motivated by the above works, in this paper, we investigate the following 𝑛 species periodic Lotka-Volterra-type cooperative systems with continuous time delays and feedback controls: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑚_𝑙=1𝑎_𝑖𝑖𝑙(𝑡)𝑥_𝑖𝑡−𝜏_𝑖𝑖𝑙(𝑡)+^𝑛𝑚_{𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗𝑙(𝑡)−𝑑_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖𝑡−𝜀_𝑖(𝑡)𝑢̇_𝑖(𝑡)=−𝑏_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑥_𝑖(𝑡)+𝛾_𝑖(𝑡)𝑥_𝑖𝑡−𝜎_𝑖(𝑡),𝑖=1,2,…,𝑛. (4) By using the technique of coincidence degree developed by Gaines and Mawhin in [36], we will establish some new sufficient conditions which guarantee that the system has at least one positive periodic solution.

2. Preliminaries

In system (4), we have that 𝑥_𝑖(𝑡)(𝑖=1,2,…𝑛) represent the density of 𝑛 cooperative 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; 𝑎_𝑖𝑖𝑙(𝑡)(𝑖=1,2,…,𝑛, 𝑙=1,2,…,𝑚) 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 cooperative coefficients between 𝑛 species 𝑥_𝑖(𝑖=1,2,…𝑛) at time 𝑡, respectively. 𝑢_𝑖(𝑡)(𝑖=1,2,…𝑛) represent the indirect feedback control variables [21] at time 𝑡, respectively. 𝛽_𝑖(𝑡), 𝑒_𝑖(𝑡), 𝑏_𝑖(𝑡), 𝑑_𝑖(𝑡), and 𝛾_𝑖(𝑡)(𝑖=1,2,…𝑛) represent the feedback control coefficients at time 𝑡, respectively. In this paper, we always assume that(H1) 𝜏_𝑖𝑗𝑙(𝑡)(𝑙=1,2,…,𝑚, 𝑖,𝑗=1,2,…,𝑛), 𝜎_𝑖(𝑡),𝜀_𝑖(𝑡),𝑟_𝑖(𝑡)  (𝑖=1,2,…𝑛) are continuous 𝜔-periodic functions with 𝜏^′_𝑖𝑗𝑙(𝑡)<1 and ∫^𝜔₀𝑟_𝑖(𝑡)𝑑𝑡>0. 𝑎_𝑖𝑗𝑙(𝑡)(𝑖,𝑗=1,2, 𝑙=1,2,…,𝑚), 𝛽_𝑖(𝑡), 𝑒_𝑖(𝑡), 𝑏_𝑖(𝑡), 𝑑_𝑖(𝑡), and 𝛾_𝑖(𝑡)(𝑖=1,2,…𝑛) are continuous, positive 𝜔-periodic functions.

From the viewpoint of mathematical biology, in this paper for system (4) we only consider the solution with the following initial conditions: 𝑥_𝑖(𝑡)=𝜙_𝑖(𝑡)[]𝑢,∀𝑡∈−𝜎,0,𝑖=1,2,…𝑛,_𝑖(𝑡)=𝜓_𝑖(𝑡),∀𝑡∈[−𝜎,0],𝑖=1,2,…𝑛, (5) where 𝜙_𝑖(𝑡),𝜓_𝑖(𝑡)(𝑖=1,2,…𝑛) are nonnegative continuous functions defined on [−𝜎,0] satisfying 𝜙_𝑖(0)>0, 𝜓_𝑖(0)>0(𝑖=1,2,…𝑛) with 𝜎=max_{𝑡∈[0,𝜔]}{𝜏_𝑖𝑗𝑙(𝑡),𝜎_𝑖(𝑡),𝜀_𝑖(𝑡)(𝑖,𝑗=1,2, 𝑙=1,2,…,𝑚)}.

In this paper, for any 𝜔-periodic continuous function 𝑓(𝑡) we denote 𝑓^𝐿=min_{𝑡∈[0,𝜔]}𝑓(𝑡),𝑓^𝑀=max_{𝑡∈[0,𝜔]}𝑓(𝑡),1𝑓=𝜔^𝜔₀𝑓(𝑡)𝑑𝑡. (6)

In order to obtain the existence of positive 𝜔-periodic solutions of system (4), we will use the continuation theorem developed by Gaines and Mawhin in [36]. For the reader’s convenience, we will introduce the continuation theorem in the following.

Let 𝑋 and 𝑍 be two normed vector spaces. Let 𝐿∶Dom𝐿⊂𝑋→𝑍 be a linear operator and let 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= Ker 𝐿 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; denote by 𝐽∶Im𝑄→Ker𝐿 an isomorphism of Im𝑄 onto Ker𝐿. 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 [35]). Suppose 𝜏∈𝐶¹(𝑅,𝑅) with 𝜏(𝑡+𝜔)≡𝜏(𝑡) and 𝜏^′(𝑡)<1, ∀𝑡∈[0,𝜔]. Then the function 𝑡−𝜏(𝑡) has a unique inverse function 𝜇(𝑡) satisfying 𝜇∈𝐶(𝑅,𝑅), 𝜇(𝑢+𝜔)=𝜇(𝑢)+𝜔,∀𝑢∈𝑅.

Lemma 2 (see [36]). 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

In order to obtain the existence of positive periodic solutions of system (4), firstly, we introduce the following lemma.

Lemma 3. Suppose that (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡), 𝑢^∗₁(𝑡), 𝑢^∗₂(𝑡),  …,𝑢^∗_𝑛(𝑡)) is an 𝜔-periodic solution of (4) and (5); then (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),…,  𝑥^∗_𝑛(𝑡),𝑢^∗₁(𝑡),𝑢^∗₂(𝑡),  …,𝑢^∗_𝑛(𝑡)) satisfies the system 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−²_𝑗=1(−1)^{𝑚𝑖+𝑗}_𝑙=1𝑎_𝑖𝑗𝑙(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗𝑙(𝑡)−𝑑_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖𝑡−𝜀_𝑖(𝑡),𝑢_𝑖(𝑡)=_𝑡^𝑡+𝜔𝛽_𝑖(𝑠)𝑥_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥_𝑖𝑠−𝜎_𝑖(𝑠)𝐺_𝑖(𝑡,𝑠)𝑑𝑠,𝑖=1,2,…,𝑛, (7) where 𝐺_𝑖()=∫𝑡,𝑠exp^𝑠_𝑡𝑏_𝑖(𝜃)𝑑𝜃exp∫^𝜔₀𝑏_𝑖(𝜃)𝑑𝜃−1,𝑖=1,2,…,𝑛. (8) The converse is also true.

Proof. By (4), (5), and the variation constants formula in ordinary differential equations, we have 𝑢_𝑖(𝑡)=^𝑡₀𝛽_𝑖(𝑠)𝑥_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)×exp−^𝑡₀𝑏_𝑖(𝜃)𝑑𝜃,𝑖=1,2,…,𝑛. (9) From (9), we obtain 𝑢_𝑖(𝑡+𝜔)=₀^𝑡+𝜔𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)×exp−₀^𝑡+𝜔𝑏_𝑖(𝜃)𝑢𝑑𝜃,𝑖=1,2,…,𝑛,_∗𝑖(𝑡)=^𝑡₀𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)×exp−^𝑡₀𝑏_𝑖(𝜃)𝑑𝜃,𝑖=1,2,…,𝑛. (10) Considering that (𝑥^∗₁(𝑡),  𝑥^∗₂(𝑡),…,𝑥^∗_𝑛(𝑡), 𝑢^∗₁(𝑡),   𝑢^∗₂(𝑡),   …,𝑢^∗_𝑛(𝑡)) is an 𝜔-periodic solution of system (4) and (5), we obtain ^𝑡₀𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)exp−^𝑡₀𝑏_𝑖(𝜃)=𝑑𝜃_0𝑡+𝜔𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)×exp−₀^𝑡+𝜔𝑏_𝑖(𝜃)𝑑𝜃,𝑖=1,2,…,𝑛. (11) Then ^𝑡₀𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)exp^𝜔₀𝑏_𝑖(𝜃)=𝑑𝜃_0𝑡+𝜔𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)=^𝑡₀𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)+_𝑡^𝑡+𝜔𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠,𝑖=1,2,…,𝑛, (12) which implies ^𝑡₀𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)𝑒^𝑏𝜔=−1_{𝑡𝑡+𝜔}𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠,𝑖=1,2,…,𝑛. (13) That is, ^𝑡₀𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑖)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠+𝜓_𝑖(0)=𝑢^∗_𝑖(𝑡)exp^𝑡₀𝑏_𝑖(𝜃)=1𝑑𝜃𝑒^𝑏𝜔−1^{𝑡𝑡+𝜔}𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp^𝑠₀𝑏_𝑖(𝜃)𝑑𝜃𝑑𝑠,𝑖=1,2,…,𝑛. (14) Hence 𝑢^∗_𝑖(𝑡)=_𝑡^𝑡+𝜔𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp∫^𝑠_𝑡𝑏_𝑖(𝜃)𝑑𝜃exp∫^𝜔₀𝑏_𝑖(𝜃)𝑑𝜃−1𝑑𝑠,𝑖=1,2,…,𝑛. (15) On the other hand, assume that (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡), …,𝑥^∗_𝑛(𝑡), 𝑢^∗₁(𝑡),𝑢^∗₂(𝑡), …,𝑢^∗_𝑛(𝑡)) is an 𝜔-periodic solution of system (7), then 𝑢^∗_𝑖(𝑡)=_𝑡^𝑡+𝜔𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp∫^𝑠_𝑡𝑏_𝑖(𝜃)𝑑𝜃exp∫^𝜔₀𝑏_𝑖(𝜃)𝑑𝜃−1𝑑𝑠,𝑖=1,2,…,𝑛. (16) By a direct calculation, we have 𝑑𝑢^∗(𝑡)𝑑𝑡=−𝑏_𝑖(𝑡)_𝑡^𝑡+𝜔𝛽_𝑖(𝑠)𝑥^∗_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥^∗_𝑖𝑠−𝜎_𝑖(𝑠)×exp∫^𝑠_𝑡𝑏_𝑖(𝜃)𝑑𝜃exp∫^𝜔₀𝑏_𝑖(𝜃)+𝛽𝑑𝜃−1𝑑𝑠_𝑖(𝑡+𝜔)𝑥^∗_𝑖(𝑡+𝜔)+𝛾_𝑖(𝑡+𝜔)𝑥^∗_𝑖𝑡+𝜔−𝜎_𝑖()×∫𝑡+𝜔exp_𝑡^𝑡+𝜔𝑏_𝑖(𝜃)𝑑𝜃exp∫^𝜔₀𝑏_𝑖(𝜃)−𝛽𝑑𝜃−1_𝑖(𝑡)𝑥^∗_𝑖(𝑡)+𝛾_𝑖(𝑡)𝑥^∗_𝑖𝑡−𝜎_𝑖(𝑡×1)exp∫^𝜔₀𝑏_𝑖(𝜃)𝑑𝜃−1=−𝑏_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)𝑥^∗_𝑖(𝑡)+𝛾_𝑖(𝑡)𝑥^∗_𝑖𝑡−𝜎_𝑖(𝑡),𝑖=1,2,…,𝑛. (17) This completes the proof.

It is easy to see that system (7) is equivalent to the following system: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−²_𝑗=1(−1)^{𝑚𝑖+𝑗}_𝑙=1𝑎_𝑖𝑗𝑙(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗𝑙(𝑡)−𝑑_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑒_𝑖(𝑡)𝑢_𝑖𝑡−𝜀_𝑖(𝑡),𝑢_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐾𝑥_𝑖𝐺_𝑖(𝑡,𝑠)𝑑𝑠,𝑖=1,2,…,𝑛, (18) where 𝐺_𝑖()=∫𝑡,𝑠exp^𝑠_𝑡𝑏_𝑖(𝜃)𝑑𝜃exp∫^𝜔₀𝑏_𝑖(𝜃),𝐾𝑥𝑑𝜃−1_𝑖=𝛽_𝑖(𝑠)𝑥_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑥_𝑖𝑠−𝜎_𝑖(𝑠),𝑖=1,2,…,𝑛. (19) It is clear that in order to prove that systems (4) and (5) have at least one 𝜔-periodic solution, we only need to prove that system (18) has at least one 𝜔-periodic solution.

Now, for convenience of statements we denote the functions 𝑎_𝑖𝑗(𝑡)=^𝑚_𝑙=1𝑎_𝑖𝑗𝑙(𝑡),𝑖,𝑗=1,2,…,𝑛. (20) The following theorem is about the existence of positive periodic solutions of system (4).

Theorem 4. Suppose that assumption (H1) holds and there exists a constant 𝜃_𝑖>0, 𝜁_𝑖>0, 𝑖=1,2,…,𝑛, such that min_{𝑡∈[0,𝜔]}^𝑚_𝑙=1𝛿_𝑖𝑖𝑙(𝑡)𝜃_𝑖−^𝑛_𝑗≠𝑖𝛿_𝑗𝑖𝑙(𝑡)𝜃_𝑗=∶𝜁_𝑖>0,𝑖=1,2,…,𝑛, (21) where 𝛿_𝑖𝑗𝑙(𝑡)=𝑎_𝑖𝑗𝑙𝜑_𝑖𝑗𝑙(𝑡)1−𝜏^′_𝑖𝑗𝑙𝜑_𝑖𝑗𝑙(𝑡),𝑖,𝑗=1,2,…,𝑛,𝑙=1,2,…,𝑚, (22) and the algebraic equation 𝑟_𝑖−𝐻_𝑖−𝑎_𝑖𝑖𝑣_𝑖+^𝑛_𝑗≠𝑖𝑎_𝑖𝑗𝑣_𝑗=0,𝑖=1,2,…,𝑛, (23) where 𝐻_𝑖=^𝜔₀𝛽_𝑖(𝑠)+𝛾_𝑖(𝑠)×𝑑_𝑖(𝑡)𝐺_𝑖(𝑡,𝑠)+𝑒_𝑖(𝑡)𝐺_𝑖𝑡−𝜀_𝑖(𝑡),𝑠𝑑𝑠,𝑖=1,2,…,𝑛, (24) has a unique positive solution. Then system (4) has at least one positive 𝜔-periodic solution.

Proof. For system (18) we introduce new variables 𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) such that 𝑥_𝑖(𝑡)=exp𝑦_𝑖(𝑡),𝑖=1,2,…,𝑛. (25) Then system (18) is rewritten in the following form: 𝑦̇_𝑖(𝑡)=𝑟_𝑖(𝑡)−²_𝑗=1(−1)^{𝑚𝑖+𝑗}_𝑙=1𝑎_𝑖𝑗𝑙(𝑡)exp𝑦_𝑗𝑡−𝜏_𝑖𝑗𝑙(𝑡)−exp−𝑦_𝑖(𝑡)𝑑_𝑖(𝑡)𝑈_𝑖(𝑡),−exp−𝑦_𝑖(𝑡)𝑒_𝑖(𝑡)𝑈_𝑖𝑡−𝜀_𝑖(𝑡)𝑈_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐾𝑒^𝑦𝑖𝐺_𝑖(𝑡,𝑠)𝑑𝑠,𝑖=1,2, (26) where 𝐾𝑒^𝑦𝑖=𝛽_𝑖(𝑠)exp𝑦_𝑖(𝑠)+𝛾_𝑖(𝑠)exp𝑦_𝑖𝑠−𝜎_𝑖(𝑠),𝑖=1,2. (27) In order to apply Lemma 2 to system (26), we introduce the normed vector spaces 𝑋 and 𝑍 as follows. Let 𝐶(𝑅,𝑅^𝑛) denote the space of all continuous functions  𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))∶𝑅→𝑅^𝑛. We take =𝑦(𝑡)𝑋=𝑍∈𝐶𝑅,𝑅^𝑛∶𝑦(𝑡)isan𝜔-periodicfunction, (28) with norm 𝑦=^𝑛_𝑖=1max_{𝑡∈[0,𝜔]}𝑦_𝑖(𝑡). (29) 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)=𝑦̇(𝑡),(𝑡)=𝐿𝑦𝑁𝑦𝑁𝑦₁(𝑡),𝑁𝑦₂(𝑡),…,𝑁𝑦_𝑛(𝑡), (30) where 𝑁𝑦_𝑖(𝑡)=𝑟_𝑖(𝑡)−^𝑚_𝑙=1𝑎_𝑖𝑖𝑙(𝑡)exp𝑦_𝑖𝑡−𝜏_𝑖𝑖𝑙(𝑡+_{𝑛𝑚𝑗≠𝑖𝑙=1})𝑎_𝑖𝑗𝑙(𝑡)exp𝑦_𝑗𝑡−𝜏_𝑖𝑗𝑙(𝑡)−exp−𝑦_𝑖(𝑡)𝑑_𝑖(𝑡)𝑢_𝑖(𝑡)−exp−𝑦_𝑖(𝑡)×𝑒_𝑖(𝑡)𝑢_𝑖𝑡−𝜀_𝑖(𝑡). (31) Further, we define continuous projectors 𝑃∶𝑋→𝑋 and 𝑄∶𝑍→𝑍 as follows: (𝑡)=1𝑃𝑦𝜔^𝜔₀𝑦(𝑡)(𝑡)=1𝑑𝑡,𝑄𝑣𝜔^𝜔₀𝑣(𝑡)𝑑𝑡. (32) We easily see 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, (33) such that 𝑣(𝑡)=𝑣₁+𝑣₂(𝑡), we have codimIm𝐿=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. (34) For convenience, we denote 𝐹(𝑡)=(𝐹₁(𝑡),𝐹₂(𝑡),…,𝐹_𝑛(𝑡)) as follows: 𝐹_𝑖(𝑡)=𝑟_𝑖(𝑡)−^𝑚_𝑙=1𝑎_𝑖𝑖𝑙(𝑡)exp𝑦_𝑖𝑡−𝜏_𝑖𝑖𝑙(𝑡+_{𝑛𝑚𝑗≠𝑖𝑙=1})𝑎_𝑖𝑗𝑙(𝑡)exp𝑦_𝑗𝑡−𝜏_𝑖𝑗𝑙(𝑡)−exp−𝑦_𝑖(𝑡)𝑑_𝑖(𝑡)𝑢_𝑖(𝑡)−exp−𝑦_𝑖(𝑡)×𝑒_𝑖(𝑡)𝑢_𝑖𝑡−𝜀_𝑖(𝑡). (35) Thus, we have (𝑡)=1𝑄𝑁𝑦𝜔^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. (36) From formulas (36), 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 the 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 2) to system (26).
Corresponding to the operator equation 𝐿𝑦(𝑡)=𝜆𝑁𝑦(𝑡) with parameter 𝜆∈(0,1), we have 𝑦̇_𝑖(𝑡)=𝜆𝐹_𝑖(𝑡),𝑖=1,2,…,𝑛, (37) where 𝐹_𝑖(𝑡)(𝑖=1,2,…,𝑛) are given in (35).
Assume that 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))∈𝑋 is a solution of system (37) for some parameter 𝜆∈(0,1). By integrating system (37) over the interval [0,𝜔], we obtain ^𝜔₀𝑟_𝑖(𝑡)−^𝑚_𝑙=1𝑎_𝑖𝑖𝑙(𝑡)exp𝑦_𝑖𝑡−𝜏_𝑖𝑖𝑙(𝑡)+^𝑛𝑚_{𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)exp𝑦_𝑗𝑡−𝜏_𝑖𝑗𝑙(𝑡)−𝑒_𝑖(𝑡)exp−𝑦_𝑖(𝑡)^{𝑡−𝜀𝑖(𝑡)+𝜔}_{𝑡−𝜀𝑖(𝑡)}𝐾𝑒^𝑦𝑖𝐺_𝑖𝑡−𝜀_𝑖(𝑡),𝑠𝑑𝑠−𝑑_𝑖(𝑡)exp−𝑦_𝑖(𝑡)_𝑡^𝑡+𝜔𝐾𝑒^𝑦𝑖𝐺_𝑖(𝑡,𝑠)𝑑𝑠𝑑𝑡=0,𝑖=1,2,…,𝑛. (38) Consequently, 𝑟_𝑖𝜔+^𝜔₀^𝑛𝑚_{𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)exp𝑦_𝑗𝑡−𝜏_𝑖𝑗𝑙(𝑡)𝑑𝑡=^𝜔₀^𝑚_𝑙=1𝑎_𝑖𝑖𝑙(𝑡)exp𝑦_𝑖𝑡−𝜏_𝑖𝑖𝑙(𝑡)+𝑑𝑡_𝜔0𝑒_𝑖(𝑡)exp−𝑦_𝑖(𝑡)×^{𝑡−𝜀𝑖(𝑡)+𝜔}_{𝑡−𝜀𝑖(𝑡)}𝐾𝑒^𝑦𝑖𝐺_𝑖𝑡−𝜀_𝑖(𝑡)+,𝑠𝑑𝑠𝑑𝑡_𝜔0𝑑_𝑖(𝑡)exp−𝑦_𝑖(𝑡)_𝑡^𝑡+𝜔𝐾𝑒^𝑦𝑖𝐺_𝑖(𝑡,𝑠)𝑑𝑠𝑑𝑡,𝑖=1,2,…,𝑛. (39) From the continuity of 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)), there exist constants 𝜉_𝑖,𝜂_𝑖∈[0,𝜔](𝑖=1,2,…,𝑛) such that 𝑦_𝑖𝜉_𝑖=max_{[]𝑡∈0,𝜔}𝑦_𝑖(𝑡),𝑦_𝑖𝜂_𝑖=min_{[]𝑡∈0,𝜔}𝑦_𝑖(𝑡)𝑖=1,2,…,𝑛. (40) By (39) and (40) we obtain 𝑟_𝑖𝜔≤^𝜔₀𝑎_𝑖𝑖(𝑡)exp𝑦_𝑖𝜉_𝑖𝑑𝑡+^𝜔₀𝐴_𝑖(𝑡)exp𝑦_𝑖𝜉_𝑖𝑑𝑡,𝑖=1,2,…,𝑛, (41) where 𝐴_𝑖(𝑡)=𝑒_𝑖(𝑡)^{𝑡−𝜀𝑖(𝑡)+𝜔}_{𝑡−𝜀𝑖(𝑡)}𝐺_𝑖𝑡−𝜀_𝑖(𝑡),𝑠𝛽_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑑𝑠+𝑑_𝑖(𝑡)_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝛽_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (42) Therefore, we further have 𝑦_𝑖𝜉_𝑖≥ln𝑟_𝑖𝑎_𝑖𝑖+𝐴_𝑖,𝑖=1,2,…,𝑛. (43) Let 𝑠_𝑖𝑗𝑙(𝑡)=𝑡−𝜏_𝑖𝑗𝑙(𝑡)(𝑖,𝑗=1,2,…,𝑛,  𝑙=1,2,…,𝑚); then from Lemma 1 and (H1) we get that function 𝑠_𝑖𝑗𝑙(𝑡) has a unique 𝜔 periodic inverse function 𝜑_𝑖𝑗𝑙(𝑡); then, for every 𝑖,𝑗=1,2,…,𝑛,𝑙=1,2,…,𝑚, we have ^𝜔₀𝑎_𝑖𝑗𝑙(𝑡)exp𝑦_𝑖𝑡−𝜏_𝑖𝑗𝑙(𝑡)=𝑑𝑡_{𝜔−𝜏𝑖𝑗𝑙(𝜔)−𝜏𝑖𝑗𝑙(0)}𝑎_𝑖𝑗𝑙𝜑_𝑖𝑗𝑙(𝑡)1−𝜏^′_𝑖𝑗𝑙𝜑_𝑖𝑗𝑙(𝑡)exp𝑦_𝑖(𝑡)𝑑𝑡. (44) One can see that 𝑎_𝑖𝑗𝑙𝜑_𝑖𝑗𝑙(𝑡)1−𝜏^′_𝑖𝑗𝑙𝜑_𝑖𝑗𝑙(𝑡)=∶𝛿_𝑖𝑗𝑙(𝑡),𝑖,𝑗=1,2,…,𝑛,𝑙=1,2,…,𝑚, (45) are 𝜔 periodic functions. Then, for every 𝑖,𝑗=1,2,…,𝑛,𝑙=1,2,…,𝑚, we have ^𝜔₀𝑎_𝑖𝑗𝑙(𝑡)exp𝑦_𝑖𝑡−𝜏_𝑖𝑗𝑙(𝑡)𝑑𝑡=^𝜔₀𝛿_𝑖𝑗𝑙(𝑡)exp𝑦_𝑖(𝑡)𝑑𝑡. (46) From (39) and (46) we further obtain ^𝑛_𝑖=1^𝜔₀^𝑚_𝑙=1𝛿_𝑖𝑖𝑙(𝑡)exp𝑦_𝑖(𝑡)+𝑑_𝑖(𝑡)exp−𝑦_𝑖(𝑡)_𝑡^𝑡+𝜔𝐾𝑒^𝑦𝑖𝐺_𝑖(𝑡,𝑠)𝑑𝑠+𝑒_𝑖(𝑡)exp−𝑦_𝑖(𝑡×_{𝑡−𝜀𝑖(𝑡)+𝜔𝑡−𝜀𝑖(𝑡)})𝐾𝑒^𝑦𝑖𝐺_𝑖𝑡−𝜀_𝑖(𝑡)=,𝑠𝑑𝑠𝑑𝑡_𝑛𝑖=1𝑟_𝑖𝜔+^𝜔₀^𝑛𝑚_{𝑗≠𝑖𝑙=1}𝛿_𝑖𝑗𝑙(𝑡)exp𝑦_𝑗(𝑡)𝑑𝑡. (47) From the above equality we have ^𝑛_𝑖=1^𝜔₀^𝑚_𝑙=1𝛿_𝑖𝑖𝑙(𝑡)−^𝑛𝑚_{𝑗≠𝑖𝑙=1}𝛿_𝑗𝑖𝑙(𝑡)exp𝑦_𝑖(𝑡≤)𝑑𝑡_𝑛𝑖=1𝑟_𝑖𝜔,_{𝜔0𝑚𝑙=1}𝛿_𝑖𝑖𝑙(𝑡)−^𝑛𝑚_{𝑗≠𝑖𝑙=1}𝛿_𝑗𝑖𝑙(𝑡)exp𝑦_𝑖(𝑡≤)𝑑𝑡_𝑛𝑖=1𝑟_𝑖𝜔,𝑖=1,2,…,𝑛. (48) From the assumptions of Theorem 4 and (48) we can obtain 𝜁_𝑖^𝜔₀exp𝑦_𝑖(𝑡)𝑑𝑡≤^𝑛_𝑖=1𝑟_𝑖𝜔,𝑖=1,2,…,𝑛. (49) Consequently, exp𝑦^𝜔_0𝑖(𝑡∑)𝑑𝑡≤^𝑛𝑖=1𝑟_𝑖𝜔𝜁_𝑖,𝑖=1,2,…,𝑛. (50) From (40) and (50), we further obtain 𝑦_𝑖𝜂_𝑖∑≤ln^𝑛𝑖=1𝑟_𝑖𝜁_𝑖,𝑖=1,2,…,𝑛. (51) On the other hand, directly from system (26) we have 𝑦̇^𝜔_0𝑖(𝑡)𝑑𝑡≤^𝜔₀𝑟_𝑖(𝑡)𝑑𝑡+^𝜔₀^𝑚_𝑙=1𝑎_𝑖𝑖𝑙(𝑡)exp𝑦_𝑖𝑡−𝜏_𝑖𝑖𝑙(𝑡)+^𝑛𝑚_{𝑗≠𝑖𝑙=1}𝑎_𝑖𝑗𝑙(𝑡)exp𝑦_𝑗𝑡−𝜏_𝑖𝑗𝑙(𝑡)𝑑𝑡≤^𝜔₀𝑟_𝑖(𝑡+)𝑑𝑡_{𝜔0𝑚𝑙=1}𝛿_𝑖𝑖𝑙(𝑡)+^𝑛_𝑗≠𝑖𝛿_𝑗𝑖𝑙(𝑡)exp𝑦_𝑖(𝑡≤)𝑑𝑡_𝜔0𝑟_𝑖(𝑡)𝑑𝑡+𝐷^𝑀_𝑖^𝜔₀exp𝑦_𝑖(𝑡≤)𝑑𝑡𝑟_𝑖𝐷𝜔+^𝑀𝑖∑^𝑛_𝑖=1𝑟_𝑖𝜔𝜁_𝑖,𝑖=1,2,…,𝑛, (52) where 𝐷_𝑖(𝑡)=^𝑚_𝑙=1𝛿_𝑖𝑖𝑙(𝑡)+^𝑛_𝑗≠𝑖𝛿_𝑗𝑖𝑙(𝑡),𝑖=1,2,…,𝑛. (53) From (43), (51), and (52), we have for any 𝑡∈[0,𝜔]𝑦_𝑖(𝑡)≤𝑦_𝑖𝜂_𝑖+^𝜔₀𝑦̇_𝑖(𝑡∑)𝑑𝑡≤ln^𝑛𝑖=1𝑟_𝑖𝜁_𝑖+𝑟_𝑖𝐷𝜔+^𝑀𝑖∑^𝑛_𝑖=1𝑟_𝑖𝜔𝜁_𝑖=∶𝑀_𝑖,𝑦𝑖=1,2,…,𝑛,_𝑖(𝑡)≥𝑦_𝑖𝜉_𝑖−^𝜔₀𝑦̇_𝑖(𝑡)𝑑𝑡≥ln𝑟_𝑖𝑎_𝑖𝑖+𝐴_𝑖−𝑟_𝑖𝐷𝜔−^𝑀𝑖∑^𝑛_𝑖=1𝑟_𝑖𝜔𝜁_𝑖=∶𝑁_𝑖,𝑖=1,2,…,𝑛. (54)(55) Therefore, from (54) and (55), we have max_{𝑡∈[0,𝜔]}𝑦_𝑖(𝑡)≤max𝑀_𝑖,𝑁_𝑖=∶𝐵_𝑖,𝑖=1,2,…,𝑛. (56) It can be seen that the constants 𝐵_𝑖(𝑖=1,2,…,𝑛) are independent of parameter 𝜆∈(0,1). For any 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)∈𝑅^𝑛, from (31) we can obtain 𝑄𝑁𝑦=𝑄𝑁𝑦₁,𝑄𝑁𝑦₂,…,𝑄𝑁𝑦_𝑛𝑄𝑁𝑦_𝑖=𝑟_𝑖−𝐻_𝑖−𝑎_𝑖𝑖exp𝑦_𝑖+^𝑛_𝑗≠𝑖𝑎_𝑖𝑗exp𝑦_𝑗,𝐻𝑖=1,2,…,𝑛,_𝑖(𝑡)=^𝜔₀𝛽_𝑖(𝑠)+𝛾_𝑖(𝑠)𝑑_𝑖(𝑡)𝐺_𝑖(𝑡,𝑠)+𝑒_𝑖(𝑡)𝐺_𝑖𝑡−𝜀_𝑖(𝑡),𝑠𝑑𝑠,𝑖=1,2,…,𝑛. (57) We consider the following algebraic equation: 𝑟_𝑖−𝐻_𝑖−𝑎_𝑖𝑖𝑣_𝑖+^𝑛_𝑗≠𝑖𝑎_𝑖𝑗𝑣_𝑗=0,𝑖=1,2,…,𝑛. (58)
From the assumption of Theorem 4, the equation has a unique positive solution 𝑣^∗=(𝑣^∗₁,𝑣^∗₂,…,𝑣^∗_𝑛). Hence, the equation 𝑄𝑁𝑦=0 has a unique solution 𝑦^∗=(𝑦^∗₁,𝑦^∗₂,…,𝑦^∗_𝑛)=(ln𝑣^∗₁,ln𝑣^∗₂,…,ln𝑣^∗_𝑛)∈𝑅^𝑛.
Choosing constant 𝐵>0 large enough such that |𝑦^∗₁|+|𝑦^∗₂|+⋯+|𝑦^∗_𝑛|<𝐵 and 𝐵>𝐵₁+𝐵₂+⋯+𝐵_𝑛, we define a bounded open set Ω⊂𝑋 as follows: Ω=𝑦∈𝑋∶𝑦<𝐵. (59) It is clear that Ω satisfies conditions (𝑎) and (𝑏) of Lemma 2. On the other hand, by directly calculating we can obtain {(𝑓deg𝐽𝑄𝑁,Ω∩Ker𝐿,0,0,…,0)}=sgn^1𝑦1𝑓¹_𝑦2⋯𝑓¹_𝑦𝑛𝑓²_𝑦1𝑓²_𝑦2⋯𝑓²_𝑦𝑛𝑓⋯⋯⋯⋯_𝑛𝑦1𝑓^𝑛_𝑦2⋯𝑓^𝑛_𝑦𝑛, (60) where 𝑓^𝑖_𝑦𝑗=−𝑎_𝑖𝑗exp𝑦^∗_𝑗𝑓,𝑖=𝑗_𝑖𝑦𝑗=𝑎_𝑖𝑗exp𝑦^∗_𝑗,𝑖≠𝑗𝑖,𝑗=1,2,…,𝑛. (61) From the assumption of Theorem 4, we have 𝑓¹_𝑦1𝑓¹_𝑦2⋯𝑓¹_𝑦𝑛𝑓²_𝑦1𝑓²_𝑦2⋯𝑓²_𝑦𝑛𝑓⋯⋯⋯⋯_𝑛𝑦1𝑓^𝑛_𝑦2⋯𝑓^𝑛_𝑦𝑛≠0. (62) From this, we finally have deg{𝐽𝑄𝑁,Ω∩Ker𝐿,(0,0,…,0)}≠0. (63) This shows that Ω satisfies condition (𝑐) of Lemma 2. Therefore, system (26) has an 𝜔-periodic solution 𝑦^∗(𝑡)=(𝑦^∗₁(𝑡),𝑦^∗₂(𝑡),…,𝑦^∗_𝑛(𝑡))∈Ω. Finally, we have system that (4) has a positive 𝜔-periodic solution. This completes the proof.

Acknowledgments

This work was supported by the Natural Science Foundation of Xinjiang (Grants no. 2012223177, 200721104) and the National Natural Science Foundation of China (Grants no. 10661010, 11261061).