Global Positive Periodic Solutions of Generalized 𝑛-Species Gilpin-Ayala Delayed Competition Systems with Impulses

Abstract

We consider the following generalized 𝑛-species Lotka-Volterra type and Gilpin-Ayala type competition 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^𝛼𝑖𝑗_𝑗(𝑡−𝜌_𝑖𝑗(𝑡))−∑^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑥^𝛽𝑖𝑗_𝑗(𝑡−𝜏_𝑖𝑗(𝑡))−∑^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)∫⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑥^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠−∑^𝑛_𝑗=1𝑓_𝑖𝑗(𝑡)∫⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑥^𝛿𝑖𝑗_𝑖(𝑡+𝜉)𝑥^𝜎𝑖𝑗_𝑗(𝑡+𝜉)𝑑𝜉],a.e, 𝑡>0, 𝑡≠𝑡_𝑘; 𝑥_𝑖(𝑡⁺_𝑘)−𝑥_𝑖(𝑡⁻_𝑘)=ℎ_𝑖𝑘𝑥_𝑖(𝑡_𝑘), 𝑖=1,2,…,𝑛, 𝑘∈𝑍₊. By applying the Krasnoselskii fixed-point theorem in a cone of Banach space, we derive some verifiable necessary and sufficient conditions for the existence of positive periodic solutions of the previously mentioned. As applications, some special cases of the previous system are examined and some earlier results are extended and improved.

1. Introduction

In the recent decades, the traditional Lotka-Volterra competition systems have been studied extensively. One of the models is the following competition system: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡),𝑖=1,2,…,𝑛. (1)

Many results concerned with the permanence, global asymptotic stability, and the existence of positive periodic solutions of system (1) are obtained; we refer to [1–10] and the reference therein. However, the Lotka-Volterra type models have often been severely criticized. One of the criticisms is that, in such a model, the per capita rate of change of the density of each species is a linear function of densities of the interacting species. In 1973, Ayala et al. [11] conducted experiments on fruit fly dynamics to test the validity of ten models of competitions. One of the models accounting best for the experimental results is given by 𝑥^′₁(𝑡)=𝑟₁𝑥₁(𝑡𝑥)1−¹(𝑡)𝐾₁^𝜃1−𝛼₁₂𝑥₂(𝑡)𝐾₂,𝑥^′₂(𝑡)=𝑟₂𝑥₂(𝑡𝑥)1−²(𝑡)𝐾₂^𝜃2−𝛼₂₁𝑥₁(𝑡)𝐾₁. (2)

In order to fit data in their experiments and to yield significantly more accurate results, Gilpin and Ayala [12] claimed that a slightly more complicated model was needed and proposed the following competition model: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑟_𝑖𝑥_𝑖(𝑡)𝑥1−^𝑖(𝑡)𝐾_𝑖^𝜃𝑖−^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗𝑥_𝑗(𝑡)𝐾_𝑗,𝑖=1,2,…,𝑛, (3)

where𝑥_𝑖is the population density of the ith species,𝑟_𝑖is the intrinsic exponential growth rate of the ith species,𝐾_𝑖is the environmental carrying capacity of species𝑖in the absence of competition,𝜃_𝑖provides a nonlinear measure of interspecific interference, and𝑎_𝑖𝑗provides a measure of interspecific interference. [13–15] obtained sufficient conditions which guarantee the global asymptotic stability of system (3). Chen [16] investigated the following𝑛-species nonautonomous Gilpin-Ayala competitive model: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝜃𝑖𝑗_𝑗(𝑡),𝑖=1,2,…,𝑛. (4)

For each𝑟≤𝑛, they established a series of criteria under which𝑟of the species of system (4) were permanent while the remaining𝑛−𝑟species were driven to extinction. In [17], Fan and Wang further studied the following delay Gilpin-Ayala type competition model: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝜃𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑖=1,2,…,𝑛. (5)

They obtained a set of easily verifiable sufficient conditions for the existence of at least one positive periodic solution of the system (5) by applying the coincidence degree theory. Recently, in [18], Chen investigated the following𝑛-species Gilpin-Ayala type competition systems: 𝑦^′_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝛼𝑖𝑗_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑥^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1⁰_{−𝜎𝑖𝑗}𝑐_𝑖𝑗(𝑡,𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (6)

He established a series of criteria under which𝑟of the species in the system (6) were permanent while the remaining𝑛−𝑟species were driven to extinction. In [19], Xia et al. considered the following almost periodic nonlinear𝑛-species competitive Lotka-Volterra model: 𝑦^′_𝑖(𝑡)=𝑦_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)𝑦^𝛼𝑖𝑖_𝑖(𝑡)𝑦^𝛼𝑖𝑗_𝑗(𝑡),𝑖=1,2,…,𝑛. (7)

By using comparison theorem and constructing suitable Lyapunov functional, they derived a set of sufficient conditions for the existence and global attractivity of a unique positive almost periodic solution of the previously mentioned model. Motivated by the previous ideas, in [20], Yan considered the following generalized periodic𝑛-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form: 𝑦^′_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝛼𝑖𝑗_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑥^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. (8)

By using a fixed point theorem in cone and the proof by contradiction, he obtained a necessary and sufficient condition for the existence of positive periodic solutions (with strictly positive components) of the system (8).

However, the ecological system is often deeply perturbed by human exploitation activities such as planting and harvesting, which makes it unsuitable to be considered continually. For having a more accurate description of such a system, we need to consider the impulsive differential equations. The theory of impulsive differential equations is not only richer than the corresponding theory of differential equations without impulses, but also represents a more natural framework for mathematical modeling of many real-world phenomena (see [21–23]). In recent years, some impulsive equations have been recently introduced in population dynamics in relation to population ecology; we refer the reader to [24–36] and the reference therein. However, to this day, only a little work has been done on the existence of positive periodic solutions to the generalized periodic𝑛-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form and impulses. Motivated by this, in this paper, we mainly consider the following𝑛-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form and impulses: 𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)𝑥^𝛼𝑖𝑗_𝑗𝑡−𝜌_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑥^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑥^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠−^𝑛_𝑗=1𝑓_𝑖𝑗(𝑡)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑥^𝛿𝑖𝑗_𝑖(𝑡+𝜉)𝑥^𝜎𝑖𝑗_𝑗(𝑡+𝜉)𝑑𝜉,a.e,𝑡>0,𝑡≠𝑡_𝑘;𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘∈𝑍₊, (9)

with initial conditions 𝑥_𝑖(𝜗)=𝜙_𝑖(𝜗),𝜗∈[−𝜏,0],𝜙_𝑖(0)𝜙>0,_𝑖∈𝐶([−𝜏,0),[0,+∞)),𝑖=1,2,…,𝑛, (10)

where𝑥_𝑖represents the density of the ith species𝑋_𝑖at time𝑡,𝑎_𝑖(𝑡)is the intrinsic growth rate of the ith species𝑋_𝑖at this time,𝑎_𝑖(𝑡)may be negative while𝑎_𝑖=(1/𝜔)∫^𝜔₀𝑎_𝑖(𝑡)𝑑𝑡>0;𝑐_𝑖𝑗(𝑡),𝑑_𝑖𝑗(𝑡),𝑒_𝑖𝑗(𝑡)(𝑖≠𝑗)denote the competitive coefficient between the ith species𝑋_𝑖and jth species𝑋_𝑗, and𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑐_𝑖𝑗(𝑡),𝑑_𝑖𝑗(𝑡),𝑒_𝑖𝑗(𝑡)are continuous𝜔-periodic functions with𝑏_𝑖(𝑡)≥0,𝑐_𝑖𝑗(𝑡)≥0,𝑑_𝑖𝑗(𝑡)≥0,𝑒_𝑖𝑗(𝑡)≥0,𝑓_𝑖𝑗(𝑡)≥0,  𝜌_𝑖𝑗(𝑡)≥0,𝜏_𝑖𝑗(𝑡)≥0,𝜂_𝑖𝑗≥0,𝜃_𝑖𝑗≥0corresponding to the time delays with𝜏=max_{𝑖,𝑗∈[1,𝑛]}{𝜌_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡),𝜂_𝑖𝑗,𝜃_𝑖𝑗}≥0;𝛼_𝑖𝑗,𝛽_𝑖𝑗,𝛾_𝑖𝑗,𝛿_𝑖𝑗,𝜎_𝑖𝑗are positive constants. Assume thatℎ_𝑖𝑘,𝑖=1,2,…,𝑛,𝑘∈𝑍₊={1,2,3,…}are constants and there exists an integer𝑞>0such that𝑡_𝑘+𝑞=𝑡_𝑘+𝜔,ℎ_{𝑖(𝑘+𝑞)}=ℎ_𝑖𝑘,𝑖=1,2,…,𝑛,where0<𝑡₁<𝑡₂<⋯<𝑡_𝑞<𝜔.

Throughout this paper, we make the following notation and assumptions: 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}.

The previously mentioned spaces are all Banach spaces. We also denote 1𝑔=𝜔^𝜔₀𝑔(𝑡)𝑑𝑡,forany𝑔∈𝑃𝐶_𝜔,Δ_𝑖𝑘=1+ℎ_𝑖𝑘,𝑖=1,2,…,𝑛,𝑘∈𝑍₊, (11)

and make the following assumptions: (𝐻₁)  𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑐_𝑖𝑗(𝑡),𝑑_𝑖𝑗(𝑡),𝑒_𝑖𝑗(𝑡),𝑓_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡),𝜌_𝑖𝑗(𝑡)∈𝑃𝐶_𝜔,𝑖,𝑗=1,2,…,𝑛, and𝜂_𝑖𝑗≥0,𝜃_𝑖𝑗≥0. (𝐻₂)  {𝑡_𝑘},𝑘∈𝑍₊satisfies0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯andlim_𝑘→+∞𝑡_𝑘=+∞. (𝐻₃)  {ℎ_𝑖𝑘}is a real sequence withΔ_𝑖𝑘=ℎ_𝑖𝑘+1>0, and∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘is an𝜔-periodic function. (𝐻₄)  𝑘_𝑖𝑗∈𝐶([−𝜂_𝑖𝑗,0],𝑅₊),𝐾_𝑖𝑗∈𝐶([−𝜃_𝑖𝑗,0],𝑅₊)are constants such that∫⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑡)𝑑𝑡=1,∫⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝑡)𝑑𝑡=1.

To conclude this section, we summarize in the following definition and lemma that will be needed in our arguments.

Definition 1. A function𝑥_𝑖∶𝑅→(0,+∞)is said to be a positive solution of the system (9) and (10), if the following conditions are satisfied: (a)𝑥_𝑖(𝑡)is absolutely continuous on each(𝑡_𝑘,𝑡_𝑘+1).(b)for each𝑘∈𝑍₊,𝑥_𝑖(𝑡⁺_𝑘)and𝑥_𝑖(𝑡⁻_𝑘)exist, and𝑥_𝑖(𝑡⁻_𝑘)=𝑥_𝑖(𝑡_𝑘).(c)𝑥_𝑖(𝑡)satisfies the first equation of the system (9) and (10) for almost everywhere (for short a.e.) in[0,∞]⧵{𝑡_𝑘}and satisfies𝑥_𝑖(𝑡⁺_𝑘)=Δ_𝑖𝑘𝑥_𝑖(𝑡_𝑘)for𝑡=𝑡_𝑘,𝑘∈𝑍₊.
Under the previously mentioned hypotheses (𝐻₁)–(𝐻₄), we consider the following 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^𝛼𝑖𝑗_𝑖𝑡−𝜌_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠−^𝑛_𝑗=1𝐹_𝑖𝑗(𝑡)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑡+𝜉)𝑦^𝜎𝑖𝑗_𝑗(𝑡+𝜉)𝑑𝜉,𝑖=1,2,…,𝑛, (12)
with initial conditions 𝑦_𝑖(𝜁)=𝜑_𝑖(𝜁),𝜁∈[−𝜏,0],𝜑_𝑖(0)𝜑>0,_𝑖∈𝐶([−𝜏,0),[0,+∞)),𝑖=1,2,…,𝑛, (13)
where 𝐵_𝑖(𝑡)=𝑏_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘,𝐶_𝑖𝑗(𝑡)=𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜌𝑖𝑗(𝑡)}1+𝜃_𝑖𝑘^𝛼𝑖𝑗,𝐷_𝑖𝑗(𝑡)=𝑑_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+𝜃_𝑖𝑘^𝛽𝑖𝑗,𝐸_𝑖𝑗(𝑡)=𝑒_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘^𝛾𝑖𝑗,𝐹_𝑖𝑗(𝑡)=𝑓_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘^{𝛿𝑖𝑗+𝜎𝑖𝑗},𝑖=𝑗=1,2,…,𝑛. (14)
By a solution𝑦(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))^𝑇of the system (12) and (13), it means an absolutely continuous function𝑦(𝑡)=(𝑦₁(𝑡),…,𝑦_𝑛(𝑡))^𝑇defined on[−𝜏,0]that satisfies (12) and (13).
The following lemma will be used in the proofs of our results. The proof of Lemma 2 is similar to that of Theorem1in [24].

Lemma 2. Suppose that (𝐻₁)–(𝐻₄) hold. Then (i)if𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a solution of (12) and (13) on[−𝜏,+∞), then𝑥_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a solution of (9) and (10) on[−𝜏,+∞);(ii)if𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a solution of (9) and (10) on[−𝜏,+∞), then𝑦_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ⁻¹_𝑖𝑘𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a solution of (12) and (13) on[−𝜏,+∞).

Proof. (i) It is easy to see that𝑥_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ_𝑖𝑘𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛)is 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^𝛼𝑖𝑗_𝑗𝑡−𝜌_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)𝑥^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑒_𝑖𝑗(𝑡)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑣)𝑥^𝛾𝑖𝑗_𝑗(𝑡+𝑣)𝑑𝑣−^𝑛_𝑗=1𝑓_𝑖𝑗(𝑡)×⁰_{−𝜉𝑖𝑗}𝐾_𝑖𝑗(𝑣)𝑥^𝛿𝑖𝑗_𝑖(𝑡+𝑣)𝑥^𝜎𝑖𝑗_𝑗×(𝑡+𝑣)𝑑𝑣=_{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−𝜌𝑖𝑗(𝑡)}Δ^𝛼𝑖𝑗_𝑖𝑘𝑦^𝛼𝑖𝑗_𝑗𝑡−𝜌_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑑_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}Δ^𝛽𝑖𝑗_𝑖𝑘𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡−_𝑛𝑗=1)𝑒_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}Δ^𝛾𝑖𝑗_𝑖𝑘⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑣)𝑦^𝛾𝑖𝑗_𝑗()−𝑡+𝑣𝑑𝑣_𝑛𝑗=1𝑓_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}Δ^{𝛿𝑖𝑗+𝜎𝑖𝑗}_𝑖𝑘×⁰_{−𝜉𝑖𝑗}𝐾_𝑖𝑗(𝑣)𝑦^𝛿𝑖𝑗_𝑖(𝑡+𝑣)𝑦^𝜎𝑖𝑗_𝑗(𝑡+𝑣)𝑑𝑣=_{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^𝛼𝑖𝑗_𝑖𝑡−𝜌_𝑖𝑗(𝑡−_𝑛𝑗=1)𝐷_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑢)𝑦^𝛾𝑖𝑗_𝑗(𝑡+𝑢)𝑑𝑢−^𝑛_𝑗=1𝐹_𝑖𝑗(𝑡)×⁰_{−𝜉𝑖𝑗}𝐾_𝑖𝑗(𝑣)𝑦^𝛿𝑖𝑗_𝑖(𝑡+𝑣)𝑦^𝜎𝑖𝑗_𝑗×(𝑡+𝑣)𝑑𝑣=0. (15)
On the other hand, for any𝑡=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; (16)
thus, 𝑥_𝑖𝑡⁺_𝑘=Δ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…. (17)
It follows from (15)–(17) that𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a solution of the system (9) and (10). Similarly, if𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a solution of the system (12) and (13), we can prove that𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a solution of the system (9) and (10).
(ii) Since𝑥_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 (16), it follows that for any𝑘=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, (18)
which implies that𝑦_𝑖(𝑡)is continuous on[−𝜏,+∞). It is easy to prove that𝑦_𝑖(𝑡)is absolutely continuous on[−𝜏,+∞). Similar to the proof of(i), we can check that𝑦_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}Δ⁻¹_𝑖𝑘𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛)are solutions of (8) on[−𝜏,+∞). Similarly, if𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a solution of the system (9) and (10), we can prove that𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a solution of the system (12) and (13). The proof of Lemma 2 is completed.

In the following section, we only discuss the existence of a periodic solution for the system (12) and (13).

The paper is organized as follows. In the next section, we give some definitions and lemmas. In Section 3, we derive a necessary and sufficient condition ensuring at least one positive periodic solution of the system, by using the Krasnoselskii fixed-point theorem in the cone of Banach space. In Section 4, as applications, we consider some particular cases of the system which have been investigated extensively in the references mentioned previously.

2. Preliminaries

We will first make some preparations and list a few preliminary results. Let𝑋={𝑦=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝐶(𝑅,𝑅^𝑛)∣𝑦_𝑖(𝑡+𝜔)=𝑦_𝑖(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 operator𝐴∶𝑋→𝑋as follows: 𝐴𝑦(𝑡)=𝐴𝑦₁(𝑡),𝐴𝑦₂(𝑡),…,𝐴𝑦_𝑛(𝑡)^𝑇, (19) where 𝐴𝑦_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑣)𝑦_𝑖(𝑣)×𝐵_𝑖(𝑣)𝑦_𝑖(𝑣)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑣)𝑦^𝛼𝑖𝑗_𝑖𝑣−𝜌_𝑖𝑗(𝑣)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑣)𝑦^𝛽𝑖𝑗_𝑗𝑣−𝜏_𝑖𝑗(𝑣)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑣)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑣+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑣)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑣+𝜉)𝑦^𝜎𝑖𝑗_𝑗×()𝐺𝑣+𝜉𝑑𝜉𝑑𝑣,𝑖=1,2,…,𝑛,_𝑖()=𝑒𝑡,𝑣^{−∫𝑣𝑡𝑎𝑖(𝑢)}𝑑𝑢1−𝑒^{−∫𝜔0𝑎𝑖(𝑢)}=𝑒𝑑𝑢^{∫𝑣𝑡+𝜔𝑎𝑖(𝑢)}𝑑𝑢𝑒^{∫𝜔0𝑎𝑖(𝑢)},𝐺()𝐺𝑑𝑢−1𝑡,𝑣=diag₁(𝑡,𝑣),𝐺₂(𝑡,𝑣),…,𝐺_𝑛(𝑡,𝑣),𝑡≤𝑣≤𝑡+𝜔. (20)

It is clear that𝐺_𝑖(𝑡+𝜔,𝑣+𝜔)=𝐺_𝑖(𝑡,𝑣),𝜕𝐺_𝑖(𝑡,𝑣)/𝜕𝑡=𝑎_𝑖(𝑡)𝐺_𝑖(𝑡,𝑣),𝐺_𝑖(𝑡,𝑡)−𝐺_𝑖(𝑡,𝑡+𝜔)=1.

In view of(𝐻₁), for1≤𝑖≤𝑛, we define 𝛼_𝑖∶=min_{0≤𝑡≤𝑠≤𝜔}𝐺_𝑖(=1𝑡,𝑣)𝑒^{∫𝜔0𝑎𝑖(𝑢)𝑑𝑢},𝛽−1_𝑖∶=max_{0≤𝑡≤𝑠≤𝜔}𝐺_𝑖(=𝑒𝑡,𝑣)^{∫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). (21)

Define𝑃as a cone in𝑋by 𝑃=𝑦=𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)^𝑇∈𝑋∶𝑦_𝑖(𝑡)≥𝜎𝑦_𝑖0,𝑡∈[0,𝜔]. (22)

We easily verify that𝑃is a cone in𝑋. For convenience of expressions, we define an operator𝑇∶𝑃→𝑋by 𝑇𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡)𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦^𝛼𝑖𝑗_𝑖𝑡−𝜌_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑡)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑡+𝜉)𝑦^𝜎𝑖𝑗_𝑗(𝑡+𝜉)𝑑𝜉,𝑖=1,2,…,𝑛. (23)

The proof of the main result in this paper is based on an application of the Krasnoselskii fixed-point theorem in cones. Firstly, we need to introduce some definitions and lemmas.

Definition 3. Let𝑋be a real Banach space and𝑃be a closed, nonempty subset of𝑋.𝑃is said to be a cone if (1)𝛼𝑥+𝛽𝑦∈𝑃for all𝑥,𝑦∈𝑃, and𝛼,𝛽>0,(2)𝑥,−𝑥∈𝑃imply𝑥=0.

Lemma 4 (see [37–39]). Let𝑃be a cone in a real Banach space𝑋. Assume thatΩ_𝑟1andΩ_𝑟2are open subsets of𝑋with0∈Ω_𝑟1⊂Ω_𝑟1⊂Ω_𝑟2,whereΩ_𝑟𝑖={𝑥∈𝑋∶‖𝑥‖<𝑟_𝑖},𝑖=1,2.Let𝑇∶𝑃∩(Ω_𝑟2⧵Ω_𝑟1)→𝑃be a continuous and completely continuous operator satisfying(1)‖𝑇𝑥‖≤‖𝑥‖, for any𝑥∈𝑃∩𝜕Ω_𝑟1;(2)The fact that there exists𝜙∈𝑃⧵{0}such that𝑦≠𝑇𝑦+𝜆𝑦, for any𝑦∈𝑃∩𝜕Ω_𝑟2and𝜆>0.
Then,𝑇has a fixed point in𝑃∩(Ω_𝑟2⧵Ω_𝑟1).The same conclusion remains valid if (1) holds for for any𝑥∈𝑃∩𝜕Ω_𝑟2and (2) holds for any𝑦∈𝑃∩𝜕Ω_𝑟1and𝜆>0.

Lemma 5. Assume that (H₁)–(H₄) hold. Then the solutions of the system (12) and (13) are defined on[−𝜏,∞)and are positive.

Proof. By Lemma 2, we only need to prove that the solutions𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛)of (12) and (13) are defined on[−𝜏,∞)and are positive on[0,∞). From (12), we have that for any𝜑_𝑖∈𝐶([−𝜏,0),𝑅⁺)(𝑖=1,2,3,…,𝑛)and𝑡>0𝑦_𝑖(𝑡)=𝜑_𝑖(0)exp^𝑡₀𝑎_𝑖(𝑣)−𝐵_𝑖(𝑣)𝑦_𝑖(𝑣)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑣)𝑦^𝛼𝑖𝑗_𝑖𝑣−𝜌_𝑖𝑗(𝑣)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑣)𝑦^𝛽𝑖𝑗_𝑗𝑣−𝜏_𝑖𝑗(𝑣)−^𝑛_𝑗=1𝐸_𝑖𝑗(𝑣)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑣+𝑠)𝑑𝑠−^𝑛_𝑗=1𝐹_𝑖𝑗(𝑣)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑣+𝜉)𝑦^𝜎𝑖𝑗_𝑗×(𝑣+𝜉)𝑑𝜉𝑑𝑣,𝑖=1,2,…,𝑛. (24)
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. Then𝐴∶𝑃→𝑃is well defined.

Proof. In view of the definitions of𝑃and𝑇, for any𝑦∈𝑃, we have 𝐴𝑦_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑣)𝑇𝑦_𝑖(𝑣)𝑑𝑣,𝐴𝑦_𝑖(𝑡+𝜔)=^𝑡+2𝜔_𝑡+𝜔𝐺_𝑖(𝑡,𝑣+𝜔)𝑇𝑦_𝑖()=𝑣+𝜔𝑑𝑣_{𝑡𝑡+𝜔}𝐺_𝑖(𝑡,𝑣)𝑇𝑦_𝑖(𝑣)𝑑𝑣=𝐴𝑦_𝑖(𝑡). (25)
Therefore,(𝐴𝑦)∈𝑋. Furthermore, for any𝑦∈𝑃, it follows from (20) that 𝐴𝑦_𝑖0≤𝛽_𝑖^𝜔₀𝑇𝑦_𝑖(𝑣)𝑑𝑣. (26)
On the other hand, for any𝑦∈𝑃, we obtain 𝐴𝑦_𝑖(𝑡)≥𝛼_𝑖^𝜔₀𝑇𝑦_𝑖(𝑣)≥𝛼𝑑𝑣_𝑖𝛽_𝑖𝐴𝑦_𝑖0≥𝜎𝐴𝑦_𝑖0. (27)
Therefore,𝐴𝑦∈𝑃.The proof of Lemma 6 is complete.

Lemma 7. The operator𝐴∶𝑃→𝑃is continuous and completely continuous.

Proof. By using a standard argument one can show that𝜓is continuous on𝑃. Now, we show that𝐴is completely continuous. Let𝑟be any positive constant and𝑆_𝑟={𝑦∈𝑋∶|𝑦_𝑖|₀≤𝑟}a bounded set. For any𝑦∈𝑆_𝑟, by (20), we have 𝐴𝑦_𝑖0≤𝛽_𝑖^𝜔₀𝑇𝑦_𝑖(𝑣)𝑑𝑣=𝛽_𝑖^𝜔₀𝑦_𝑖(𝑣)𝐵_𝑖(𝑣)𝑦_𝑖(𝑣)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑣)𝑦^𝛼𝑖𝑗_𝑖𝑣−𝜌_𝑖𝑗(𝑣+_𝑛𝑗=1)𝐷_𝑖𝑗(𝑣)𝑦^𝛽𝑖𝑗_𝑗𝑣−𝜏_𝑖𝑗(𝑣)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑣)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑣+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑣)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑣+𝜉)𝑦^𝜎𝑖𝑗_𝑗×(𝑣+𝜉)𝑑𝜉𝑑𝑣≤𝜔𝛽_𝑖𝑟𝐵_𝑖𝑟+^𝑛_𝑗=1𝐶_𝑖𝑗𝑟^𝛼𝑖𝑗+𝐷_𝑖𝑗𝑟^𝛽𝑖𝑗+𝐸_𝑖𝑗𝑟^𝛾𝑖𝑗+𝐹_𝑖𝑗𝑟^{𝛿𝑖𝑗+𝜎𝑖𝑗}∶=𝑅_𝑖. (28)
Therefore, for any𝑦∈𝑆_𝑟, we obtain 𝐴𝑦=^𝑛_𝑖=1𝐴𝑦_𝑖0≤^𝑛_𝑖=1𝑅_𝑖∶=𝑅, (29)
which implies that𝐴(𝑆_𝑟)is a uniformly bounded set. On the other hand, in view of the definitions of𝐴and𝑇, we have 𝑑𝐴𝑦_𝑖(𝑡)𝑑𝑡=𝐺_𝑖(𝑡,𝑡+𝜔)𝑇𝑦_𝑖(𝑡+𝜔)−𝐺_𝑖(𝑡,𝑡)𝑇𝑦_𝑖(𝑡)+_𝑡^𝑡+𝜔𝑑𝐺_𝑖(𝑡,𝑣)𝑑𝑡𝑇𝑦_𝑖(𝑣)𝑑𝑣=−𝑇𝑦_𝑖(𝑡)+𝑎_𝑖(𝑡)_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑣)𝑇𝑦_𝑖(𝑣)𝑑𝑣=𝑎_𝑖(𝑡)𝐴𝑦_𝑖(𝑡)−𝑇𝑦_𝑖(𝑡). (30)
Again, from (20), we obtain 𝑑𝐴𝑦_𝑖(𝑡)𝑑𝑡≤𝑎^𝑀_𝑖𝑅_𝑖×+𝑟𝐵_𝑖𝑟+^𝑛_𝑗=1𝐶_𝑖𝑗𝑟^𝛼𝑖𝑗+𝐷_𝑖𝑗𝑟^𝛽𝑖𝑗+𝐸_𝑖𝑗𝑟^𝛾𝑖𝑗+𝐹_𝑖𝑗𝑟^{𝛿𝑖𝑗+𝜎𝑖𝑗}∶=𝑅_𝑖≤𝑀∶=max_{[]𝑖∈1,𝑛}𝑅_𝑖, (31)
which implies that𝑑[(𝐴𝑦)_𝑖(𝑡)]/𝑑𝑡, for any𝑦∈𝑆_𝑟, is also uniformly bounded. Hence,𝐴(𝑆_𝑟)⊂𝑋is a family of uniformly bounded and equi-continuous functions. By the well-known Ascoli-Arzela theorem, we know that the operator𝜓is completely continuous. The proof of Lemma 7 is complete.

Lemma 8. Assume that (H₁)–(H₄) hold. The existence of positive𝜔-periodic solution of the system (12) and (13) is equivalent to that of nonzero fixed point of𝐴in𝑃.

Proof. Assume that𝑦=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝑋is a periodic solution of (12) and (13). 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^𝛼𝑖𝑗_𝑖𝑡−𝜌_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑡)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑡+𝜉)𝑦^𝜎𝑖𝑗_𝑗×(𝑡+𝜉)𝑑𝜉,𝑖=1,2,…,𝑛. (32)
Integrating the previous equation over[𝑡,𝑡+𝜔], we can have 𝑦_𝑖(𝑣)𝑒^{−∫𝑣0𝑎𝑖(𝑢)𝑑𝑢}_𝑡^𝑡+𝜔=−_𝑡^𝑡+𝜔𝑒^{−∫𝑣0𝑎𝑖(𝑢)𝑑𝑢}𝑦_𝑖(𝑣)×𝐵_𝑖(𝑣)𝑦_𝑖(𝑣)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑣)𝑦^𝛼𝑖𝑗_𝑖𝑣−𝜌_𝑖𝑗(𝑣+_𝑛𝑗=1)𝐷_𝑖𝑗(𝑣)𝑦^𝛽𝑖𝑗_𝑗𝑣−𝜏_𝑖𝑗(𝑣)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑣)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑣+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑣)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑣+𝜉)𝑦^𝜎𝑖𝑗_𝑗×(𝑣+𝜉)𝑑𝜉𝑑𝑣,𝑖=1,2,…,𝑛. (33)
Therefore, we have 𝑦_𝑖(𝑡)𝑒^{−∫𝑡0𝑎𝑖(𝑢)𝑑𝑢}1−𝑒^{−∫𝜔0𝑎𝑖(𝑢)𝑑𝑢}=_𝑡^𝑡+𝜔𝑒^{−∫𝑣0𝑎𝑖(𝑢)𝑑𝑢}𝑦_𝑖(𝑣)×𝐵_𝑖(𝑣)𝑦_𝑖(𝑣)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑣)𝑦^𝛼𝑖𝑗_𝑖𝑣−𝜌_𝑖𝑗(𝑣)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑣)𝑦^𝛽𝑖𝑗_𝑗𝑣−𝜏_𝑖𝑗(𝑣)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑣)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑣+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑣)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑣+𝜉)𝑦^𝜎𝑖𝑗_𝑗×(𝑣+𝜉)𝑑𝜉𝑑𝑣,𝑖=1,2,…,𝑛, (34)
which can be transformed into 𝑦_𝑖(𝑡)=_𝑡^𝑡+𝜔𝑒^{−∫𝑣0𝑎𝑖(𝑢)𝑑𝑢}𝑒^{−∫𝑡0𝑎𝑖(𝑢)𝑑𝑢}1−𝑒^{−∫𝜔0𝑎𝑖(𝑢)𝑑𝑢}𝑦_𝑖(𝑣)×𝐵_𝑖(𝑣)𝑦_𝑖(𝑣)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑣)𝑦^𝛼𝑖𝑗_𝑖𝑣−𝜌_𝑖𝑗(𝑣)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑣)𝑦^𝛽𝑖𝑗_𝑗𝑣−𝜏_𝑖𝑗(𝑣)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑣)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑣+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑣)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑣+𝜉)𝑦^𝜎𝑖𝑗_𝑗×(𝑣+𝜉)𝑑𝜉𝑑𝑣=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑣)𝑦_𝑖(𝑣)×𝐵_𝑖(𝑣)𝑦_𝑖(𝑣)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑣)𝑦^𝛼𝑖𝑗_𝑖𝑣−𝜌_𝑖𝑗(𝑣+_𝑛𝑗=1)𝐷_𝑖𝑗(𝑣)𝑦^𝛽𝑖𝑗_𝑗𝑣−𝜏_𝑖𝑗(𝑣)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑣)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑣+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑣)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑣+𝜉)𝑦^𝜎𝑖𝑗_𝑗×()=𝑣+𝜉𝑑𝜉𝑑𝑣_{𝑡𝑡+𝜔}𝐺_𝑖(𝑡,𝑣)𝑇𝑦_𝑖(𝑣)𝑑𝑣,𝑖=1,2,…,𝑛. (35)
Thus,𝑦_𝑖(𝑡)(𝑖=1,2,…,𝑛)is a periodic solution for system (12) and (13).
If𝑦=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝑋, and𝐴𝑦=((𝐴𝑦)₁,(𝐴𝑦)₂,…,(𝐴𝑦)_𝑛)^𝑇=𝑦with𝑦≠0, then for any𝑡=𝑡_𝑘derivative the two sides of (20) about𝑡, 𝑑𝐴𝑦_𝑖(𝑡)𝑑𝑡=𝐺_𝑖(𝑡,𝑡+𝜔)𝑇𝑦_𝑖(𝑡+𝜔)−𝐺_𝑖(𝑡,𝑡)𝑇𝑦_𝑖(𝑡)+_𝑡^𝑡+𝜔𝑑𝐺_𝑖(𝑡,𝑣)𝑑𝑡𝑇𝑦_𝑖(𝑣)𝑑𝑣=−𝑇𝑦_𝑖(𝑡)+𝑎_𝑖(𝑡)_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑣)𝑇𝑦_𝑖(𝑣)𝑑𝑣=𝑎_𝑖(𝑡)𝑦_𝑖(𝑡)−𝑇𝑦_𝑖(𝑡)=𝑦_𝑖(𝑡)𝑎_𝑖(𝑡)−𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦^𝛼𝑖𝑗_𝑖𝑡−𝜌_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠−^𝑛_𝑗=1𝐹_𝑖𝑗(𝑡)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑡+𝜉)𝑦^𝜎𝑖𝑗_𝑗()=𝑡+𝜉𝑑𝜉𝑑𝑦_𝑖(𝑡).𝑑𝑡 (36)
Hence,𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇∈𝑋is a positive𝜔-periodic solution of (12) and (13). Thus we complete the proof of Lemma 8.

3. Existence of Periodic Solution of the System

Now, we are at the position to study the existence of positive periodic solutions of system (9) and (10). We mainly apply the Krasnoselskii fixed-point theorem in the cone of Banach space under some conditions to prove the main Theorem 9.

Theorem 9. Assume (H₁)–(H₄). System (9) and (10) has at least one positive𝜔-periodic solution if and only if the condition 𝑙₀=min_{1≤𝑖≤𝑛}𝐵_𝑖+^𝑛_𝑗=1𝐶_𝑖𝑗+𝐷_𝑖𝑗+𝐸_𝑖𝑗+𝐹_𝑖𝑗>0 (37) holds.

Proof (sufficiency). Let 𝐿₀=max_{[]𝑖∈1,𝑛}𝑏_𝑖+^𝑛_𝑗=1𝐶_𝑖𝑗+𝐷_𝑖𝑗+𝐸_𝑖𝑗+𝐹_𝑖𝑗, (38) by condition (37), we know that𝐿₀≥𝑙₀>0. Choose a constant𝐿≥𝐿₀such that1/𝜔𝛽_𝑖𝐿<1. Let𝑟=1/𝜔𝛽_𝑖𝐿and Ω_𝑟=𝑦(𝑡)=𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡)^𝑇∈𝑋∶𝑦_𝑖0<𝑟,𝑖=1,2,…,𝑛. (39)
For any𝑦=𝑦(𝑡)∈𝑃∩𝜕Ω_𝑟,𝜎|𝑦_𝑖|₀≤𝑦_𝑖(𝑡)≤|𝑦_𝑖|₀, from (20), we obtain 𝐴𝑦_𝑖0≤𝛽_𝑖^𝜔₀𝑇𝑦_𝑖(𝑡)𝑑𝑡=𝛽_𝑖^𝜔₀𝑦_𝑖(𝑡)𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦^𝛼𝑖𝑗_𝑖𝑡−𝜌_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑡)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑡+𝜉)𝑦^𝜎𝑖𝑗_𝑗(𝑡+𝜉)𝑑𝜉𝑑𝑡≤𝜔𝛽_𝑖𝑦_𝑖0𝐵_𝑖𝑟+^𝑛_𝑗=1𝐶_𝑖𝑗𝑟^𝛼𝑖𝑗+𝐷_𝑖𝑗𝑟^𝛽𝑖𝑗+𝐸_𝑖𝑗𝑟^𝛾𝑖𝑗+𝐹_𝑖𝑗𝑟^{𝛿𝑖𝑗+𝜎𝑖𝑗}≤𝜔𝛽_𝑖𝑦_𝑖0𝐵_𝑖+^𝑛_𝑗=1𝐶_𝑖𝑗+𝐷_𝑖𝑗+𝐸_𝑖𝑗+𝐹_𝑖𝑗𝑟≤𝜔𝛽_𝑖𝑦_𝑖0𝐿₀𝑟≤𝑦_𝑖0. (40)
Hence, for any𝑦=𝑦(𝑡)∈𝑃∩𝜕Ω_𝑟,𝜎|𝑦_𝑖|₀≤𝑦_𝑖(𝑡)≤|𝑦_𝑖|₀, we have 𝐴𝑦=^𝑛_𝑗=1𝐴𝑦_𝑖0≤^𝑛_𝑗=1𝑦_𝑖0=𝑦, (41) which implies that condition (1) in Lemma 4 is satisfied.
On the other hand, we choose0<𝑙≤𝑙₀such that1/𝜔𝜎𝛼_𝑖𝑙>1. Let𝑅=1/𝜔𝜎𝛼_𝑖𝑙>1and suppose𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇∈𝑃/{0}. We show that, for any𝑦=𝑦(𝑡)∈𝑃∩𝜕Ω_𝑅and𝜆>0,𝑦≠𝜓𝑦+𝜆𝑢. Otherwise, there exist𝑦₀=𝑦₀(𝑡)∈𝑃∩𝜕Ω_𝑅and𝜆₀>0, such that𝑦₀=𝜓𝑦₀+𝜆₀𝑢. Let𝑢_𝑖0≠0(1≤𝑖₀≤𝑛); since𝑦_𝑖0(𝑡)≥𝜎|𝑦_𝑖0|₀, it follows that 𝑦_𝑖0=𝐴𝑦_𝑖0(𝑡)+𝜆₀𝑢_𝑖0=_𝑡^𝑡+𝜔𝐺_𝑖0(𝑡,𝑣)𝑇𝑦_𝑖0(𝑣)𝑑𝑣+𝜆₀𝑢_𝑖0≥𝜎𝛼_𝑖𝑦_𝑖00^𝜔₀𝐵_𝑖(𝑡)𝑦_𝑖(𝑡)+^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)𝑦^𝛼𝑖𝑗_𝑖𝑡−𝜌_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐷_𝑖𝑗(𝑡)𝑦^𝛽𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+^𝑛_𝑗=1𝐸_𝑖𝑗(𝑡)×⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠+^𝑛_𝑗=1𝐹_𝑖𝑗(𝑡)×⁰_{−𝜃𝑖𝑗}𝐾_𝑖𝑗(𝜉)𝑦^𝛿𝑖𝑗_𝑖(𝑡+𝜉)𝑦^𝜎𝑖𝑗_𝑗×(𝑡+𝜉)𝑑𝜉𝑑𝑡+𝜆₀𝑢_𝑖0≥𝜔𝜎𝛼_𝑖𝑦_𝑖00𝐵_𝑖𝑅+^𝑛_𝑗=1𝐶_𝑖𝑗𝑅^𝛼𝑖𝑗+𝐷_𝑖𝑗𝑅^𝛽𝑖𝑗+𝐸_𝑖𝑗𝑅^𝛾𝑖𝑗+𝐹_𝑖𝑗𝑅^{𝛿𝑖𝑗+𝜎𝑖𝑗}≥𝜔𝜎𝛼_𝑖𝑦_𝑖0𝐵_𝑖+^𝑛_𝑗=1𝐶_𝑖𝑗+𝐷_𝑖𝑗+𝐸_𝑖𝑗+𝐹_𝑖𝑗𝑅+𝜆₀𝑢_𝑖0≥𝜔𝜎𝛼_𝑖𝑙₀𝑅𝑦_𝑖0+𝜆₀𝑢_𝑖0≥𝑦_𝑖00+𝜆₀𝑢_𝑖0>𝑦_𝑖00, (42)
which is a contradiction. This proves that condition (2) in Lemma 4 is also satisfied. By Lemmas 4 and 8, system (12) and (13) has at least one positive omega-periodic solution. From Lemma 2, system (9) and (10) has at least one positive𝜔-periodic solution.
(Necessity). Suppose that (37) does not hold. Then, there exists at least an𝑖₀(1≤𝑖₀≤𝑛)such that 𝐵_𝑖0=0,𝐶_𝑖0𝑗=𝐷_𝑖0𝑗=𝐸_𝑖0𝑗=𝐹_𝑖0𝑗=0,𝑗∈[1,𝑛]. (43)
If system (12) and (13) has a positive𝜔-periodic solution𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡),…,𝑦_𝑛(𝑡))^𝑇, then we have 𝑑𝑦_𝑖0(𝑡)𝑑𝑡=𝑎_𝑖0(𝑡)𝑦_𝑖0(𝑡). (44)
Integrating the previous equation over[𝑡,𝑡+𝜔],we can have 𝑦0=ln^𝑖0(𝑡+𝜔)𝑦_𝑖0(𝑡)=^𝜔₀𝑎_𝑖0(𝑡)𝑑𝑡>0, (45)
which is a contradiction. The proof of Theorem 9 is complete.

4. Applications

In this section, we apply the results obtained in the previous section to some𝑛-species competition systems which are mentioned in Section 1. If we consider the environmental or biological factors, the assumption of the periodic oscillation of the parameters and impulse functions seems realistic and reasonable in view of any seasonal phenomena which they might be subject to, for example, mating habits, availability of food, weather conditions, and so forth.

Application 1. We consider the following three classes of periodic𝑛-species competition systems with impulses: 𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗(𝑡),𝑡∈𝑅,𝑡≠𝑡_𝑘,𝑖=1,2,…,𝑛,Δ𝑥_𝑖𝑡_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘∈𝑍₊,𝑖=1,2,…,𝑛;𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝜃𝑖𝑗_𝑗(𝑡),𝑡∈𝑅,𝑡≠𝑡_𝑘,𝑖=1,2,…,𝑛,Δ𝑥_𝑖𝑡_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘∈𝑍₊,𝑖=1,2,…,𝑛;𝑑𝑥_𝑖(𝑡)𝑑𝑡=𝑥_𝑖(𝑡)𝑏_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥^𝜃𝑖𝑗_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑡∈𝑅,𝑡≠𝑡_𝑘,𝑖=1,2,…,𝑛,Δ𝑥_𝑖𝑡_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘∈𝑍₊,𝑖=1,2,…,𝑛, (46)(47)(48)

which are special cases of system (9), where𝑏_𝑖(𝑡),𝑎_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡),𝜃_𝑖𝑗,ℎ_𝑖𝑘are the same as in (𝐻₁)–(𝐻₄). We denote 𝐴_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+ℎ_𝑖𝑘,𝐴∗_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+ℎ_𝑖𝑘, (49)

from Theorem 9, we have the following results.

Corollary 10. Assume that (H₁)–(H₄) hold. The systems (46) and (47) have at least one positive𝜔-periodic solution if and only if the following condition min_{1≤𝑖≤𝑛}𝐴_𝑖𝑗>0 (50) holds.

Corollary 11. Assume that (H₁)–(H₄) hold. The system (48) has at least one positive𝜔-periodic solution if and only if the following condition min_{1≤𝑖≤𝑛}𝐴∗_𝑖𝑗>0 (51) holds.

Application 2. We consider the following𝑛-species Gilpin-Ayala type competition systems with impulses: 𝑦^′_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(𝑠)𝑦^𝛾𝑖𝑗_𝑗(𝑡+𝑠)𝑑𝑠,a.e,𝑡>0,𝑡≠𝑡_𝑘𝑦,𝑖=1,2,…,𝑛,_𝑖𝑡⁺_𝑘−𝑦_𝑖𝑡⁻_𝑘=ℎ_𝑖𝑘𝑦_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘∈𝑍₊, (52)

which is a special case of system (9), where𝑟_𝑖(𝑡),𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),𝑐_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡)∈𝐶(𝑅,𝑅₊)are𝜔-periodic,𝑘_𝑖𝑗∈𝐶([−𝜂_𝑖𝑗,0],𝑅₊)are constants such that∫⁰_{−𝜂𝑖𝑗}𝑘_𝑖𝑗(𝑡)𝑑𝑡=1, and𝛼_𝑖𝑗≥0,𝛽_𝑖𝑗≥0,𝛾_𝑖𝑗≥0,𝑖=1,2,…,𝑛. We denote 𝐴_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+ℎ_𝑖𝑘,𝐵_𝑖𝑗(𝑡)=𝑏_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+ℎ_𝑖𝑘,𝐶_𝑖𝑗(𝑡)=𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑖𝑗}1+ℎ_𝑖𝑘, (53)

from Theorem 9, we have the following result.

Corollary 12. Assume that (H₁)–(H₄) and the condition 𝑅₂=min_{1≤𝑖≤𝑛}^𝑛_𝑗=1𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗>0 (54)
hold, the system (52) has at least one positive𝜔-periodic solution.

Application 3. We study the following periodic nonlinear𝑛-species competitive Lotka-Volterra models with impulses: 𝑦^′_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)𝑦^𝛾𝑖𝑗_𝑖(𝑡)𝑦^𝛿𝑖𝑗_𝑗(𝑡),a.e,𝑡>0,𝑡≠𝑡_𝑘𝑦,𝑖=1,2,…,𝑁,_𝑖𝑡⁺_𝑘−𝑦_𝑖𝑡⁻_𝑘=ℎ_𝑖𝑘𝑦_𝑖𝑡_𝑘,𝑖=1,2,…,𝑁,𝑘∈𝑍₊, (55)

which is a special case of system (9), where𝑟_𝑖(𝑡),𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),𝑐_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡)∈𝐶(𝑅,𝑅₊)are𝜔-periodic, and𝛼_𝑖𝑗≥0,𝛽_𝑖𝑗≥0,𝛾_𝑖𝑗≥0,𝛿_𝑖𝑗≥0. We denote 𝐴_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+ℎ_𝑖𝑘^𝛼𝑖𝑗,𝐵_𝑖𝑗(𝑡)=𝑏_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+ℎ_𝑖𝑘^𝛽𝑖𝑗,𝐶_𝑖𝑗(𝑡)=𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+ℎ_𝑖𝑘^{𝛾𝑖𝑗+𝛿𝑖𝑗}, (56)

from Theorem 9, we have the following result.

Corollary 13. Assume that (H₁)–(H₄) and the condition 𝑅₃=min_{1≤𝑖≤𝑁}^𝑁_𝑗=1𝐴_𝑖𝑗+𝐵_𝑖𝑗+𝐶_𝑖𝑗>0 (57)
hold, the system (55) has at least one positive𝜔-periodic solution.

Application 4. We study the following generalized periodic𝑛-species Gilpin-Ayala type competition models in periodic environments with deviating arguments of the form and impulses: 𝑦^′_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,Δ𝑦_𝑖𝑡_𝑘=𝜃_𝑘𝑦𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘∈𝑍₊,𝑡=𝑡_𝑘, (58)

which is a special case of system (9), where𝑟_𝑖(𝑡),𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),𝑐_𝑖𝑗(𝑡),𝜏_𝑖𝑗(𝑡)∈𝐶(𝑅,𝑅₊)are𝜔-periodic,

𝐾_𝑖𝑗∈𝐶([−𝜎_𝑖𝑗,0],𝑅₊)are constants such that∫⁰_{−𝜎𝑖𝑗}𝐾_𝑖𝑗(𝑡)𝑑𝑡=1, and𝛼_𝑖𝑗≥0,𝛽_𝑖𝑗≥0,𝛾_𝑖𝑗≥0,𝛿_𝑖𝑗≥0,𝑖=1,2,…,𝑛,1+𝐻_𝑖𝑘>0. We denote 𝐴_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+ℎ_𝑖𝑘^𝛼𝑖𝑗,𝐵_𝑖𝑗(𝑡)=𝑏_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+ℎ_𝑖𝑘^𝛽𝑖𝑗,𝐶_𝑖𝑗(𝑡)=𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑖𝑗(𝑡)}1+ℎ_𝑖𝑘^{𝛾𝑖𝑗+𝛿𝑖𝑗}, (59)

from Theorem 9, we have the following result.

Corollary 14. Assume that (H₁)–(H₄) and the condition 𝑅₄=min_1≤𝑖≤2𝐴_𝑖+^𝑛_𝑖=1𝐵_𝑖𝑗+𝐶_𝑖𝑗>0 (60)
hold, the system (58) has at least one positive𝜔-periodic solution.

Application 5. We study the following two-species competitive systems with impulses: 𝑦^′₁(𝑡)=𝑦₁(𝑡)𝑟₁(𝑡)−𝑎₁(𝑡)𝑦₁(𝑡)−^𝑛_𝑗=1𝑏_1𝑗(𝑡)𝑦^𝛼1𝑗₁𝑡−𝜏_𝑗(𝑡)−^𝑛_𝑗=1𝑐_1𝑗(𝑡)𝑦^𝛽2𝑗₂𝑡−𝜌_𝑗(𝑡),𝑦^′₂(𝑡)=𝑦₂(𝑡)𝑟₂(𝑡)−𝑎₂(𝑡)𝑦₂(𝑡)−^𝑛_𝑗=1𝑏_2𝑗(𝑡)𝑦^𝛼2𝑗₂𝑡−𝜂_𝑗(𝑡)−^𝑛_𝑗=1𝑐_2𝑗(𝑡)𝑦^𝛽1𝑗₁𝑡−𝜎_𝑗(𝑡),𝑡≠𝑡_𝑘,Δ𝑦_𝑖𝑡_𝑘=ℎ_𝑖𝑘𝑦_𝑖𝑡_𝑘,𝑖=1,2,𝑘∈𝑍₊,𝑡=𝑡_𝑘, (61)

which is a special case of system (9), where𝑟_𝑖(𝑡),𝑎_𝑖𝑗(𝑡),𝑏_𝑖𝑗(𝑡),𝑐_𝑖𝑗(𝑡),𝜏_𝑗(𝑡),𝜌_𝑗(𝑡),𝜂_𝑗(𝑡),𝜎_𝑗(𝑡)∈𝐶(𝑅,𝑅₊)(𝑖=1,2)are𝜔-periodic. We denote 𝐴_𝑖(𝑡)=𝑎_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘,𝐵_𝑖𝑗(𝑡)=𝑏_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑗(𝑡)}1+𝜃_𝑖𝑘^𝛼𝑖𝑗(),𝐶𝑖=1,2_1𝑗(𝑡)=𝐶_1𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜌𝑗(𝑡)}1+𝜃_𝑖𝑘^𝛽2𝑗𝐶_2𝑗(𝑡)=𝑐_2𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜎𝑗(𝑡)}1+𝜃_𝑖𝑘^𝛽1𝑗, (62)

from Theorem 9, we have the following result.

Corollary 15. Assume that (H₁)–(H₄) and the condition 𝑅₄=min_1≤𝑖≤2𝐴_𝑖+^𝑛_𝑖=1𝐵_𝑖𝑗+𝐶_𝑖𝑗>0 (63)
hold, the system (61) has at least one positive𝜔-periodic solution.

Remark 16. We apply the main results obtained in the previous section to study some examples which have some biological implications; from the previous corollaries, we see that, under the appropriate conditions, the impulsive perturbations do not affect the existence of periodic solution of systems. However, if the impulsive perturbations are unbounded, some properties of the solution of systems could be changed significantly, which will be our further work.

Acknowledgments

The research was supported by the National Natural Science Foundation of China (10971229, 11161015, 11371367 and 11361012), the China Postdoctoral Science Foundation (2012M512162 and 2013T60934), Hunan Provincial Natural Science-Hengyang United Foundation of China (11JJ9002, 12JJ9001, 13JJ4098), the Education Foundation of Hunan province (nos. 12C0541, and 13C084), the Science Foundation of Hengyang Normal University (no. 11B36), and the Construct Program of the Key Discipline in Hunan province.