Permanence in Multispecies Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulses

Abstract

This paper studies multispecies nonautonomous Lotka-Volterra competitive systems with delays and fixed-time impulsive effects. The sufficient conditions of integrable form on the permanence of species are established.

1. Introduction

In this paper, we consider the nonautonomous 𝑛-species Lotka-Volterra type competitive systems with delays and impulses 𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…, (1.1) where 𝑥_𝑖(𝑡) represents the population density of the 𝑖th species at time 𝑡, the functions 𝑎_𝑖(𝑡), 𝑏_𝑖(𝑡), 𝑎_𝑖𝑗(𝑡), and 𝜏_𝑖𝑗(𝑡)(𝑖,𝑗=1,2,…,𝑛) are bounded and continuous functions defined on 𝑅₊=[0,+∞), 𝑎_𝑖𝑗(𝑡)≥0, 𝑏_𝑖(𝑡)≥0, 𝜏_𝑖𝑗(𝑡)≥0 for all 𝑡∈𝑅₊, and impulsive coefficients ℎ_𝑖𝑘 for any 𝑖=1,2,…,𝑛 and 𝑘=1,2,… are positive constants.

In particular, when the delays 𝜏_𝑖𝑗(𝑡)≡0 for all 𝑡∈𝑅₊ and 𝑖,𝑗=1,2,…,𝑛, then the system (1.1) degenerate into the following nondelayed non-autonomous 𝑛-species Lotka-volterra system𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑥_𝑗(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…, (1.2) where 𝑏_𝑖𝑖(𝑡)=𝑏_𝑖(𝑡)+𝑎_𝑖𝑖(𝑡) and 𝑏_𝑖𝑗(𝑡)=𝑎_𝑖𝑗(𝑡) for 𝑖,𝑗=1,2,…,𝑛 and 𝑖≠𝑗. For system (1.2), the author establish some new sufficient condition on the permanence of species and global attractivity in [1].

As we well know, systems like (1.1) and (1.2) without impulses are very important in the models of multispecies populations dynamics. Many important results on the permanence, extinction, global asymptotical stability for the two species or multi-species non-autonomous Lotka-Volterra systems and their special cases of periodic and almost periodic systems can be found in [2–14] and the references therein.

However, owing to many natural and man-made factors (e.g., fire, flooding, crop-dusting, deforestation, hunting, harvesting, etc.), the intrinsic discipline of biological species or ecological environment usually undergoes some discrete changes of relatively short duration at some fixed times. Such sudden changes can often be characterized mathematically in the form of impulses. In the last decade, much work has been done on the ecosystem with impulsive(see [1, 15–21] and the reference therein). Specially, the following system is considered in [22]:𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_{𝑗=1,𝑗≠𝑖}⁰_−∞𝑘_𝑗(𝑠)𝑥_𝑗(𝑡+𝑠)d𝑠,𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘=1,2,…. (1.3) The author establish some new sufficient conditions on the permanence of species and global attractivity for system (1.3). However, the effect of discrete delays on the possibility of species survival has been an important subject in population biology. We find that infinite delays are considered in the system (1.3). In this paper, it is very meaningful that discrete delays are proposed in the impulsive system (1.1).

2. Preliminaries

Let 𝜏=sup{𝜏_𝑖𝑗(𝑡),𝑡≥0,𝑖,𝑗=1,2,…,𝑛}. We define 𝐶^𝑛[−𝜏,0] the Banach space of bounded continuous function 𝜙∶[−𝜏,0]→𝑅^𝑛 with the supremum norm defined by:𝜙_𝑐=sup_{−𝜏≤𝑠≤0}𝜙(𝑠), (2.1) where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛), and |𝜙(𝑠)|=∑^𝑛_𝑖=1|𝜙_𝑖(𝑠)|. Define 𝐶^𝑛₊[−𝜏,0]={𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)∈𝐶^𝑛[−𝜏,0]∶𝜙_𝑖(𝑠)≥0, and 𝜙_𝑖(0)≥0 for all 𝑠∈[−𝜏,0] and 𝑖=1,2,…,𝑛}. Motivated by the biological background of system (1.1), we always assume that all solutions (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1.1) satisfy the following initial condition:𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠)∀𝑠∈[−𝜏,0],𝑖=1,2,…,𝑛, (2.2) where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)∈𝐶^𝑛₊[−𝜏,0].

It is obvious that the solution (𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1.1) with initial condition (2.2) is positive, that is, 𝑥_𝑖(𝑡)>0(𝑖=1,2,…,𝑛) on the interval of the existence and piecewise continuous with points of discontinuity of the first kind 𝑡_𝑘(𝑘∈𝑁) at which it is left continuous, that is, the following relations are satisfied:𝑥_𝑖𝑡⁻_𝑘=𝑥_𝑖𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑖=1,2,…,𝑛,𝑘∈𝑁. (2.3)

For system (1.1), we introduce the following assumptions:(H₁) functions 𝑎_𝑖(𝑡),𝑏_𝑖(𝑡),𝑎_𝑖𝑗(𝑡) and 𝜏_𝑖𝑗(𝑡) are bounded continuous on [0,+∞], and 𝑏_𝑖(𝑡), 𝑎_𝑖𝑗(𝑡) and 𝜏_𝑖𝑗(𝑡)(𝑖,𝑗=1,2,…,𝑛) are nonnegative for all 𝑡≥0.(H₂) for each 1≤𝑖≤𝑛, there are positive constants 𝜔_𝑖>0 such that liminf_𝑡→∞^{𝑡+𝜔𝑖}_𝑡𝑏_𝑖(𝑠)d𝑠>0, (2.4) and the functions ℎ_𝑖𝑡,𝜇=_{𝑡≤𝑡𝑘<𝑡+𝜇}lnℎ_𝑖𝑘 (2.5) are bounded for all 𝑡∈𝑅₊ and 𝜇∈[0,𝜔_𝑖].

First, we consider the following impulsive logistic system𝑥^′(𝑡)=𝑥(𝑡)𝛼(𝑡)−𝛽(𝑡)𝑥(𝑡),𝑡≠𝑡_𝑘,𝑥𝑡⁺_𝑘=ℎ_𝑘𝑥𝑡_𝑘,𝑘=1,2,…, (2.6) where 𝛼(𝑡) and 𝛽(𝑡) are bounded and continuous functions defined on 𝑅₊, 𝛽(𝑡)≥0 for all 𝑡∈𝑅₊, and impulsive coefficients ℎ_𝑘 for any 𝑘=1,2,… are positive constants. We have the following results.

Lemma 2.1. Suppose that there is a positive constant 𝜔 such that liminf_{𝑡→∞𝑡}^𝑡+𝑤𝛽(𝑠)d𝑠>0,liminf_{𝑡→∞𝑡}^𝑡+𝑤𝛼(𝑠)d𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑘>0, (2.7) and function ℎ𝑡,𝜇=_{𝑡≤𝑡𝑘<𝑡+𝜔}lnℎ_𝑘 (2.8) is bounded on 𝑡∈𝑅₊ and 𝜇∈[0,𝜔]. Then we have(a)there exist positive constants 𝑚 and 𝑀 such that 𝑚≤liminf_𝑡→∞𝑥(𝑡)≤limsup_𝑡→∞𝑥(𝑡)≤𝑀, (2.9) for any positive solution 𝑥(𝑡) of system (2.6);(b)lim_𝑡→∞(𝑥⁽¹⁾(𝑡)−𝑥⁽²⁾(𝑡))=0 for any two positive solutions 𝑥⁽¹⁾(𝑡) and 𝑥⁽²⁾(𝑡) of system (2.6).

The proof of Lemma 2.1 can be found as Lemma  2.1 in [1] by Hou et al.

On the assumption (H₂), we firstly have the following result.

Lemma 2.2. If assumption (H₂) holds, then there exist constants 𝑑>0 and 𝐷>0 such that for any 𝑡₂≥𝑡₁≥0_{𝑡1≤𝑡𝑘<𝑡2}lnℎ_𝑖𝑘≤𝑑𝑡₂−𝑡₁+𝐷,𝑖=1,2,…,𝑛. (2.10)

The proof of Lemma 2.2 is simple, we hence omit it here.

3. Main Results

Let 𝑥_𝑖0(𝑡) be some fixed positive solution of the following impulsive logistic systems as the subsystems of system (1.1):𝑥^′_𝑖(𝑡)=𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘=1,2,…. (3.1) On the permanence of all species 𝑥_𝑖(𝑖=1,2,…,𝑛) for system (1.1), we have the following result.

Theorem 3.1. Suppose that assumptions (H₁)-(H₂) hold. If there exist positive constants 𝜔_𝑖 such that for each 1≤𝑖≤𝑛: liminf_𝑡→∞^{𝑡+𝜔𝑖}_𝑡𝑎_𝑖(𝑠)−^𝑛_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑥_𝑗0𝑠−𝜏_𝑖𝑗(𝑠)d𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔𝑖}lnℎ_𝑖𝑘>0, (3.2) and the functions ℎ_𝑖𝑡,𝜇=_{𝑡≤𝑡𝑘<𝑡+𝜇}lnℎ_𝑖𝑘 (3.3) are bounded for all 𝑡∈𝑅₊ and 𝜇∈[0,𝜔_𝑖]. Then the system (1.1) is permanent, that is, there are positive constants 𝛾>0 and 𝑀>0 such that 𝛾≤liminf_𝑡→∞𝑥_𝑖(𝑡)≤limsup_𝑡→∞𝑥_𝑖(𝑡)≤𝑀,𝑖=1,2,…,𝑛, (3.4) for any positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1.1).

Proof. Let 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) be any positive solution of system (1.1). We first prove that the components 𝑥_𝑖(𝑖=1,2,…,𝑛) of system (1.1) are bounded. From assumption (H₁) and the 𝑖th equation of system (1.1), we have 𝑥^′_𝑖(𝑡)≤𝑥_𝑖(𝑡)𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡),𝑡≠𝑡_𝑘,𝑥_𝑖𝑡⁺_𝑘=ℎ_𝑖𝑘𝑥_𝑖𝑡_𝑘,𝑘∈𝑁. (3.5) by the comparison theorem of impulsive differential equation, we have 𝑥_𝑖(𝑡)≤𝑦_𝑖(𝑡),∀𝑡≥0, (3.6) where 𝑦_𝑖(𝑡) is the solution of (3.1) with initial value 𝑦_𝑖(0)=𝑥_𝑖(0). From the condition (3.2), we directly have liminf_𝑡→∞^{𝑡+𝜔𝑖}_𝑡𝑎_𝑖(𝑠)d𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔𝑖}lnℎ_𝑖𝑘>0,𝑖=1,2,…,𝑛. (3.7) Hence, from conclusion (a) of Lemma 2.1, we can obtain a constant 𝑀_𝑖1>0, and there is a 𝑇_𝑖1>0 such that 𝑦_𝑖(𝑡)<𝑀_𝑖1 for all 𝑡≥𝑇_𝑖1. Let 𝑀=max_{1≤𝑖≤𝑛}{𝑀_𝑖1} and 𝑇₁=max_{1≤𝑖≤𝑛}{𝑇_𝑖1}, we have 𝑥_𝑖(𝑡)≤𝑀,∀𝑡≥𝑇₁,𝑖=1,2,…,𝑛. (3.8) Hence, we finally have limsup_𝑡→∞𝑥(𝑡)≤𝑀. (3.9)

Next, we prove that there is a constant 𝛾>0 such that liminf_𝑡→∞𝑥(𝑡)≥𝛾,𝑖=1,2,…,𝑛. (3.10) For any 𝑡₁ and 𝑡₂ directly from system (1.1), we have 𝑥_𝑖𝑡₁=𝑥_𝑖𝑡₂exp^𝑡1_𝑡2𝑎_𝑖(𝑡)−𝑏_𝑖(𝑡)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑥_𝑗𝑡−𝜏_𝑖𝑗(𝑡)d𝑡+_{𝑡2≤𝑡𝑘≤𝑡1}lnℎ_𝑖𝑘. (3.11) From condition (3.2), we can choose constants 0<𝜀<1 small enough and 𝑇₂>0 large enough such that ^{𝑡+𝜔𝑖}_𝑡𝑎_𝑖(𝑠)−𝑏_𝑖(𝑠)+𝑎_𝑖𝑖(𝑠)𝜀−^𝑛_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑥_𝑗0𝑠−𝜏_𝑖𝑗(𝑠)+𝜀d𝑠+_{𝑡≤𝑡𝑘<𝑡+𝜔𝑖}lnℎ_𝑖𝑘>𝜀, (3.12) for all 𝑡≥𝑇₂ and 𝑖=1,2,…,𝑛. Considering (3.5), by the comparison theorem of impulsive differential equation and the conclusion (b) of Lemma 2.1., we obtain for the above 𝜀≥0 that there is a 𝑇₃>𝑇₂ such that 𝑥_𝑖(𝑡)≤𝑥_𝑖0(𝑡)+𝜀∀𝑡≥𝑇₃,𝑖=1,2,…,𝑛, (3.13) where 𝑥_𝑖0(𝑡) is a globally uniformly attractive positive solution of system (3.1).

Claim 1. There is a constant 𝜂>0 such that limsup_𝑡→∞𝑥_𝑖(𝑡)>𝜂(𝑖=1,2,…,𝑛) for any positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1.1). In fact, if Claim 1 is not true, then there is an integer 𝑘∈{1,2,…,𝑛} and a positive solution 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)) of system (1.1) such that limsup_𝑡→∞𝑥_𝑘(𝑡)<𝜀. (3.14) Hence, there is a constant 𝑇₄>𝑇₃ such that 𝑥_𝑘(𝑡)<𝜀∀𝑡≥𝑇₄. (3.15) On the other hand, by (3.13) there is a 𝑇₅≥𝑇₄ such that 𝑥_𝑖(𝑡)≤𝑥_𝑖0(𝑡)+𝜀∀𝑡≥𝑇₅, (3.16) where 𝑖=1,2,…,𝑛 and 𝑖≠𝑘. By (3.11) and (3.16), we obtain 𝑥_𝑘(𝑡)=𝑥_𝑘𝑇₅+𝜏exp^𝑡_𝑇5+𝜏𝑎_𝑘(𝑠)−𝑏_𝑘(𝑠)𝑥_𝑘(𝑡)−^𝑛_𝑗=1𝑎_𝑘𝑗(𝑠)𝑥_𝑗𝑠−𝜏_𝑖𝑗(𝑠)d𝑠+_{𝑇5+𝜏≤𝑡𝑘≤𝑡}lnℎ_𝑘𝑘≥𝑥_𝑘𝑇₅+𝜏exp^𝑡_𝑇5+𝜏𝑎_𝑘(𝑠)−𝑏_𝑘(𝑠)+𝑎_𝑘𝑘(𝑠)𝜀−^𝑛_{𝑗=1,𝑗≠𝑘}𝑎_𝑖𝑗(𝑠)𝑥_𝑗0𝑠−𝜏_𝑖𝑗(𝑠)+𝜀d𝑠+_{𝑇5+𝜏≤𝑡𝑘≤𝑡}lnℎ_𝑘𝑘, (3.17) for all 𝑡≥𝑇₅+𝜏. Thus, from (3.12) we finally obtain lim_𝑡→∞𝑥_𝑘(𝑡)=∞, which lead to a contradiction.

Claim 2. There is a constant 𝛾>0 such that liminf_𝑡→∞𝑥_𝑖(𝑡)>𝛾(𝑖=1,2,…,𝑛) for any positive solution of system (1.1).
If Claim 2 is not true, then there is an integer 𝑘∈{1,2,…,𝑛} and a sequence of initial function {𝜙_𝑚}⊂𝐶₊[−𝜏,0] such that liminf_𝑡→∞𝑥_𝑘𝑡,𝜙_𝑚<𝜂𝑚²∀𝑚=1,2,…, (3.18) where constant 𝜂 is given in Claim 1. By Claim 1, for every m there are two time sequences 𝑠_𝑞^(𝑚) and 𝑡_𝑞^(𝑚), satisfying: 0<𝑠₁^(𝑚)<𝑡₁^(𝑚)<𝑠₂^(𝑚)<𝑡₂^(𝑚)<⋯<𝑠_𝑞^(𝑚)<𝑡_𝑞^(𝑚)<⋯,lim_𝑞→∞𝑠_𝑞^(𝑚)=∞, (3.19) such that 𝑥_𝑘𝑠_𝑞^(𝑚),𝜙_𝑚≥𝜂𝑚,𝑥_𝑘𝑡_𝑞^(𝑚),𝜙_𝑚≤𝜂𝑚²,𝜂𝑚²≤𝑥_𝑘𝑡,𝜙_𝑚≤𝜂𝑚∀𝑡∈𝑠_𝑞^(𝑚),𝑡_𝑞^(𝑚). (3.20)(3.21) From the above proof, there is a constant 𝑇^(𝑚)≥𝑇₂ such that 𝑥_𝑖(𝑡,𝜙_𝑚)<𝑀(𝑖=1,2,…,𝑛) for all 𝑡≥𝑇^(𝑚). Further, there is an integer 𝐾₁^(𝑚)>0 such that 𝑠_𝑞^(𝑚)>𝑇^(𝑚) for all 𝑞>𝐾₁^(𝑚). From (3.11) and lemma 2.2., we can obtain 𝑥_𝑘𝑡_𝑞^(𝑚),𝜙_𝑚≥𝑥_𝑘𝑠_𝑞^(𝑚),𝜙_𝑚exp^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)}𝑎_𝑘(𝑠)−𝑏_𝑘(𝑠)𝑀−^𝑛_𝑗=1𝑎_𝑘𝑗(𝑠)𝑀d𝑠+_{𝑠𝑞(𝑚)≤𝑡𝑘≤𝑡𝑞(𝑚)}lnℎ_𝑘𝑘≥𝑥_𝑘𝑠_𝑞^(𝑚),𝜙_𝑚exp−𝑟₁+𝑑𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)−𝐷, (3.22) where 𝑟₁=sup_𝑡≥0{|𝑎_𝑖(𝑡)|+𝑏_𝑖(𝑡)𝑀+∑^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑀}. Consequently, from (3.20) we have 𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)≥ln𝑚−𝐷𝑟₁+𝑑∀𝑞>𝐾₁^(𝑚). (3.23) By (3.12), there is a large enough 𝑃>0 such that for all 𝑡≥𝑇₂, 𝑎≥𝑃 and 𝑎∈[𝑙𝑤_𝑖,(𝑙+1)𝑤_𝑖) and 𝑖=1,2,…,𝑛, then, we obtain _𝑡^𝑡+𝑎𝑎_𝑖(𝑠)−𝑏_𝑖(𝑠)+𝑎_𝑖𝑖(𝑠)𝜀−^𝑛_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑥_𝑗0𝑠−𝜏_𝑖𝑗(𝑠)+𝜀d𝑠+_{𝑡≤𝑡𝑘<𝑡+𝑎}lnℎ_𝑖𝑘=^{𝑡+𝑙𝑤𝑖}_𝑡𝑎_𝑖(𝑠)−𝑏_𝑖(𝑠)+𝑎_𝑖𝑖(𝑠)𝜀−^𝑛_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑥_𝑗0𝑠−𝜏_𝑖𝑗(𝑠)+𝜀d𝑠+_{𝑡≤𝑡𝑘<𝑡+𝑙𝑤𝑖}lnℎ_𝑖𝑘+^𝑡+𝑎_{𝑡+𝑙𝑤𝑖}𝑎_𝑖(𝑠)−𝑏_𝑖(𝑠)+𝑎_𝑖𝑖(𝑠)𝜀−^𝑛_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑥_𝑗0𝑠−𝜏_𝑖𝑗(𝑠)+𝜀d𝑠+_{𝑡+𝑙𝑤𝑖≤𝑡𝑘<𝑡+𝑎}lnℎ_𝑖𝑘>𝑙𝜀−𝑟₂𝑤_𝑖, (3.24) where 𝑟₂=sup_𝑡≥0{|𝑎_𝑖(𝑡)|+[𝑏_𝑖(𝑡)+𝑎_𝑖𝑖(𝑡)]𝜀+∑^𝑛_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)[𝑥_𝑗0(𝑠−𝜏_𝑖𝑗(𝑠))+𝜀]}. So, we choose 𝐿=2+(𝑟₂𝑤_𝑖/𝜀) such that for all 𝑙>𝐿, we have _𝑡^𝑡+𝑎𝑎_𝑖(𝑠)−𝑏_𝑖(𝑠)+𝑎_𝑖𝑖(𝑠)𝜀−^𝑛_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑥_𝑗0𝑠−𝜏_𝑖𝑗(𝑠)+𝜀d𝑠+_{𝑡≤𝑡𝑘<𝑡+𝑎}lnℎ_𝑖𝑘>𝜀. (3.25) From (3.23), there is an integer 𝑁₀ such that for any 𝑚>𝑁₀ and 𝑞>𝐾₁^(𝑚), we have 𝜂𝑚<𝜀,𝑡_𝑞^(𝑚)−𝑠_𝑞^(𝑚)>2𝑄, (3.26) where constant 𝑄>𝑃+𝜏.
So, when 𝑚>𝑁₀ and 𝑞>𝐾₁^(𝑚), for any 𝑡∈[𝑠_𝑞^(𝑚)+𝑄+𝜏,𝑡_𝑞^(𝑚)], from (3.11), (3.21), (3.25), and (3.26) we can obtain 𝑥_𝑘𝑡_𝑞^(𝑚),𝜙_𝑚=𝑥_𝑘𝑠_𝑞^(𝑚)+𝑄+𝜏,𝜙_𝑚×exp^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)+𝑄+𝜏}𝑎_𝑘(𝑡)−𝑏_𝑘(𝑡)𝑥_𝑘𝑡,𝜙_𝑚−^𝑛_𝑗=1𝑎_𝑘𝑗(𝑡)𝑥_𝑗0𝑡−𝜏_𝑘𝑗(𝑡),𝜙_𝑚d𝑡+_{𝑠𝑞(𝑚)+𝑄+𝜏≤𝑡𝑘≤𝑡𝑞(𝑚)}lnℎ_𝑘𝑘 (3.27) Consequently, from (3.20) and (3.25) it follows 𝜂𝑚²≥𝜂𝑚²×exp^{𝑡𝑞(𝑚)}_{𝑠𝑞(𝑚)+𝑄+𝜏}𝑎_𝑘(𝑡)−𝑏_𝑘(𝑡)+𝑎_𝑘𝑘(𝑡)𝜀−^𝑛_{𝑗=1,𝑗≠𝑘}𝑎_𝑘𝑗(𝑡)𝑥_𝑗0𝑡−𝜏_𝑘𝑗(𝑡)+𝜀d𝑡+_{𝑠𝑞(𝑚)+𝑄+𝜏≤𝑡𝑘≤𝑡𝑞(𝑚)}lnℎ_𝑘𝑘>𝜂𝑚². (3.28) This leads to a contradiction. Therefore, Claim 2 is true. This completes the proof.
When system (1.1) degenerates into the periodic case, then we can assume that there is a constant 𝜔>0 and an integer 𝑞>0 such that 𝑎_𝑖(𝑡+𝜔)=𝑎_𝑖(𝑡), 𝑏_𝑖(𝑡+𝜔)=𝑏_𝑖(𝑡), 𝑎_𝑖𝑗(𝑡+𝜔)=𝑎_𝑖𝑗(𝑡), 𝑡_𝑘+𝑞=𝑡_𝑘+𝜔 and ℎ_{𝑖𝑘+𝑞}=ℎ_𝑖𝑘 for all 𝑡∈𝑅₊, 𝑘=1,2,… and 𝑖,𝑗=1,2,…,𝑛. From Remarks2.3 and  2.4 in [1], we can see the fixed positive solution 𝑥_𝑗0 of system (3.1) can be chosen to be the 𝜔-periodic solution of system (3.1). Therefore, as a consequence of Theorem 3.1. we have the following result.

Corollary 3.2. Suppose that system (1.1) is 𝜔-periodic and for each 𝑖=1,2,…,𝑛, ^𝜔₀𝑏_𝑖(𝑠)d𝑠>0,_𝜔0𝑎_𝑖(𝑠)−^𝑛_𝑗≠𝑖𝑎_𝑖𝑗(𝑠)𝑥_𝑗0𝑠−𝜏_𝑖𝑗(𝑠)d𝑠+^𝑞_𝑘=1lnℎ_𝑖𝑘>0. (3.29) Then, system (1.1) is permanent.

4. Numerical Example

In this section, we will give an example to demonstrate the effectiveness of our main results. We consider the following two species competitive system with delays and impulses:𝑥^′₁(𝑡)=𝑥₁(𝑡)𝑎₁(𝑡)−𝑏₁(𝑡)𝑥₁(𝑡)−𝑎₁₁(𝑡)𝑥₁𝑡−𝜏₁₁(𝑡)−𝑎₁₂(𝑡)𝑥₂𝑡−𝜏₁₂(𝑡),𝑥^′₂(𝑡)=𝑥₂(𝑡)𝑎₂(𝑡)−𝑏₂(𝑡)𝑥₂(𝑡)−𝑎₂₁(𝑡)𝑥₁𝑡−𝜏₂₁(𝑡)−𝑎₂₂(𝑡)𝑥₂𝑡−𝜏₂₂(𝑡),𝑡≠𝑡_𝑘𝑥₁𝑡⁺_𝑘=ℎ_1𝑘𝑥₁𝑡_𝑘,𝑥₂𝑡⁺_𝑘=ℎ_2𝑘𝑥₂𝑡_𝑘,𝑘=1,2,…. (4.1) We take 𝑎₁(𝑡)=2, 𝑎₂(𝑡)=𝑏₁(𝑡)=𝑏₂(𝑡)=𝑎₁₁(𝑡)=𝑎₁₂(𝑡)=𝑎₂₂(𝑡)=1, 𝑎₂₁=1−|sin(𝜋/2)𝑡|, 𝜏_𝑖𝑗(𝑡)=2, ℎ_1𝑘=𝑒⁻¹, ℎ_2𝑘=𝑒, 𝑡_𝑘=𝑘. Obviously, system (4.1) is periodic with period 𝜔=2. For 𝑞=2, we have 𝑡_𝑘+𝑞=𝑡_𝑘+𝜔, ℎ_1𝑘+𝑞=ℎ_1𝑘 and ℎ_2𝑘+𝑞=ℎ_2𝑘 for all 𝑘=1,2,…. Consider the following impulsive logistic systems as the subsystems of system (4.1):𝑥^′₁(𝑡)=𝑥₁(𝑡)2−𝑥₁(𝑡),𝑥^′₂(𝑡)=𝑥₂(𝑡)1−𝑥₂(𝑡),𝑥𝑡≠𝑘₁𝑡⁺=𝑒⁻¹𝑥₁𝑡_𝑘,𝑥₂𝑡⁺=𝑒𝑥₂𝑡_𝑘,𝑡=𝑘. (4.2) According to the formula in [1], we can obtain that subsystem (4.2) has a unique globally asymptotically stable positive 2-periodic solution (𝑥₁₀(𝑡),𝑥₂₀(𝑡)), which can be expressed in following form:𝑥₁₀(𝑡)=2𝑥₁₀𝑥₁₀+2−𝑥₁₀𝑒^{−2(𝑡−𝑘)}[)𝑥,𝑡∈𝑘,𝑘+1,𝑘=0,1,2,…,₂₀(𝑡)=𝑥₂₀𝑥₂₀+1−𝑥₂₀𝑒^{−(𝑡−𝑘)},𝑡∈[𝑘,𝑘+1),𝑘=0,1,2,…, (4.3) where 𝑥₁₀=(2(𝑒^−0.2−𝑒⁻²)/(1−𝑒⁻²)) and 𝑥₂₀=(𝑒−𝑒⁻¹)/(1−𝑒⁻¹). Since^𝜔₀𝑎₁(𝑡)−𝑎₁₂(𝑡)𝑥₂₀𝑡−𝜏₁₂(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_1𝑘=2¹₀𝑥2−²⁰𝑥₂₀+1−𝑥₂₀𝑒^{−(𝑡−2)}d𝑡+²_𝑘=1lnℎ_1𝑘≈1.5244,_𝜔0𝑎₂(𝑡)−𝑎₂₁(𝑡)𝑥₁₀𝑡−𝜏₂₁(𝑡)d𝑡+^𝑞_𝑘=1lnℎ_2𝑘=2¹₀𝜋1−1−sin2𝑡2𝑥₁₀𝑥₁₀+2−𝑥₁₀𝑒^{−2(𝑡−2)}d𝑡+²_𝑘=1lnℎ_2𝑘≈3.8398, (4.4) we obtain that all conditions in Corollary 3.2 for system (1.1) holds. Therefore, from Theorem 3.1. we see that system (1.1) is permanent (see Figure 1).

Figure 1 

Time series of 𝑥1(𝑡) and 𝑥2(𝑡).

Acknowledgments

This paper was supported by the National Sciences Foundation of China (11071283), the Sciences Foundation of Shanxi (2009011005-3), the Young foundation of Shanxi province (no. 2011021001-1), research project supported by Shanxi Scholarship Council of China (2011-093), the Major Subject Foundation of Shanxi, and Doctoral Scientific Research fund of Xinjiang Medical University.