Existence and Stability of Positive Periodic Solutions for a Neutral Multispecies Logarithmic Population Model with Feedback Control and Impulse

Abstract

We investigate a neutral multispecies logarithmic population model with feedback control and impulse. By applying the contraction mapping principle and some inequality techniques, a set of easily applicable criteria for the existence, uniqueness, and global attractivity of positive periodic solution are established. The conditions we obtained are weaker than the previously known ones and can be easily reduced to several special cases. We also give an example to illustrate the applicability of our results.

1. Introduction

As is known to all, ecosystem in the real world is continuously distributed by unpredictable forces which can result in changes in the biological parameters such as survival rates. Of practical interest in ecology is the question of whether or not an ecosystem can withstand those unpredictable disturbances which persist for a finite period of time. In recent years, the qualitative behaviors of the population dynamics with feedback control has attracted the attention of many mathematicians and biologists [1–5]. On the other hand, there are some other perturbations in the real world such as fires and floods, which are not suitable to be considered continually. These perturbations bring sudden changes to the system. Systems with such sudden perturbations involving impulsive differential equations have attracted the interest of many researchers in the past twenty years [6–10], since they provide a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. Such processes are often investigated in various fields of science and technology such as physics, population dynamics, ecology, biological systems, and optimal control; for details, see [11–13]. However, to the best of the author’s knowledge, to this day, no scholar considered the neutral multispecies logarithmic population model with feedback control and impulse.

The aim of this paper is to investigate the existence, uniqueness, and global attractivity of the positive periodic solution for the following neutral multispecies logarithmic population system with feedback control and impulse: 𝑑𝑁_𝑖(𝑡)𝑑𝑡=𝑁_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)ln𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)ln𝑁_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ln𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑑_𝑖𝑗(𝑡)𝑑ln𝑁_𝑗𝑡−𝛿_𝑖𝑗(𝑡)𝑑𝑡−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝜎_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽_𝑖(𝑡)ln𝑁_𝑖(𝑡)+𝜗_𝑖(𝑡)ln𝑁_𝑖𝑡−𝛾_𝑖(𝑡)𝑁,𝑡≥0,_𝑖𝑡⁺_𝑘=𝑒^{(1+𝜃𝑖𝑘)}𝑁_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…, (1) where 𝑢_𝑖(𝑡) denote indirect feedback control variables. For the ecological justification of (1) and the similar types, refer to [14–20].

For the sake of generality and convenience, we always make the following fundamental assumptions:(𝐻₁)𝑟_𝑖(𝑡), 𝑎_𝑖𝑗(𝑡), 𝑏_𝑖𝑗(𝑡), 𝑐_𝑖𝑗(𝑡), 𝑑_𝑖𝑗(𝑡), 𝑒_𝑖(𝑡), 𝑓_𝑖(𝑡), 𝜏_𝑖𝑗(𝑡), 𝛿_𝑖𝑗(𝑡)∈𝐶²(𝑅,𝑅), 𝜎_𝑖(𝑡), 𝛾_𝑖(𝑡), 𝛼_𝑖(𝑡), 𝛽_𝑖(𝑡), and 𝜂_𝑖(𝑡) are continuous nonnegative 𝜔-periodic functions with ∫^𝜔₀𝑟_𝑖(𝑡)>0, 𝑎_𝑖𝑖(𝑡)>0, 𝛿^′_𝑖𝑗(𝑡)<1, 𝜏=max_{𝑡∈[0,𝜔]}{𝜏_𝑖𝑗(𝑡),𝛿_𝑖𝑗(𝑡),𝜎_𝑖(𝑡),𝛾_𝑖(𝑡)}, and ∫^∞₀𝐾_𝑖𝑗(𝑠)𝑑𝑠=1, ∫₀^+∞𝑠𝐾_𝑖𝑗(𝑠)𝑑𝑠<+∞, 𝑖,𝑗=1,2,…,𝑛;(𝐻₂)0<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯ are fixed impulsive points with lim_𝑘→∞𝑡_𝑘=+∞;(𝐻₃){𝜃_𝑖𝑘} is a real sequence, 𝜃_𝑖𝑘+1>0, and ∏_{0<𝑡𝑘<𝑡}(1+𝜃_𝑖𝑘) is an 𝜔-periodic function.

In the following section, some definitions and some useful lemmas are listed. In the third section, by applying the contraction mapping principle, some sufficient conditions which ensure the existence and uniqueness of positive periodic solution of system (1) are established, and then we get a few sufficient conditions ensuring the global attractivity of the positive periodic solution by employing some inequality techniques. Finally, we give an example to show our results.

2. Preliminaries

In order to obtain the existence and uniqueness of a periodic solution for system (1), we first give some definitions and lemmas.

Definition 1. A function 𝑁_𝑖∶𝑅→(0,∞) (𝑖=1,2,…,𝑛) is said to be a positive solution of (1), if the following conditions are satisfied:(a)𝑁_𝑖(𝑡) is absolutely continuous on each (𝑡_𝑘,𝑡_𝑘+1);(b)for each 𝑘∈𝑍₊, 𝑁_𝑖(𝑡⁺_𝑘) and 𝑁_𝑖(𝑡⁻_𝑘) exist, and 𝑁_𝑖(𝑡⁻_𝑘)=𝑁_𝑖(𝑡_𝑘);(c)𝑁_𝑖(𝑡) satisfies the first equation of (1) for almost everywhere (for short a.e.) in [0,∞]⧵{𝑡_𝑘} and satisfies 𝑁_𝑖(𝑡⁺_𝑘)=(1+𝜃_𝑖𝑘)𝑁_𝑖(𝑡_𝑘) for 𝑡=𝑡_𝑘, 𝑘∈𝑍₊={1,2,…}.

Definition 2. System (1) is said to be globally attractive, if there exists a positive solution (𝑁_𝑖(𝑡),𝑢_𝑖(𝑡)) of (1) such that lim_𝑡→+∞|𝑁_𝑖(𝑡)−𝑁^∗_𝑖(𝑡)|=0, lim_𝑡→+∞|𝑢_𝑖(𝑡)−𝑢^∗_𝑖(𝑡)|=0, for any other positive solution (𝑁^∗_𝑖(𝑡),𝑢^∗_𝑖(𝑡)) of the system (1).

We can easily get the following lemma.

Lemma 3. 𝑅₊^2𝑛={(𝑁_𝑖(𝑡),𝑢_𝑖(𝑡))∶𝑁_𝑖(0)>0,𝑢_𝑖(0)>0,𝑖=1,2,…,𝑛} is the positive invariable region of the system (1).

Proof. In view of biological population, we obtain 𝑁_𝑖(0)>0, 𝑢_𝑖(0)>0. By the system (1), we have 𝑁_𝑖(𝑡)=𝑁_𝑖(0)×exp^𝑡₀𝑟_𝑖𝜂−^𝑛_𝑗=1𝑎_𝑖𝑗𝜂ln𝑁_𝑗𝜂−^𝑛_𝑗=1𝑏_𝑖𝑗𝜂ln𝑁_𝑗𝜂−𝜏_𝑖𝑗𝜂−^𝑛_𝑗=1𝑐_𝑖𝑗𝜂^𝑡_−∞𝐾_𝑖𝑗𝜂−𝑠ln𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑑_𝑖𝑗𝜂𝑑ln𝑁_𝑗𝜂−𝛿_𝑖𝑗𝜂𝑑𝑡−𝑒_𝑖𝜂𝑢_𝑖𝜂−𝑓_𝑖𝜂𝑢_𝑖𝜂−𝜎_𝑖𝜂𝑑𝜂,𝑡∈0,𝑡₁𝑁,𝑖=1,2,…,𝑛,_𝑖(𝑡)=𝑁_𝑖𝑡_𝑘×exp^𝑡_𝑡𝑘𝑟_𝑖𝜂−^𝑛_𝑗=1𝑎_𝑖𝑗𝜂ln𝑁_𝑗𝜂−^𝑛_𝑗=1𝑏_𝑖𝑗𝜂ln𝑁_𝑗𝜂−𝜏_𝑖𝑗𝜂−^𝑛_𝑗=1𝑐_𝑖𝑗𝜂^𝑡_−∞𝐾_𝑖𝑗𝜂−𝑠ln𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑑_𝑖𝑗𝜂𝑑ln𝑁_𝑗𝜂−𝛿_𝑖𝑗𝜂𝑑𝑡−𝑒_𝑖𝜂𝑢_𝑖𝜂−𝑓_𝑖𝜂𝑢_𝑖𝜂−𝜎_𝑖𝜂,𝑡𝑑𝜂𝑡∈_𝑘,𝑡_𝑘+1𝑁,𝑖=1,2,…,𝑛,_𝑖𝑡⁺_𝑘=𝑒^{(1+𝑝𝑖𝑘)}𝑁_𝑖𝑡_𝑘𝑢>0,𝑘∈𝑁,𝑖=1,2,…,𝑛,_𝑖(𝑡)=_𝑡^𝑡+𝜔𝐺(𝑡,𝑠)𝛽_𝑖(𝑠)ln𝑁_𝑖(𝑠)+𝜗_𝑖(𝑠)ln𝑁_𝑖𝑠−𝛾_𝑖(𝑠)𝜙𝑑𝑠∶=_𝑖ln𝑁_𝑖(𝑡), (2) where 𝐺_𝑖()=∫𝑡,𝑠exp^𝑠_𝑡𝛼_𝑖(𝜉)𝑑𝜉exp∫^𝑠_𝑡𝛼_𝑖(𝜉)𝑑𝜉−1. (3) Then the solution of the system (1) is positive.
Under the above hypotheses (𝐻₁)--(𝐻₃), we consider the neutral nonimpulsive system: 𝑑𝑦_𝑖(𝑡)𝑑𝑡=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)ln𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)ln𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ln𝑦_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝐷_𝑖𝑗(𝑡)𝑑ln𝑦_𝑗𝑡−𝛿_𝑖𝑗(𝑡)𝑑𝑡−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝜎_𝑖(𝑡),𝑑𝑢_𝑖(𝑡)𝑑𝑡=−𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)+𝛽^∗_𝑖(𝑡)ln𝑦_𝑖(𝑡)+𝜗^∗_𝑖(𝑡)ln𝑦_𝑖𝑡−𝛾_𝑖(𝑡), (4) where 𝐴_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘,𝐵_𝑖𝑗(𝑡)=𝑏_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+𝜃_𝑖𝑘,𝐶_𝑖𝑗(𝑡)=𝑐_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘,𝐷_𝑖𝑗(𝑡)=𝑑_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝛿𝑖𝑗(𝑡)}1+𝜃_𝑖𝑘,𝛽^∗_𝑖(𝑡)=𝛽_𝑖(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘,𝜗^∗_𝑖(𝑡)=𝜃_𝑖(𝑡)_{0<𝑡𝑘<𝑡−𝛾𝑖(𝑡)}1+𝜃_𝑖𝑘. (5) By a solution (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡)) of (4), it means an absolutely continuous function (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡)) defined on [−𝜏,0] that satisfies (4) a.e., for 𝑡≥0, and 𝑦(𝜉)=𝜑(𝜉), 𝑦^′(𝜉)=𝜑^′(𝜉) on [−𝜏,0].

The following lemma will be used in the proofs of our results, and the proof of the lemma is similar to that of Theorem 1 in [6].

Lemma 4. Suppose that (𝐻₁)--(𝐻₄) hold. Then(i)if (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡)) is a solution of (4) on [−𝜏,+∞), then (𝑁_𝑖(𝑡),𝑢_𝑖(𝑡))=(∏_{0<𝑡𝑘<𝑡}𝑒^{(1+𝜃𝑖𝑘)}𝑦_𝑖(𝑡),𝑢_𝑖(𝑡)) is a solution of (1) on [−𝜏,+∞),(ii)if (𝑁_𝑖(𝑡),𝑢_𝑖(𝑡)) is a solution of (1) on [−𝜏,+∞), then (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡))=(∏_{0<𝑡𝑘<𝑡}(𝑒^{(1+𝜃𝑖𝑘)})⁻¹𝑁_𝑖(𝑡),𝑢_𝑖(𝑡)) is a solution of (4) on [−𝜏,+∞).

Proof. (i) It is easy to see that (𝑁_𝑖(𝑡),𝑢_𝑖(𝑡))=(∏_{0<𝑡𝑘<𝑡}𝑒^{(1+𝜃𝑖𝑘)}𝑦_𝑖(𝑡),𝑢_𝑖(𝑡)) is absolutely continuous on every interval (𝑡_𝑘,𝑡_𝑘+1], 𝑡≠𝑡_𝑘, 𝑘=1,2,…, 𝑁^′_𝑖(𝑡)−𝑁_𝑖(𝑡)×𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)ln𝑁_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)ln𝑁_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ln𝑁_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑑_𝑖𝑗(𝑡)𝑑ln𝑁_𝑗𝑡−𝛿_𝑖𝑗(𝑡)𝑑𝑡−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝜎_𝑖(𝑡)=_{0<𝑡𝑘<𝑡}𝑒^{(1+𝜃𝑖𝑘)}𝑦^′_𝑖(𝑡)−_{0<𝑡𝑘<𝑡}𝑒^{(1+𝜃𝑖𝑘)}𝑦_𝑖(𝑡)×𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘ln𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝜏𝑖𝑗(𝑡)}1+𝜃_𝑖𝑘ln𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝑐_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑗(𝑡−𝑠)_{0<𝑡𝑘<𝑡}1+𝜃_𝑖𝑘ln𝑦_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝑑_𝑖𝑗(𝑡)_{0<𝑡𝑘<𝑡−𝛿𝑖𝑗(𝑡)}1+𝜃_𝑖𝑘𝑑𝑦_𝑗𝑡−𝛿_𝑖𝑗(𝑡)𝑑𝑡−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝜎_𝑖(𝑡)=_{0<𝑡𝑘<𝑡}𝑒^(1+𝜃𝑘)×𝑦^′(𝑡)(𝑡)×𝑟−𝑦_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)ln𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)ln𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)×^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ln𝑦_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝐷_𝑖𝑗(𝑡)𝑑ln𝑦_𝑗𝑡−𝛿_𝑖𝑗(𝑡)𝑑𝑡−𝑒_𝑖(𝑡)𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝑢_𝑖𝑡−𝜎_𝑖(𝑡)𝑢=0,_′𝑖(𝑡)+𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)−𝛽_𝑖(𝑡)ln𝑁_𝑖(𝑡)−𝜗_𝑖(𝑡)ln𝑁_𝑖𝑡−𝛾_𝑖(𝑡)=𝑢^′_𝑖(𝑡)+𝛼_𝑖(𝑡)𝑢_𝑖(𝑡)−𝛽^∗_𝑖(𝑡)ln𝑦_𝑖(𝑡)−𝜗^∗_𝑖(𝑡)ln𝑦_𝑖𝑡−𝛾_𝑖(𝑡)=0. (6) On the other hand, for any 𝑡=𝑡_𝑘, 𝑘=1,2,…, 𝑁_𝑖𝑡⁺_𝑘=lim_{𝑡→𝑡+𝑘0<𝑡𝑗<𝑡}𝑒^{(1+𝜃𝑖𝑘)}𝑦_𝑖(𝑡)=_{0<𝑡𝑗≤𝑡𝑘}𝑒^{(1+𝜃𝑖𝑘)}𝑦_𝑖𝑡_𝑘,𝑁_𝑖𝑡_𝑘=_{0<𝑡𝑗<𝑡𝑘}𝑒^{(1+𝜃𝑖𝑘)}𝑦_𝑖𝑡_𝑘. (7) Thus 𝑁𝑡⁺_𝑘=𝑒^{(1+𝜃𝑖𝑘)}𝑁𝑡_𝑘,𝑘=1,2,…. (8) It follows from (6)–(8) that (𝑁_𝑖(𝑡),𝑢_𝑖(𝑡)) is a solution of (1).
(ii) Since 𝑁_𝑖(𝑡)=∏_{0<𝑡𝑘<𝑡}𝑒^{(1+𝜃𝑖𝑘)}𝑦_𝑖(𝑡) is absolutely continuous on every interval (𝑡_𝑘,𝑡_𝑘+1], 𝑡≠𝑡_𝑘, 𝑘=1,2,…, and in view of (8), it follows that for any 𝑘=1,2,…, 𝑦_𝑖𝑡⁺_𝑘=_{0<𝑡𝑗≤𝑡𝑘}𝑒^{(1+𝜃𝑖𝑘)−1}𝑁_𝑖𝑡⁺_𝑘=_{0<𝑡𝑗<𝑡𝑘}𝑒^{(1+𝜃𝑖𝑘)−1}𝑁_𝑖𝑡_𝑘=𝑦_𝑖𝑡_𝑘,𝑦_𝑖𝑡⁻_𝑘=_{0<𝑡𝑗<𝑡𝑘}𝑒^{(1+𝜃𝑖𝑘)−1}𝑁_𝑖𝑡⁻_𝑘=_{0<𝑡𝑗≤𝑡−𝑘}𝑒^{(1+𝜃𝑖𝑘)−1}𝑁_𝑖𝑡⁻_𝑘=𝑦_𝑖𝑡_𝑘, (9) 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 (i), we can check that (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡))=(∏_{0<𝑡𝑘<𝑡}(𝑒^{(1+𝜃𝑖𝑘)})⁻¹𝑁_𝑖(𝑡),𝑢_𝑖(𝑡)) are solutions of (4) on [−𝜏,+∞). The proof of Lemma 4 is completed.

Lemma 5. (𝑦_𝑖(𝑡),𝑢_𝑖(𝑡)) is a 𝜔-periodic solution of (4) if and only if 𝑦_𝑖(𝑡) is a 𝜔-periodic solution of the following system: 𝑑𝑦_𝑖(𝑡)𝑑𝑡=𝑦_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)ln𝑦_𝑗(𝑡)−^𝑛_𝑗=1𝐵_𝑖𝑗(𝑡)ln𝑦_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−^𝑛_𝑗=1𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)ln𝑦_𝑗(𝑠)−𝑑𝑠_𝑛𝑗=1𝐷_𝑖𝑗(𝑡)𝑑ln𝑦_𝑗𝑡−𝛿_𝑖𝑗(𝑡)𝑑𝑡−𝑒_𝑖(𝑡)𝜙_𝑖ln𝑦_𝑖(𝑡)−𝑓_𝑖(𝑡)𝜙_𝑖ln𝑦_𝑖(𝑡)𝑡−𝜎_𝑖(𝑡), (10) where 𝜙_𝑖ln𝑦_𝑖(𝑡)∶=_𝑡^𝑡+𝜔𝐺_𝑖(𝑡,𝑠)𝛽^∗_𝑖(𝑠)ln𝑦_𝑖(𝑠)+𝜗^∗_𝑖(𝑠)ln𝑦_𝑖𝑠−𝛾_𝑖(𝑠)𝑑𝑠, (11) and 𝐺_𝑖(𝑡,𝑠) is defined by (3).

Proof. The proof of Lemma 5 is similar to that of Lemma 2.2 in [2], and we omit the details here.

Obviously, the existence, uniqueness, and global attractivity of positive periodic solution of system (1) is equivalent to the existence, uniqueness, and global attractivity of periodic solution of system (10).

Lemma 6. Assume that 𝑢(𝑡), 𝜏(𝑡) are all continuously differentiable 𝜔-periodic functions and 𝑎(𝑡) is a nonnegative continuous 𝜔-periodic function such that ∫^𝜔₀𝑎(𝑡)𝑑𝑡>0; then ^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑏(𝑠)𝑢^′((𝑠(𝑡)𝑢((𝑡−𝑠−𝜏))𝑑𝑠=𝑐𝑡−𝜏))_𝑡−∞𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑎(𝑠)𝑐(𝑠)+𝑐^′(𝑠)𝑢(𝑠−𝜏(𝑠))𝑑𝑠, (12) where 𝑐(𝑡)=𝑏(𝑡)/(1−𝜏^′(𝑡)).

Proof. As ^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑏(𝑠)𝑢^′((𝑠=𝑠−𝜏))𝑑𝑠_𝑡−∞𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑐(𝑠)𝑑𝑢(𝑠−𝜏(𝑠))=𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑐(𝑠)𝑢(𝑠−𝜏(𝑠))^𝑡_−∞−^𝑡_−∞𝑢(𝑠−𝜏(𝑠))𝑑𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑐(𝑠)=𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑐(𝑠)𝑢(𝑠−𝜏(𝑠))^𝑡_−∞−^𝑡_−∞𝑢(𝑠−𝜏(𝑠))𝑎(𝑠)𝑐(𝑠)+𝑐^′(𝑠)𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑑𝑠. (13) Denote 𝑚=𝑒^{−∫𝜔0𝑎(𝑡)𝑑𝑡}; then from 𝑎(𝑡)≥0, ∫^𝜔₀𝑎(𝑡)𝑑𝑡>0, it follows that 𝑚<1. Also, when 𝑡≥𝑠 without loss of generality, we may assume that 𝑠+𝑛𝜔≤𝑡≤𝑠+(𝑛+1)𝜔; thus 𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑐(𝑠)𝑢(𝑠−𝜏(𝑠))≤𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}‖𝑐‖‖𝑢‖=𝑒^{−∑𝑛−1𝑗=1∫𝑠+(𝑗+1)𝜔𝑠+𝑗𝜔𝑎(𝜉)𝑑𝜉−∫𝑡𝑠+𝑛𝜔𝑎(𝜉)𝑑𝜉}‖𝑐‖‖𝑢‖=𝑚^𝑛𝑒^{−∫𝑡𝑠+𝑛𝜔𝑎(𝜉)𝑑𝜉}‖𝑐‖‖𝑢‖≤𝑚^𝑛‖𝑐‖‖𝑢‖. (14) Therefore, lim_{𝑠→−∞}𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑐(𝑠)𝑢(𝑠−𝜏(𝑠))=0, (15) and so from (13)–(15) it follows that ^𝑡_−∞𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑏(𝑠)𝑢^′((𝑠(𝑡)𝑢((𝑡−𝑠−𝜏))𝑑𝑠=𝑐𝑡−𝜏))_𝑡−∞𝑒^{−∫𝑡𝑠𝑎(𝜉)𝑑𝜉}𝑎(𝑠)𝑐(𝑠)+𝑐^′(𝑠)𝑢(𝑠−𝜏(𝑠))𝑑𝑠. (16) The proof Lemma 6 is complete.

3. Main Theorem

In this section, by using contraction principle and some inequality techniques, several conditions on the existence, uniqueness, and global attractivity of periodic solution for system (1) are presented.

Let 𝑦_𝑖(𝑡)=𝑒^{ℎ𝑖𝑥𝑖(𝑡)}; the system (10) can be reduced to 𝑑𝑥_𝑖𝑑𝑡=−𝐴_𝑖𝑖(𝑡)𝑥_𝑖(𝑡)−ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑥_𝑗(𝑠)𝑑𝑠−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑡)1−𝛿^′_𝑖𝑗(𝑡)𝑥^′_𝑗𝑡−𝛿_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝜙_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖(𝑡)𝜙_𝑖𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+ℎ_𝑖⁻¹𝑟_𝑖(𝑡), (17) where ℎ_𝑖>0  (𝑖=1,2,…,𝑛) are 𝑛 positive real numbers.

Obviously, the existence, uniqueness, and global attractivity of positive periodic solution of system (10) is equivalent to the existence, uniqueness, and global attractivity of periodic solution of system (17).

For the rest of this paper, we will devote ourselves to study the existence, uniqueness, and global attractivity of periodic solution of (17). We denote Γ¹_𝑖(𝑡)∶=𝑒_𝑖(𝑡)𝜙_𝑖1(𝑡),Γ²_𝑖(𝑡)∶=𝑓_𝑖(𝑡)𝜙_𝑖1𝑡−𝜎_𝑖(𝑡,Δ_𝑖)(𝑡)∶=ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)+𝐶_𝑖𝑗(𝑡)+𝐴_𝑖𝑖(𝑠)𝐷_𝑖𝑗(𝑡)+𝐷^′_𝑖𝑗(𝑡)+Γ¹_𝑖(𝑡)+Γ²_𝑖(𝑡),𝑖=1,2,…,𝑛. (18) Our first result on the global existence of a periodic solution of system (1) is stated in the following theorem.

Theorem 7. In addition to (𝐻₁)--(𝐻₃), assume further that there exist positive constants ℎ_𝑖  (𝑖=1,2,…,𝑛) and a positive constant 𝑀<1 such that(𝐻₄)sup_{𝑡∈[0,𝜔]}max_{𝑖∈[1,𝑛]}{∑^𝑛_𝑗=1(ℎ_𝑗/ℎ_𝑖)𝐷_𝑖𝑗(𝑡)+∫^𝑡_−∞exp{−∫^𝑡_𝑠𝐴_𝑖𝑖(𝜉)𝑑𝜉}Δ_𝑖(𝑠)𝑑𝑠}≤𝑀.Then, system (1) has a unique 𝜔-periodic solution with strictly positive components, where Δ_𝑖(𝑡) is defined by (18).

Proof. From the above analysis, to finish the proof of Theorem 7, it is enough to prove under the conditions of Theorem 7 that system (17) has a unique 𝜔-periodic solution. Let Ω=𝑢(𝑡)∣𝑢∈𝐶𝑅^𝑛,𝑅,𝑢(𝑡+𝜔)=𝑢(𝑡); (19) under the norm ‖𝑢‖=max_{1≤𝑖≤𝑛}max_{𝑡∈[0,𝜔]}{|𝑢_𝑖(𝑡)|}, Ω is a Banach space. For any 𝑢(𝑡)=(𝑢₁(𝑡),…,𝑢_𝑛(𝑡))^𝑇∈Ω, we consider the periodic solution 𝑥_𝑢(𝑡) of periodic differential equation 𝑑𝑥_𝑖𝑑𝑡=−𝐴_𝑖𝑖(𝑡)𝑥_𝑖(𝑡)−ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)𝑢_𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑢_𝑗(𝑠)𝑑𝑠−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑡)1−𝛿^′_𝑖𝑗(𝑡)𝑢^′_𝑗𝑡−𝛿_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝜙_𝑖𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝜙_𝑖𝑢_𝑖(𝑡)𝑡−𝜎_𝑖(𝑡)+ℎ_𝑖⁻¹𝑟_𝑖(𝑡),𝑖=1,2,…,𝑛. (20) Since 𝐴_𝑖𝑖(𝑡)>0, we know that the linear system of system (20) 𝑑𝑥_𝑖𝑑𝑡=−𝐴_𝑖𝑖(𝑡)𝑥_𝑖(𝑡),𝑖=1,2,…,𝑛 (21) admits exponential dichotomies on 𝑅, and so system (20) has a unique periodic solution 𝑥_𝑢(𝑡), which can be expressed as 𝑥_𝑢(𝑡)=𝑥_1𝑢(𝑡),…,𝑥_𝑛𝑢(𝑡)^𝑇=^𝑡_−∞exp−^𝑡_𝑠𝐴₁₁(𝜉)𝑑𝜉𝐹_1𝑢(𝑠)𝑑𝑠,…,_𝑡−∞exp−^𝑡_𝑠𝐴_𝑛𝑛(𝜉)𝑑𝜉𝐹_𝑛𝑢(𝑠)𝑑𝑠^𝑇, (22) where 𝐹_𝑖𝑢(𝑡)=−ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑥_𝑗(𝑠)𝑑𝑠−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑡)1−𝛿^′_𝑖𝑗(𝑡)𝑥^′_𝑗𝑡−𝛿_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝜙_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖(𝑡)𝜙_𝑖𝑥_𝑖(𝑡)𝑡−𝜎_𝑖(𝑡)+ℎ_𝑖⁻¹𝑟_𝑖(𝑡); (23) its proof is similar to that of Theorem 1 in [18]; here we omit it.
Now, by using Lemma 6, 𝑥_𝑖𝑢(𝑡) can also be expressed as 𝑥_𝑖𝑢(𝑡)=−^𝑛_𝑗=1ℎ_𝑗ℎ_𝑖𝐷_𝑖𝑗(𝑡)𝑢_𝑗𝑡−𝛿_𝑖𝑗(𝑡+_𝑡−∞)exp−^𝑡_𝑠𝐴_𝑖𝑖(𝜉)𝑑𝜉𝐺_𝑖𝑢(𝑠)𝑑𝑠,𝑖=1,2,…,𝑛, (24) where 𝐺_𝑖𝑢(𝑡)=−ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)𝑢_𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗×𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑢_𝑗(𝑠)𝑑𝑠+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐴_𝑖𝑖(𝑠)𝐷_𝑖𝑗(𝑡)+𝐷^′_𝑖𝑗(𝑡)𝑢_𝑗𝑡−𝛿_𝑖𝑗(𝑡)+𝑒_𝑖(𝑡)𝜙_𝑖𝑢_𝑖(𝑡)−𝑓_𝑖(𝑡)𝜙_𝑖𝑢_𝑖(𝑡)𝑡−𝜎_𝑖(𝑡)+ℎ_𝑖⁻¹𝑟_𝑖(𝑡). (25) Now we define mapping 𝑇∶Ω→Ω, 𝑇𝑢(𝑡)=𝑥_𝑢(𝑡). Following this we will prove that 𝑇 is a contraction mapping; that is, there exists a constant 𝛽∈(0,1), such that ‖𝑇𝑢−𝑇𝑣‖≤𝛽‖𝑢−𝑣‖, for all 𝑢,𝑣∈Ω. In fact, for any 𝑢(𝑡)=(𝑢₁(𝑡),…,𝑢_𝑛(𝑡))^𝑇 and 𝑣(𝑡)=(𝑣₁(𝑡),…,𝑣_𝑛(𝑡))^𝑇, we have 𝐺_𝑖𝑢(𝑡)−𝐺_𝑖𝑣(𝑡)≤ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)𝑢_𝑗(𝑡)−𝑣_𝑗(𝑡)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)𝑢_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−𝑣_𝑗𝑡−𝜏_𝑖𝑗(𝑡)+𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑢_𝑗(𝑠)−𝑣_𝑗(𝑠+𝐴)𝑑𝑠_𝑖𝑖(𝑠)𝐷_𝑖𝑗(𝑡)+𝐷^′_𝑖𝑗(𝑡×𝑢_𝑗)𝑡−𝛿_𝑖𝑗(𝑡)−𝑣_𝑗𝑡−𝛿_𝑖𝑗(𝑡)+𝑒_𝑖(𝑡)𝜙_𝑖1‖𝑢−𝑣‖+𝑓_𝑖(𝑡)𝜙_𝑖2𝑡−𝜎_𝑖(𝑡)‖‖≤ℎ𝑢−𝑣_{𝑖𝑛−1𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)+𝐶_𝑖𝑗(𝑡)+𝐴_𝑖𝑖(𝑠)𝐷_𝑖𝑗(𝑡)+𝐷^′_𝑖𝑗(𝑡)+Γ¹_𝑖(𝑡)+Γ²_𝑖(𝑡)‖𝑢−𝑣‖=Δ_𝑖(𝑡)‖𝑢−𝑣‖. (26) Hence, ‖𝑇𝑢−𝑇𝑣‖=sup_{[]𝑡∈0,𝜔}max^𝑛_𝑗=1ℎ_𝑗ℎ₁𝐷_1𝑗(𝑡)×𝑢_𝑗𝑡−𝛿_1𝑗(𝑡)−𝑣_𝑗𝑡−𝛿_1𝑗(𝑡)+^𝑡_−∞exp−^𝑡_𝑠𝐴₁₁(𝜉)×𝐺𝑑𝜉_1𝑢(𝑠)−𝐺_1𝑣(𝑠)𝑑𝑠,…,_𝑛𝑗=1ℎ_𝑗ℎ_𝑛𝐷_𝑛𝑗(𝑡)×𝑢_𝑗𝑡−𝛿_𝑛𝑗(𝑡)−𝑣_𝑗𝑡−𝛿_𝑛𝑗(𝑡)+^𝑡_−∞exp−^𝑡_𝑠𝐴_𝑛𝑛(𝜉)×𝐺𝑑𝜉_𝑛𝑢(𝑠)−𝐺_𝑛𝑣(𝑠)𝑑𝑠≤sup_{[]𝑡∈0,𝜔}max^𝑛_𝑗=1ℎ_𝑗ℎ₁𝐷_1𝑗(𝑡)+^𝑡_−∞exp−^𝑡_𝑠𝐴₁₁(𝜉)𝑑𝜉Δ₁(𝑠)×‖‖𝑑𝑠𝑢−𝑣,…,_𝑛𝑗=1ℎ_𝑗ℎ_𝑛𝐷_𝑛𝑗(𝑡)+^𝑡_−∞exp−^𝑡_𝑠𝐴_𝑛𝑛(𝜉)𝑑𝜉Δ_𝑛(𝑠)×‖‖.𝑑𝑠𝑢−𝑣 (27) It follows from (𝐻₄) that ‖𝑇𝑢−𝑇𝑣‖≤‖𝑢−𝑣‖ for all 𝑢,𝑣∈Ω. That is, 𝑇 is a contraction mapping. Hence, there exists a unique fixed point 𝑥^∗(𝑡)=(𝑥^∗₁(𝑡),…,𝑥^∗_𝑛(𝑡))^𝑇∈Ω; that is, 𝑇𝑥^∗(𝑡)=𝑥^∗(𝑡). Therefore, 𝑥^∗(𝑡) is the unique periodic solution of system (17). It follows from (1), (4), (10), and (17) that (𝑁^∗(𝑡),𝑢^∗(𝑡))^𝑇=(𝑁^∗₁(𝑡),…,𝑁^∗_𝑛(𝑡),𝑢^∗₁(𝑡),…,𝑢^∗_𝑛(𝑡))^𝑇 is the unique positive periodic solution of system (1). The proof of Theorem 7 is completed.

Our next theorem is concerned with the global stability of periodic solution for system (1).

Theorem 8. In addition to (𝐻₁)--(𝐻₄), suppose further that the following condition holds:(𝐻₅)exp{−∫^𝑡₀𝐴_𝑖𝑖(𝜉)𝑑𝜉}→0, as 𝑡→+∞, 𝑖=1,2,…,𝑛.Then system (1) has a unique periodic solution which is globally attractive.

Proof. Let 𝑁^∗(𝑡)=(𝑁^∗₁(𝑡),𝑁^∗₂(𝑡),…,𝑁^∗_𝑛(𝑡))^𝑇 be the unique positive periodic solution of system (1), whose existence and uniqueness are guaranteed by Theorem 7, and let 𝑁(𝑡)=(𝑁₁(𝑡),𝑁₂(𝑡),…,𝑁_𝑛(𝑡))^𝑇 be any other solution of system (1). Let 𝑁^∗_𝑖(𝑡)=exp{∏_{0<𝑡𝑘<𝑡}ℎ_𝑖(1+𝑝_𝑖𝑘)𝑥^∗_𝑖(𝑡)}, 𝑁_𝑖(𝑡)=exp{∏_{0<𝑡𝑘<𝑡}×ℎ_𝑖(1+𝑝_𝑖𝑘)𝑥_𝑖(𝑡)}; then, similar to (17), we have 𝑑𝑥^∗_𝑖𝑑𝑡=−𝐴_𝑖𝑖(𝑡)𝑥^∗_𝑖(𝑡)−ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)𝑥^∗_𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)𝑥^∗_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑥^∗_𝑗(𝑠)𝑑𝑠−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑡)1−𝛿^′_𝑖𝑗(𝑡)𝑥_𝑗^′∗𝑡−𝛿_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝜙_𝑖𝑥^∗_𝑖(𝑡)−𝑓_𝑖(𝑡)𝜙_𝑖𝑥^∗_𝑖𝑡−𝜎_𝑖(𝑡)+ℎ_𝑖⁻¹𝑟_𝑖(𝑡),𝑑𝑥_𝑖𝑑𝑡=−𝐴_𝑖𝑖(𝑡)𝑥_𝑖(𝑡)−ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑥_𝑗(𝑠)𝑑𝑠−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑡)1−𝛿^′_𝑖𝑗(𝑡)𝑥^′_𝑗𝑡−𝛿_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝜙_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖(𝑡)𝜙_𝑖𝑥_𝑖𝑡−𝜎_𝑖(𝑡)+ℎ_𝑖⁻¹𝑟_𝑖(𝑡). (28) Let 𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡)=𝑤_𝑖(𝑡); then 𝑑𝑤_𝑖𝑑𝑡=−𝐴_𝑖𝑖(𝑡)𝑤_𝑖(𝑡)−ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑡)𝑤_𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑡)𝑤_𝑗𝑡−𝜏_𝑖𝑗(𝑡)−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐶_𝑖𝑗(𝑡)^𝑡_−∞𝐾_𝑖𝑗(𝑡−𝑠)𝑤_𝑗(𝑠)𝑑𝑠−ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑡)1−𝛿^′_𝑖𝑗(𝑡)𝑤^′_𝑗𝑡−𝛿_𝑖𝑗(𝑡)−𝑒_𝑖(𝑡)𝜙_𝑖𝑤_𝑖(𝑡)−𝑓_𝑖(𝑡)𝜙_𝑖𝑤_𝑖𝑡−𝜎_𝑖(𝑡). (29) Multiply both sides of (29) with exp{∫^𝑡₀𝐴_𝑖𝑖(𝜉)𝑑𝜉}, and then integrate from 0 to 𝑡 to obtain ^𝑡₀𝑤_𝑖(𝑢)exp^𝑢₀𝐴_𝑖𝑖(𝜉)𝑑𝜉^′𝑑𝑢=−^𝑡₀ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑢)𝑤_𝑗(𝑢)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑢)𝑤_𝑗𝑢−𝜏_𝑖𝑗(𝑢)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐶_𝑖𝑗(𝑢)^𝑡_−∞𝐾_𝑖𝑗(𝑢−𝑠)𝑤_𝑗(𝑠)𝑑𝑠+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑢)1−𝛿^′_𝑖𝑗(𝑢)𝑤^′_𝑗𝑢−𝛿_𝑖𝑗(𝑢)+𝑒_𝑖(𝑢)𝜙_𝑖𝑤_𝑖(𝑢)+𝑓_𝑖(𝑢)𝜙_𝑖𝑤_𝑖𝑢−𝜎_𝑖(𝑡)×exp^𝑢₀𝐴_𝑖𝑖(𝜉)𝑑𝜉𝑑𝑢,𝑖=1,2,…,𝑛; (30) then 𝑤_𝑖(𝑡)=𝑤_𝑖(0)exp−^𝑡₀𝐴_𝑖𝑖(𝜉)−𝑑𝜉_𝑡0ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑢)𝑤_𝑗(𝑢)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐵_𝑖𝑗(𝑢)𝑤_𝑗𝑢−𝜏_𝑖𝑗(𝑢)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐶_𝑖𝑗(𝑢)×^𝑡_−∞𝐾_𝑖𝑗(𝑢−𝑠)𝑤_𝑗(𝑠)𝑑𝑠+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑢)1−𝛿^′_𝑖𝑗(𝑢)×𝑤^′_𝑗𝑢−𝛿_𝑖𝑗(𝑢)+𝑒_𝑖(𝑢)𝜙_𝑖𝑤_𝑖(𝑢)+𝑓_𝑖(𝑢)𝜙_𝑖𝑤_𝑖𝑢−𝜎_𝑖(𝑡)×exp−^𝑡_𝑢𝐴_𝑖𝑖(𝜉)𝑑𝜉𝑑𝑢,𝑖=1,2,…,𝑛. (31) Let 𝐷_0𝑖𝑗(𝑡)=𝐷_𝑖𝑗(𝑡)(1−𝛿^′_𝑖𝑗(𝑡)); we see that ^𝑡₀𝐷_0𝑖𝑗(𝑢)𝑤^′_𝑗𝑢−𝛿_𝑖𝑗(𝑢)exp−^𝑡_𝑢𝐴_𝑖𝑖(𝜉)=𝑑𝜉𝑑𝑢_𝑡0𝐷_0𝑖𝑗(𝑢)𝑤^′_𝑗𝑢−𝛿_𝑖𝑗(𝑢)1−𝛿^′_𝑖𝑗(𝑢)1−𝛿_𝑖𝑗(𝑢)×exp−^𝑡_𝑢𝐴_𝑖𝑖(𝜉)=𝑑𝜉𝑑𝑢_𝑡0𝐷_0𝑖𝑗(𝑢)exp−∫^𝑡_𝑢𝐴_𝑖𝑖(𝜉)𝑑𝜉1−𝛿^′_𝑖𝑗(𝑢)×𝑤^′_𝑗𝑢−𝛿_𝑖𝑗(𝑢)1−𝛿^′_𝑖𝑗(𝑢)=𝐷𝑑𝑢_0𝑖𝑗(𝑡)1−𝛿^′_𝑖𝑗(𝑡)𝑤_𝑗𝑡−𝛿_𝑖𝑗(𝑡)−𝐷_0𝑖𝑗(0)1−𝛿^′_𝑖𝑗(0)𝑤_𝑗−𝛿_𝑖𝑗(0)exp−^𝑡₀𝐴_𝑖𝑖(𝜉)−𝑑𝜉_𝑡0𝐴_𝑖𝑖(𝑢)𝐷_𝑖𝑗(𝑢)+𝐷^′_𝑖𝑗(𝑢)×exp−^𝑡_𝑢𝐴_𝑖𝑖(𝜉)𝑑𝜉𝑤_𝑗𝑢−𝛿_𝑖𝑗(𝑢)=𝐷𝑑𝑢_𝑖𝑗(𝑡)𝑤_𝑗𝑡−𝛿_𝑖𝑗(𝑡)−𝐷_𝑖𝑗(0)𝑤_𝑗−𝛿_𝑖𝑗(0)exp−^𝑡₀𝐴_𝑖𝑖(𝜉)−𝑑𝜉_𝑡0𝐴_𝑖𝑖(𝑢)𝐷_𝑖𝑗(𝑢)+𝐷^′_𝑖𝑗(𝑢−)×exp^𝑡_𝑢𝐴_𝑖𝑖(𝜉)𝑑𝜉𝑤_𝑗𝑢−𝛿_𝑖𝑗(𝑢)𝑑𝑢. (32) Substituting (32) into (31), we get 𝑤_𝑖(𝑡)=𝑤_𝑖(0)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(0)𝑤_𝑗−𝛿_𝑖𝑗(0)×exp−^𝑡₀𝐴_𝑖𝑖(𝜉)𝑑𝜉+^𝑡₀ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐴_𝑖𝑖(𝑢)𝐷_𝑖𝑗(𝑢)+𝐷^′_𝑖𝑗(𝑢)𝑤_𝑗𝑢−𝛿_𝑖𝑗(𝑢)−𝐵_𝑖𝑗(𝑢)𝑤_𝑗𝑢−𝜏_𝑖𝑗(𝑢)−𝐶_𝑖𝑗(𝑢)^𝑡_−∞𝐾_𝑖𝑗(𝑢−𝑠)𝑤_𝑗(𝑠)𝑑𝑠−ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑢)𝑤_𝑗(𝑢)−𝑒_𝑖(𝑢)𝜙_𝑖𝑤_𝑖(𝑢)−𝑓_𝑖(𝑢)𝜙_𝑖𝑤_𝑖𝑢−𝜎_𝑖(𝑡)×exp−^𝑡_𝑢𝐴_𝑖𝑖(𝜉)𝑑𝜉𝑑𝑢+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑡)𝑤_𝑗𝑡−𝛿_𝑖𝑗(𝑡); (33) therefore, we have ‖𝑤‖≤𝑤_𝑖(0)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(0)𝑤_𝑗−𝛿_𝑖𝑗(0)×exp−^𝑡₀𝐴_𝑖𝑖(𝜉)+𝑑𝜉_𝑡0ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐴_𝑖𝑖(𝑢)𝐷_𝑖𝑗(𝑢)+𝐷^′_𝑖𝑗(𝑢×𝐵_𝑖𝑗)(𝑢)+𝐶_𝑖𝑗(𝑢)+ℎ_𝑖^𝑛−1_{𝑗=1,𝑗≠𝑖}ℎ_𝑗𝐴_𝑖𝑗(𝑢)+Γ¹_𝑖(𝑢)+Γ²_𝑖(𝑢)×exp−^𝑡_𝑢𝐴_𝑖𝑖(𝜉)𝑑𝜉𝑑𝑢+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑡‖𝑤‖=𝑤_𝑖)(0)+ℎ_𝑖^𝑛−1_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(0)𝑤_𝑗−𝛿_𝑖𝑗(0)×exp−^𝑡_𝑢𝐴_𝑖𝑖(𝜉)+𝑑𝜉_𝑛𝑗=1ℎ_𝑗ℎ_𝑖𝐷_𝑖𝑗(𝑡)+^𝑡₀exp−^𝑡_𝑢𝐴_𝑖𝑖(𝜉)𝑑𝜉Δ_𝑖(𝑢)𝑑𝑢‖𝑤‖, (34) where Δ_𝑖(𝑡) is defined by (18). From (𝐻₄), we have ‖𝑤‖≤𝑤_𝑖(0)+ℎ_𝑖⁻¹∑^𝑛_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(0)𝑤_𝑗−𝛿_𝑖𝑗(0)exp−∫^𝑡₀𝐴_𝑖𝑖(𝜉)𝑑𝜉1−ℎ_𝑖⁻¹∑^𝑛_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(𝑡)−∫^𝑡₀Δ_𝑖(𝑢)exp−∫^𝑡_𝑢𝐴_𝑖𝑖(𝜉)≤𝑤𝑑𝜉𝑑𝑢_𝑖(0)+ℎ_𝑖⁻¹∑^𝑛_𝑗=1ℎ_𝑗𝐷_𝑖𝑗(0)𝑤_𝑗−𝛿_𝑖𝑗(0)exp−∫^𝑡₀𝐴_𝑖𝑖(𝜉)𝑑𝜉.1−𝑀 (35) From (𝐻₅), we have ‖𝑤‖=max_{[]𝑖∈0,𝑛}𝑤_𝑖(𝑡)=max_{[]𝑖∈0,𝑛}𝑥^∗_𝑖(𝑡)−𝑥_𝑖(𝑡)=0,as𝑡⟶+∞; (36) thus, 𝑥_𝑖(𝑡)→𝑥^∗_𝑖(𝑡), as 𝑡→+∞, 𝑖=1,2,…,𝑛. Hence, the positive 𝜔-periodic solution of (17) is globally attractive; accordingly, 𝑁_𝑖(𝑡)=exp{∏_{0<𝑡𝑘<𝑡}ℎ_𝑖(1+𝑝_𝑖𝑘)𝑥_𝑖(𝑡)}→exp{∏_{0<𝑡𝑘<𝑡}ℎ_𝑖(1+𝑝_𝑖𝑘)𝑥^∗_𝑖(𝑡)}=𝑁^∗_𝑖(𝑡), 𝑢_𝑖(𝑡)=(𝜙_𝑖ln𝑁_𝑖)(𝑡)→(𝜙_𝑖ln𝑁^∗_𝑖)(𝑡)=𝑢^∗_𝑖(𝑡) as 𝑡→+∞, 𝑖=1,2,…,𝑛, and by Definition 2, the positive 𝜔-periodic solution of (1) is globally attractive. The proof of Theorem 8 is completed.

Remark 9. If 𝑒_𝑖(𝑡)=𝑓_𝑖(𝑡)=𝛼_𝑖(𝑡)=𝛽_𝑖(𝑡)=𝜗_𝑖(𝑡)=0, 𝜃_𝑖𝑘+1=0, 𝑖=1,2,…,𝑛, 𝑘=1,2,…, then system (1) is studied by [3]. Hence, Theorems 7 and 8 generalize the corresponding results in [3].

Remark 10. If 𝜃_𝑖𝑘+1=0, 𝑖=1,2,…,𝑛, 𝑘=1,2,…, then system (1) is studied by [4]. Hence Theorems 7 and 8 also generalize the corresponding results in [4].

4. Example

Consider the following impulsive model: 𝑑𝑁(𝑡)(𝑡𝑟(𝑡)(𝑡)(𝑡)(𝑡)((𝑡(𝑡)𝑑𝑡=𝑁)−𝑎ln𝑁−𝑏ln𝑁𝑡−𝜏))−𝑐^𝑡_−∞𝐾()(𝑠)(𝑡)((𝑡𝑡−𝑠ln𝑁𝑑𝑠−𝑑𝑑ln𝑁𝑡−𝛿))(𝑡)𝑢(𝑡)(𝑡)𝑢((𝑡𝑑𝑡−𝑒−𝑓𝑡−𝜎)),𝑡≠𝑡_𝑘,𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)(𝑡)(𝑡)(𝑡)(𝑡)𝑁𝑡𝑑𝑡=−𝛼+𝛽ln𝑁+𝜃ln𝑁𝑡−𝛾,𝑡≥0,_+𝑘=𝑒^(1+𝑝𝑘)𝑁𝑡_𝑘,𝑘=1,2,…, (37) where 𝑟(𝑡), 𝑎(𝑡), 𝑏(𝑡), 𝑐(𝑡), 𝑑(𝑡), 𝑒(𝑡), 𝑓(𝑡), 𝛿(𝑡)∈𝐶²(𝑅,𝑅), 𝜎(𝑡), 𝛾(𝑡), 𝛼(𝑡), 𝛽(𝑡), 𝜃(𝑡) are all nonnegative 𝜔-periodic continuous functions with ∫^𝜔₀𝑟(𝑡)>0, 𝑎(𝑡)>0, 𝛿^′(𝑡)<1 and 𝑝_𝑘 is a real sequence, and ∏_{0<𝑡𝑘<𝑡}(1+𝑝_𝑘) is a positive 𝜔-periodic function with 𝑘=1,2,…. Furthermore, ∫^∞₀𝐾(𝑠)𝑑𝑠=1, ∫₀^+∞𝑠𝐾(𝑠)𝑑𝑠<+∞.

We denote 𝐴(𝑡)=𝑎(𝑡)_{0<𝑡𝑘<𝑡}1+𝑝_𝑘,𝐵(𝑡)=𝑏(𝑡)_{0<𝑡𝑘(𝑡)<𝑡−𝜏}1+𝑝_𝑘,𝐶(𝑡)=𝑐(𝑡)_{0<𝑡𝑘<𝑡}1+𝑝_𝑘,𝐷(𝑡)=𝑑(𝑡)_{0<𝑡𝑘<𝑡−𝛿(𝑡))}1+𝑝_𝑘,𝐷₀(𝑡)=𝐷(𝑡)1−𝛿^′(𝑡),𝛽^∗(𝑡)=𝛽(𝑡)_{0<𝑡𝑘<𝑡}1+𝑝_𝑘𝜃^∗(𝑡)=𝜃(𝑡)_{0<𝑡𝑘(𝑡)<𝑡−𝛾}1+𝑝_𝑘,Γ¹(𝑡)(𝑡)(𝑡),Γ∶=𝑒𝜙1₂(𝑡)(𝑡)(𝑡,Δ(𝑡)(𝑡)(𝑡)(𝑡)𝐷(𝑡)+𝐷∶=𝑓𝜙1(𝑡−𝜎))∶=𝐵+𝐶+𝐴^′(𝑡)+Γ¹(𝑡)+Γ²(𝑡). (38) Similar to Theorems 7 and 8, we can get the following results.

Corollary 11. In addition to conditions (𝐻₁)--(𝐻₃), assume further that there exists a positive constant 𝑀<1 such that(𝐻₆)𝐷(𝑡)+∫^𝑡_−∞exp{−∫^𝑡_𝑠𝐴(𝜉)𝑑𝜉}Δ(𝑠)𝑑𝑠≤𝑀. Then, (37) has a unique 𝜔-periodic solution with strictly positive components, where Δ(𝑡) is defined by (38).

Corollary 12. In addition to conditions (𝐻₁)--(𝐻₃) and (𝐻₆), suppose further that the following condition holds:(𝐻₇)exp{−∫^𝑡₀𝐴(𝜉)𝑑𝜉}→0, as 𝑡→+∞.Then, system (37) has a unique periodic solution which is globally attractive.

Remark 13. The results in the work show that by means of appropriate impulsive perturbations and feedback control we can control the dynamics of these equations.

Acknowledgments

The authors are thankful to the referees and editor for the improvement of the paper. This work was supported by the Construct Program of the Key Discipline in Hunan Province, NSF of China (nos. 10971229, 11161015, and 11371367), PSF of China (no. 2012M512162), NSF of Hunan Province (nos. 11JJ900, 12JJ9001, and 13JJ4098), and the Education Foundation of Hunan province (nos. 12C0541 and 13C084).