Dynamics of a Nonautonomous Leslie-Gower Type Food Chain Model with Delays

Abstract

A nonautonomous Leslie-Gower type food chain model with time delays is investigated. It is proved the general nonautonomous system is permanent and globally asymptotically stable under some appropriate conditions. Furthermore, if the system is periodic one, some sufficient conditions are established, which guarantee the existence, uniqueness, and global asymptotic stability of a positive periodic solution of the system. The conditions for the permanence, global stability of system, and the existence, uniqueness of positive periodic solution depend on delays; so, time delays are profitless.

1. Introduction

Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The dynamic relationship between predators and their preys has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance [1]. Food-chain predator-prey system, as one of the most important predator-prey system, has been extensively studied by many scholars, many excellent results concerned with the permanent property and positive periodic solution of the system, see [2–4] and the references cited therein. Recently, Nindjin and Aziz-Alaoui [5] proposed the following autonomous delayed predator-prey model with modified Leslie-Gower functional response 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑎₁−𝑎₁₁𝑥₁𝑡−𝜏₁₁−𝑎₁₂𝑥₂(𝑡)𝑥₁(𝑡)+𝑑₁,𝑥̇₂(𝑡)=𝑥₂(𝑡)−𝑎₂+𝑎₂₁𝑥₁𝑡−𝜏₂₁𝑥₁𝑡−𝜏₂₁+𝑑₁−𝑎₂₂𝑥₂𝑡−𝜏₂₂−𝑎₂₃𝑥₃(𝑡)𝑥₂(𝑡)+𝑑₂,𝑥̇₃(𝑡)=𝑥₃(𝑡)𝑎₃−𝑎₃₃𝑥₃𝑡−𝜏₃₃𝑥₂𝑡−𝜏₃₂+𝑑₃, (1) system (1) represents an ecological situation where a prey population 𝑥₁ is the only food for a predator 𝑥₂. This specialist predator 𝑥₂, in turns, serves as the prey of a top-predator 𝑥₃. The interaction between species 𝑥₂ and its prey 𝑥₁ has been modeled by the Volterra scheme. But, the interaction between species 𝑥₃ and its prey 𝑥₂ has been modeled by a modified version of Leslie-Gower scheme. About Leslie-Gower scheme one could refer to [6–13] and the references cited therein.

In [5], the authors showed that the system is uniformly persistent under some appropriate conditions and obtained sufficient conditions for global stability of the positive equilibrium of system (1).

We note that any biological or environment parameters are naturally subject to fluctuation in time, and if a model is to take into account such fluctuation then the model must be nonautonomous. On the other hand, time delays occur so often in nature, a number of models can be formulated as systems of differential equations with time delays (see, e.g., [2, 14, 15] and the references cited therein). Motivated by above considerations, in this paper, we consider the following general nonautonomous Leslie-Gower type food chain model with time delays of the form 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑎₁(𝑡)−𝑎₁₁(𝑡)𝑥₁𝑡−𝜏₁₁(𝑡)−𝑎₁₂(𝑡)𝑥₂(𝑡)𝑥₁(𝑡)+𝑑₁(𝑡),𝑥̇₂(𝑡)=𝑥₂(𝑡)−𝑎₂(𝑡)+𝑎₂₁(𝑡)𝑥₁𝑡−𝜏₂₁(𝑡)𝑥₁𝑡−𝜏₂₁(𝑡)+𝑑₁(𝑡)−𝑎₂₂(𝑡)𝑥₂𝑡−𝜏₂₂(𝑡−𝑎)²³(𝑡)𝑥₃(𝑡)𝑥₂(𝑡)+𝑑₂(𝑡),𝑥̇₃(𝑡)=𝑥₃(𝑡)𝑎₃(𝑡)−𝑎₃₃(𝑡)𝑥₃𝑡−𝜏₃₃(𝑡)𝑥₂𝑡−𝜏₃₂(𝑡)+𝑑₃(𝑡), (1.2) where 𝜏_𝑖𝑖(𝑡), 𝑖=1,2,3 denote the time delays due to negative feedbacks of the prey, specialist predator and top-predator, respectively. 𝜏₂₁(𝑡) is a time delay due to gestation, that is, mature adult predators can only contribute to the reproduction of predator biomass. 𝜏₃₂(𝑡) can be regarded as a gestation period.

In this paper, for system (1.2) we always assume that for all 𝑖,𝑗=1,2,3(𝐻₁)𝑎_𝑖(𝑡),𝑎_𝑖𝑗(𝑡),𝑑_𝑖(𝑡) are continuous and bounded above and below by positive constants on [0,+∞), and (𝐻₂)𝜏_𝑖𝑗(𝑡) are continuous and differentiable bounded functions on [0,+∞), and 𝜏̇_𝑖𝑗(𝑡) is uniformly continuous with respect to 𝑡 on [0,+∞) and inf_{𝑡∈[0,+∞)}{1−𝜏̇_𝑖𝑗(𝑡)}>0.

Let 𝜏=sup{𝜏_𝑖𝑗(𝑡)∶𝑡∈[0,+∞),𝑖,𝑗=1,2,3}, then we have 0≤𝜏<+∞. Let 𝜎_𝑖𝑗(𝑡)=𝑡−𝜏_𝑖𝑗(𝑡), then the function 𝜎⁻¹_𝑖𝑗(𝑡) is the inverse function of the function 𝜎_𝑖𝑗(𝑡), 𝑖,𝑗=1,2,3. Motivated by the application of system (1.2) to population dynamics, we assume that solutions of system (1.2) satisfies the following initial conditions 𝑥_𝑖(𝜃)=𝜙_𝑖(𝜃)≥0,𝜃∈[−𝜏,0],𝜙_𝑖(0)𝜙>0,_𝑖∈𝐶[−𝜏,0],𝑅₊,𝑖=1,2,3. (1.3)

It is well known that by the fundamental theory of functional differential equation [16], one can prove the solution of system (1.2) with initial conditions (1.3) exists and remains positive for 𝑡≥0, we call such a solution the positive solution of system (1.2).

The organization of this paper is as follows. In Section 2, by using comparison theorem and further developing the analytical technique of [2, 14], we obtain a set of sufficient conditions, which ensure the permanence of the system (1.2). In Section 3, by constructing a suitable Lyapunov function, we establish a set of sufficient conditions, which ensure the global stability of the system (1.2). In Section 4, we will explore the existence and stability of the solutions of the periodic system (1.2). At last, the conclusion ends with brief remarks.

2. Permanence

In this section, we establish a permanent result for system (1.2).

Here, for any bounded function {𝑓(𝑡)}𝑓^𝑢=lim_𝑡→+∞sup𝑓(𝑡),𝑓^𝑙=lim_𝑡→+∞inf𝑓(𝑡). (2.1)

Definition 2.1. System (1.2) is said to be permanent, if there are positive constants 𝑚_𝑖 and 𝑀_𝑖, such that each positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)) of system (1.2) satisfies 𝑚_𝑖≤lim_𝑡→+∞inf𝑥_𝑖(𝑡)≤lim_𝑡→+∞sup𝑥_𝑖(𝑡)≤𝑀_𝑖,𝑖=1,2,3. (2.2)

Theorem 2.2. Assume that (𝐻₁) and (𝐻₂) hold assume further that(𝐻₃)𝑎^𝑙₁>𝑎^𝑢₁₂𝑀₂/𝑑^𝑙₁, 𝑎^𝑙₂₁𝑚₁/(𝑚₁+𝑑^𝑢₁)>𝑎^𝑢₂+𝑎^𝑢₂₃𝑀₃/𝑑^𝑙₂hold. Then system (1.2) is permanent.

Proof. From the first equation of the system (1.2) it follows that 𝑥̇₁(𝑡)≤𝑥₁(𝑡)𝑎₁(𝑡)−𝑎₁₁(𝑡)𝑥₁𝑡−𝜏₁₁(𝑡)≤𝑥₁(𝑡)𝑎^𝑢₁−𝑎^𝑙₁₁𝑥₁𝑡−𝜏₁₁(𝑡). (2.3)
Let 𝜏=sup{𝜏_𝑖𝑗(𝑡)∶𝑡∈[0,+∞),𝑖,𝑗=1,2,3}, by (𝐻₂), then we have 0≤𝜏<+∞. Taking 𝑀₁=(𝑎^𝑢₁/𝑎^𝑙₁₁)(1+ℎ₁), where 0<ℎ₁<exp{𝑎^𝑢₁𝜏}−1. Firstly, suppose 𝑥₁(𝑡) is not oscillatory about 𝑀₁. That is, there exists a 𝑇₁>0, for 𝑡>𝑇₁ such that𝑥₁(𝑡)<𝑀₁, (2.4) or 𝑥₁(𝑡)>𝑀₁. (2.5)
If (2.4) holds, then our aim is obtained. Suppose (2.5) holds, then for 𝑡≥𝑇₁+𝜏, we obtain𝑥̇₁(𝑡)<−ℎ₁𝑎^𝑢₁𝑥₁(𝑡), (2.6) thus 𝑥₁(𝑡)<𝑥₁(0)exp[−ℎ₁𝑎^𝑢₁𝑡]→0, as 𝑡→+∞, which is contradiction with (2.5). Hence there must exist 𝑇^∗₁>𝑇₁+𝜏 such that 𝑥₁(𝑡)<𝑀₁ for 𝑡>𝑇^∗₁. Secondly now assume that 𝑥₁(𝑡) is oscillatory about 𝑀₁ for 𝑡≥𝑇₁, that is, there exists a time sequence {𝑡_𝑛} such that 𝜏<𝑡₁<𝑡₂<⋯<𝑡_𝑛<⋯ is a sequence of zeros of 𝑥₁(𝑡_𝑛)−𝑀₁ with lim_𝑛→∞𝑡_𝑛=+∞ and 𝑥₁(𝑡_𝑛)=𝑀₁. Set 𝑡_𝑛̃ a point where 𝑥₁(𝑡) attends its maximum in (𝑡_𝑛,𝑡_𝑛+1). Thus we get 𝑥₁(𝑡_𝑛̃)≥𝑥₁(𝑡_𝑛)=𝑀₁. Then it follows from (2.3) that 0=𝑥̇₁(𝑡)_𝑡=𝑡_𝑛̃≤𝑥₁𝑡_𝑛̃𝑎^𝑢₁−𝑎^𝑙₁₁𝑥₁𝑡_𝑛̃−𝜏₁₁𝑡_𝑛̃, (2.7) which leads to 𝑥₁𝑡_𝑛̃−𝜏₁₁𝑡_𝑛̃≤𝑎^𝑢₁𝑎^𝑙₁₁. (2.8) Integrating the both sides of (2.3) from 𝑡_𝑛̃−𝜏₁₁(𝑡_𝑛̃) to 𝑡_𝑛̃, it follows that 𝑥ln¹𝑡_𝑛̃𝑥₁𝑡_𝑛̃−𝜏₁₁𝑡_𝑛̃≤𝑡_𝑛̃𝑡̃_{𝑛−𝜏11(}𝑡̃_𝑛)𝑎^𝑢₁−𝑎^𝑙₁₁𝑥₁𝑡−𝜏₁₁(𝑡)𝑑𝑡≤𝑎^𝑢₁𝜏₁₁𝑡_𝑛̃. (2.9) From (2.8) and (2.9) we have 𝑥₁𝑡_𝑛̃≤𝑎^𝑢₁𝑎^𝑙₁₁exp𝑎^𝑢₁𝜏. (2.10)
Since 𝑥₁(𝑡_𝑛̃) is an arbitrary local maximum of 𝑥₁(𝑡), we can see that there exists a 𝑇₂>𝑇₁ such that for all 𝑡≥𝑇₂𝑥₁(𝑡)≤𝑎^𝑢₁𝑎^𝑙₁₁exp𝑎^𝑢₁𝜏∶=𝑀₁. (2.11) Thus lim_𝑡→+∞sup𝑥₁(𝑡)≤𝑀₁. (2.12) For 𝑡≥𝑇₂+𝜏, from (2.11) and the second equation of the system (1.2) it follows that 𝑥̇₂(𝑡)≤𝑥₂(𝑡)−𝑎^𝑙₂+𝑎^𝑢₂₁𝑀₁𝑀₁+𝑑^𝑙₁−𝑎^𝑙₂₂𝑥₂𝑡−𝜏₂₂(𝑡). (2.13) Similar to the argument above, from (2.13) we obtain lim_𝑡→+∞sup𝑥₂(𝑡)≤−𝑎^𝑙₂+𝑎^𝑢₂₁𝑀₁/𝑀₁+𝑑^𝑙₁𝑎^𝑙₂₂exp−𝑎^𝑙₂+𝑎^𝑢₂₁𝑀₁𝑀₁+𝑑^𝑙₁𝜏∶=𝑀₂. (2.14) Similarly, from the third equation of the system (1.2), we have 𝑥̇₃(𝑡)≤𝑥₃(𝑡)𝑎^𝑢₃−𝑎^𝑙₃₃𝑥₃𝑡−𝜏₃₃(𝑡)𝑀₂+𝑑^𝑢₃, (2.15) and so lim_𝑡→+∞sup𝑥₃(𝑡)≤𝑎^𝑢₃𝑀₂+𝑑^𝑢₃𝑎^𝑙₃₃exp𝑎^𝑢₃𝜏∶=𝑀₃. (2.16) Condition (𝐻₃) of Theorem 2.2 also implies that we could choose 𝜀>0 small enough such that 𝑎^𝑙₁>𝑎^𝑢₁₂𝑀₂+𝜀𝑑^𝑙₁,𝑎^𝑙₂₁𝑚₁−𝜀𝑚₁−𝜀+𝑑^𝑢₁>𝑎^𝑢₂+𝑎^𝑢₂₃𝑀₃+𝜀𝑑^𝑙₂, (2.17) hence, for 𝜀>0 satisfies (2.17), from (2.12), (2.14), and (2.16), we know that there exists 𝑇₃>𝑇₂+𝜏 such that for 𝑖=1,2,3 and 𝑡≥𝑇₃𝑥_𝑖(𝑡)≤𝑀_𝑖+𝜀. (2.18) From the first equation and (2.18) it follows that for 𝑡≥𝑇₃+𝜏, 𝑥̇₁(𝑡)=𝑥₁(𝑡)𝑎₁(𝑡)−𝑎₁₁(𝑡)𝑥₁𝑡−𝜏₁₁(𝑡)−𝑎₁₂(𝑡)𝑥₂(𝑡)𝑥₁(𝑡)+𝑑₁(𝑡)≥𝑥₁(𝑡)𝑎^𝑙₁−𝑎^𝑢₁₁𝑀₁−𝑎+𝜀^𝑢12𝑀₂+𝜀𝑑^𝑙₁. (2.19) Note that 𝑎^𝑢₁/𝑎^𝑙₁₁≤𝑀₁ implies that 𝑎^𝑙₁−𝑎^𝑢₁₁𝑀₁−𝑎+𝜀^𝑢12𝑀₂+𝜀𝑑^𝑙₁≤𝑎^𝑙₁−𝑎^𝑢₁₁𝑀₁+𝜀≤𝑎^𝑢₁−𝑎^𝑙₁₁𝑀₁+𝜀≤0. (2.20) Now we consider the following two cases.Case 1. Suppose 𝑎^𝑙₁−𝑎^𝑢₁₁(𝑀₁+𝜀)−𝑎^𝑢₁₂(𝑀₂+𝜀)/𝑑^𝑙₁=0, then for 𝑡≥𝑇₃+𝜏, from the positivity of the solution and (2.19) it follows that 𝑥̇₁(𝑡)=0. (2.21) Then from (2.12) and (2.21) it follows that lim_𝑡→+∞𝑥₁(𝑡)=∶𝑝₁<𝑎^𝑙₁/𝑎^𝑢₁₁, then there exists 𝑇₄>𝑇₃+𝜏 such that for 𝑡≥𝑇₄𝑥₁(𝑡)<𝑝₁+𝑎^𝑙₁/𝑎^𝑢₁₁−𝑝₁2<𝑎^𝑙₁𝑎^𝑢₁₁<𝑀₁. (2.22) From the positivity of the solution, (2.19) and (2.22) it follows that 𝑥̇₁(𝑡)≥𝑥₁(𝑡)𝑎^𝑙₁−𝑎^𝑢₁₁𝑎^𝑙₁/𝑎^𝑢₁₁+𝑝₁2−𝑎^𝑢₁₂𝑀₂+𝜀𝑑^𝑙₁>𝑥₁(𝑡)𝑎^𝑙₁−𝑎^𝑢₁₁𝑀₁−𝑎+𝜀^𝑢12𝑀₂+𝜀𝑑^𝑙₁=0,𝑡≥𝑇₄𝑥+𝜏,₁(𝑡)≥𝑥₁𝑇₄+𝜏exp𝑎^𝑙₁−𝑎^𝑢₁₁𝑎^𝑙₁/𝑎^𝑢₁₁+𝑝₁2−𝑎^𝑢₁₂𝑀₂+𝜀𝑑^𝑙₁𝑡−𝑇₄+𝜏, (2.23) then we can see that 𝑥₁(𝑡)→+∞ as 𝑡→+∞, which is contradiction with (2.12). Hence we have lim_𝑡→+∞𝑥₁(𝑡)≥𝑎^𝑙₁/𝑎^𝑢₁₁, which implies that there exists 𝑇^∗₄>𝑇₄+𝜏 such that 𝑥₁(𝑡)≥𝑎^𝑙₁/2𝑎^𝑢₁₁ for 𝑡≥𝑇^∗₄.Case 2. Suppose 𝑎^𝑙₁−𝑎^𝑢₁₁(𝑀₁+𝜀)−𝑎^𝑢₁₂(𝑀₂+𝜀)/𝑑^𝑙₁<0, from (2.19), for 𝑡≥𝑇₃+𝜏, it follows that 𝑥̇₁(𝑡)≥𝑥₁(𝑡)𝑎^𝑙₁−𝑎^𝑢₁₂𝑀₂+𝜀𝑑^𝑙₁−𝑎^𝑢₁₁𝑥₁𝑡−𝜏₁₁(𝑡). (2.24)
Let𝑚₁=𝑎^𝑙₁−𝑎^𝑢₁₂𝑀₂+𝜀/𝑑^𝑙₁𝑎^𝑢₁₁1−𝜎₁, (2.25) where 0<𝜎₁<1−exp{(𝑎^𝑙₁−𝑎^𝑢₁₁(𝑀₁+𝜀)−𝑎^𝑢₁₂(𝑀₂+𝜀)/𝑑^𝑙₁)𝜏}.Firstly, suppose 𝑥₁(𝑡) is not oscillatory about 𝑚₁. That is, there exists a 𝑇₅>0, for 𝑡>𝑇₅ such that 𝑥₁(𝑡)>𝑚₁, (2.26) or 𝑥₁(𝑡)<𝑚₁. (2.27)
If (2.26) holds, then our aim is obtained. Suppose (2.27) holds, then for 𝑡≥𝑇₅+𝜏, we obtain𝑥̇₁(𝑡)≥𝜎₁𝑎^𝑙₁−𝑎^𝑢₁₂𝑀₂+𝜀𝑑^𝑙₁𝑥₁(𝑡). (2.28) thus there must exist 𝑇^∗₅>𝑇₅+𝜏 such that 𝑥₁(𝑡)>𝑚₁ for 𝑡>𝑇^∗₅, which is a contradiction. Hence, (2.27) could not hold. Secondly now assume that 𝑥₁(𝑡) is oscillatory about 𝑚₁ for 𝑡≥𝑇₃+𝜏, that is, there exists a time sequence {𝑡_𝑛} such that 𝜏<𝑡₁<𝑡₂<⋯<𝑡_𝑛<⋯ is a sequence of zeros of 𝑥₁(𝑡_𝑛)−𝑚₁ with lim_𝑛→∞𝑡_𝑛=+∞ and 𝑥₁(𝑡_𝑛)=𝑚₁. Set 𝑡_𝑛̂ be a point where 𝑥₁(𝑡) attends its minimum in (𝑡_𝑛,𝑡_𝑛+1). Thus, we get 𝑥₁(𝑡_𝑛̂)≤𝑥₁(𝑡_𝑛)=𝑚₁. Then it follows from (2.24) that 0=𝑥̇₁(𝑡)_𝑡=𝑡_𝑛̂≥𝑥₁𝑡_𝑛̂𝑎^𝑙₁−𝑎^𝑢₁₂𝑀₂+𝜀𝑑^𝑙₁−𝑎^𝑢₁₁𝑥₁𝑡_𝑛̂−𝜏₁₁𝑡_𝑛̂, (2.29) which implies that 𝑥₁𝑡_𝑛̂−𝜏₁₁𝑡_𝑛̂≥𝑎^𝑙₁−𝑎^𝑢₁₂𝑀₂+𝜀/𝑑^𝑙₁𝑎^𝑢₁₁. (2.30) Integrating (2.24) on the interval [𝑡_𝑛̂−𝜏₁₁(𝑡_𝑛̂),𝑡_𝑛̂], we have 𝑥ln¹𝑡_𝑛̂𝑥₁𝑡_𝑛̂−𝜏₁₁𝑡_𝑛̂≥𝑡_𝑛̂𝑡̂_{𝑛−𝜏11(}𝑡̂_𝑛)𝑎^𝑙₁−𝑎^𝑢₁₂𝑀₂+𝜀𝑑^𝑙₁−𝑎^𝑢₁₁𝑥₁𝑡−𝜏₁₁(𝑡)≥𝑡̂𝑑𝑡_𝑛𝑡̂_{𝑛−𝜏11(}𝑡̂_𝑛)𝑎^𝑙₁−𝑎^𝑢₁₂𝑀₂+𝜀𝑑^𝑙₁−𝑎^𝑢₁₁𝑀₁=𝑎+𝜀𝑑𝑡_𝑙1−𝑎^𝑢₁₁𝑀₁−𝑎+𝜀^𝑢12𝑀₂+𝜀𝑑^𝑙₁𝜏₁₁𝑡_𝑛̂. (2.31) From (2.30) and (2.31), we get that 𝑥₁𝑡_𝑛̂≥𝑎^𝑙₁−𝑎^𝑢₁₂𝑀₂+𝜀/𝑑^𝑙₁𝑎^𝑢₁₁exp𝑎^𝑙₁−𝑎^𝑢₁₁𝑀₁−𝑎+𝜀^𝑢12𝑀₂+𝜀𝑑^𝑙₁𝜏. (2.32)
Since 𝑥₁(𝑡_𝑛̂) is an arbitrary local minimum of 𝑥₁(𝑡), we can have that there exists a 𝑞₁≤𝑚₁ such that for all 𝑡≥𝑇^∗₆𝑥₁(𝑡)≥𝑞₁, (2.33) where 0<𝑞₁≤𝑎^𝑙₁−𝑎^𝑢₁₂𝑀₂/𝑑^𝑙₁𝑎^𝑢₁₁exp𝑎^𝑙₁−𝑎^𝑢₁₁𝑀₁−𝑎^𝑢₁₂𝑀₂𝑑^𝑙₁𝜏. (2.34) Taken 𝑚₁=min{𝑎^𝑙₁/2𝑎^𝑢₁₁,𝑞₁}, thus, we have lim_𝑡→+∞inf𝑥₁(𝑡)≥𝑚₁. (2.35) Thus, for 𝜀>0 satisfies (2.17), it follows from (2.35) that there exists large enough 𝑇₆ such that for 𝑡≥𝑇₆𝑥₁(𝑡)≥𝑚₁−𝜀. (2.36)
For 𝑡≥𝑇₆+𝜏, by using (2.36), from the second equation of system (1.2) it follows that𝑥̇₂(𝑡)=𝑥₂(𝑡)−𝑎₂(𝑡)+𝑎₂₁(𝑡)𝑥₁𝑡−𝜏₂₁(𝑡)𝑥₁𝑡−𝜏₂₁(𝑡)+𝑑₁(𝑡)−𝑎₂₂(𝑡)𝑥₂𝑡−𝜏₂₂(𝑡)−𝑎₂₃(𝑡)𝑥₃(𝑡)𝑥₂(𝑡)+𝑑₂(𝑡)≥𝑥₂(𝑡)−𝑎^𝑢₂+𝑎^𝑙₂₁𝑚₁−𝜀𝑚₁−𝜀+𝑑^𝑢₁−𝑎^𝑢₂₂𝑀₂−𝑎+𝜀^𝑢23𝑀₃+𝜀𝑑^𝑙₂,𝑥̇₂(𝑡)≥𝑥₂(𝑡)−𝑎^𝑢₂+𝑎^𝑙₂₁𝑚₁−𝜀𝑚₁−𝜀+𝑑^𝑢₁−𝑎^𝑢₂₃𝑀₃+𝜀𝑑^𝑙₂−𝑎₂₂(𝑡)𝑥₂𝑡−𝜏₂₂(𝑡); (2.37) from (2.37), by a procedure similar to the discussion above, we can verify that lim_𝑡→+∞inf𝑥₂(𝑡)≥𝑚₂, (2.38) where 𝑚₂=min−𝑎^𝑢₂+𝑎^𝑙₂₁𝑚₁/𝑚₁+𝑑^𝑢₁2𝑎^𝑢₂₂,𝑞₂,0<𝑞₂≤−𝑎^𝑢₂+𝑎^𝑙₂₁𝑚₁/𝑚₁+𝑑^𝑢₁−𝑎^𝑢₂₃𝑀₃/𝑑^𝑙₂𝑎^𝑢₂₂exp−𝑎^𝑢₂+𝑎^𝑙₂₁𝑚₁𝑚₁+𝑑^𝑢₁−𝑎^𝑢₂₂𝑀₂−𝑎^𝑢₂₃𝑀₃𝑑^𝑙₂𝜏. (2.39) From (2.38), we see that there exists large enough 𝑇₇ such that for 𝑡≥𝑇₇𝑥₂(𝑡)≥𝑚₂−𝜀. (2.40) Substituting (2.18) to the last equation of system (1.2), it follows that 𝑥̇₃(𝑡)≥𝑥₃(𝑡)𝑎^𝑙₃−𝑎^𝑢₃₃𝑀₃+𝜀𝑑^𝑙₃,𝑥̇₃(𝑡)≥𝑥₃(𝑡)𝑎^𝑙₃−𝑎^𝑢₃₃𝑥₃𝑡−𝜏₃₃(𝑡)𝑑^𝑙₃; (2.41) from (2.41), similar to the argument of (2.35), we also have lim_𝑡→+∞inf𝑥₃(𝑡)≥𝑚₃, (2.42) where 𝑚₃𝑎=min^𝑙3𝑑^𝑙₃2𝑎^𝑢₃₃,𝑞₃,0<𝑞₃≤𝑎^𝑙₃𝑑^𝑙₃𝑎^𝑢₃₃exp𝑎^𝑙₃−𝑎^𝑢₃₃𝑀₃𝑑^𝑙₃𝜏. (2.43) Consequently, (2.12), (2.14), (2.16), (2.35), (2.38), and (2.42) show that under the assumption (𝐻₁)-(𝐻₃), for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)) of system (1.2), one has 𝑚_𝑖≤lim_𝑡→+∞inf𝑥_𝑖(𝑡)≤lim_𝑡→+∞sup𝑥_𝑖(𝑡)≤𝑀_𝑖,𝑖=1,2,3, (2.44) where 𝑚_𝑖 and 𝑀_𝑖, 𝑖=1,2,3 are independent of the solution of system (1.2), thus system (1.2) is permanent. This completes the proof of Theorem 2.2.

3. Global Stability

Now we study the global stability of the positive solution of system (1.2). We say a positive solution of system (1.2) is globally asymptotically stable if it attracts all other positive solution of the system.

The following lemma is from [17], and will be employed in establishing the global stability of positive solution of system (1.2).

Lemma 3.1. Let ℎ be a real number and 𝑓 be a nonnegative function defined on [ℎ;+∞) such that 𝑓 is integrable on [ℎ;+∞) and is uniformly continuous on [ℎ;+∞), then lim_𝑡→+∞𝑓(𝑡)=0.

Theorem 3.2. In addition to (𝐻₁)-(𝐻₃), assume further that(𝐻₄)there exist constants 𝜆_𝑖>0 such that liminf_𝑡→+∞𝐴_𝑖(𝑡)>0,𝑖=1,2,3, (3.1) where 𝐴₁(𝑡)=𝜆₁𝑎₁₁(𝑡)−𝑎₁₂(𝑡)𝑀₂𝑑²₁(𝑡)−𝑎₁₁𝜎⁻¹₁₁(𝑡)𝑀₁1−𝜏̇₁₁𝜎⁻¹₁₁(𝑡)^{𝜎−111(𝜎−111}_𝜎^(𝑡))−1_11(𝑡)𝑎₁₁(𝑠)−𝑎𝑑𝑠₁(𝑡)+𝑎₁₁(𝑡)𝑀₁+𝑎₁₂(𝑡)𝑀₂𝑑₁(𝑡)^𝜎−111_𝑡^(𝑡)𝑎₁₁(𝑠)𝑑𝑠−𝜆₂𝑎₂₁𝜎⁻¹₂₁(𝑡)𝑑₁𝜎⁻¹₂₁(𝑡)1−𝜏̇₂₁𝜎⁻¹₂₁(𝑡)+𝑎₂₁𝜎⁻¹₂₁(𝑡)𝑀₂𝑑₁𝜎⁻¹₂₁(𝑡)1−𝜏̇₂₁𝜎⁻¹₂₁(𝑡)^{𝜎−122(𝜎−121}_𝜎^(𝑡))−1_21(𝑡)𝑎₂₂(𝑠),𝐴𝑑𝑠₂(𝑡)=𝜆₂𝑎₂₂(𝑡)−𝑎₂₃(𝑡)𝑀₃𝑑²₂(𝑡)−𝑎₂₂𝜎⁻¹₂₂(𝑡)𝑀₂1−𝜏̇₂₂𝜎⁻¹₂₂(𝑡)^{𝜎−122(𝜎−122}_𝜎^(𝑡))−1_22(𝑡)𝑎₂₂(𝑠)−𝑎𝑑𝑠₂(𝑡)+𝑎₂₂(𝑡)𝑀₂+𝑎₂₁(𝑡)+𝑎₂₃(𝑡)𝑑₂(𝑡)𝑀₃^𝜎−122_𝑡^(𝑡)𝑎₂₂(𝑠)𝑑𝑠−𝜆₁𝑎₁₂(𝑡1)𝑑₁(𝑡)+^𝜎−111_𝑡^(𝑡)𝑎₁₁(𝑠)𝑑𝑠−𝜆₃𝑎₃₃𝜎⁻¹₃₂(𝑡)𝑀₃𝑑²₃𝜎⁻¹₃₂(𝑡)1−𝜏̇₃₂𝜎⁻¹₃₂(𝑡)+𝑎₃₃𝜎⁻¹₃₂(𝑡)𝑀²₃𝑑²₃𝜎⁻¹₃₂(𝑡)1−𝜏̇₃₂𝜎⁻¹₃₂(𝑡)^{𝜎−133(𝜎−132𝜎(𝑡))−1}₃₂^(𝑡)𝑎₃₃(𝑠)𝑑₃(𝑠),𝐴𝑑𝑠₃(𝑡)=𝜆₃𝑎₃₃(𝑡)𝑀₂+𝑑₃(𝑡)−𝑎₃(𝑡)+𝑎₃₃(𝑡)𝑀₃𝑑₃(𝑡)^𝜎−133_𝑡^(𝑡)𝑎₃₃(𝑠)𝑑₃(𝑠)−𝑎𝑑𝑠₃₃𝜎⁻¹₃₃(𝑡)𝑀₃𝑑₃𝜎⁻¹₃₃(𝑡)1−𝜏̇₃₃𝜎⁻¹₃₃(𝑡)^{𝜎−133(𝜎−133}_𝜎^(𝑡))−1_33(𝑡)𝑎₃₃(𝑠)𝑑₃(𝑠)𝑑𝑠−𝜆₂𝑎₂₃(𝑡1)𝑑₂(𝑡)+^𝜎−122_𝑡^(𝑡)𝑎₂₂(𝑠)𝑑𝑠. (3.2) Then for any positive solutions (𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)) and (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),𝑥^∗₃(𝑡)) of system (1.2), one has lim_𝑡→+∞𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)=0,𝑖=1,2,3. (3.3)

Proof. For two arbitrary nontrivial solutions (𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)) and (𝑥^∗₁(𝑡),𝑥^∗₂(𝑡),𝑥^∗₃(𝑡)) of system (1.2), we have from Theorem 2.2 that there exist positive constants 𝑇>𝑇₇ and 𝑀_𝑖, 𝑚_𝑖(𝑖=1,2,3) such that for all 𝑡≥𝑇 and 𝑖=1,2,3𝑚_𝑖≤𝑥_𝑖(𝑡)≤𝑀_𝑖. (3.4) We define 𝑉₁₁(𝑡)=ln𝑥₁(𝑡)−ln𝑥^∗₁(𝑡). (3.5) Calculating the upper right derivative of 𝑉₁₁(𝑡) along the solution of system (1.2), for 𝑡≥𝑇+𝜏, it follow that 𝐷⁺𝑉₁₁(𝑡)=𝑥̇₁(𝑡)𝑥₁(𝑡)−𝑥̇^∗₁(𝑡)𝑥^∗₁(𝑡)sgn𝑥₁(𝑡)−𝑥^∗₁(𝑡)=sgn𝑥₁(𝑡)−𝑥^∗₁(𝑡)×−𝑎₁₁(𝑡)𝑥₁𝑡−𝜏₁₁(𝑡)−𝑥^∗₁𝑡−𝜏₁₁(𝑡)−𝑎₁₂(𝑡)𝑥₂(𝑡)𝑥₁(𝑡)+𝑑₁(𝑡)+𝑎₁₂(𝑡)𝑥^∗₂(𝑡)𝑥^∗₁(𝑡)+𝑑₁(𝑡)=sgn𝑥₁(𝑡)−𝑥^∗₁(𝑡)×−𝑎₁₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡)−𝑎₁₂(𝑡)𝑥₁(𝑡)+𝑑₁(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡)+𝑎₁₂(𝑡)𝑥^∗₂(𝑡)𝑥₁(𝑡)+𝑑₁(𝑡)𝑥^∗₁(𝑡)+𝑑₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝑎₁₁(𝑡)^𝑡_{𝑡−𝜏11(𝑡)}𝑥̇₁(𝑢)−𝑥̇^∗₁(𝑢)𝑑𝑢≤−𝑎₁₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡+𝑎)¹²(𝑡)𝑑₁(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡+𝑎₁₂)(𝑡)𝑀₂𝑑²₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝑎₁₁(𝑡)^𝑡_{𝑡−𝜏11(𝑡)}𝑥̇₁(𝑢)−𝑥̇^∗₁(𝑢)𝑑𝑢. (3.6) On substituting (1.2) into (3.6), we derive that 𝐷⁺𝑉₁₁(𝑡)≤−𝑎₁₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡+𝑎)¹²(𝑡)𝑑₁(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡+𝑎)¹²(𝑡)𝑀₂𝑑²₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝑎₁₁(𝑡)^𝑡_{𝑡−𝜏11(𝑡)}𝑎₁(𝑢)−𝑎₁₁(𝑢)𝑥^∗₁𝑢−𝜏₁₁(𝑢)−𝑎₁₂(𝑢)𝑑₁(𝑢)𝑥^∗₂(𝑢)𝑥₁(𝑢)+𝑑₁(𝑢)𝑥^∗₁(𝑢)+𝑑₁(𝑢)𝑥₁(𝑢)−𝑥^∗₁(𝑢)−𝑎₁₁(𝑢)𝑥₁(𝑢)𝑥₁𝑢−𝜏₁₁(𝑢)−𝑥^∗₁𝑢−𝜏₁₁(𝑢−𝑎₁₂)(𝑢)𝑥₁(𝑢)𝑥₁(𝑢)+𝑑₁(𝑢)𝑥₂(𝑢)−𝑥^∗₂(𝑢)𝑑𝑢. (3.7) It follows (3.4) and (3.7) that for 𝑡>𝑇+𝜏𝐷⁺𝑉₁₁(𝑡)≤−𝑎₁₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡+𝑎)¹²(𝑡)𝑑₁(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡+𝑎₁₂)(𝑡)𝑀₂𝑑²₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝑎₁₁(𝑡)^𝑡_{𝑡−𝜏11(𝑡)}𝑎₁(𝑢)+𝑎₁₁(𝑢)𝑀₁+𝑎₁₂(𝑢)𝑀₂𝑑₁(𝑢)𝑥₁(𝑢)−𝑥^∗₁(𝑢)+𝑎₁₁(𝑢)𝑀₁𝑥₁𝑢−𝜏₁₁(𝑢)−𝑥^∗₁𝑢−𝜏₁₁(𝑢)+𝑎₁₂(𝑢)𝑥₂(𝑢)−𝑥^∗₂(𝑢)𝑑𝑢. (3.8)
Let 𝜎₁₁(𝑡)=𝑡−𝜏₁₁(𝑡), by (𝐻₂), we can obtain the inverse function of the function 𝜎₁₁(𝑡) denoted by 𝜎⁻¹₁₁(𝑡).
Define𝑉₁₂(𝑡)=^𝜎−111_𝑡^(𝑡)𝑡_𝜎11(𝑠)𝑎₁₁(𝑠)×𝑎₁(𝑢)+𝑎₁₁(𝑢)𝑀₁+𝑎₁₂(𝑢)𝑀₂𝑑₁(𝑢)𝑥₁(𝑢)−𝑥^∗₁(𝑢)+𝑎₁₁(𝑢)𝑀₁𝑥₁𝑢−𝜏₁₁(𝑢)−𝑥^∗₁𝑢−𝜏₁₁(𝑢)+𝑎₁₂(𝑢)𝑥₂(𝑢)−𝑥^∗₂(𝑢)𝑑𝑢𝑑𝑠. (3.9) We obtain from (3.8) and (3.9) for 𝑡>𝑇+𝜏𝐷⁺𝑉₁₁(𝑡)+𝑉̇₁₂(𝑡)≤−𝑎₁₁(𝑡)−𝑎₁₂(𝑡)𝑀₂𝑑²₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡+𝑎)¹²(𝑡)𝑑₁(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡+_{𝜎−111𝑡(𝑡)})𝑎₁₁(𝑠)×𝑎𝑑𝑠₁(𝑡)+𝑎₁₁(𝑡)𝑀₁+𝑎₁₂(𝑡)𝑀₂𝑑₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝑎₁₁(𝑡)𝑀₁𝑥₁𝑡−𝜏₁₁(𝑡)−𝑥^∗₁𝑡−𝜏₁₁(𝑡)+𝑎₁₂(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡). (3.10) We now define 𝑉₁(𝑡)=𝑉₁₁(𝑡)+𝑉₁₂(𝑡)+𝑉₁₃(𝑡), (3.11) where 𝑉₁₃(𝑡)=𝑀₁^𝑡_{𝑡−𝜏11(𝑡)}^{𝜎−111(𝜎−111}_𝜎^(𝑙))_−111(𝑙)𝑎₁₁(𝑠)𝑎₁₁𝜎⁻¹₁₁(𝑙)1−𝜏̇₁₁𝜎⁻¹₁₁(𝑙)𝑥₁(𝑙)−𝑥^∗₁(𝑙)𝑑𝑠𝑑𝑙. (3.12) It then follows from (3.10) that for 𝑡>𝑇+𝜏𝐷⁺𝑉₁(𝑡)≤−𝑎₁₁(𝑡)−𝑎₁₂(𝑡)𝑀₂𝑑²₁(𝑡)𝑥₁(𝑡)−𝑥^∗₁(𝑡)+𝑎₁₂(𝑡)×1𝑑₁(𝑡)+^𝜎−111_𝑡^(𝑡)𝑎₁₁(𝑠)𝑑𝑠𝑥₂(𝑡)−𝑥^∗₂(𝑡+𝑎₁)(𝑡)+𝑎₁₁(𝑡)𝑀₁+𝑎₁₂(𝑡)𝑀₂𝑑₁(𝑡)^𝜎−111_𝑡^(𝑡)𝑎₁₁(𝑠)𝑑𝑠𝑥₁(𝑡)−𝑥^∗₁(𝑡+𝑎₁₁)𝜎⁻¹₁₁(𝑡)𝑀₁1−𝜏̇₁₁𝜎⁻¹₁₁(𝑡)^{𝜎−111(𝜎−111}_𝜎^(𝑡))−1_11(𝑡)𝑎₁₁(𝑠)𝑑𝑠𝑥₁(𝑡)−𝑥^∗₁(𝑡). (3.13) Similarly, we define 𝑉₂(𝑡)=𝑉₂₁(𝑡)+𝑉₂₂(𝑡)+𝑉₂₃(𝑡), (3.14) where 𝑉₂₁(𝑡)=ln𝑥₂(𝑡)−ln𝑥^∗₂(𝑡,𝑉₂₂)(𝑡)=^𝑡_{𝑡−𝜏21(𝑡)}𝑎₂₁𝜎⁻¹₂₁(𝑠)𝑑₁𝜎⁻¹₂₁(𝑠)1−𝜏̇₂₁𝜎⁻¹₂₁(𝑠)𝑥₁(𝑠)−𝑥^∗₁(𝑠)𝑑𝑠+^𝜎−122_𝑡^(𝑡)𝑡_𝜎22(𝑠)𝑎₂₂(𝑠)×𝑎₂(𝑢)+𝑎₂₂(𝑢)𝑀₂+𝑎₂₁(𝑢)+𝑎₂₃(𝑢)𝑀₃𝑑₂(𝑢)𝑥₂(𝑢)−𝑥^∗₂(𝑢)+𝑎₂₃(𝑢)𝑥₃(𝑢)−𝑥^∗₃(𝑢)+𝑎₂₂(𝑢)𝑀₂𝑥₂𝑢−𝜏₂₂(𝑢)−𝑥^∗₂𝑢−𝜏₂₂(𝑢)+𝑎₂₁(𝑢)𝑀₂𝑑₁(𝑢)𝑥₁𝑢−𝜏₂₁(𝑢)−𝑥^∗₁𝑢−𝜏₂₁(𝑢)𝑉𝑑𝑢𝑑𝑠,₂₃(𝑡)=𝑀₂^𝑡_{𝑡−𝜏22(𝑡)}^{𝜎−122(𝜎−122}_𝜎^(𝑙))_−122(𝑙)𝑎₂₂(𝑠)𝑎₂₂𝜎⁻¹₂₂(𝑙)1−𝜏̇₂₂𝜎⁻¹₂₂(𝑙)𝑥₂(𝑙)−𝑥^∗₂(𝑙)𝑑𝑠𝑑𝑙+𝑀₂^𝑡_{𝑡−𝜏21(𝑡)}^{𝜎−122(𝜎−121}_𝜎^(𝑙))_−121(𝑙)𝑎₂₂(𝑠)𝑎₂₁𝜎⁻¹₂₁(𝑙)𝑑₁𝜎⁻¹₂₁(𝑙)1−𝜏̇₂₁𝜎⁻¹₂₁(𝑙)𝑥₁(𝑙)−𝑥^∗₁(𝑙)𝑑𝑠𝑑𝑙. (3.15) Calculating the upper right derivative of 𝑉₂(𝑡) along solutions of (1.2), we derive that for 𝑡>𝑇+𝜏𝐷⁺𝑉₂(𝑡)≤−𝑎₂₂(𝑡)−𝑎₂₃(𝑡)𝑀₃𝑑²₂(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡)+𝑎₂₃(𝑡1)𝑑₂(𝑡)+^𝜎−122_𝑡^(𝑡)𝑎₂₂(𝑠)𝑑𝑠𝑥₃(𝑡)−𝑥^∗₃(𝑡+𝑎₂₁)𝜎⁻¹₂₁(𝑡)𝑑₁𝜎⁻¹₂₁(𝑡)1−𝜏̇₂₁𝜎⁻¹₂₁(𝑡𝑥)¹(𝑡)−𝑥^∗₁(𝑡+𝑎₂)(𝑡)+𝑎₂₂(𝑡)𝑀₂+𝑎₂₁(𝑡)+𝑎₂₃(𝑡)𝑀₃𝑑₂(𝑡)×^𝜎−122_𝑡^(𝑡)𝑎₂₂(𝑠)𝑑𝑠𝑥₂(𝑡)−𝑥^∗₂(𝑡+𝑎₂₂)𝜎⁻¹₂₂(𝑡)𝑀₂1−𝜏̇₂₂𝜎⁻¹₂₂(𝑡)^{𝜎−122(𝜎−122}_𝜎^(𝑡))−1_22(𝑡)𝑎₂₂(𝑠)𝑑𝑠𝑥₂(𝑡)−𝑥^∗₂(𝑡+𝑎₂₁)𝜎⁻¹₂₁(𝑡)𝑀₂𝑑₁𝜎⁻¹₂₁(𝑡)1−𝜏̇₂₁𝜎⁻¹₂₁(𝑡)^{𝜎−122(𝜎−121}_𝜎^(𝑡))−1_21(𝑡)𝑎₂₂(𝑠)𝑑𝑠𝑥₁(𝑡)−𝑥^∗₁(𝑡). (3.16) Similarly, we define 𝑉₃(𝑡)=𝑉₃₁(𝑡)+𝑉₃₂(𝑡)+𝑉₃₃(𝑡), (3.17) where 𝑉₃₁(𝑡)=ln𝑥₃(𝑡)−ln𝑥^∗₃(𝑡,𝑉₃₂)(𝑡)=𝑀₃^𝑡_{𝑡−𝜏32(𝑡)}𝑎₃₃𝜎⁻¹₃₂(𝑠)𝑑²₃𝜎⁻¹₃₂(𝑠)1−𝜏̇₃₂𝜎⁻¹₃₂(𝑠𝑥)²(𝑠)−𝑥^∗₂(𝑠+)𝑑𝑠_{𝜎−133𝑡(𝑡)𝑡𝜎33(𝑠)}𝑎₃₃(𝑠)𝑑₃(𝑠)×𝑎₃(𝑢)+𝑎₃₃(𝑢)𝑀₃𝑑₃(𝑢)𝑥₃(𝑢)−𝑥^∗₃(𝑢+𝑎₃₃)(𝑢)𝑀₃𝑑₃(𝑢)𝑥₃𝑢−𝜏₃₃(𝑢)−𝑥^∗₃𝑢−𝜏₃₃(𝑢)+𝑎₃₃(𝑢)𝑀²₃𝑑²₃(𝑢)𝑥₂𝑢−𝜏₃₂(𝑢)−𝑥^∗₂𝑢−𝜏₃₂(𝑢𝑉)𝑑𝑢𝑑𝑠,₃₃(𝑡)=𝑀₃^𝑡_{𝑡−𝜏33(𝑡)}^{𝜎−133(𝜎−133}_𝜎^(𝑙))_−133(𝑙)𝑎₃₃(𝑠)𝑎₃₃𝜎⁻¹₃₃(𝑙)𝑑₃(𝑠)𝑑₃𝜎⁻¹₃₃(𝑙)1−𝜏̇₃₃𝜎⁻¹₃₃(𝑙)𝑥₃(𝑙)−𝑥^∗₃(𝑙)𝑑𝑠𝑑𝑙+𝑀²₃^𝑡_{𝑡−𝜏32(𝑡)}^{𝜎−133(𝜎−132}_𝜎^(𝑙))_−132(𝑙)𝑎₃₃(𝑠)𝑎₃₃𝜎⁻¹₃₂(𝑙)𝑑₃(𝑠)𝑑²₃𝜎⁻¹₃₂(𝑙)1−𝜏̇₃₂𝜎⁻¹₃₂(𝑙)𝑥₂(𝑙)−𝑥^∗₂(𝑙)𝑑𝑠𝑑𝑙. (3.18) Calculating the upper right derivative of 𝑉₃(𝑡) along solutions of (1.2), we derive that for 𝑡>𝑇+𝜏𝐷⁺𝑉₃(𝑡)𝑎≤−³³(𝑡)𝑀₂+𝑑₃(𝑡)𝑥₃(𝑡)−𝑥^∗₃(𝑡)+𝑀₃𝑎₃₃𝜎⁻¹₃₂(𝑡)𝑑²₃𝜎⁻¹₃₂(𝑡)1−𝜏̇₃₂𝜎⁻¹₃₂(𝑡)𝑥₂(𝑡)−𝑥^∗₂(𝑡+𝑎₃)(𝑡)+𝑎₃₃(𝑡)𝑀₃𝑑₃(𝑡)^𝜎−133_𝑡^(𝑡)𝑎₃₃(𝑠)𝑑₃(𝑠)𝑑𝑠𝑥₃(𝑡)−𝑥^∗₃(𝑡+𝑎₃₃)𝜎⁻¹₃₃(𝑡)𝑀₃𝑑₃𝜎⁻¹₃₃(𝑡)1−𝜏̇₃₃𝜎⁻¹₃₃(𝑡)^{𝜎−133(𝜎−133}_𝜎^(𝑡))−1_33(𝑡)𝑎₃₃(𝑠)𝑑₃(𝑠)𝑑𝑠𝑥₃(𝑡)−𝑥^∗₃(𝑡+𝑎₃₃)𝜎⁻¹₃₂(𝑡)𝑀²₃𝑑²₃𝜎⁻¹₃₂(𝑡)1−𝜏̇₃₂𝜎⁻¹₃₂(𝑡)^{𝜎−133(𝜎−132𝜎(𝑡))−1}₃₂^(𝑡)𝑎₃₃(𝑠)𝑑₃(𝑠)𝑑𝑠𝑥₂(𝑡)−𝑥^∗₂(𝑡). (3.19) We now define a Lyapunov functional as 𝑉(𝑡)=³_𝑖=1𝜆_𝑖𝑉_𝑖(𝑡). (3.20) It then follows from (3.13), (3.16), (3.19), and (3.20) that for 𝑡>𝑇+𝜏𝐷⁺𝑉(𝑡)≤−³_𝑖=1𝐴_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡). (3.21) By the hypothes is (𝐻₄), there exist enough small positive constants 𝛼_𝑖, 𝑖=1,2,3 and a large enough constant 𝑇^∗≥𝑇+𝜏, such that for all 𝑖=1,2,3 and 𝑡≥𝑇^∗𝐴_𝑖(𝑡)≥𝛼_𝑖>0. (3.22) Integrating both sides of (3.21) on interval [𝑇^∗,𝑡]𝑉(𝑡)+³_𝑖=1^𝑡_𝑇∗𝐴_𝑖(𝑠)𝑥_𝑖(𝑠)−𝑥^∗_𝑖(𝑠)𝑑𝑠≤𝑉𝑇^∗. (3.23) It follows from (3.22) and (3.23) that 𝑉(𝑡)+³_𝑖=1𝛼_𝑖^𝑡_𝑇∗𝑥_𝑖(𝑠)−𝑥^∗_𝑖(𝑠)𝑑𝑠≤𝑉𝑇^∗. (3.24) Therefore, 𝑉(𝑡) is bounded on [𝑇^∗,+∞) and also ^𝑡_𝑇∗𝑥_𝑖(𝑠)−𝑥^∗_𝑖(𝑠)𝑑𝑠<∞,𝑖=1,2,3. (3.25) By Theorem 2.2, we know that |𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|, 𝑖=1,2,3 are bounded on [𝑇^∗,+∞). On the other hand, it is easy to see that 𝑥̇_𝑖(𝑡), 𝑖=1,2,3 are bounded for 𝑡≥𝑇^∗. Therefore, |𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)|, 𝑖=1,2,3 are uniformly continuous on [𝑇^∗,+∞). By Lemma 3.1, one can conclude that lim_𝑡→+∞𝑥_𝑖(𝑡)−𝑥^∗_𝑖(𝑡)=0,𝑖=1,2,3. (3.26) This completes the proof of Theorem 3.2.

Remark 3.3. In the proof of the stability of system (1.2), we construct the Lyapunov functional, which need the condition (𝐻₂). Similar method and conditon can be found in the [2, 14].

4. Existence and Stability of the Positive Periodic Solutions

In this section, we suppose that all the coefficients in system (1.2) are continuous and positive 𝜔-periodic functions, then the system (1.2) is an 𝜔-periodic system for this case.

We let the following denote the unique solution of periodic system (1.2) for initial value 𝑍⁰={𝑥⁰₁,𝑥⁰₂,𝑥⁰₃}: 𝑍𝑡,𝑍⁰=𝑥₁𝑡,𝑍⁰,𝑥₂𝑡,𝑍⁰,𝑥₃𝑡,𝑍⁰𝑍,for𝑡>0,0,𝑍⁰=𝑍⁰. (4.1)

Now define Poincâre transformation 𝐴∶𝑅³₊→𝑅³₊ is 𝐴𝑍⁰=𝑍𝜔,𝑍⁰. (4.2) In this way, the existence of periodic solution of system (1.2) will be equal to the existence of the fixed point 𝐴.

Theorem 4.1. Assume that the conditions of (𝐻₁)-(𝐻₃) hold, then system (1.2) with initial condition (1.3) has at least one positive 𝜔-periodic solution.

Proof. If assumption (𝐻₁)-(𝐻₃) are satisfied, then from Theorem 2.2 we have that there exist positive constants 𝑇>𝑇₇ and 𝑀_𝑖,𝑚_𝑖(𝑖=1,2,3) such that for all 𝑡≥𝑇 and 𝑖=1,2,3𝑚_𝑖≤𝑥_𝑖(𝑡)≤𝑀_𝑖. (4.3) Let 𝐾=𝑥₁(𝑡),𝑥₂(𝑡),𝑥₃(𝑡)∣𝑚_𝑖≤𝑥_𝑖(𝑡)≤𝑀_𝑖,𝑖=1,2,3, (4.4) then the compact region 𝐾⊂𝑅³₊ is a positive invariant set of system (1.2), and 𝐾 is also a close bounded convex set. So we have 𝑍⁰∈𝐾⟹𝑍𝑡,𝑍⁰∈𝐾, (4.5) also 𝑍(𝜔,𝑍⁰)∈𝐾, thus 𝐴𝐾⊂𝐾. The operator 𝐴 is continuous because the solution is continuous about the initial value. Using the fixed point theorem of Brower, we can obtain that 𝐴 has at least one fixed point in 𝐾, then there exists at least one strictly positive 𝜔-periodic solution of system (1.2). This ends the proof of Theorem 4.1.

By constructing similar Lyapunov functional to those of Theorem 3.2, and using Theorem 4.1, we have the following theorem.

Theorem 4.2. Assume that the conditions of (𝐻₁)-(𝐻₄) hold, then system (1.2) has a unique positive 𝜔-periodic solution which is globally asymptotically stable.

5. Concluding Remarks

In this paper, a nonautonomous Leslie-Gower type food chain model with time delays is investigated, which is based on the Holling type II and a Leslie-Gower modified functional response. By using comparison theorem, we prove the system is permanent under some appropriate conditions. Further, by constructing the suitable Lyapunov functional, we show that the system is globally asymptotically stable under some appropriate conditions. If the system is periodic one, some sufficient conditions are established, which guarantee the existence, uniqueness and global asymptotic stability of a positive periodic solution of the system. Our results have showed that the permanence, global stability of system and the existence, uniqueness of positive periodic solution depend on delays, and so time delays are profitless.

Acknowledgments

The authors are grateful to the Associate Editor, Manuel De la Sen, and two referees for a number of helpful suggestions that have greatly improved our original submission. This work is supported by the National Natural Science Foundation of China (no. 70901016) and Excellent Talents Program of Liaoning Educational Committee (no. 2008RC15).