Global Positive Periodic Solutions of Generalized 𝑛-Species Competition Systems with Multiple Delays and Impulses

Abstract

By applying the fixed point theorem in a cone of Banach space, we obtain an easily verifiable necessary and sufficient condition for the existence of positive periodic solutions of two kinds of generalized 𝑛-species competition systems with multiple delays and impulses as follows: 𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)[𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−∑^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥_𝑖(𝑡−𝜏_𝑖𝑗(𝑡))−∑^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑥_𝑗(𝑡−𝛾_𝑖𝑗(𝑡))−∑^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)∫⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑥_𝑗(𝑡+𝑠)𝑑𝑠],a.e.,𝑡>0,𝑡≠𝑡_𝑘,𝑘∈𝑍₊,𝑖=1,2,…,𝑛;𝑥_𝑖(𝑡⁺_𝑘)−𝑥_𝑖(𝑡⁻_𝑘)=𝜃_𝑖𝑘𝑥_𝑖(𝑡_𝑘𝑍),𝑖=1,2,…,𝑛,𝑘∈₊;and𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)[𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)+∑^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥_𝑖(𝑡−𝜏_𝑖𝑗(𝑡))−∑^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑥_𝑗(𝑡−𝛾_𝑖𝑗∑(𝑡))−_𝑛𝑗=1𝑒_𝑖𝑗(𝑡)∫⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑥_𝑗(𝑡+𝑠)𝑑𝑠],a.e.,𝑡>0,𝑡≠𝑡_𝑘,𝑘∈𝑍₊,𝑖=1,2,…,𝑛;𝑥_𝑖(𝑡⁺_𝑘)−𝑥_𝑖(𝑡⁻_𝑘)=𝜃_𝑖𝑘𝑥_𝑖(𝑡_𝑘),𝑖=1,2,…,𝑛,𝑘∈𝑍₊. It improves and generalizes a series of the well-known sufficiency theorems in the literature about the problems mentioned previously.

1. Introduction

Let 𝑅=(−∞,+∞), 𝑅₊=[0,+∞), 𝑅₋=(−∞,0], 𝑅^𝑛={(𝑥₁,…,𝑥_𝑛)^𝑇∶𝑥_𝑖≥0,1≤𝑖≤𝑛}, 𝐽⊂𝑅, 𝑍₊={1,2,3,…}, respectively, and let 𝜔>0 be a constant and 𝐶_𝜔={𝑥∣𝑥∈𝐶(𝑅,𝑅),𝑥(𝑡+𝜔)=𝑥(𝑡)}, with the norm defined by |𝑥|₀=max_{𝑡∈[0,𝜔]}|𝑥(𝑡)|; 𝐶¹_𝜔={𝑥∣𝑥∈𝐶¹(𝑅,𝑅),𝑥(𝑡+𝜔)=𝑥(𝑡)}, with the norm defined by ‖𝑥‖  =max_{𝑡∈[0,𝜔]}{|𝑥|₀,|𝑥^′|₀}; 𝑃𝐶={𝑥∣𝑥∶𝑅→𝑅⁺,lim_𝑠→𝑡𝑥(𝑠)=𝑥(𝑡),if𝑡≠𝑡_𝑘,lim_𝑡→𝑡⁻_𝑘𝑥(𝑡)=𝑥(𝑡_𝑘),lim_𝑡→𝑡⁺_𝑘𝑥(𝑡)exists,𝑘∈𝑍₊}; 𝑃𝐶¹={𝑥∣𝑥∶𝑅→𝑅⁺,𝑥^′∈𝑃𝐶}; 𝑃𝐶_𝜔={𝑥∣𝑥∈𝑃𝐶,𝑥(𝑡+𝜔)=𝑥(𝑡)}, with the norm defined by |𝑥|₀=max_{𝑡∈[0,𝜔]}|𝑥(𝑡)|; 𝑃𝐶¹_𝜔={𝑥∣𝑥∈𝑃𝐶¹,𝑥(𝑡+𝜔)=𝑥(𝑡)}, with the norm defined by ‖𝑥‖  =max_{𝑡∈[0,𝜔]}{|𝑥|₀,|𝑥^′|₀}; 𝑔^𝑀=max_{𝑡∈[0,𝜔]}{𝑔(𝑡)},𝑔^𝐿=min_{𝑡∈[0,𝜔]}{𝑔(𝑡)}, for any 𝑔(𝑡)∈𝑃𝐶_𝜔.

The theory of impulsive delay differential equations is emerging as an important area of investigation, since it is a lot richer than the corresponding theory of nonimpulsive delay differential equations. Many evolution processes in nature are characterized by the fact that at certain moments of time they experience abrupt change of state. That was the reason for the development of the theory of impulsive differential equations and impulsive delay differential equations, see the monographs [1–4]. In recent years, the existence theory of positive periodic solutions of delay differential equations with impulsive effects or without impulsive effects has been an object of active research, which is referred to as [5–12]. However, only a little work has been done on the existence of positive periodic solutions to the high-dimension impulsive differential equations based on the theory in cones. Motivated by this, in this paper, we mainly consider the following two classes of 𝑛-species Lotka-Volterra competitive systems with multiple delays and impulses: 𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥_𝑖𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑥_𝑗(𝑡+𝑠)𝑑𝑠,a.e.,𝑡>0,𝑡≠𝑡_𝑘,𝑘∈𝑍₊𝑥,𝑖=1,2,…,𝑛,_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝜃_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛;𝑘∈𝑍₊;𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)+^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥_𝑖𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑥_𝑗(𝑡+𝑠)𝑑𝑠,a.e.,𝑡>0,𝑡≠𝑡_𝑘,𝑘∈𝑍₊𝑥,𝑖=1,2,…,𝑛,_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝜃_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛;𝑘∈𝑍₊, (1)(2) where 𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑐_𝑖𝑗(𝑡),𝑑_𝑖𝑗(𝑡),𝑒_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡),𝛾_𝑖𝑗(𝑡)∈𝑃𝐶_𝜔, 𝑖,𝑗=1,2,…,𝑛, with 𝑎_𝑖=(1/𝜔)∫^𝜔₀𝑎_𝑖(𝑡)𝑑𝑡>0. Moreover, 𝑓_𝑖𝑗∈𝑃𝐶,𝜎_𝑖𝑗>0 is a constant, and ∫⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑑𝑠=1. Furthermore, Δ𝑥_𝑖(𝑡_𝑘)=𝑥_𝑖(𝑡⁺_𝑘)−𝑥_𝑖(𝑡_𝑘), 𝑖=1,2,…,𝑛 (here 𝑥(𝑡⁺_𝑘) represents the right limit of 𝑥(𝑡) at the point 𝑡_𝑘); 𝜃_𝑘=(𝜃_1𝑘,𝜃_2𝑘,…,𝜃_𝑛𝑘)∈𝐶(𝑅^𝑛₊,𝑅^𝑛₋); that is, 𝑥_𝑖(𝑡) changes decreasingly suddenly at times 𝑡_𝑘. We assume that there exists an integer 𝑝>0 such that 𝑡_𝑘+𝑝=𝑡_𝑘+𝜔, 𝜃_{𝑖(𝑘+𝑝)}=𝜃_𝑖𝑘, 𝑖=1,2,…,𝑛, where 0<𝑡₁<𝑡₂<⋯<𝑡_𝑝<𝜔.

References [13, 14], G. Seifert investigated the following periodic single-species population growth models with periodic delay: 𝑦^′(𝑡)=𝑦(𝑡)𝑟(𝑡)−𝑎(𝑡)𝑦(𝑡)−𝑏(𝑡)𝑦(𝑡−𝜏(𝑡)). (3) They had assumed that the net birth 𝑟(𝑡), the self-inhibition rate 𝑎(𝑡), and the delay 𝜏(𝑡) are continuously differentiable 𝜔-periodic functions, and 𝑟(𝑡)>0, 𝑎(𝑡)>0, 𝑏(𝑡)≥0, 𝜏(𝑡)≥0, for 𝑡∈𝑅. The negative feedback term −𝑏(𝑡)𝑦(𝑡−𝜏(𝑡)) in the average growth rate of species has a negative time delay (the sign of the time delay term is negative), which can be regarded as the deleterious effect of time delay on a species growth rate. They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (3). But the discussion of global attractivity is only confined to the special case when the periodic delay is constant.

In [15], Freedman and Wu proposed the following periodic single-species population growth models with periodic delay: 𝑦^′(𝑡)=𝑦(𝑡)𝑟(𝑡)−𝑎(𝑡)𝑦(𝑡)+𝑏(𝑡)𝑦(𝑡−𝜏(𝑡)). (4) They had assumed that the net birth 𝑟(𝑡), the self-inhibition rate 𝑎(𝑡), and the delay 𝜏(𝑡) are continuously differentiable 𝜔-periodic functions, and 𝑟(𝑡)>0, 𝑎(𝑡)>0, 𝑏(𝑡)≥0, 𝜏(𝑡)≥0 for 𝑡∈𝑅. The positive feedback term 𝑏(𝑡)𝑦(𝑡−𝜏(𝑡)) in the average growth rate of species has a positive time delay (the sign of the time delay term is positive), which is a delay due to gestation. They had established sufficient conditions which guarantee that system (4) has a positive periodic solution which is globally asymptotically stable.

References [16, 17] have studied the following two-species competitive system without delay: 𝑦^′₁(𝑡)=𝑦₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑦₁(𝑡)−𝑐₁(𝑡)𝑦₂(𝑡,𝑦_′2)(𝑡)=𝑦₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦₂(𝑡)−𝑐₂(𝑡)𝑦₁(𝑡). (5) They had derived sufficient conditions for the existence and global attractivity of positive periodic solutions of system (5) by using differential inequalities and topological degree, respectively. In fact, in many practical situations the time delay occurs so often. A more realistic model should include some of the past states of the system. Motivated by the previous ideas, Liu et al. [18] considered two corresponding periodic Lotka-Volterra competitive systems involving multiple delays: 𝑦^′₁(𝑡)=𝑦₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑦₁(𝑡)+^𝑛_𝑖=1𝑏_1𝑖(𝑡)𝑦₁𝑡−𝜏_𝑖(𝑡)−^𝑚_𝑗=1𝑐_1𝑗(𝑡)𝑦₂𝑡−𝜌_𝑗(𝑡),𝑦^′₂(𝑡)=𝑦₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦₂(𝑡)+^𝑚_𝑗=1𝑏_2𝑗(𝑡)𝑦₂𝑡−𝜂_𝑗(𝑡)−^𝑛_𝑖=1𝑐_2𝑖(𝑡)𝑦₁𝑡−𝜎_𝑖(𝑡),𝑦^′₁(𝑡)=𝑦₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑦₁(𝑡)−^𝑛_𝑖=1𝑏_1𝑖(𝑡)𝑦₁𝑡−𝜏_𝑖(𝑡−_𝑚𝑗=1)𝑐_1𝑗(𝑡)𝑦₂𝑡−𝜌_𝑗(𝑡),𝑦^′₂(𝑡)=𝑦₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦₂(𝑡)−^𝑚_𝑗=1𝑏_2𝑗(𝑡)𝑦₂𝑡−𝜂_𝑗(𝑡)−^𝑛_𝑖=1𝑐_2𝑖(𝑡)𝑦₁𝑡−𝜎_𝑖(𝑡), (6) where 𝑏_1𝑖(𝑡),𝑏_2𝑗(𝑡)∈𝐶(𝑅,[0,+∞)), 𝑎₁(𝑡),𝑎₂(𝑡),𝑐_1𝑗(𝑡),𝑐_2𝑖(𝑡)∈𝐶(𝑅,[0,+∞)),𝜏_𝑖(𝑡),𝜌_𝑗(𝑡),𝜂_𝑗(𝑡),𝜎_𝑖(𝑡)∈𝐶¹(𝑅,[0,+∞))(𝑖=1,2,…,𝑛;𝑗=1,2,…,𝑚) are 𝜔-periodic functions. Here, the intrinsic growth rates 𝑟_𝑘(𝑡)∈𝐶(𝑅,𝑅) are 𝜔-periodic functions with ∫^𝜔₀𝑟_𝑘(𝑡)𝑑𝑡>0, 𝑘=1,2. They had derived the same criteria for the existence and globally asymptotic stability of positive periodic solutions of the previous two competitive systems by using Gaines and Mawhin's continuation theorem of coincidence degree theory and by means of a suitable Lyapunov functional. However, to this day, no scholars had done works on the existence of positive periodic solution of (1) and (2). One could easily see that systems (3)–(6) are all special cases of system (1) and (2).

Throughout the paper, we make the following assumptions:(H₁)𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑐_𝑖𝑗(𝑡),𝑑_𝑖𝑗(𝑡),𝑒_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡),𝛾_𝑖𝑗(𝑡)∈𝑃𝐶_𝜔, 𝑖,𝑗=1,2,…,𝑛, with 𝑎_𝑖=(1/𝜔)∫^𝜔₀𝑎_𝑖(𝑡)𝑑𝑡>0, 𝜏=max_{1≤𝑖≤𝑛}max_{1≤𝑗≤𝑛}{𝜏_𝑖𝑗(𝑡),𝛾_𝑖𝑗(𝑡)};(H₂){𝑡_𝑘},𝑘∈𝑍₊ satisfies 0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯ and lim_𝑘→+∞𝑡_𝑘=+∞;(H₃){𝜃_𝑖𝑘} is a real sequence with Δ_𝑖𝑘=𝜃_𝑖𝑘+1>0, 𝑖=1,2,…,𝑛, 𝑘∈𝑍₊;(H₄)∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘 is a 𝜔-periodic function.

To conclude this section, we summarize in the following a few definitions and lemmas that will be needed in our arguments.

Definition 1 (see [1]). A function 𝑥_𝑖∶𝑅→(0,+∞) is said to be a positive solution of (1) and (2), if the following conditions are satisfied: (a)𝑥_𝑖(𝑡) is absolutely continuous on each (𝑡_𝑘,𝑡_𝑘+1); (b) for each 𝑘∈𝑍₊, 𝑥_𝑖(𝑡⁺_𝑘) and 𝑥_𝑖(𝑡⁻_𝑘) exist and 𝑥_𝑖(𝑡⁻_𝑘)=𝑥_𝑖(𝑡_𝑘); (c)𝑥_𝑖(𝑡) satisfies the first equation of (1) and (2) for almost everywhere (for short a.e.) in [0,∞]⧵{𝑡_𝑘} and satisfies 𝑥_𝑖(𝑡⁺_𝑘)=Δ_𝑖𝑘𝑥_𝑖(𝑡_𝑘) for 𝑡=𝑡_𝑘, 𝑘∈𝑍₊={1,2,…}.
Under the previous hypotheses (H₁)–(H₄), we consider the following two classes of nonimpulsive Lotka-Volterra competitive systems: 𝑑𝑦_𝑖(𝑡)𝑑𝑡=𝑦_𝑖(𝑡)𝑎_𝑖(𝑡)−𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)×⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝑡+𝑠)𝑑𝑠,a.e.,𝑡≥0,𝑑𝑦_𝑖(𝑡)𝑑𝑡=𝑦_𝑖(𝑡)𝑎_𝑖(𝑡)−𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)×⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝑡+𝑠)𝑑𝑠,a.e.,𝑡≥0, (7)(8) where 𝐵_𝑖(𝑡)=𝑏_𝑖(𝑡)_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘,𝐶_𝑖𝑗(𝑡)=𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}Δ_𝑖𝑘,𝐷_𝑖𝑗(𝑡)=𝑑_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝛾𝑖𝑗(𝑡)}Δ_𝑖𝑘,𝐸_𝑖𝑗(𝑡)=𝑒_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑖𝑗}Δ_𝑖𝑘. (9) By a solution 𝑦(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))^𝑇 of (7) and (8), it means an absolutely continuous function 𝑦(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))^𝑇 defined on [−𝜏,0] that satisfies (7) and (8).

The following lemmas will be used in the proofs of our results. The proof of the first lemma is similar to that of Theorem 1 in [5].

Lemma 2. Suppose that (𝐻₁)–(𝐻₄) hold. Then(1)if 𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) are solutions of (7) and (8) on [−𝜏,+∞), then 𝑥_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) are solutions of (1) and (2) on [−𝜏,+∞);(2)if 𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛) are solutions of (1) and (2) on [−𝜏,+∞), then 𝑦_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ⁻¹_𝑖𝑘𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛) are solutions of (7) and (8) on [−𝜏,+∞).

Proof. (1) It is easy to see that 𝑥_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) are absolutely continuous on every interval (𝑡_𝑘,𝑡_𝑘+1],𝑡≠𝑡_𝑘,𝑘=1,2,…, 𝑥^′_𝑖(𝑡)−𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥_𝑖𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)×⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑥_𝑗(𝑡+𝑠)𝑑𝑠=_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦^′_𝑖(𝑡)−_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)×𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)×_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}Δ_𝑖𝑘𝑦_𝑖𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)×_{0<𝑡𝑘<𝑡−𝛾𝑖𝑗(𝑡)}Δ_𝑖𝑘𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_{𝑗=10<𝑡𝑘<𝑡−𝜎𝑖𝑗}Δ_𝑖𝑘𝑒_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗()=𝑡+𝑠𝑑𝑠_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦^′_𝑖(𝑡)−𝑦_𝑖(𝑡)×𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}Δ_𝑖𝑘𝑦_𝑖×𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)×_{0<𝑡𝑘<𝑡−𝛾𝑖𝑗(𝑡)}Δ_𝑖𝑘𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_{𝑗=10<𝑡𝑘<𝑡−𝜎𝑖𝑗}Δ_𝑖𝑘𝑒_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗×()=𝑡+𝑠𝑑𝑠_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦^′_𝑖(𝑡)−𝑦_𝑖(𝑡)×𝑎_𝑖(𝑡)−𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖×𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝑡+𝑠)𝑑𝑠=0. (10) On the other hand, for any 𝑡=𝑡_𝑘, 𝑘=1,2,…, 𝑥_𝑖𝑡⁺_𝑘=lim_{𝑡→𝑡+𝑘0<𝑡𝑗<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)=_{0<𝑡𝑗≤𝑡𝑘}Δ_𝑖𝑘𝑦_𝑖𝑡_𝑘,𝑥_𝑖𝑡_𝑘=_{0<𝑡𝑗<𝑡𝑘}Δ_𝑖𝑘𝑦_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛; (11) thus 𝑥_𝑖𝑡⁺_𝑘=Δ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…. (12) It follows from (10)–(12) that 𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛) are solutions of (1). Similarly, if 𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) are solutions of (8), we can prove that 𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛) are solutions of (2).
(2) Since 𝑥_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡) is absolutely continuous on every interval (𝑡_𝑘,𝑡_𝑘+1],𝑡≠𝑡_𝑘,𝑘=1,2,…, and in view of (12), it follows that for any 𝑘=1,2,…, 𝑦_𝑖𝑡⁺_𝑘=_{0<𝑡𝑗≤𝑡𝑘}Δ⁻¹_𝑖𝑘𝑥_𝑖𝑡⁺_𝑘=_{0<𝑡𝑗<𝑡𝑘}Δ⁻¹_𝑖𝑘𝑥_𝑖𝑡_𝑘=𝑦_𝑖𝑡_𝑘,𝑦_𝑖𝑡⁻_𝑘=_{0<𝑡𝑗<𝑡𝑘}Δ⁻¹_𝑖𝑘𝑥_𝑖𝑡⁻_𝑘=_{0<𝑡𝑗≤𝑡−𝑘}Δ⁻¹_𝑖𝑘𝑥_𝑖𝑡⁻_𝑘=𝑦_𝑖𝑡_𝑘, (13) which implies that 𝑦_𝑖(𝑡) is continuous on [−𝜏,+∞). It is easy to prove that 𝑦_𝑖(𝑡) is absolutely continuous on [−𝜏,+∞). Similar to the proof of (1), we can check that 𝑦_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ⁻¹_𝑖𝑘𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛) are solutions of (7) on [−𝜏,+∞). Similarly, if 𝑥_𝑖(𝑡), 𝑖=1,2,…,𝑛, is a solution of (2), we can prove that 𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) are solutions of (8). The proof of Lemma 2 is completed.

In the following section, we only discuss the existence of a periodic solution for (7) and (8).

Definition 3 (see [19]). Let 𝑋 be a real Banach space and 𝐸 a closed, nonempty subset of 𝑋. 𝐸 is said to be a cone if(1)𝛼𝑥+𝛽𝑦∈𝐸 for all 𝑥,𝑦∈𝐸, and 𝛼,𝛽>0,(2)𝑥,−𝑥∈𝐸 imply 𝑥=0.

Lemma 4 (see Krasnoselskii [20], Deimling [21], and Guo and Lakshmikantham [22]). Let 𝐸 be a cone in a real Banach space 𝑋. Assume that Ω_𝑟1 and Ω_𝑟2 are open subsets of 𝑋 with 0∈Ω_𝑟1⊂Ω_𝑟1⊂Ω_𝑟2, where Ω_𝑟𝑖={𝑥∈𝑋∶‖𝑥‖<𝑟_𝑖},𝑖=1,2. Let 𝑇∶𝐸∩(Ω_𝑟2⧵Ω_𝑟1)→𝐸 be a continuous and completely continuous operator satisfying(1)‖𝑇𝑥‖≤‖𝑥‖, for any 𝑥∈𝐸∩𝜕Ω_𝑟1;(2)there exists 𝜙∈𝐸⧵{0} such that 𝑦≠𝑇𝑦+𝜆𝑦, for any 𝑦∈𝐸∩𝜕Ω_𝑟2 and 𝜆>0.
Then 𝑇 has fixed points in 𝐸∩(Ω_𝑟2⧵Ω_𝑟1). The same conclusion remains valid if (1) holds for any 𝑥∈𝐸∩𝜕Ω_𝑟2  and (2) holds for any 𝑦∈𝐸∩𝜕Ω_𝑟1 and 𝜆>0.

The paper is organized as follows. In Section 2, firstly, we give some definitions and lemmas. Secondly, we derive a necessary and sufficient condition for at least one positive periodic solution of (1) which is established by using the fixed-point theorem in the cone of Banach space under some conditions. In the following section, we also get a necessary and sufficient condition for at least one positive periodic solution of (2) that is also established by applying the fixed-point theorem in the cone of Banach space under some conditions. Finally, as applications, we study some particular cases of system (1) and (2) which have been investigated extensively in the references mentioned earlier.

2. Existence of Periodic Solution of (1)

We establish the existence of positive periodic solutions of (1) by applying Lemma 4. We will first make some preparations and list later a few preliminary results. Let 𝑋={𝑦=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝐶(𝑅,𝑅^𝑛)∣𝑦(𝑡+𝜔)=𝑦(𝑡)} with the norm ‖𝑦‖=∑^𝑛_𝑖=1|𝑦_𝑖|₀, |𝑦_𝑖|₀=sup_{𝑡∈[0,𝜔]}|𝑦_𝑖(𝑡)|. It is easy to verify that (𝑋,‖⋅‖) is a Banach space.

We define an operator 𝜓∶𝑋→𝑋 as follows: 𝜓𝑦(𝑡)=𝜓𝑦₁(𝑡),𝜓𝑦₂(𝑡),…,𝜓𝑦_𝑛(𝑡)^𝑇, (14) where 𝜓𝑦_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝜉)𝑦_𝑖(𝜉)×𝐵_𝑖(𝜉)𝑦_𝑖(𝜉)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝜉)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝜉)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝜉)𝑦_𝑗𝜉−𝛾_𝑖𝑗(𝜉)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝜉)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝜉+𝑠)𝑑𝑠𝑑𝜉,𝑖=1,2,…,𝑛, (15) where 𝐺_𝑖()=𝑒𝑡,𝜉^{−∫𝜉𝑡𝑎𝑖(𝑢)}𝑑𝑢1−𝑒^{−∫𝜔0𝑎𝑖(𝑢)}=𝑒𝑑𝑢^{∫𝜉𝑡+𝜔𝑎𝑖(𝑢)}𝑑𝑢𝑒^{∫𝜔0𝑎𝑖(𝑢)},𝐺()𝐺𝑑𝑢−1𝑡,𝜉=diag₁(𝑡,𝜉),𝐺₂(𝑡,𝜉),…,𝐺_𝑛(𝑡,𝜉),𝑡≤𝜉≤𝑡+𝜔. (16) It is clear that 𝐺_𝑖(𝑡+𝜔,𝜉+𝜔)=𝐺_𝑖(𝑡,𝜉), 𝜕𝐺_𝑖(𝑡,𝜉)/𝜕𝑡=𝑎_𝑖(𝑡)𝐺_𝑖(𝑡,𝜉), 𝐺_𝑖(𝑡,𝑡)−𝐺_𝑖(𝑡,𝑡+𝜔)=1. In view of (H₁), we also define for 1≤𝑖≤𝑛𝛼_𝑖∶=min_{0≤𝑡≤𝜉≤𝜔}𝐺_𝑖(=1𝑡,𝜉)𝑒^{∫𝜔0𝑎𝑖(𝑢)𝑑𝑢},𝛽−1_𝑖∶=max_{0≤𝑡≤𝜉≤𝜔}𝐺_𝑖(=𝑒𝑡,𝜉)^{∫𝜔0𝑎𝑖(𝑢)𝑑𝑢}𝑒^{∫𝜔0𝑎𝑖(𝑢)𝑑𝑢},−1𝛼=min_{1≤𝑖≤𝑛}𝛼_𝑖,𝛽=max_{1≤𝑖≤𝑛}𝛽_𝑖𝛼,𝜎=𝛽∈(0,1). (17) Define 𝐸 to be a cone in 𝑋 by 𝐸=𝑦=𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)^𝑇∈𝑋∶𝑦_𝑖(𝑡)≥𝜎𝑦_𝑖0,𝑡∈[0,𝜔]. (18) We easily verify that 𝐸 is a cone in 𝑋. For convenience of expressions, we define an operator 𝜑∶𝐸→𝑋 by 𝜑𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡)𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡+_𝑛𝑗=1)𝐸_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝑡+𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (19) The proof of the main result in this paper is based on an application of Krasnoselskii fixed-point theorem in cones. To make use of fixed point theorem in the cone, firstly, we need to introduce some definitions and lemmas.

Lemma 5. Assume that (𝐻₁)–(𝐻₄) hold. Then the solutions of (1) are defined on [−𝜏,∞) and are positive.

Proof. By Lemma 2, we only need to prove that the solutions 𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) of (7) are defined on [−𝜏,∞) and are positive on [0,∞). From (7), we have that for any 𝑢_𝑖∈𝐶([−𝜏,0),𝑅⁺), 𝑖=1,2,3,…,𝑛, and 𝑡>0, 𝑦_𝑖(𝑡)=𝑢_𝑖(0)exp^𝑡₀𝑎_𝑖(𝜉)−𝐵_𝑖(𝜉)𝑦_𝑖(𝜉)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝜉)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝜉)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝜉)𝑦_𝑗𝜉−𝛾_𝑖𝑗(𝜉)−^𝑛_𝑗=1𝐸_𝑖𝑗(𝜉)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗×(𝜉+𝑠)𝑑𝑠𝑑𝜉,𝑖=1,2,…,𝑛. (20) Therefore, 𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) are defined on [−𝜏,∞) and are positive on [0,∞). The proof of Lemma 5 is complete.

Lemma 6. Assume that (𝐻₁)–(𝐻₄) hold. Then 𝜓∶𝐸→𝐸 is well defined.

Proof. In view of the definitions of 𝐸 and 𝐹, for any 𝑦∈𝐸, we have 𝜓𝑦_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝜉)𝜑𝑦_𝑖(𝜉)𝑑𝜉,𝜓𝑦_𝑖(𝑡+𝜔)=^𝑡+2𝜔_𝑡+𝜔𝐺_𝑖(𝑡,𝜉+𝜔)𝜑𝑦_𝑖()=𝜉+𝜔𝑑𝜉_{𝑡𝑡+𝜔}𝐺_𝑖(𝑡,𝜉)𝜑𝑦_𝑖(𝜉)=𝑑𝜉𝜓𝑦_𝑖(𝑡). (21) Therefore, (𝜓𝑦)∈𝑋. Furthermore, for any 𝑦∈𝐸, it follows from (15) that 𝜓𝑦_𝑖0≤𝛽_𝑖^𝜔₀𝜑𝑦_𝑖(𝜉)𝑑𝜉. (22) On the other hand, we obtain 𝜓𝑦_𝑖(𝑡)≥𝛼_𝑖^𝜔₀𝜑𝑦_𝑖(𝜉)≥𝛼𝑑𝜉_𝑖𝛽_𝑖𝜓𝑦_𝑖0≥𝜎𝜓𝑦_𝑖0. (23) Therefore, 𝜓𝑦∈𝐸. The proof of Lemma 6 is complete.

Lemma 7. The operator 𝜓∶𝐸→𝐸 is continuous and completely continuous.

Proof. By using a standard argument one can show that 𝜓 is continuous on 𝐸. Now, we show that 𝜓 is completely continuous. Let 𝑟 be any positive constant and 𝑆_𝑟={𝑦∈𝑋∶|𝑦_𝑖|₀≤𝑟} a bounded set. For any 𝑦∈𝑆_𝑟, by (15) we have 𝜓𝑦_𝑖0≤𝛽_𝑖^𝜔₀𝜑𝑦_𝑖(𝜉)𝑑𝜉=𝛽_𝑖^𝜔₀𝑦_𝑖(𝑡)𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝐹_𝑖𝑗(𝑠)𝑦_𝑗(𝑡+𝑠)𝑑𝑠𝑑𝜉≤𝜔𝛽_𝑖𝑟²𝐵^𝑀_𝑖+^𝑛_𝑗=1𝐶^𝑀_𝑖𝑗+𝐷^𝑀_𝑖𝑗+𝐸^𝑀_𝑖𝑗∶=𝑅_𝑖. (24) Therefore, for any 𝑦∈𝑆_𝑟, we obtain 𝜓𝑦=^𝑛_𝑖=1𝜓𝑦_𝑖0≤^𝑛_𝑖=1𝑅_𝑖∶=𝑅, (25) which implies that 𝜓(𝑆_𝑟) is a uniformly bounded set. On the other hand, in view of the definitions of 𝜓 and 𝜑, we have 𝑑𝜓𝑦_𝑖(𝑡)𝑑𝑡=𝐺_𝑖(𝑡,𝑡+𝜔)𝜑𝑦_𝑖(𝑡+𝜔)−𝐺_𝑖(𝑡,𝑡)𝜑𝑦_𝑖(𝑡)+_𝑡^𝑡+𝜔𝑑𝐺_𝑖(𝑡,𝜉)𝑑𝑡𝜑𝑦_𝑖(𝜉)𝑑𝜉=−𝜑𝑦_𝑖(𝑡)+𝑎_𝑖(𝑡)_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝜉)𝜑𝑦_𝑖(𝜉)𝑑𝜉=𝑎_𝑖(𝑡)𝜓𝑦_𝑖(𝑡)−𝜑𝑦_𝑖(𝑡). (26) Again, from (15), we obtain 𝑑𝜓𝑦_𝑖(𝑡)𝑑𝑡≤𝑎^𝑀_𝑖𝑅_𝑖+𝑟²𝐵^𝑀_𝑖+^𝑛_𝑗=1𝐶^𝑀_𝑖𝑗+𝐷^𝑀_𝑖𝑗+𝐸^𝑀_𝑖𝑗∶=𝑅_𝑖≤𝑀∶=max_{[]𝑖∈1,𝑛}𝑅_𝑖, (27) which implies that 𝑑[(𝜓𝑦)_𝑖(𝑡)]/𝑑𝑡, for any 𝑦∈𝑆_𝑟, is also uniformly bounded. Hence, 𝜓(𝑆_𝑟)⊂𝑋 is a family of uniformly bounded and equicontinuous functions. By the well-known Ascoli-Arzela theorem, we know that the operator 𝜓 is completely continuous. The proof of Lemma 7 is complete.

Lemma 8. Assume that (𝐻₁)–(𝐻₄) hold. The existence of positive 𝜔-periodic solution of (7) is equivalent to that of nonzero fixed point of 𝜓 in 𝐸.

Proof. Assume that 𝑦=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝑋 is a periodic solution of (7). Then, we have 𝑦_𝑖(𝑡)𝑒^{−∫𝑡0𝑎𝑖(𝑢)𝑑𝑢′}=−𝑒^{−∫𝑡0𝑎𝑖(𝑢)𝑑𝑢}𝑦_𝑖(𝑡)𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝑡+𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (28) Integrating the previous equation over [𝑡,𝑡+𝜔], we can have 𝑦_𝑖(𝜉)𝑒^{−∫𝜉0𝑎𝑖(𝑢)𝑑𝑢}_𝑡^𝑡+𝜔=−_𝑡^𝑡+𝜔𝑒^{−∫𝜉0𝑎𝑖(𝑢)𝑑𝑢}𝑦_𝑖(𝜉)×𝐵_𝑖(𝜉)𝑦_𝑖(𝜉)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝜉)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝜉)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝜉)𝑦_𝑗𝜉−𝛾_𝑖𝑗(𝜉)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝜉)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝜉+𝑠)𝑑𝑠𝑑𝜉,𝑖=1,2,…,𝑛. (29) Therefore, we have 𝑦_𝑖(𝑡)𝑒^{−∫𝑡0𝑎𝑖(𝑢)𝑑𝑢}1−𝑒^{−∫𝜔0𝑎𝑖(𝑢)𝑑𝑢}=_𝑡^𝑡+𝜔𝑒^{−∫𝜉0𝑎𝑖(𝑢)𝑑𝑢}𝑦_𝑖(𝜉)×𝐵_𝑖(𝜉)𝑦_𝑖(𝜉)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝜉)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝜉)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝜉)𝑦_𝑗𝜉−𝛾_𝑖𝑗(𝜉)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝜉)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝜉+𝑠)𝑑𝑠𝑑𝜉,𝑖=1,2,…,𝑛, (30) which can be transformed into 𝑦_𝑖(𝑡)=_𝑡^𝑡+𝜔𝑒^{−∫𝜉0𝑎𝑖(𝑢)𝑑𝑢}𝑒^{−∫𝑡0𝑎𝑖(𝑢)𝑑𝑢}1−𝑒^{−∫𝜔0𝑎𝑖(𝑢)𝑑𝑢}𝑦_𝑖(𝜉)×𝐵_𝑖(𝜉)𝑦_𝑖(𝜉)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝜉)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝜉)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝜉)𝑦_𝑗𝜉−𝛾_𝑖𝑗(𝜉)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝜉)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗()=𝜉+𝑠𝑑𝑠𝑑𝜉_{𝑡𝑡+𝜔}𝐺_𝑖(𝑡,𝜉)𝑦_𝑖(𝜉)×𝐵_𝑖(𝜉)𝑦_𝑖(𝜉)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝜉)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝜉)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝜉)𝑦_𝑗𝜉−𝛾_𝑖𝑗(𝜉+_𝑛𝑗=1)𝐸_𝑖𝑗(𝜉)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗()=𝜉+𝑠𝑑𝑠𝑑𝜉_{𝑡𝑡+𝜔}𝐺_𝑖(𝑡,𝜉)𝜑𝑦_𝑖(𝜉)𝑑𝜉,𝑖=1,2,…,𝑛. (31) Thus, 𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛) is a periodic solution for (15).
If 𝑦=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝑋 and 𝜓𝑦=((𝜓𝑦)₁,(𝜓𝑦)₂,…,(𝜓𝑦)_𝑛)^𝑇=𝑦 with 𝑦≠0, then for any 𝑡=𝑡_𝑘 derivative the two sides of (15) about 𝑡, 𝑑𝜓𝑦_𝑖(𝑡)𝑑𝑡=𝐺_𝑖(𝑡,𝑡+𝜔)𝜑𝑦_𝑖(𝑡+𝜔)−𝐺_𝑖(𝑡,𝑡)𝜑𝑦_𝑖(𝑡)+_𝑡^𝑡+𝜔𝑑𝐺_𝑖(𝑡,𝜉)𝑑𝑡𝜑𝑦_𝑖(𝜉)𝑑𝜉=−𝜑𝑦_𝑖(𝑡)+𝑎_𝑖(𝑡)_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝜉)𝜑𝑦_𝑖(𝜉)𝑑𝜉=𝑎_𝑖(𝑡)𝑦_𝑖(𝑡)−𝜑𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑎_𝑖(𝑡)−𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗()=𝑡+𝑠𝑑𝑠𝑑𝑦_𝑖(𝑡).𝑑𝑡 (32) Hence, 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝑋 is a positive 𝜔-periodic solution of (7). Thus we complete the proof of Lemma 8.

Our main result of this paper is as follows.

Theorem 9. Assume (𝐻₁)–(𝐻₄). Then condition 𝑚₀=min_{1≤𝑖≤𝑛}𝐵_𝑖+^𝑛_𝑗=1𝐶_𝑖𝑗+𝐷_𝑖𝑗+𝐸_𝑖𝑗>0 (33) is necessary and sufficient for system (1) to have at least one positive 𝜔-periodic solution.

Proof. (Sufficiency) Let 𝑀₀=max_{[]𝑖∈1,𝑛}𝐵_𝑖+^𝑛_𝑗=1𝐶_𝑖𝑗+𝐷_𝑖𝑗+𝐸_𝑖𝑗, (34) by condition (33), we know that 𝑀₀≥𝑚₀>0. Choose a constant 𝑀≥𝑀₀ such that 1/𝜔𝛽_𝑖𝑀<1. Let 𝑟₁=1/𝜔𝛽_𝑖𝑀 and Ω_𝑟1=𝑦(𝑡)=𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)^𝑇∈𝑋∶𝑦_𝑖0<𝑟₁,𝑖=1,2,…,𝑛. (35) For any 𝑦=𝑦(𝑡)∈𝐸⋂𝜕Ω_𝑟1, 𝜎|𝑦_𝑖|₀≤𝑦_𝑖(𝑡)≤|𝑦_𝑖|₀, from (15), we obtain 𝜓𝑦_𝑖0≤𝛽_𝑖^𝜔₀𝜑𝑦_𝑖(𝜉)𝑑𝜉=𝛽_𝑖^𝜔₀𝑦_𝑖(𝑡)𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝑡+𝑠)𝑑𝑠𝑑𝜉≤𝜔𝛽_𝑖𝑦_𝑖0𝐵_𝑖+^𝑛_𝑗=1𝐶_𝑖𝑗+𝐷_𝑖𝑗+𝐸_𝑖𝑗𝑟₁≤𝜔𝛽_𝑖𝑦_𝑖0𝑀₀𝑟₁≤𝑦_𝑖0. (36) Hence, for any 𝑦=𝑦(𝑡)∈𝐸⋂𝜕Ω_𝑟1, 𝜎|𝑦_𝑖|₀≤𝑦_𝑖(𝑡)≤|𝑦_𝑖|₀, we have 𝜓𝑦=^𝑛_𝑗=1𝜓𝑦_𝑖0≤^𝑛_𝑗=1𝑦_𝑖0=𝑦, (37) which implies that condition (1) in Lemma 4 is satisfied.
On the other hand, we choose 0<𝑚≤𝑚₀ such that 1/𝜔𝜎𝛼_𝑖𝑚>1. Let 𝑟₂=1/𝜔𝜎𝛼_𝑖𝑚>1 and suppose that 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈𝐸/{0}. We show that for any 𝑦=𝑦(𝑡)∈𝐸⋂𝜕Ω_𝑟2 and 𝜆>0, 𝑦≠𝜓𝑦+𝜆𝑢. Otherwise, there exist 𝑦₀=𝑦₀(𝑡)∈𝐸⋂𝜕Ω_𝑟2 and 𝜆₀>0, such that 𝑦₀=𝜓𝑦₀+𝜆₀𝑢. Let 𝑢_𝑖0≠0(1≤𝑖₀≤𝑛), since 𝑦_𝑖0(𝑡)≥𝜎|𝑦_𝑖0|₀, it follows that 𝑦_𝑖0=𝜓𝑦_𝑖0(𝑡)+𝜆₀𝑢_𝑖0=_𝑡^𝑡+𝜔𝐺_𝑖0(𝑡,𝜉)𝜑𝑦_𝑖0(𝜉)𝑑𝜉+𝜆₀𝑢_𝑖0≥𝜔𝜎𝛼_𝑖𝑦_𝑖00^𝜔₀𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡+_𝑛𝑗=1)𝐸_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗×(𝑡+𝑠)𝑑𝑠𝑑𝑡+𝜆₀𝑢_𝑖0≥𝜔𝜎𝛼_𝑖𝑦_𝑖00𝐵_𝑖+^𝑛_𝑗=1𝐶_𝑖𝑗+𝐷_𝑖𝑗+𝐸_𝑖𝑗𝑟₂+𝜆₀𝑢_𝑖0≥𝜔𝜎𝛼_𝑖𝑦_𝑖00𝑚₀𝑟₂+𝜆₀𝑢_𝑖0≥𝑦_𝑖00+𝜆₀𝑢_𝑖0>𝑦_𝑖00, (38) which is a contradiction. This proves that condition (2) in Lemma 4 is also satisfied. By Lemmas 4 and 8, system (7) has at least one positive 𝜔-periodic solution. From Lemma 2, system (1) has at least one positive 𝜔-periodic solution.
(Necessity) Suppose that (33) does not hold. Then there exists at least an 𝑖₀(1≤𝑖₀≤𝑛) such that 𝐵_𝑖0=0,𝐶_𝑖0𝑗=𝐷_𝑖0𝑗=𝐸_𝑖0𝑗=0,𝑗∈[1,𝑛]. (39) If system (7) has a positive 𝜔-periodic solution 𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇, then we have 𝑑𝑦_𝑖0(𝑡)𝑑𝑡=𝑎_𝑖0(𝑡)𝑦_𝑖0(𝑡). (40) Integrating the previos equation over [𝑡,𝑡+𝜔], we can have 𝑦0=ln^𝑖0(𝑡+𝜔)𝑦_𝑖0(𝑡)=^𝜔₀𝑎_𝑖0(𝑡)𝑑𝑡>0, (41) which is a contradiction. The proof of Theorem 9 is complete.

3. Existence of Periodic Solution of (2)

Now, we are at the position to study the existence of positive periodic solutions of (2). By carrying out similar arguments as in Section 2, it is not difficult to derive sufficient criteria for the existence of positive periodic solutions of (2). For simplicity, we prefer to list later the corresponding criteria for (2) without proof since the proofs are very similar to those in Section 2.

For (𝑡,𝜉)∈𝑅², 1≤𝑖≤𝑛, we define 𝐺_𝑖()=𝑒𝑡,𝜉^{−∫𝜉𝑡𝑎𝑖(𝑢)}𝑑𝑢1−𝑒^{−∫𝜔0𝑎𝑖(𝑢)}=𝑒𝑑𝑢^{∫𝜉𝑡+𝜔𝑎𝑖(𝑢)}𝑑𝑢𝑒^{∫𝜔0𝑎𝑖(𝑢)},𝐺()𝐺𝑑𝑢−1𝑡,𝜉=diag₁(𝑡,𝜉),𝐺₂(𝑡,𝜉),…,𝐺_𝑛(𝑡,𝜉),𝑡≤𝜉≤𝑡+𝜔. (42) It is clear that 𝐺_𝑖(𝑡+𝜔,𝜉+𝜔)=𝐺_𝑖(𝑡,𝑠), 𝜕𝐺_𝑖(𝑡,𝜉)/𝜕𝑡=𝑎_𝑖(𝑡)𝐺_𝑖(𝑡,𝜉), 𝐺_𝑖(𝑡,𝑡)−𝐺_𝑖(𝑡,𝑡+𝜔)=1. In view of (H₁), we also define for 1≤𝑖≤𝑛𝛼_𝑖∶=min_{0≤𝑡≤𝜉≤𝜔}𝐺_𝑖(=1𝑡,𝜉)𝑒^{∫𝜔0𝑎𝑖(𝑢)𝑑𝑢},𝛽−1_𝑖∶=max_{0≤𝑡≤𝜉≤𝜔}𝐺_𝑖(=𝑒𝑡,𝜉)^{∫𝜔0𝑎𝑖(𝑢)𝑑𝑢}𝑒^{∫𝜔0𝑎𝑖(𝑢)𝑑𝑢},−1𝛼=min_{1≤𝑖≤𝑛}𝛼_𝑖,𝛽=max_{1≤𝑖≤𝑛}𝛽_𝑖𝛼,𝜎=𝛽∈(0,1). (43) Let 𝑋={𝑦=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝑃𝐶(𝑅,𝑅^𝑛)∣𝑦(𝑡+𝜔)=𝑦(𝑡)} with the norm ‖𝑦‖=∑^𝑛_𝑖=1|𝑦_𝑖|₀, |𝑦_𝑖|₀=sup_{𝑡∈[0,𝜔]}|𝑦_𝑖(𝑡)|. It is easy to verify that (𝑋,‖⋅‖) is a Banach space. Define 𝑃 to be a cone in 𝑋 by 𝑃=𝑦=𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)^𝑇∈𝑋∶𝑦_𝑖(𝑡)≥𝛿𝑦_𝑖0,𝑡∈[0,𝜔]. (44) We easily verify that 𝑃 is a cone in 𝑋. We define an operator 𝐴∶𝑃→𝑋 as follows: 𝐴𝑦(𝑡)=𝐴𝑦₁(𝑡),𝐴𝑦₂(𝑡),…,𝐴𝑦_𝑛(𝑡)^𝑇, (45) where 𝐴𝑦_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝜉)𝜙𝑦_𝑖(𝜉)𝑑𝑠, (46) where ∅𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡)𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦_𝑖𝜉−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦_𝑗𝑡−𝛾_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)⁰_{−𝜎𝑖𝑗}𝑓_𝑖𝑗(𝑠)𝑦_𝑗(𝑡+𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (47) The proof of the following lemmas and theorems is similar to those in the Section 2, so we all omit the details here.

Lemma 10. Assume that (𝐻₁)–(𝐻₄) hold. Then the solutions of (2) are defined on [−𝜏,∞) and are positive.

Lemma 11. Assume that (𝐻₁)–(𝐻₄) hold. Then 𝐴∶𝐸→𝐸 is well defined.

Lemma 12. The operator 𝐴∶𝐸→𝐸 is continuous and completely continuous.

Lemma 13. Assume that (𝐻₁)–(𝐻₄) hold. The existence of positive 𝜔-periodic solution of (8) is equivalent to that of nonzero fixed point of 𝐴 in 𝑃.

Theorem 14. Assume (𝐻₁)–(𝐻₄). Moreover, if the condition 𝑙₀=min_{1≤𝑖≤𝑛}𝐵_𝑖+^𝑛_𝑗=1−𝐶_𝑖𝑗+𝐷_𝑖𝑗+𝐸_𝑖𝑗>0 (48) holds, then the system (2) has at least one positive 𝜔-periodic solution.

4. Applications

In this section, as some applications of our main results, we will consider some special cases of systems (1) and (2), which have been investigated extensively in the literature.

Application 1. We consider the following periodic single-species population growth models with periodic delay and impulse: 𝑦^′(𝑡)=𝑦(𝑡)𝑟(𝑡)−𝑎(𝑡)𝑦(𝑡)−𝑏(𝑡)𝑦(𝑡−𝜏(𝑡)),𝑡∈𝑅,𝑡≠𝑡_𝑘,Δ𝑦𝑡_𝑘=𝜃_𝑘𝑦𝑡_𝑘,𝑘∈𝑍₊, (49) which is a special case of system (1), and where 𝑎(𝑡),𝑏(𝑡),𝜏(𝑡)∈𝐶(𝑅,𝑅₊) are 𝜔-periodic. Thus from Theorem 9 we have the following.

Theorem 15. Assume that (𝐻₁)–(𝐻₄) hold. Then condition: 𝑅₁=𝐴+𝐵>0 (50) is necessary and sufficient for system (49) to have at least one positive 𝜔-periodic solution, where 𝐴(𝑡)=𝑎(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝐵(𝑡)=𝑏(𝑡)_{0<𝑡𝑘(𝑡)<𝑡−𝜏}1+𝜃_𝑘. (51)

Application 2. We consider the following periodic single-species population growth models with periodic delay and impulse: 𝑦^′(𝑡)=𝑦(𝑡)𝑟(𝑡)−𝑎(𝑡)𝑦(𝑡)+𝑏(𝑡)𝑦(𝑡−𝜏(𝑡)),𝑡∈𝑅,𝑡≠𝑡_𝑘,Δ𝑦𝑡_𝑘=𝜃_𝑘𝑦𝑡_𝑘,𝑘∈𝑍₊, (52) which is a special case of system (2), and where 𝑎(𝑡),𝑏(𝑡),𝜏(𝑡)∈𝐶(𝑅,𝑅₊) are 𝜔-periodic. Thus from Theorem 14 we have the following.

Theorem 16. Assume that (𝐻₁)–(𝐻₄) hold. Moreover, if the condition 𝑅₂=𝐴−𝐵>0 (53) holds, the system (52) has at least one positive 𝜔-periodic solution, where 𝐴(𝑡)=𝑎(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝐵(𝑡)=𝑏(𝑡)_{0<𝑡𝑘(𝑡)<𝑡−𝜏}1+𝜃_𝑘. (54)

Application 3. We study the following two-species competitive system with impulses: 𝑦^′₁(𝑡)=𝑦₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑦₁(𝑡)−𝑐₁(𝑡)𝑦₂(𝑡,𝑦_′2)(𝑡)=𝑦₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦₂(𝑡)−𝑐₂(𝑡)𝑦₁(𝑡),𝑡≠𝑡_𝑘,Δ𝑦_𝑖𝑡_𝑘=𝜃_𝑘𝑦𝑡_𝑘,𝑖=1,2,𝑘∈𝑍₊,𝑡=𝑡_𝑘, (55) which is a special case of system (1), and where 𝑎_𝑖(𝑡),𝑐_𝑖(𝑡)∈𝐶(𝑅,𝑅₊) are 𝜔-periodic. Thus from Theorem 9 we have the following.

Theorem 17. Assume that (𝐻₁)–(𝐻₄) hold. Then condition 𝑅₃=min_1≤𝑖≤2𝐴_𝑖+𝐶_𝑖>0 (56) is necessary and sufficient for system (55) to have at least one positive 𝜔-periodic solution, where 𝐴_𝑖(𝑡)=𝑎_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝐶_𝑖(𝑡)=𝑐_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝑖=1,2. (57)

Application 4. We study the following two-species competitive system with impulses: 𝑦^′₁(𝑡)=𝑦₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑦₁(𝑡)−^𝑛_𝑗=1𝑏_1𝑗(𝑡)𝑦₁𝑡−𝜏_𝑗(𝑡)−^𝑛_𝑗=1𝑐_1𝑗(𝑡)𝑦₂𝑡−𝜌_𝑗(𝑡),𝑦^′₂(𝑡)=𝑦₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦₂(𝑡)−^𝑛_𝑗=1𝑏_2𝑗(𝑡)𝑦₂𝑡−𝜂_𝑗(𝑡)−^𝑛_𝑗=1𝑐_2𝑗(𝑡)𝑦₁𝑡−𝜎_𝑗(𝑡),𝑡≠𝑡_𝑘,Δ𝑦_𝑖𝑡_𝑘=𝜃_𝑘𝑦𝑡_𝑘,𝑖=1,2,𝑘∈𝑍₊,𝑡=𝑡_𝑘, (58) which is a special case of system (1), and where 𝑎_𝑖(𝑡),𝑏_𝑖𝑗(𝑡),𝑐_𝑖𝑗(𝑡)∈𝐶(𝑅,𝑅₊), 𝑖=1,2, are 𝜔-periodic. Thus from Theorem 9 we have the following.

Theorem 18. Assume that (𝐻₁)–(𝐻₄) hold. Then condition 𝑅₄=min_1≤𝑖≤2𝐴_𝑖+^𝑛_𝑖=1𝐵_𝑖𝑗+𝐶_𝑖𝑗>0 (59) is necessary and sufficient for system (58) to have at least one positive 𝜔-periodic solution, where 𝐴_𝑖(𝑡)=𝑎_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝐵_1𝑗(𝑡)=𝑏_1𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑗(𝑡)}1+𝜃_𝑘,𝐵_2𝑗(𝑡)=𝑏_2𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜂𝑗(𝑡)}1+𝜃_𝑘,𝐶_1𝑗(𝑡)=𝐶_1𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜌𝑗(𝑡)}1+𝜃_𝑘,𝐶_2𝑗(𝑡)=𝑐_2𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑗(𝑡)}1+𝜃_𝑘,𝑖=1,2. (60)

Application 5. We study the following two-species competitive system with impulses: 𝑦^′₁(𝑡)=𝑦₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑦₁(𝑡)+^𝑛_𝑗=1𝑏_1𝑗(𝑡)𝑦₁𝑡−𝜏_𝑗(𝑡)−^𝑛_𝑗=1𝑐_1𝑗(𝑡)𝑦₂𝑡−𝜌_𝑗(𝑡),𝑦^′₂(𝑡)=𝑦₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦₂(𝑡)+^𝑛_𝑗=1𝑏_2𝑗(𝑡)𝑦₂𝑡−𝜂_𝑗(𝑡)−^𝑛_𝑗=1𝑐_2𝑗(𝑡)𝑦₁𝑡−𝜎_𝑗(𝑡),𝑡≠𝑡_𝑘,Δ𝑦_𝑖𝑡_𝑘=𝜃_𝑘𝑦𝑡_𝑘,𝑖=1,2,𝑘∈𝑍₊,𝑡=𝑡_𝑘, (61) which is a special case of system (2), and where 𝑎_𝑖(𝑡),𝑏_𝑖𝑗(𝑡),𝑐_𝑖𝑗(𝑡)∈𝐶(𝑅,𝑅₊), 𝑖=1,2, are 𝜔-periodic. Thus from Theorem 14 we have the following.

Theorem 19. Assume that (𝐻₁)–(𝐻₄) hold. Moreover, if the condition 𝑅₅=min_1≤𝑖≤2𝐴_𝑖+^𝑛_𝑖=1−𝐵_𝑖𝑗+𝐶_𝑖𝑗>0 (62) holds, the system (61) has at least one positive 𝜔-periodic solution, where 𝐴_𝑖(𝑡)=𝑎_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑘,𝐵_1𝑗(𝑡)=𝑏_1𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑗(𝑡)}1+𝜃_𝑘,𝐵_2𝑗(𝑡)=𝑏_2𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜂𝑗(𝑡)}1+𝜃_𝑘,𝐶_1𝑗(𝑡)=𝐶_1𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜌𝑗(𝑡)}1+𝜃_𝑘,𝐶_2𝑗(𝑡)=𝑐_2𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑗(𝑡)}1+𝜃_𝑘,𝑖=1,2. (63)

Application 6. We investigate the following n-species competitive systems with impulses: 𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)×⁰_{−𝜎𝑖𝑗}𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡+𝑠)𝑑𝑠,𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝜃_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘∈𝑍₊, (64) which is a special case of system (1), and where 𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑐_𝑖𝑗(𝑡),𝑑_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡)∈𝐶(𝑅,𝑅₊), 𝑖=1,2,…,𝑛, are 𝜔-periodic. Moreover, 𝐾_𝑖𝑗∈𝑃𝐶,𝜎_𝑖𝑗>0 is a constant and ∫⁰_{−𝜎𝑖𝑗}𝐾_𝑖𝑗(𝑠)𝑑𝑠=1. Thus from Theorem 9 we have the following.

Theorem 20. Assume that (𝐻₁)–(𝐻₄) hold. Then condition 𝑅₆=min_1≤𝑖≤2𝐵_𝑖+^𝑛_𝑖=1𝐶_𝑖𝑗+𝐷_𝑖𝑗>0 (65) is necessary and sufficient for system (64) to have at least one positive 𝜔-periodic solution, where 𝐵_𝑖(𝑡)=𝑏_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘,𝐶_𝑖𝑗(𝑡)=𝐶_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+𝜃_𝑖𝑘,𝐷_𝑖𝑗(𝑡)=𝑑_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑖𝑗}1+𝜃_𝑖𝑘. (66)

Remark 21. We apply the main results obtained in the previous section to study some examples which have some biological implications. A very basic and important ecological problem associated with the study of population is that of the existence of positive periodic solutions which play the role played by the equilibrium of the autonomous models, meaning that the species is in an equilibrium state. From Theorems 15–20, we see that under the appropriate conditions, the impulsive perturbations do not affect the existence of periodic solution of systems.

Acknowledgments

This work was supported by the Construct Program of the Key Discipline in Hunan province, NSF of China (nos. 10971229, 11161015), PSF of China (2012M512162), and NSF of Hunan province (nos. 11JJ9002, 12JJ9001, and 13JJ4098).