Note on the Persistence of a Nonautonomous Lotka-Volterra Competitive System with Infinite Delay and Feedback Controls

Abstract

We study a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls. We establish a series of criteria under which a part of 𝑛-species of the systems is driven to extinction while the remaining part of the species is persistent. Particularly, as a special case, a series of new sufficient conditions on the persistence for all species of system are obtained. Several examples together with their numerical simulations show the feasibility of our main results.

1. Introduction

In this paper, we consider the following nonautonomous 𝑛-species Lotka-Volterra competitive system with infinite delay and feedback controls: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)₀^+∞𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡−𝑠)𝑑𝑠−𝑏_𝑖(𝑡)₀^+∞𝐻_𝑖(𝑠)𝑢_𝑖(),𝑢̇𝑡−𝑠𝑑𝑠_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+𝑑_𝑖(𝑡)₀^+∞𝑅_𝑖(𝑠)𝑥_𝑖(𝑡−𝑠)𝑑𝑠,𝑖=1,2,…,𝑛, (1) where 𝑥_𝑖(𝑡)(𝑖=1,2,…,𝑛) is the density of the 𝑖th species at time 𝑡 and 𝑢_𝑖(𝑡)(𝑖=1,2,…,𝑛) is the indirect control variable.

In particular, when the coefficients 𝑏_𝑖(𝑡)≡0, 𝑐_𝑖(𝑡)≡0, and 𝑑_𝑖(𝑡)≡0 for all 𝑡∈𝑅 and 𝑖=1,2,…,𝑛, the system (1) will degenerate into the following pure delay type system: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)₀^+∞𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡−𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (2)

As is well known, systems such as (2) without feedback controls are very important mathematical models of multispecies populations dynamics. This is a generalization from Ahmad [1] about two-species system without delays to 𝑛-species system of infinite delay. Systems without delays such as [1] have attracted the interest of many researchers (see, e.g., [2–5]), and systems with delays have been studied extensively in the past twenty years, and some good results on the permanence, extinction and persistence or uniform persistence, global stability, and almost periodic solution have been developed (see [6–18]). In [19], Montes de Oca and Pérez provided for us a very interesting work for system (2), who showed that if the coefficients are bounded and continuous and satisfy certain inequalities, then any solution with initial function of system (2) in an appropriate space will have 𝑛−1 of its components tenting to zero, while the remaining one will stabilize at a certain solution of a logistic differential equation. And for more works about single species dynamic behaviors of infinite delay, one could refer to [20, 21].

On the other hand, as was pointed out by Fan and Wang [22], feedback control is the basic mechanism by which systems, whether mechanical, electrical, or biological, maintain their equilibrium or homeostasis. Many scholars have done works on the ecosystem with feedback controls (see, e.g., [23–29] and the references cited therein). In [23], Shi et al. proposed the feedback control system (1). By using the method of multiple Lyapunov functionals and by developing a new analysis technique, Shi et al. established the sufficient conditions which guarantee part species 𝑥_𝑟+1,𝑥_𝑟+2,…,𝑥_𝑟+𝑛 of the 𝑛-species driven to extinction. But in the paper [23], they did not discuss the survival problems for the remaining species. The main aim of this paper is to study the persistence of the remaining species 𝑥₁,𝑥₂,…,𝑥_𝑟 of system (1). By the new method motivated by work [11, 27, 28], we will establish new sufficient conditions for which surplus species 𝑥₁,𝑥₂,…,𝑥_𝑟 of system (1) remain persistent.

The organization of the paper is as follows. In the next section, some assumptions and lemmas are introduced. In Section 3, we state and prove our main results. Finally, several examples with their numerical simulations are presented to show the feasibility of the main results.

2. Preliminaries

Throughout this paper, for system (1), we introduce the following hypotheses.(H₁)𝑟_𝑖(𝑡), 𝑎_𝑖𝑗(𝑡), 𝑏_𝑖(𝑡), 𝑐_𝑖(𝑡), and 𝑑_𝑖(𝑡)(𝑖,𝑗=1,2,…,𝑛) are bounded and continuous, defined on [0,∞). Furthermore, 𝑎_𝑖𝑗(𝑡)(𝑖≠𝑗), 𝑏_𝑖(𝑡), 𝑐_𝑖(𝑡), and 𝑑_𝑖(𝑡) are nonnegative on [0,∞), and 0<𝑎^𝑙_𝑖𝑖≤𝑎_𝑖𝑖(𝑡)≤𝑎^𝑢_𝑖𝑖<∞. Here, we denote 𝑓^𝑙=inf_𝑡≥0𝑓(𝑡) and 𝑓^𝑢=sup_𝑡≥0𝑓(𝑡).(H₂)𝐾_𝑖𝑗∶[0,∞)→[0,∞), 𝐻_𝑖∶[0,∞)→[0,∞), and 𝑅_𝑖∶[0,∞)→[0,∞), 𝑖, 𝑗=1,2,…,𝑛, are piecewise continuous and satisfy ₀^+∞𝐾_𝑖𝑗(𝑠)𝑑𝑠=1,𝒦_𝑖𝑗=₀^+∞𝑠𝐾_𝑖𝑗(𝑠)𝑑𝑠<∞,_0+∞𝐻_𝑖(𝑠)𝑑𝑠=1,ℋ_𝑖=₀^+∞𝑠𝐻_𝑖(𝑠)𝑑𝑠<∞,_0+∞𝑅_𝑖(𝑠)𝑑𝑠=1,ℛ_𝑖=₀^+∞𝑠𝑅_𝑖(𝑠)𝑑𝑠<∞. (3)(H₃)There exists a positive constant 𝜔 such that for each 𝑖=1,2,…,𝑛liminf_{𝑡→∞𝑡}^𝑡+𝜔𝑟_𝑖(𝑠)𝑑𝑠>0. (4)(H₄)There exist positive constants 𝜆 and 𝛾 such that for each 𝑖=1,2,…,𝑛liminf_{𝑡→∞𝑡}^𝑡+𝜆𝑐_𝑖(𝑠)𝑑𝑠>0,liminf_{𝑡→∞𝑡}^𝑡+𝛾𝑑_𝑖(𝑠)𝑑𝑠>0. (5)

We will consider system (1) together with the initial conditions 𝑥_𝑖(𝜃)=𝜙_𝑖(𝜃),𝑢_𝑖(𝜃)=𝜓_𝑖(𝜃),𝜃≤0,𝑖=1,2,…,𝑛, (6) where 𝜙_𝑖,𝜓_𝑖∈𝐵𝐶⁺, 𝑖=1,2,…,𝑛, and 𝐵𝐶⁺=],[∶𝜑(0).𝜑∈𝐶[(−∞,00,+∞)]>0,𝜑isbounded (7) It is easy to verify that solutions of (1) satisfying the initial condition (6) are well defined for all 𝑡≥0 and satisfy 𝑥_𝑖(𝑡)>0,𝑢_𝑖(𝑡)>0,∀𝑡≥0. (8)

We now introduce several lemmas which will be useful in the proofs of the main results.

We consider the following nonautonomous linear equation: 𝑥̇(𝑡)=𝑎(𝑡)−𝑏(𝑡)𝑥(𝑡), (9) where nonnegative functions 𝑎(𝑡) and 𝑏(𝑡) are bounded and continuous, defined on [0,+∞). We have the following results.

Lemma 1 (see [30]). Suppose that there exist positive constants 𝜂₁ and 𝜂₂ such that liminf_𝑡→∞^𝑡+𝜂1_𝑡𝑎(𝑠)𝑑𝑠>0,liminf_𝑡→∞^𝑡+𝜂2_𝑡𝑏(𝑠)𝑑𝑠>0. (10) Then, there exist positive constants 𝑀≥𝑚 such that 𝑚≤liminf_𝑡→∞𝑥(𝑡)≤limsup_𝑡→∞𝑥(𝑡)≤𝑀, (11) for any positive solution 𝑥(𝑡) of (9).

Lemma 2 (see [23]). Suppose that assumptions (𝐻₁)–(𝐻₄) hold; then there exist constants 𝑥_𝑖>0 and 𝑢_𝑖>0 such that limsup_𝑡→∞𝑥_𝑖(𝑡)<𝑥_𝑖,limsup_𝑡→∞𝑢_𝑖(𝑡)<𝑢_𝑖,𝑖=1,2,…,𝑛, (12) for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡),𝑢₁(𝑡), 𝑢₂(𝑡),…,𝑢_𝑛(𝑡)) of system (1).

Remark 3. If all parameters 𝑟_𝑖(𝑡), 𝑎_𝑖𝑗(𝑡), 𝑏_𝑖(𝑡), 𝑐_𝑖(𝑡), and 𝑑_𝑖(𝑡)(𝑖,𝑗=1,2,…,𝑛) of system (1) have the positive lower bound on [0,+∞), then, from Lemma 2.2 in [23], we can choose 𝑥_𝑖=𝑟^𝑢_𝑖𝑎^𝑙_𝑖𝑖∫₀^+∞𝑘_𝑖𝑖(𝑠)exp−𝑟^𝑢_𝑖𝑠,𝑑𝑠𝑢_𝑖=𝑑^𝑢_𝑖𝑐^𝑙_𝑖𝑥_𝑖. (13)

Lemma 4 (see [6]). Let 𝑥(𝑡)∶𝑅→𝑅 be a nonnegative and bounded continuous function, and let 𝑘(𝑠)∶[0,+∞)→[0,+∞) be an integral function satisfying ∫₀^+∞𝑘(𝑠)𝑑𝑠=1. Then liminf_𝑡→∞𝑥(𝑡)≤liminf_𝑡→∞0^+∞𝑘(𝑠)𝑥(𝑡−𝑠)𝑑𝑠≤limsup_𝑡→∞0^+∞𝑘(𝑠)𝑥(𝑡−𝑠)𝑑𝑠≤limsup_𝑡→∞𝑥(𝑡). (14)

3. Main Results

In this section, we discuss the persistence of part species 𝑥_𝑖(1≤𝑖≤𝑟) of system (1), where integer 𝑟∈{1,2,…,𝑛}. Let functions 𝐴_𝑖𝑗(𝑡)=₀^+∞𝑎_𝑖𝑗(𝑡+𝑠)𝐾_𝑖𝑗(𝑠)𝐵𝑑𝑠,_𝑖(𝑡)=₀^+∞𝑏_𝑖(𝑡+𝑠)𝐻_𝑖(𝑠)𝐷𝑑𝑠,_𝑖(𝑡)=₀^+∞𝑑_𝑖(𝑡+𝑠)𝑅_𝑖(𝑠)𝑑𝑠,𝑖,𝑗=1,2,…,𝑛. (15)

Lemma 5. Suppose that assumptions (𝐻₁)–(𝐻₄) hold and there exists an integer 1≤𝑟<𝑛 such that for any 𝑘>𝑟 there exists an integer 𝑖_𝑘<𝑘 such that limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟_𝑘(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟_𝑖𝑘(𝑠)𝑑𝑠<liminf_𝑡→∞𝐴_𝑘𝑗(𝑡)𝐴_𝑖𝑘𝑗(𝑡)∀𝑗≤𝑘,liminf_𝑡→∞𝐵_𝑘(𝑡)𝑐_𝑘(𝑡)>limsup_𝑡→∞𝐴_𝑖𝑘𝑘(𝑡)𝐷_𝑘(𝑡)limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟_𝑘(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟_𝑖𝑘(𝑠)−𝐴𝑑𝑠^𝑘𝑘(𝑡)𝐷_𝑘(𝑡),limsup_𝑡→∞𝐵_𝑖𝑘(𝑡)𝑐_𝑖𝑘(𝑡)<liminf_𝑡→∞𝐴_𝑘𝑖𝑘(𝑡)𝐷_𝑖𝑘(𝑡)liminf_𝑡→∞∫_𝑡^𝑡+𝑤𝑟_𝑖𝑘(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟_𝑘(𝑠)−𝐴𝑑𝑠^{𝑖𝑘𝑖𝑘}(𝑡)𝐷_𝑖𝑘(𝑡). (16) Then for each 𝑖=𝑟+1,…,𝑛 we have lim_𝑡→∞𝑥_𝑖(𝑡)=0,lim_𝑡→∞𝑢_𝑖(𝑡)=0,^∞₀𝑥_𝑖(𝑡)𝑑𝑡<∞, (17) for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡),𝑢₁(𝑡), 𝑢₂(𝑡),…,𝑢_𝑛(𝑡)) of system (1).

The proof of the extinction of part species 𝑥_𝑟+1,𝑥_𝑟+2,…,𝑥_𝑟+𝑛 of system (1) could be found in [23] and we hence omit it here.

On the persistence of part species 𝑥_𝑖(1≤𝑖≤𝑟) of system (1), we state and prove the following results.

Theorem 6. Suppose that all assumptions of Lemma 5 hold and there exists a positive constant 𝜂>0 such that liminf_{𝑡→∞𝑡}^𝑡+𝜂𝑟_𝑖(𝑠)−^𝑟_𝑗≠𝑖𝐴_𝑖𝑗(𝑠)𝑥_𝑗−𝐵_𝑖(𝑠)𝑢_𝑖𝑑𝑠>0,∀𝑖≤𝑟. (18) Then, for each 𝑖=1,2,…,𝑟, there exist positive constants 𝑚 and 𝑀, with 𝑚<𝑀, such that 𝑚≤liminf_𝑡→∞𝑥_𝑖(𝑡)≤limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑀,𝑚≤liminf_𝑡→∞𝑢_𝑖(𝑡)≤limsup_𝑡→∞𝑢_𝑖(𝑡)≤𝑀, (19) for any positive solution (𝑥(𝑡),𝑢(𝑡))=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡),𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡)) of system (1).

Proof. Let (𝑥(𝑡),𝑢(𝑡))=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡), 𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡)) be any positive solution of system (1). By Lemma 2, let 𝑀=max_{1≤𝑖≤𝑟}{𝑥_𝑖,𝑢_𝑖}; for each 𝑖=1,2,…,𝑟, we have limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑀 and limsup_𝑡→∞𝑢_𝑖(𝑡)≤𝑀. So, we only need to prove that there exists a positive constant 𝑚 such that liminf_𝑡→∞𝑥_𝑖(𝑡)≥𝑚 and liminf_𝑡→∞𝑢_𝑖(𝑡)≥𝑚 for all 𝑖≤𝑟.
First of all, assumption (18) implies that there are positive constants 𝛼₀ and 𝑇₀ such that _𝑡^𝑡+𝜂𝑟_𝑖(𝑠)−^𝑟_𝑗≠𝑖𝐴_𝑖𝑗(𝑠)𝑥_𝑗−𝐵_𝑖(𝑠)𝑢_𝑖𝑑𝑠≥𝛼₀, (20) for all 𝑡≥𝑇₀ and 𝑖≤𝑟.
By Lemmas 2 and 5, we obtain that, for any constant 𝜀>0, there is a 𝑇(𝜀)>𝑇₀ such that, for all 𝑡≥𝑇(𝜀), 𝑥_𝑖(𝑡)≤𝑥_𝑖+𝜀,𝑢_𝑖(𝑡)≤𝑢_𝑖𝑥+𝜀,∀𝑖≤𝑟,_𝑖(𝑡)≤𝜀,𝑢_𝑖(𝑡)≤𝜀,∀𝑖>𝑟. (21)
Now, for any 𝑖≤𝑟, we define the Lyapunov function as follows: 𝑊_𝑖(𝑡)=𝑥_𝑖(𝑡)exp−^𝑛_𝑗=10^+∞𝐾_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃+𝑠)𝑥_𝑗(𝜃)𝑑𝜃𝑑𝑠−₀^+∞𝐻_𝑖(𝑠)^𝑡_𝑡−𝑠𝑏_𝑖(𝜃+𝑠)𝑢_𝑖(𝜃)𝑑𝜃𝑑𝑠. (22) By assumptions (H₁) and (H₂) and Lemma 2, we have ^𝑛_𝑗=10^+∞𝐾_𝑖𝑗(𝑠)^𝑡_𝑡−𝑠𝑎_𝑖𝑗(𝜃+𝑠)𝑥_𝑗(𝜃)≤𝑑𝜃𝑑𝑠_𝑛𝑗=1𝑎^𝑢_𝑖𝑗0^+∞𝑠𝐾_𝑖𝑗(𝑠)𝑑𝑠sup_𝑡∈𝑅𝑥_𝑗(𝑡)<∞,_0+∞𝐻_𝑖(𝑠)^𝑡_𝑡−𝑠𝑏_𝑖(𝜃+𝑠)𝑢_𝑖(𝜃)𝑑𝜃𝑑𝑠≤𝑏^𝑢_𝑖0^+∞𝑠𝐻_𝑖(𝑠)𝑑𝑠sup_𝑡∈𝑅𝑢_𝑖(𝑡)<∞. (23) So we see that 𝑊_𝑖(𝑡) has definition for all 𝑡≥0. From (23), we can obtain that for any 𝑖≤𝑟 there is a positive constant 𝑑_𝑖<1, and 𝑑_𝑖 may be dependent on the positive solution of system (1) such that 𝑑_𝑖𝑥_𝑖(𝑡)≤𝑊_𝑖(𝑡)≤𝑥_𝑖(𝑡),∀𝑡≥0. (24) Calculating the derivative of 𝑊_𝑖(𝑡) with respect to 𝑡, we have 𝑊̇_𝑖(𝑡)=𝑊_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)₀^+∞𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡−𝑠)𝑑𝑠−𝑏_𝑖(𝑡)₀^+∞𝐻_𝑖(𝑠)𝑢_𝑖()−𝑡−𝑠𝑑𝑠_{𝑛𝑗=10+∞}𝐾_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡+𝑠)𝑑𝑠𝑥_𝑗(𝑡)+^𝑛_𝑗=10^+∞𝐾_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡)𝑥_𝑗()−𝑡−𝑠𝑑𝑠_0+∞𝐻_𝑖(𝑠)𝑏_𝑖(𝑡+𝑠)𝑑𝑠𝑢_𝑖(𝑡)+₀^+∞𝐻_𝑖(𝑠)𝑏_𝑖(𝑡)𝑢_𝑖(𝑡−𝑠)𝑑𝑠=𝑊_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=10^+∞𝐾_𝑖𝑗(𝑠)𝑎_𝑖𝑗(𝑡+𝑠)𝑑𝑠𝑥_𝑗(𝑡)−₀^+∞𝐻_𝑖(𝑠)𝑏_𝑖(𝑡+𝑠)𝑑𝑠𝑢_𝑖(𝑡)=𝑊_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝐴_𝑖𝑗(𝑡)𝑥_𝑗(𝑡)−𝐵_𝑖(𝑡)𝑢_𝑖(𝑡),∀𝑖≤𝑟,𝑡≥0. (25) Let 𝛽_𝑖(𝑡,𝜀)=𝑟_𝑖(𝑡)−∑^𝑛_𝑗≠𝑖𝐴_𝑖𝑗(𝑡)𝜀−∑^𝑟_𝑗≠𝑖𝐴_𝑖𝑗(𝑡)𝑥_𝑗−𝐵_𝑖(𝑡)(𝑢_𝑖+𝜀). From (21), for all 𝑡≥𝑇(𝜀)>0, we have 𝑊̇_𝑖(𝑡)≥𝑊_𝑖(𝑡)𝛽_𝑖(𝑡,𝜀)−𝐴_𝑖𝑖(𝑡)𝑥_𝑖(𝑡)≥𝑊_𝑖(𝑡)𝛽_𝑖(𝑡,𝜀)−𝐴_𝑖𝑖(𝑡)𝑑_𝑖⁻¹𝑊_𝑖(𝑡). (26) Obviously, from inequality (20), we can find enough small positive constants 𝛿_𝑖 and 𝜀₀ such that _𝑡^𝑡+𝜂𝛽_𝑖𝑠,𝜀₀−𝐴_𝑖𝑖(𝑠)𝑑_𝑖⁻¹𝛿_𝑖1𝑑𝑠>2𝛼₀, (27) for all 𝑡≥𝑇₁=𝑇(𝜀₀). So for the above 𝜀₀, when 𝑡≥𝑇₁=𝑇(𝜀₀), 𝑊̇_𝑖(𝑡)≥𝑊_𝑖(𝑡)𝛽_𝑖𝑡,𝜀₀−𝐴_𝑖𝑖(𝑡)𝑑_𝑖⁻¹𝑊_𝑖(𝑡). (28) Consider the auxiliary equation 𝑊̇_𝑖(𝑡)=𝑊_𝑖(𝑡)𝛽_𝑖𝑡,𝜀₀−𝐴_𝑖𝑖(𝑡)𝑑_𝑖⁻¹𝑊_𝑖(𝑡); (29) then by (28), we obtain that 𝑊_𝑖(𝑡)≥𝑊^∗_𝑖(𝑡),∀𝑡≥𝑇₁, (30) where 𝑊^∗_𝑖(𝑡) is the solution of (29) with the initial condition 𝑊_𝑖(𝑇₁)=𝑊^∗_𝑖(𝑇₁). If 𝑊^∗_𝑖(𝑡)<𝛿_𝑖 for all 𝑡≥𝑇₁, then 𝑊^∗_𝑖(𝑡) is defined on [𝑇₁,+∞). Integrating inequality (29) from 𝑇₁ to 𝑡, we obtain 𝑊^∗_𝑖(𝑡)=𝑊^∗_𝑖𝑇₁exp^𝑡_𝑇1𝛽_𝑖𝑠,𝜀₀−𝐴_𝑖𝑖(𝑠)𝑑_𝑖⁻¹𝑊^∗_𝑖(𝑠)𝑑𝑠≥𝑊^∗_𝑖𝑇₁exp^𝑡_𝑇1𝛽_𝑖𝑠,𝜀₀−𝐴_𝑖𝑖(𝑠)𝑑_𝑖⁻¹𝛿_𝑖𝑑𝑠, (31) for all 𝑡≥𝑇₁. Putting 𝑡=𝑇₁+𝑚𝜂, 𝑚=1,2,…, then, from (27) and (31), we have 𝑊^∗_𝑖𝑇₁+𝑚𝜂≥𝑊^∗_𝑖𝑇₁1exp2𝑚𝛼₀,𝑚=1,2,…. (32) Letting 𝑚→+∞, we have 𝑊^∗_𝑖(𝑇₁+𝑚𝜂)→+∞, a contradiction. Hence, there is a 𝑡_𝑖≥𝑇₁ such that 𝑊^∗_𝑖(𝑡_𝑖)>𝛿_𝑖. Now, we prove that 𝑊^∗_𝑖(𝑡)≥𝛿_𝑖exp−𝛽_𝑖𝛿_𝑖,𝜀₀𝜂,∀𝑡≥𝑡_𝑖, (33) where 𝛽_𝑖(𝛿_𝑖,𝜀₀)=sup_𝑡≥0{|𝛽_𝑖(𝑡,𝜀₀)|+𝐴_𝑖𝑖(𝑡)𝑑_𝑖⁻¹𝛿_𝑖}, and the definition of 𝛽_𝑖(𝑡,𝜀₀) implies 0<𝛽_𝑖(𝛿_𝑖,𝜀₀)<∞. In fact, if (33) is not true, then there are 𝑡₁ and 𝑡₂, 𝑡₁<𝑡₂, such that 𝑊^∗_𝑖𝑡₂<𝛿_𝑖exp−𝛽_𝑖𝛿_𝑖,𝜀₀𝜂,𝑊^∗_𝑖𝑡₁=𝛿_𝑖,𝑊^∗_𝑖(𝑡)<𝛿_𝑖,∀𝑡∈𝑡₁,𝑡₂. (34) Choosing the integer 𝑚≥0 such that 𝑡₂∈(𝑡₁+𝑚𝜂,𝑡₁+(𝑚+1)𝜂], then, by (27) and (29), it follows that 𝛿_𝑖exp−𝛽_𝑖𝛿_𝑖,𝜀₀𝜂>𝑊^∗_𝑖𝑡₂=𝑊^∗_𝑖𝑡₁exp^𝑡2_𝑡1𝛽_𝑖𝑡,𝜀₀−𝐴_𝑖𝑖(𝑡)𝑑_𝑖⁻¹𝑊^∗_𝑖(𝑡)𝑑𝑡≥𝛿_𝑖exp^𝑡1_𝑡^+𝑚𝜂₁+^𝑡2_{𝑡1+𝑚𝜂}𝛽_𝑖𝑡,𝜀₀−𝐴_𝑖𝑖(𝑡)𝑑_𝑖⁻¹𝛿_𝑖𝑑𝑡≥𝛿_𝑖exp^𝑡2_{𝑡1+𝑚𝜂}𝛽_𝑖𝑡,𝜀₀−𝐴_𝑖𝑖(𝑡)𝑑_𝑖⁻¹𝛿_𝑖𝑑𝑡≥𝛿_𝑖exp−𝛽_𝑖𝛿_𝑖,𝜀₀𝜂, (35) which is a contradiction.
From (24), (30), and (33), we can obtain that 𝑥_𝑖(𝑡)≥𝛿_𝑖exp−𝛽_𝑖𝛿_𝑖,𝜀₀𝜂∀𝑡≥𝑡_𝑖. (36) Finally, we define the constants 𝑚_𝑖=𝛿_𝑖exp(−𝛽_𝑖(𝛿_𝑖,𝜀₀)𝜂) and 𝑇=max_𝑖≤𝑟{𝑡_𝑖}; then we have 𝑥_𝑖(𝑡)≥𝑚_𝑖∀𝑡≥𝑇,𝑖≤𝑟. (37) Letting 𝑚^∗_𝑖={inf_{𝑡∈[0,𝑇]}𝑥_𝑖(𝑡)>0} and 𝑚^∗=min_{1≤𝑖≤𝑟}{𝑚_𝑖,𝑚^∗_𝑖}, we have liminf_𝑡→∞𝑥_𝑖(𝑡)≥𝑚^∗, (38) for all 𝑖≤𝑟.
Further, by Lemma 4 and (38), we can choose constants 𝜖>0 and 𝑇^∗>0 such that for all 𝑖≤𝑟 and 𝑡≥𝑇^∗₀^+∞𝑅_𝑖(𝑠)𝑥_𝑖(𝑡−𝑠)𝑑𝑠≥𝑚^∗−𝜖>0. (39) Considering the second equation of system (1), from (39), for any 𝑡≥𝑇^∗, we obtain 𝑢̇_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+𝑑_𝑖(𝑡)₀^+∞𝑅_𝑖(𝑠)𝑥_𝑖(𝑡−𝑠)𝑑𝑠≥−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+𝑑_𝑖(𝑡)𝑚^∗−𝜖. (40) We consider the following auxiliary equation: 𝑣̇_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑣_𝑖(𝑡)+𝑑_𝑖(𝑡)𝑚^∗−𝜖. (41) Then by assumption (H₄) and applying Lemma 1 there exists a constant 𝑢_𝑖>0 such that liminf_𝑡→∞𝑣_𝑖(𝑡)>𝑢_𝑖, (42) for any positive solution 𝑣_𝑖(𝑡) of (41). Let 𝑣^∗_𝑖(𝑡) be the solution of (41) with the initial condition 𝑣^∗_𝑖(𝑇^∗)=𝑢_𝑖(𝑇^∗); then by the comparison theorem we have 𝑢_𝑖(𝑡)≥𝑣^∗_𝑖(𝑡)∀𝑡≥𝑇^∗. (43) Thus, we finally obtain liminf_𝑡→∞𝑢_𝑖(𝑡)≥𝑢_𝑖. (44) Let 𝑚=min_{1≤𝑖≤𝑟}{𝑚^∗,𝑢_𝑖}; from (38) and (44), we obtain that liminf_𝑡→∞𝑥_𝑖(𝑡)≥𝑚 and liminf_𝑡→∞𝑢_𝑖(𝑡)≥𝑚. This completes the proof of Theorem 6.

As consequences of Theorem 6 we have the following corollaries.

Corollary 7. If, in system (1), 𝑏_𝑖(𝑡)=𝑐_𝑖(𝑡)=𝑑_𝑖(𝑡)=0(𝑖=1,2,…,𝑛) for all 𝑡≥0, then system (1) will be reduced to the following 𝑛-species competitive system with infinite delay: 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)×₀^+∞𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡−𝑠)𝑑𝑠,𝑖=1,2,…,𝑛. (45) Suppose that assumptions (H₁)–(H₃) hold and there exists an integer 1≤𝑟<𝑛 such that for any 𝑘>𝑟 there exists an integer 𝑖_𝑘<𝑘 such that limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟_𝑘(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟_𝑖𝑘(𝑠)𝑑𝑠<liminf_𝑡→∞𝐴_𝑘𝑗(𝑡)𝐴_𝑖𝑘𝑗(𝑡),∀𝑗≤𝑘. (46) Furthermore, there exists a positive constant 𝜂>0 such that liminf_{𝑡→∞𝑡}^𝑡+𝜂𝑟_𝑖(𝑠)−^𝑟_𝑗≠𝑖𝐴_𝑖𝑗(𝑠)𝑥_𝑗𝑑𝑠>0,∀𝑖≤𝑟. (47) Then, for each 𝑖=1,2,…,𝑟, there exist positive constants 𝑚≤𝑀 such that 𝑚≤liminf_𝑡→∞𝑥_𝑖(𝑡)≤limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑀, (48) and for each 𝑖=𝑟+1,…,𝑛 we have lim_𝑡→∞𝑥_𝑖(𝑡)=0,^∞₀𝑥_𝑖(𝑡)𝑑𝑡<∞, (49) for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (45).

Proof. From the condition, limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟_𝑘(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟_𝑖𝑘(𝑠)𝑑𝑠<liminf_𝑡→∞𝐴_𝑘𝑗(𝑡)𝐴_𝑖𝑘𝑗(𝑡),∀𝑗≤𝑘. (50) And the assumptions (H₁)–(H₃) hold; from Corollary 7 in [23], for each 𝑖=𝑟+1,…,𝑛 we have lim_𝑡→∞𝑥_𝑖(𝑡)=0,^∞₀𝑥_𝑖(𝑡)𝑑𝑡<∞. (51) Further condition liminf_{𝑡→∞𝑡}^𝑡+𝜂𝑟_𝑖(𝑠)−^𝑟_𝑗≠𝑖𝐴_𝑖𝑗(𝑠)𝑥_𝑗𝑑𝑠>0,∀𝑖≤𝑟 (52) holds, so we see, from Theorem 6, for each 𝑖=1,2,…,𝑟, that there exist positive constants 𝑚≤𝑀 such that 𝑚≤liminf_𝑡→∞𝑥_𝑖(𝑡)≤limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑀. (53)

Remark 8. When 𝑟=1, the conditions of Corollary 7 will reduce to the assumptions that (H₁)–(H₃) hold and for any 𝑘>1 such that limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟_𝑘(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₁(𝑠)𝑑𝑠<liminf_𝑡→∞𝐴_𝑘𝑗(𝑡)𝐴_1𝑗(𝑡),∀𝑗≤𝑘. (54) We have that there exist positive constants 𝑚≤𝑀 such that 𝑚≤liminf_𝑡→∞𝑥₁(𝑡)≤limsup_𝑡→∞𝑥₁(𝑡)≤𝑀, (55) and for each 𝑖=2,…,𝑛 we have lim_𝑡→∞𝑥_𝑖(𝑡)=0,^∞₀𝑥_𝑖(𝑡)𝑑𝑡<∞. (56) In comparison with the assumptions (1.5) together with Proposition 2.2 given by Montes de Oca and Pérez [19], we can see that our assumptions in Corollary 7 are weaker.

Remark 9. When 𝑟=𝑛, from Corollary 7 we can easily obtain a criterion on the persistence of all species (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (45).

Remark 10. The conclusion of Corollary 7 improves that of Proposition 2.2 given by Montes de Oca and Pérez [19].

Corollary 11. Suppose that (H₁)–(H₄) hold and there exists a positive constant 𝜂>0 such that liminf_{𝑡→∞𝑡}^𝑡+𝜂𝑟_𝑖(𝑠)−^𝑛_𝑗≠𝑖𝐴_𝑖𝑗(𝑠)𝑥_𝑗−𝐵_𝑖(𝑠)𝑢_𝑖𝑑𝑠>0,∀𝑖=1,2,…,𝑛. (57) Then, for each 𝑖=1,2,…,𝑛, there exist positive constants 𝑚 and 𝑀, with 𝑚<𝑀, such that 𝑚≤liminf_𝑡→∞𝑥_𝑖(𝑡)≤limsup_𝑡→∞𝑥_𝑖(𝑡)<𝑀,𝑚≤liminf_𝑡→∞𝑢_𝑖(𝑡)≤limsup_𝑡→∞𝑢_𝑖(𝑡)<𝑀, (58) for any positive solution (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡),𝑢₁(𝑡), 𝑢₂(𝑡),…,𝑢_𝑛(𝑡)) of system (1).

Remark 12. From Corollary 11 we can easily obtain a criterion on the persistence of all species (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡),𝑢₁(𝑡),𝑢₂(𝑡),…,𝑢_𝑛(𝑡)) of system (1).

4. Examples

In this section, we will give several examples to illustrate the conclusions of Corollary 7, Theorem 6, and Corollary 11. In the first part we will illustrate the conclusions of Corollary 7, in the second we will illustrate the conclusions of Theorem 6, and in the last we will illustrate the conclusions of Corollary 11.

Example 1. Consider the system 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−³_𝑗=1𝑎_𝑖𝑗(𝑡)₀^+∞𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡−𝑠)𝑑𝑠,𝑖=1,2,3, (59) where 𝑟₁(𝑡)=5+3sin𝑡,𝑟₂(𝑡)𝑟=3+3cos𝑡,₃(𝑡)𝑎=2+2sin𝑡,₁₁(𝑡)=4,𝑎₁₂(𝑡)=5,𝑎₁₃(𝑡)𝑎=6,₂₁(𝑡)=3,𝑎₂₂(𝑡)=4,𝑎₂₃(𝑡)𝑎=5,₃₁(𝑡)=2,𝑎₃₂(𝑡)=3,𝑎₃₃(𝑡)𝐾=4,₁₁(𝑡)=𝑒^−𝑡,𝐾₁₂(𝑡)=2𝑒^−2𝑡,𝐾₁₃(𝑡)=3𝑒^−3𝑡,𝐾₂₁(𝑡)=𝑒^−𝑡,𝐾₂₂(𝑡)=2𝑒^−2𝑡,𝐾₂₃(𝑡)=3𝑒^−3𝑡,𝐾₃₁(𝑡)=𝑒^−𝑡,𝐾₃₂(𝑡)=2𝑒^−2𝑡,𝐾₃₃(𝑡)=3𝑒^−3𝑡. (60) Obviously, we have that the period of system (59) is 𝜔=2𝜋. By calculating, we obtain limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₂(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₁(𝑠)=3𝑑𝑠5<liminf_𝑡→∞𝐴₂₁(𝑡)𝐴₁₁(𝑡)=34,limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₂(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₁(𝑠)=3𝑑𝑠5<liminf_𝑡→∞𝐴₂₂(𝑡)𝐴₁₂(𝑡)=45,limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₃(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₂(𝑠)=2𝑑𝑠5<liminf_𝑡→∞𝐴₃₁(𝑡)𝐴₂₁(𝑡)=23,limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₃(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₂(𝑠)=2𝑑𝑠5<liminf_𝑡→∞𝐴₃₂(𝑡)𝐴₂₂(𝑡)=34,limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₃(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₂(𝑠)=2𝑑𝑠5<liminf_𝑡→∞𝐴₃₃(𝑡)𝐴₂₃(𝑡)=45. (61) We can choose 𝑟=1, since all conditions of Corollary 7 hold; therefore, species 𝑥₂ and 𝑥₃ in system (59) are extinct, and only species 𝑥₁ is persistent (see Figure 1). However, conditions (1.5) of Proposition 2.2 given by Montes de Oca and Pérez [19] do not apply in this example.

Figure 1 

Dynamic behaviors of system (59). Here, we take the initial conditions 𝑥1(𝜃)≡𝑥2(𝜃)≡𝑥3(𝜃)=0.3 for all 𝜃∈(−∞,0].

Example 2. Consider the system 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−³_𝑗=1𝑎_𝑖𝑗(𝑡)₀^+∞𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡−𝑠)𝑑𝑠−𝑏_𝑖(𝑡)₀^+∞𝐻_𝑖(𝑠)𝑢_𝑖(),𝑢̇𝑡−𝑠𝑑𝑠_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+𝑑_𝑖(𝑡)₀^+∞𝑅_𝑖(𝑠)𝑥_𝑖(𝑡−𝑠)𝑑𝑠,𝑖=1,2,3, (62) where 𝑟₁(𝑡)=2+sin𝑡,𝑎₁₁(𝑡)𝑎=4+cos𝑡,₁₂(𝑡)=15,𝑎₁₃(𝑡)=16,𝑏₁(𝑡)=110,𝑟₂(𝑡)𝑎=3+cos𝑡,₂₁(𝑡)=14,𝑎₂₂(𝑡)𝑎=6+sin𝑡,₂₃(𝑡)=27,𝑏₂(𝑡)=13,𝑟₃(𝑡)1=1+4cos𝑡,𝑎₃₁(𝑡)=52,𝑎₃₂(𝑡)=52,𝑎₃₃(𝑡)𝑏=2+sin𝑡,₃(𝑡)=25,𝑐₁(𝑡)𝑐=4+2sin𝑡,₂(𝑡)=6+2sin𝑡,𝑐₃(𝑡)𝑑=5+2sin𝑡,₁(𝑡)1=1+2cos𝑡,𝑑₂(𝑡)1=1+3𝑑cos𝑡,₃(𝑡)1=1+4𝐾cos𝑡,_𝑖𝑗(𝑡)=𝐻_𝑖(𝑡)=𝑅_𝑖(𝑡)=𝑒^−𝑡,𝑖,𝑗=1,2,3. (63) Obviously, we have that the period of system (62) is 𝜔=2𝜋. By calculating, we obtain limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₃(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₁(𝑠)=1𝑑𝑠2<liminf_𝑡→∞𝐴₃₁(𝑡)𝐴₁₁(𝑡)=58+√2,limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₃(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₁(𝑠)=1𝑑𝑠2<liminf_𝑡→∞𝐴₃₂(𝑡)𝐴₁₂(𝑡)=252,limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₃(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₁(𝑠)=1𝑑𝑠2<liminf_𝑡→∞𝐴₃₃(𝑡)𝐴₁₃(𝑡)=12−3√2,limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₃(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₂(𝑠)=1𝑑𝑠3<liminf_𝑡→∞𝐴₃₁(𝑡)𝐴₂₁(𝑡)=10,limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₃(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₂(𝑠)=1𝑑𝑠3<liminf_𝑡→∞𝐴₃₂(𝑡)𝐴₂₂(𝑡)=512+√2,limsup_𝑡→∞∫_𝑡^𝑡+𝑤𝑟₃(𝑠)𝑑𝑠∫_𝑡^𝑡+𝑤𝑟₂(𝑠)=1𝑑𝑠3<liminf_𝑡→∞𝐴₃₃(𝑡)𝐴₂₃(𝑡)7=7−4√2. (64) From Remark 3, we can choose 𝑟=2, 𝑥₁=4, 𝑥₂=4, 𝑥₃=45/16, 𝑢₁=3, 𝑢₂=4/3, 𝑢₃=75/64, and 𝜂=2𝜋 such that liminf_{𝑡→∞𝑡}^𝑡+2𝜋𝑟₁(𝑠)−𝐴₁₂(𝑠)𝑥₂−𝐴₁₃(𝑠)𝑥₃−𝐵₁(𝑠)𝑢₁𝑑𝑠=0.8625𝜋>0,liminf_{𝑡→∞𝑡}^𝑡+2𝜋𝑟₂(𝑠)−𝐴₂₁(𝑠)𝑥₁−𝐴₂₃(𝑠)𝑥₃−𝐵₂(𝑠)𝑢₂𝑑𝑠≈1.504𝜋>0. (65) All the conditions of Theorem 6 hold; therefore, species 𝑥₁ and 𝑥₂ coexist, and species 𝑥₃ in system (62) is extinct (see Figure 2).

Figure 2 

Dynamic behaviors of system (62). Here, we take the initial conditions 𝑥1(𝜃)≡𝑥2(𝜃)≡𝑥3(𝜃)≡𝑢1(𝜃)≡𝑢2(𝜃)≡𝑢2(𝜃)=0.3 for all 𝜃∈(−∞,0].

Example 3. Consider the system 𝑥̇_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑟_𝑖(𝑡)−³_𝑗=1𝑎_𝑖𝑗(𝑡)₀^+∞𝐾_𝑖𝑗(𝑠)𝑥_𝑗(𝑡−𝑠)𝑑𝑠−𝑏_𝑖(𝑡)₀^+∞𝐻_𝑖(𝑠)𝑢_𝑖(),𝑢̇𝑡−𝑠𝑑𝑠_𝑖(𝑡)=−𝑐_𝑖(𝑡)𝑢_𝑖(𝑡)+𝑑_𝑖(𝑡)₀^+∞𝑅_𝑖(𝑠)𝑥_𝑖(𝑡−𝑠)𝑑𝑠,𝑖=1,2,3, (66) where 𝑟₃(𝑡)=4+cos𝑡,𝑎₃₁(𝑡)=12,𝑎₃₂(𝑡)=27,𝑎₃₃(𝑡)=7+sin𝑡, (67) and the coefficients and the other kernels are as Example 2. In this case, we can choose 𝑥₁=4, 𝑥₂=4, 𝑥₃=5, 𝑢₁=3, 𝑢₂=4/3, 𝑢₃=25/12, and 𝜂=2𝜋 such that liminf_{𝑡→∞𝑡}^𝑡+2𝜋𝑟₁(𝑠)−𝐴₁₂(𝑠)𝑥₂−𝐴₁₃(𝑠)𝑥₃−𝐵₁(𝑠)𝑢₁=2𝑑𝑠15𝜋>,liminf_{𝑡→∞𝑡}^𝑡+2𝜋𝑟₂(𝑠)−𝐴₂₁(𝑠)𝑥₁−𝐴₂₃(𝑠)𝑥₃−𝐵₂(𝑠)𝑢₂=𝑑𝑠1663𝜋>0,liminf_{𝑡→∞𝑡}^𝑡+2𝜋𝑟₃(𝑠)−𝐴₃₁(𝑠)𝑥₁−𝐴₃₂(𝑠)𝑥₂−𝐵₃(𝑠)𝑢₃=1𝑑𝑠21𝜋>0. (68) All conditions of Corollary 11 hold, so all the species 𝑥₁, 𝑥₂, and 𝑥₃ are persistent (see Figure 3).

Figure 3 

Dynamic behaviors of system (66). Here, we take the initial conditions 𝑥1(𝜃)≡𝑥2(𝜃)≡𝑥3(𝜃)≡𝑢1(𝜃)≡𝑢2(𝜃)≡𝑢3(𝜃)=0.3 for all 𝜃∈(−∞,0].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work is supported by the Foundation of Fujian Education Bureau (nos. JA12369 and JA13361).