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Discrete Dynamics in Nature and Society
Volume 2008, Article ID 310425, 22 pages
http://dx.doi.org/10.1155/2008/310425
Research Article

Dynamic Behaviors of a General Discrete Nonautonomous System of Plankton Allelopathy with Delays

College of Mathematics and Computer Science, Fuzhou University, Fuzhou, Fujian 350002, China

Received 6 May 2008; Revised 8 September 2008; Accepted 22 November 2008

Academic Editor: Juan Jose Nieto

Copyright © 2008 Yaoping Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the dynamic behaviors of a general discrete nonautonomous system of plankton allelopathy with delays. We first show that under some suitable assumption, the system is permanent. Next, by constructing a suitable Lyapunov functional, we obtain a set of sufficient conditions which guarantee the global attractivity of the two species. After that, by constructing an extinction-type Lyapunov functional, we show that under some suitable assumptions, one species will be driven to extinction. Finally, two examples together with their numerical simulations show the feasibility of the main results.

1. Introduction

The aim of this paper is to investigate the dynamic behaviors of the following general discrete nonautonomous system of plankton allelopathy with delay: 𝑁1(𝑘+1)=𝑁1(𝑘)exp[𝑟1(𝑘)𝑚𝑙=0𝑎1𝑙(𝑘)𝑁1(𝑘𝑙)𝑚𝑙=0𝑏1𝑙(𝑘)𝑁2(𝑘𝑙)𝑚𝑙=0𝑐1𝑙(𝑘)𝑁1(𝑘)𝑁2𝑁(𝑘𝑙)],2(𝑘+1)=𝑁2(𝑘)exp[𝑟2(𝑘)𝑚𝑙=0𝑎2𝑙(𝑘)𝑁2(𝑘𝑙)𝑚𝑙=0𝑏2𝑙(𝑘)𝑁1(𝑘𝑙)𝑚𝑙=0𝑐2𝑙(𝑘)𝑁2(𝑘)𝑁1(𝑘𝑙)](1.1)together with the initial condition𝑁𝑖(𝑙)0,𝑁𝑖(0)>0,𝑖=1,2;𝑙=0,1,,𝑚,(1.2)where 𝑚 is a positive integer, 𝑁𝑖(𝑘) represent the densities of population 𝑖 at the 𝑘th generation, 𝑟𝑖(𝑘) are the intrinsic growth rate of population 𝑖 at the 𝑘th generation, 𝑎𝑖𝑙(𝑘) measure the intraspecific influence of the (𝑘𝑙)th generation of population 𝑖 on the density of own population, 𝑏𝑖𝑙(𝑘) stand for the interspecific influence of the (𝑘𝑙)th generation of population 𝑖 on the density of own population, and 𝑐𝑖𝑙(𝑘) stand for the effect of toxic inhibition of population 𝑖 by population 𝑗 at the (𝑘𝑙)th generation, 𝑖,𝑗=1,2 and 𝑖𝑗. Also, {𝑟𝑖(𝑘)},{𝑎𝑖𝑙(𝑘)},{𝑏𝑖𝑙(𝑘)} and {𝑐𝑖𝑙(𝑘)} are all bounded nonnegative sequences defined for 𝑘𝑁, denoted by the set of all nonnegative integers, and 𝑙{0,1,,𝑚} such that0<𝑟𝐿𝑖𝑟𝑖(𝑘)𝑟𝑀𝑖,0<𝑎𝐿𝑖𝑙𝑎𝑖𝑙(𝑘)𝑎𝑀𝑖𝑙,0<𝑏𝐿𝑖𝑙𝑏𝑖𝑙(𝑘)𝑏𝑀𝑖𝑙,0<𝑐𝐿𝑖𝑙𝑐𝑖𝑙(𝑘)𝑐𝑀𝑖𝑙,(1.3)here, for any bounded sequence {𝑓(𝑘)}, define𝑓𝑀=sup𝑘𝑁𝑓(𝑘),𝑓𝐿=inf𝑘𝑁𝑓(𝑘).(1.4)

As was pointed out by Chattopadhyay [1] the effects of toxic substances on ecological communities are an important problem from an environmental point of view. Chattopadhyay [1] and Maynard-Smith [2] proposed the following two species Lotka-Volterra competition system, which describes the changes of size and density of phytoplankton:𝑑𝑥1(𝑡)𝑑𝑡=𝑥1𝑟(𝑡)1𝑎11𝑥1(𝑡)𝑎12𝑥2(𝑡)𝑏1𝑥1(𝑡)𝑥2,(𝑡)𝑑𝑥2(𝑡)𝑑𝑡=𝑥2𝑟(𝑡)2𝑎21𝑥1(𝑡)𝑎22𝑥2(𝑡)𝑏2𝑥1(𝑡)𝑥2,(𝑡)(1.5)where 𝑥1(𝑡) and 𝑥2(𝑡) denote the population density of two competing species at time 𝑡 for a common pool of resources. The terms 𝑏1𝑥1(𝑡)𝑥2(𝑡) and 𝑏2𝑥1(𝑡)𝑥2(𝑡) denote the effect of toxic substances. Here, they made the assumption that each species produces a substance toxic to the other, only when the other is present. Noticing that the production of the toxic substance allelopathic to the competing species will not be instantaneous, but delayed by different discrete time lags required for the maturity of both species, thus, Mukhopadhyay et al. [3] also incorporated the discrete time delay into the above system. Tapaswi and Mukhopadhyay [4] also studied a two-dimensional system that arises in plankton allelopathy involving discrete time delays and environmental fluctuations. They assumed that the environmental parameters are assumed to be perturbed by white noise characterized by a Gaussian distribution with mean zero and unit spectral density. They focus on the dynamic behavior of the stochastic system and the fluctuations in population. For more works on system (1.5), one could refer to [13, 524] and the references cited therein.

Since the discrete time models governed by difference equations are more appropriate than the continuous ones when the populations have nonoverlapping generations, and discrete time models can also provide efficient computational models of continuous models for numerical simulations, corresponding to system (1.5), Huo and Li [25] argued that it is necessary to study the following discrete two species competition system:𝑥1(𝑘+1)=𝑥1𝑟(𝑘)exp1(𝑘)𝑎11(𝑘)𝑥1(𝑘)𝑎12(𝑘)𝑥2(𝑘)𝑏1(𝑘)𝑥1(𝑘)𝑥2,𝑥(𝑘)2(𝑘+1)=𝑥2𝑟(𝑘)exp2(𝑘)𝑎21(𝑘)𝑥1(𝑘)𝑎22(𝑘)𝑥2(𝑘)𝑏2(𝑘)𝑥1(𝑘)𝑥2,(𝑘)(1.6)where 𝑥1(𝑘) and 𝑥2(𝑘) are the population sizes of the two competitors at generation 𝑘,𝑏1(𝑘) and 𝑏2(𝑘) have respectively, shown that each species produces a toxic substance to the other but the other only is present. In [25], sufficient conditions were obtained to guarantee the permanence of the above system, they also investigated the existence and stability property of the positive periodic solution of system (1.6). Recently, Li and Chen [26] further investigated the dynamic behaviors of the system (1.6). For general nonatonomous case, they obtain a set of sufficient conditions which guarantee the extinction of species 𝑥2 and the global stability of species 𝑥1 when species 𝑥2 is eventually extinct. For periodic case, the other set of sufficient conditions, which concerned with the average condition of the coefficients of he system, were obtained to ensure the eventual extinction of species 𝑥2 and the global stability of positive periodic solution of species 𝑥1 when species 𝑥2 is eventually extinct. For more works on discrete population dynamics, one could refer to [7, 10, 2545].

Liu and Chen [32] argued that for a more realistic model, both seasonality of the changing environment and some of the past states, that is, the effects of time delays, should be taken into account in a model of multiple species growth. They proposed and studied the system (1.1), which is more general than system (1.6). By applying the coincidence degree theory, they obtained a set of sufficient conditions for the existence of at least one positive periodic solution of system (1.1)-(1.2). Zhang and Fang [46] also investigated the periodic solution of the system (1.1), they showed that under some suitable assumption, system (1.1) could admit at least two positive periodic solution. As we can see, the works [32, 46] are all concerned with the positive periodic solution of the system. However, since few things in the nature are really periodic, it is nature to study the general nonautonomous system (1.1), in this case, it is impossible to study the periodic solution of the system, however, such topics as permanence, extinction, and stability become the most important things. In this paper, we will further investigate the dynamics behaviors of the system (1.1). More precisely, by developing the analysis technique of Liu [31] and Muroya [35, 36], we study the permanence, global attractivity and extinction of system (1.1)-(1.2).

The organization of this paper is as follows. We study the persistence property of the system in Section 2 and the stability property in Section 3. Then in Section 4, by constructing a suitable Lyapunov functional, sufficient conditions which ensure the extinction of species 𝑁2 of system (1.1)-(1.2) are studied. In Section 5, two examples together with their numeric simulations show the feasibility of main results. For more relevant works, one could refer to [2, 3, 59, 12, 13, 2730, 33, 34, 3745] and the references cited therein.

2. Permanence

In this section, we study the persistent property of system (1.1)-(1.2).

Lemma 2.1. For any positive solution {(𝑁1(𝑘),𝑁2(𝑘))} of system (1.1)-(1.2), limsup𝑘𝑁𝑖(𝑘)𝐵𝑖,𝑖=1,2,(2.1)where 𝐵𝑖def=𝑟exp𝑀𝑖1𝑎𝐿𝑖0,𝑖=1,2.(2.2)

Proof. Let {(𝑁1(𝑘),𝑁2(𝑘))} be any positive solution of system (1.1)-(1.2), in view of the system (1.1) for all 𝑘𝑁, we have𝑁𝑖(𝑘+1)𝑁𝑖𝑟(𝑘)exp𝑖(𝑘)𝑎𝑖0(𝑘)𝑁𝑖(𝑘),𝑖=1,2.(2.3)Applying Lemma 2.1 of Yang [44] to (2.3), we can obtainlimsup𝑘𝑁𝑖𝑟(𝑘)exp𝑀𝑖1𝑎𝐿𝑖0def=𝐵𝑖,𝑖=1,2.(2.4)This completes the proof of Lemma 2.1.

Lemma 2.2. Assume that Δ11def=𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵2Δ>0,21def=𝑟𝐿2𝑚𝑙=1𝑎𝑀2𝑙𝐵2𝑚𝑙=0𝑏𝑀2𝑙+𝑐𝑀2𝑙𝐵2𝐵1>0(2.5)hold, where 𝐵1 and 𝐵2 are defined in (2.2). Then for any positive solution {(𝑁1(𝑘),𝑁2(𝑘))} of system (1.1)-(1.2), liminf𝑘𝑁𝑖(𝑘)𝐴𝑖,𝑖=1,2,(2.6)where 𝐴𝑖=Δ𝑖1𝑎𝑀𝑖0exp[Δ𝑖2Δ],𝑖=1,2,12=𝑟𝐿1𝑚𝑙=0𝑎𝑀1𝑙𝐵1𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵2,Δ22=𝑟𝐿2𝑚𝑙=0𝑎𝑀2𝑙𝐵2𝑚𝑙=0𝑏𝑀2𝑙+𝑐𝑀2𝑙𝐵2𝐵1.(2.7)

Proof. In view of (2.5), we can choose a constant 𝜀>0 small enough such that𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2𝑟+𝜀>0,(2.8)𝐿2𝑚𝑙=1𝑎𝑀2𝑙𝐵2+𝜀𝑚𝑙=0𝑏𝑀2𝑙+𝑐𝑀2𝑙𝐵2𝐵+𝜀1+𝜀>0.(2.9)𝜀>0,In view of (2.1), for above 𝑘0𝑁 there exists an integer 𝑁𝑖(𝑘)𝐵𝑖+𝜀𝑘𝑘0,𝑖=1,2.(2.10) such that𝑙0𝑘0+𝑚We consider the following two cases.
Case (i). We assume that there exists an integer 𝑁1(𝑙0+1)𝑁1(𝑙0). such that 𝑁1𝑙0+1=𝑁1𝑙0exp[𝑟1𝑙0𝑚𝑙=0𝑎1𝑙𝑙0𝑁1𝑙0𝑙𝑚𝑙=0𝑏1𝑙𝑙0𝑁2𝑙0𝑙𝑚𝑙=0𝑐1𝑙𝑙0𝑁1𝑙0𝑁2𝑙0]𝑙𝑁1𝑙0exp{𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀𝑎𝑀10𝑁1𝑙0}.(2.11) Note that𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀𝑎𝑀10𝑁1𝑙00.(2.12)So we can obtain𝑁1𝑙0𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀𝑎𝑀10>0.(2.13)It follows from (2.8) that𝑁1𝑙0𝑟+1𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀𝑎𝑀10×exp{𝑟𝐿1𝑚𝑙=0𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀}.(2.14)Then we have𝑁1𝜀=𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀𝑎𝑀10×exp{𝑟𝐿1𝑚𝑙=0𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀}.(2.15)Let𝐵1=𝑟exp𝑀11𝑎𝐿10𝑟𝑀1𝑎𝐿10𝑟𝐿1𝑎𝑀10,(2.16)Note that𝑟𝐿1𝑎𝑀10𝐵10,thus 𝜀>0, and so, for above 𝑟𝐿1𝑚𝑙=0𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀<𝑟𝐿1𝑎𝑀10𝐵1+𝜀<𝑟𝐿1𝑎𝑀10𝐵10(2.17)𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀𝑎𝑀10𝑁1𝜀.(2.18)or𝑁1(𝑘)𝑁1𝜀𝑘𝑙0.(2.19)
We can claim that𝑝0𝑙0By way of contradiction, assume that there exists an integer 𝑁1(𝑝0)<𝑁1𝜀. such that 𝑝0𝑙0+2. Then 𝑝0𝑙0+2 Let 𝑁1(𝑝0)<𝑁1𝜀. be the smallest integer such that 𝑁1(𝑝01)>𝑁1(𝑝0). Then 𝑁1(𝑝0)𝑁1𝜀, The above argument produces that 𝑁1(𝑘+1)>𝑁1(𝑘) a contradiction. Thus (2.19) proved.
Case (ii). We assume that 𝑘𝑘0+𝑚, for all lim𝑘𝑁1(𝑘) then 𝑁1. exists, denoted by 𝑁1𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀𝑎𝑀10.(2.20) We can claim that𝑁1<𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀𝑎𝑀10.(2.21)By the way of contradiction, assume thatlim𝑘[𝑟1(𝑘)𝑚𝑙=0𝑎1𝑙(𝑘)𝑁1(𝑘𝑙)𝑚𝑙=0𝑏1𝑙(𝑘)𝑁2(𝑘𝑙)𝑚𝑙=0𝑐1𝑙(𝑘)𝑁1(𝑘)𝑁2(𝑘𝑙)]=0,(2.22)Taking limit in the first equation of (1.1) giveslim𝑘[𝑟1(𝑘)𝑚𝑙=0𝑎1𝑙(𝑘)𝑁1(𝑘𝑙)𝑚𝑙=0𝑏1𝑙(𝑘)𝑁2(𝑘𝑙)𝑚𝑙=0𝑐1𝑙(𝑘)𝑁1(𝑘)𝑁2(𝑘𝑙)]𝑟𝐿1𝑚𝑙=1𝑎𝑀1𝑙𝐵1+𝜀𝑎𝑀10𝑁1𝑚𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵+𝜀2+𝜀>0.(2.23)which is a contradiction since𝑁1𝑁1𝜀.(2.24)The claim is thus proved.
From (2.20), we see thatliminf𝑘𝑁1(𝑘)𝑁1𝜀.(2.25)Combining Cases (i) and (ii), we see that𝜀0,Setting lim𝜀0𝑁1𝜀=Δ11𝑎𝑀10Δexp12def=𝐴1.(2.26) it follows thatliminf𝑘𝑁1(𝑘)𝐴1.(2.27)
So we can easily see thatliminf𝑘𝑁2(𝑘)𝐴2,(2.28)From the second equation of (1.1), similar to above analysis, we have𝐴2where 𝜂>0 is defined in (2.6). This completes the proof of Lemma 2.2.

It immediately follows from Lemmas 2.1 and 2.2 that the following theorem holds.

Theorem 2.3. Assume that (2.5) hold, then system (1.1)-(1.2) is permanent.

3. Global attractivity

This section devotes to study the stability property of the positive solution of system (1.1)-(1.2).

Theorem 3.1. Assume that there exists a constant 𝑎min𝐿𝑖0,2𝐵𝑖𝑎𝑀𝑖02𝑗=1,𝑗𝑖[𝑚𝑙=1𝑎𝑀𝑖𝑙𝑏+(𝑚+1)𝑀𝑗+2𝐵𝑗𝑐𝑀]>𝜂,𝑖=1,2,(H0) such that 𝑖,𝑗=1,2,𝑖𝑗,𝐵𝑖where, for 𝐵𝑗 and 𝑏𝑀𝑗𝑏=max𝑀𝑗𝑙𝑙=0,1,,𝑚,𝑐𝑀𝑖𝑐=max𝑀𝑖𝑙,𝑐𝑙=0,1,,𝑚𝑀𝑐=max𝑀𝑖,𝑖=1,2(3.1) are defined in (2.2), {(𝑁1(𝑘),𝑁2(𝑘))}then for any two positive solutions {(𝑁1(𝑘),𝑁2(𝑘))} and lim𝑘𝑁𝑖(𝑘)𝑁𝑖(𝑘)=0,𝑖=1,2.(3.2) of system (1.1)-(1.2), 𝑉11|||(𝑘)=ln𝑁1(𝑘)ln𝑁1|||(𝑘).(3.3)

Proof. First, let𝑉11|||(𝑘+1)=ln𝑁1(𝑘+1)ln𝑁1||||||(𝑘+1)ln𝑁1(𝑘)ln𝑁1(𝑘)𝑎10𝑁(𝑘)1(𝑘)𝑁1|||+(𝑘)𝑚𝑙=1𝑎1𝑙|||𝑁(𝑘)1(𝑘𝑙)𝑁1|||+(𝑘𝑙)𝑚𝑙=0𝑏1𝑙|||𝑁(𝑘)2(𝑘𝑙)𝑁2|||+(𝑘𝑙)𝑚𝑙=0𝑐1𝑙|||𝑁(𝑘)1(𝑘)𝑁2(𝑘𝑙)𝑁1(𝑘)𝑁2|||.(𝑘𝑙)(3.4)Then from the first equation of (1.1), we have|||ln𝑁1(𝑘)ln𝑁1|||=1(𝑘)𝜃1|||𝑁(𝑘)1(𝑘)𝑁1|||(𝑘),(3.5)Noticing that by mean-value theory0<𝜃1(𝑘)max{𝑁1(𝑘),𝑁1(𝑘)}.where |||ln𝑁1(𝑘)ln𝑁1(𝑘)𝑎10𝑁(𝑘)1(𝑘)𝑁1|||=|||(𝑘)ln𝑁1(𝑘)ln𝑁1||||||(𝑘)ln𝑁1(𝑘)ln𝑁1|||+|||(𝑘)ln𝑁1(𝑘)ln𝑁1(𝑘)𝑎10𝑁(𝑘)1(𝑘)𝑁1|||=|||(𝑘)ln𝑁1(𝑘)ln𝑁1|||1(𝑘)𝜃1|||1(𝑘)𝜃1(𝑘)𝑎10||||||𝑁(𝑘)1(𝑘)𝑁1|||.(𝑘)(3.6) Then𝑉11|||(𝑘+1)ln𝑁1(𝑘)ln𝑁1|||1(𝑘)𝜃1|||1(𝑘)𝜃1(𝑘)𝑎10||||||𝑁(𝑘)1(𝑘)𝑁1|||+(𝑘)𝑚𝑙=1𝑎1𝑙|||𝑁(𝑘)1(𝑘𝑙)𝑁1|||+(𝑘𝑙)𝑚𝑙=0𝑏1𝑙|||𝑁(𝑘)2(𝑘𝑙)𝑁2|||+(𝑘𝑙)𝑚𝑙=0𝑐1𝑙|||𝑁(𝑘)1(𝑘)𝑁2(𝑘𝑙)𝑁1(𝑘)𝑁2|||.(𝑘𝑙)(3.7)Substituting (3.6) into (3.4) leads toΔ𝑉111𝜃1|||1(𝑘)𝜃1(𝑘)𝑎10||||||𝑁(𝑘)1(𝑘)𝑁1|||+(𝑘)𝑚𝑙=1𝑎1𝑙|||𝑁(𝑘)1(𝑘𝑙)𝑁1|||+(𝑘𝑙)𝑚𝑙=0𝑏1𝑙|||𝑁(𝑘)2(𝑘𝑙)𝑁2|||+(𝑘𝑙)𝑚𝑙=0𝑐1𝑙|||𝑁(𝑘)1(𝑘)𝑁2(𝑘𝑙)𝑁1(𝑘)𝑁2|||.(𝑘𝑙)(3.8)So it follows that𝜀>0,According to (2.1), for any constant 𝑘0𝑁 there exists an integer 𝑁1(𝑘)𝐵1+𝜀,𝑁2(𝑘)𝐵2+𝜀𝑘𝑘0.(3.9) such that𝑘𝑘0+𝑚,𝑙=0,1,,𝑚,So for all |||𝑁1(𝑘)𝑁2(𝑘𝑙)𝑁1(𝑘)𝑁2|||=|||𝑁(𝑘𝑙)1(𝑘)𝑁2(𝑘𝑙)𝑁1(𝑘)𝑁2(𝑘𝑙)+𝑁1(𝑘)𝑁2(𝑘𝑙)𝑁1(𝑘)𝑁2|||=|||𝑁(𝑘𝑙)1𝑁(𝑘)2(𝑘𝑙)𝑁2(𝑘𝑙)+𝑁2𝑁(𝑘𝑙)1(𝑘)𝑁1|||𝐵(𝑘)1|||𝑁+𝜀2(𝑘𝑙)𝑁2|||+𝐵(𝑘𝑙)2|||𝑁+𝜀1(𝑘)𝑁1|||.(𝑘)(3.10) it follows that𝑘𝑘0+𝑚,So for all Δ𝑉111(𝜃1|||1(𝑘)𝜃1(𝑘)𝑎10|||(𝑘)𝑚𝑙=0𝐵2𝑐+𝜀1𝑙|||𝑁(𝑘))1(𝑘)𝑁1|||+(𝑘)𝑚𝑙=1𝑎1𝑙|||𝑁(𝑘)1(𝑘𝑙)𝑁1|||+(𝑘𝑙)𝑚𝑙=0𝑏1𝑙𝐵(𝑘)+1𝑐+𝜀1𝑙|||𝑁(𝑘)2(𝑘𝑙)𝑁2|||.(𝑘𝑙)(3.11) it follows from (3.4) that𝑉12(𝑘)=𝑚𝑙=1𝑘1𝑠=𝑘𝑙𝑎1𝑙|||𝑁(𝑠+𝑙)1(𝑠)𝑁1|||+(𝑠)𝑚𝑙=0𝑘1𝑠=𝑘𝑙𝑏1𝑙𝐵(𝑠+𝑙)+1𝑐+𝜀1𝑙|||𝑁(𝑠+𝑙)2(𝑠)𝑁2|||,(𝑠)(3.12)Next, letΔ𝑉12=𝑉12(𝑘+1)𝑉12=(𝑘)𝑚𝑘𝑙=1𝑠=𝑘+1𝑙𝑎1𝑙|||𝑁(𝑠+𝑙)1(𝑠)𝑁1|||(𝑠)𝑚𝑙=1𝑘1𝑠=𝑘𝑙𝑎1𝑙|||𝑁(𝑠+𝑙)1(𝑠)𝑁1|||+(𝑠)𝑚𝑘𝑙=0𝑠=𝑘+1𝑙𝑏1𝑙𝐵(𝑠+𝑙)+1𝑐+𝜀1𝑙|||𝑁(𝑠+𝑙)2(𝑠)𝑁2|||(𝑠)𝑚𝑙=0𝑘1𝑠=𝑘𝑙𝑏1𝑙𝐵(𝑠+𝑙)+1𝑐+𝜀1𝑙|||𝑁(𝑠+𝑙)2(𝑠)𝑁2|||=(𝑠)𝑚𝑙=1𝑎1𝑙|||𝑁(𝑘+𝑙)1(𝑘)𝑁|||(𝑘)𝑚𝑙=1𝑎1𝑙|||𝑁(𝑘)1(𝑘𝑙)𝑁1|||+(𝑘𝑙)𝑚𝑙=0𝑏1𝑙𝐵(𝑘+𝑙)+1𝑐+𝜀1𝑙|||𝑁(𝑘+𝑙)2(𝑘)𝑁2|||(𝑘)𝑚𝑙=0𝑏1𝑙𝐵(𝑘)+1𝑐+𝜀1𝑙|||𝑁(𝑘)2(𝑘𝑙)𝑁2|||.(𝑘𝑙)(3.13)and we can obtain𝑉1Now, we define 𝑉1(𝑘)=𝑉11(𝑘)+𝑉12(𝑘).(3.14) by𝑘𝑘0+𝑚,So for all Δ𝑉1=Δ𝑉11+Δ𝑉121(𝜃1|||1(𝑘)𝜃1(𝑘)𝑎10|||(𝑘)𝑚𝑙=0𝐵2𝑐+𝜀1𝑙(𝑘)𝑚𝑙=1𝑎1𝑙|||𝑁(𝑘+𝑙))1(𝑘)𝑁1|||+(𝑘)𝑚𝑙=0𝑏1𝑙𝐵(𝑘+𝑙)+1𝑐+𝜀1𝑙|||𝑁(𝑘+𝑙)2(𝑘)𝑁2|||.(𝑘)(3.15) it follows from (3.6) and (3.9) that𝑉2(𝑘)=𝑉21(𝑘)+𝑉22(𝑘),(3.16)Similar to above arguments, we can define𝑉21|||(𝑘)=ln𝑁2(𝑘)ln𝑁2|||,𝑉(𝑘)22(𝑘)=𝑚𝑙=1𝑘1𝑠=𝑘𝑙𝑎2𝑙|||𝑁(𝑠+𝑙)2(𝑠)𝑁2|||+(𝑠)𝑚𝑙=0𝑘1𝑠=𝑘𝑙𝑏2𝑙𝐵(𝑠+𝑙)+2𝑐+𝜀2𝑙|||𝑁(𝑠+𝑙)1(𝑠)𝑁1|||.(𝑠)(3.17)where𝑘𝑘0+𝑚,Then for all Δ𝑉2=Δ𝑉21+Δ𝑉221(𝜃2|||1(𝑘)𝜃2(𝑘)𝑎20|||(𝑘)𝑚𝑙=0𝐵1𝑐+𝜀2𝑙(𝑘)𝑚𝑙=1𝑎2𝑙|||𝑁(𝑘+𝑙))2(𝑘)𝑁2|||+(𝑘)𝑚𝑙=0𝑏2𝑙𝐵(𝑘+𝑙)+2𝑐+𝜀2𝑙|||𝑁(𝑘+𝑙)1(𝑘)𝑁1|||,(𝑘)(3.18) we can obtain𝜃2(𝑘)where 𝑁2(𝑘) lies between 𝑁2(𝑘). and 𝑉
Now, we define 𝑉(𝑘)=𝑉1(𝑘)+𝑉2(𝑘).(3.19) by𝑉(𝑘)0It is easy to see that 𝑘𝑍 for all 𝑉(𝑘0+𝑚)<+. and 𝜀>0 For the arbitrariness of 𝜀>0 and by (H0), we can choose 𝑖=1,2, small enough such that for 𝑎min𝐿𝑖0,2𝐵𝑖+𝜀𝑎𝑀𝑖02𝑗=1,𝑗𝑖[𝑚𝑙=1𝑎𝑖𝑙𝑏(𝑘+𝑙)+(𝑚+1)𝑀𝑗𝐵+2𝑗𝑐+𝜀𝑀]>𝜂.(3.20)𝑘𝑘0+𝑚,So for all Δ𝑉2𝑖=1{1𝜃𝑖|||1(𝑘)𝜃𝑖(𝑘)𝑎𝑖0|||(𝑘)2𝑗=1,𝑗𝑖[𝑚𝑙=1𝑎𝑖𝑙(𝑘+𝑙)+𝑚𝑙=0𝑏𝑗𝑙𝐵(𝑘+𝑙)+𝑗𝑐+𝜀𝑖𝑙(𝑘)+𝑐𝑗𝑙×|||𝑁(𝑘+𝑙)]}𝑖(𝑘)𝑁𝑖|||(𝑘)2𝑖=1𝑎{min𝐿𝑖0,2𝐵𝑖+𝜀𝑎𝑀𝑖02𝑗=1,𝑗𝑖[𝑚𝑙=1𝑎𝑖𝑙𝑏(𝑘+𝑙)+(𝑚+1)𝑀𝑗𝐵+2𝑗𝑐+𝜀𝑀×|||𝑁]}𝑖(𝑘)𝑁𝑖|||(𝑘)𝜂2𝑖=1|||𝑁𝑖(𝑘)𝑁𝑖|||,(𝑘)(3.21) it follows from (3.15) and (3.18) that𝑘𝑝=𝑘0+𝑚𝑉(𝑝+1)𝑉(𝑝)𝜂𝑘𝑝=𝑘02+𝑚𝑖=1|||𝑁𝑖(𝑝)𝑁𝑖|||(𝑝),(3.22)So we have𝑉(𝑘+1)+𝜂𝑘𝑝=𝑘02+𝑚𝑖=1|||𝑁𝑖(𝑝)𝑁𝑖|||𝑘(𝑝)𝑉0+𝑚.(3.23)which implies𝑘𝑝=𝑘02+𝑚𝑖=1|||𝑁𝑖(𝑝)𝑁𝑖|||𝑉𝑘(𝑝)0+𝑚𝜂.(3.24)It follows that𝑘=𝑘02+𝑚𝑖=1|||𝑁𝑖(𝑘)𝑁𝑖|||𝑉𝑘(𝑘)0+𝑚𝜂<+,(3.25)Thenlim𝑘2𝑖=1|𝑁𝑖(𝑘)𝑁𝑖(𝑘)|=0,which implies that lim𝑘𝑁𝑖(𝑘)𝑁𝑖(𝑘)=0,𝑖=1,2.(3.26) that is, 𝑁2This completes the proof of Theorem 3.1.

4. Extinction of species 𝑁2.

This section devotes to study the extinction of the species {(𝑁1(𝑘),𝑁2(𝑘))}

Lemma 4.1. For any positive solution 𝜎>0 of system (1.1)-(1.2), there exists a constant liminf𝑘+𝑁1(𝑘)+𝑁2(𝑘)>𝜎.(4.1) such that 𝐵>0

Proof. By (2.1), there exists a constant 𝑁𝑖(𝑘)<𝐵𝑘>𝑘0,𝑖=1,2.(4.2) such that𝑘>𝑘0+𝑚,𝑖=1,2,In view of (1.1) for all 𝑁𝑖(𝑘)𝑁𝑖(𝑘+1)exp{𝑟𝐿𝑖+𝑚𝑙=0𝑎𝑀𝑖𝑙+𝑏𝑀𝑖𝑙+𝑐𝑀𝑖𝑙𝐵𝐵}.(4.3) it follows that𝑁𝑖(𝑘𝑙)𝑁𝑖(𝑘)exp{𝑙[𝑟𝐿𝑖+𝑚𝑙=0𝑎𝑀𝑖𝑙+𝑏𝑀𝑖𝑙+𝑐𝑀𝑖𝑙𝐵𝐵]},𝑙=0,1,,𝑚.(4.4)So we have𝐶1=max{exp{𝑙[𝑟𝐿𝑖+𝑚𝑙=0𝑎𝑀𝑖𝑙+𝑏𝑀𝑖𝑙+𝑐𝑀𝑖𝑙𝐵𝐶𝐵]}𝑖=1,2,𝑙=0,1,,𝑚},2𝑑=max𝑀𝑖𝑙𝐶1,𝑖=1,2,𝑙=0,1,,𝑚(4.5)Let𝑑𝑀𝑖𝑙=max{𝑎𝑀𝑖𝑙,𝑏𝑀𝑖𝑙+𝑐𝑀𝑖𝑙𝐵},where 𝑖=1,2,𝑙=0,1,,𝑚.𝑘>𝑘0+𝑚,𝑖,𝑗=1,2,𝑖𝑗, For all 𝑁𝑖(𝑘+1)𝑁𝑖(𝑘)exp{𝑟𝐿𝑖𝑚𝑙=0𝑎𝑀𝑖𝑙𝐶1𝑁𝑖(𝑘)𝑚𝑙=0𝑏𝑀𝑖𝑙+𝑐𝑀𝑖𝑙𝐵𝐶1𝑁𝑗(𝑘)}𝑁𝑖(𝑘)exp{𝑟𝐿𝑖𝑚𝑙=0𝑑𝑀𝑖𝑙𝐶1𝑁𝑖(𝑘)+𝑁𝑗}(𝑘)𝑁𝑖𝑟(𝑘)expmin𝐿1,𝑟𝐿2(𝑚+1)𝐶2𝑁1(𝑘)+𝑁2,(𝑘)(4.6) it follows from (1.1) and (4.4) that𝑁1(𝑘+1)+𝑁2𝑁(𝑘+1)1(𝑘)+𝑁2𝑟(𝑘)expmin𝐿1,𝑟𝐿2(𝑚+1)𝐶2𝑁1(𝑘)+𝑁2(𝑘).(4.7)so we have𝑥(𝑘)=𝑁1(𝑘)+𝑁2(𝑘),Let 𝑟𝑥(𝑘+1)𝑥(𝑘)expmin𝐿1,𝑟𝐿2(𝑚+1)𝐶2𝑟𝑥(𝑘)=𝑥(𝑘)exp{min𝐿1,𝑟𝐿2[1(𝑚+1)𝐶2𝑟min𝐿1,𝑟𝐿2𝑥(𝑘)]}def𝑟.=𝑥(𝑘)exp1𝑎𝑥(𝑘)(4.8) then we have𝑘>𝑘0,Note that for all 𝑥(𝑘)=𝑁1(𝑘)+𝑁2(𝑘)<2𝐵,liminf𝑘+1𝑥(𝑘)𝑎exp𝑟(12𝑎𝐵)>0.(4.9) so similar to the proof of Lemma 2.2 of Chen [27], we have𝜎>0Then, there is a positive constant liminf𝑘+𝑁1(𝑘)+𝑁2(𝑘)=liminf𝑘+1𝑥(𝑘)𝑎exp𝑟(12𝑎𝐵)>𝜎.(4.10) such that𝑚𝑙=0𝑟𝐿1𝑏𝐿2𝑙𝑟𝑀2𝑎𝑀1𝑙>0,𝑚𝑙=0𝑟𝐿1𝑎𝐿2𝑙𝑟𝑀2𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1>0,(H1)This completes the proof of Lemma 4.1.

Theorem 4.2. Assume that 𝐵1where {(𝑁1(𝑘),𝑁2(𝑘))} is defined in (2.2). Let 𝑁2(𝑘)0 be any positive solution of system (1.1)-(1.2), then 𝑘+. as 𝑙=0,1,,𝑚,

Corollary 4.3. Assume that for all 𝑟𝑀2𝑟𝐿1−min{𝑏𝐿2𝑙𝑎𝑀1𝑙,𝑎𝐿2𝑙𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1}<0() the following inequalities 𝐵1hold, where {(𝑁1(𝑘),𝑁2(𝑘))} is defined in (2.2). Let 𝑁2(𝑘)0 be any positive solution of system (1.1)-(1.2), then 𝑘+. as 𝜀>0

Proof of Corollary 4.3. Obviously, if condition (H1*) holds, one could easily see that condition (H1) holds, thus, the conclusion of Corollary 4.3 follows from Theorem 4.2. The proof is complete.

Proof of Theorem 4.2. It follows from (H1) that we can choose a constant 𝑚𝑙=0𝑟𝐿1𝑏𝐿2𝑙𝑟𝑀2𝑎𝑀1𝑙>0,𝑚𝑙=0𝑟𝐿1𝑎𝐿2𝑙𝑟𝑀2𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1+𝜀>0.(4.11) small enough such thatΔ𝜀=min{𝑚𝑙=0𝑟𝐿1𝑏𝐿2𝑙𝑟𝑀2𝑎𝑀1𝑙,𝑚𝑙=0𝑟𝐿1𝑎𝐿2𝑙𝑟𝑀2𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1+𝜀}>0.(4.12)Set𝜀>0For above 𝐾𝑁 from (2.1), there is an integer 𝑖=1,2, such that for 𝑁𝑖(𝑘)𝐵𝑖+𝜀𝑘>𝐾.(4.13)𝐾1>𝐾Lemma 4.1 also implies that there exists 𝑁1(𝑘)+𝑁2𝜎(𝑘)2𝑘𝐾1.(4.14) such that𝑁𝑢(𝑘)=𝑟𝐿12(𝑘)𝑁𝑟𝑀21(𝑘)exp{𝑚𝑙=0𝑘1𝑠=𝑘𝑙𝑟𝐿1𝑎𝐿2𝑙𝑁2(𝑠)+𝑏𝐿2𝑙𝑁1+(𝑠)𝑚𝑙=0𝑘1𝑠=𝑘𝑙𝑟𝑀2𝑎𝑀1𝑙𝑁1(𝑠)+𝑏𝑀1𝑙𝑁2(𝑠)+𝑐𝑀1𝑙𝐵1𝑁+𝜀2(𝑠)}.(4.15)Set𝑘>𝐾2>𝐾1+𝑚,So for all 𝑢(𝑘+1)𝑢(𝑘)=exp{𝑟𝐿1𝑟2(𝑘)𝑚𝑙=0𝑟𝐿1𝑎2𝑙(𝑘)𝑁2(𝑘𝑙)+𝑏2𝑙(𝑘)𝑁1(𝑘𝑙)+𝑐2𝑙(𝑘)𝑁2(𝑘)𝑁1(𝑘𝑙)(4.16)𝑟𝑀2𝑟1(𝑘)+𝑚𝑙=0𝑟𝑀2𝑎1𝑙(𝑘)𝑁1(𝑘𝑙)+𝑏1𝑙(𝑘)𝑁2(𝑘𝑙)+𝑐1𝑙(𝑘)𝑁1(𝑘)𝑁2(𝑘𝑙)(1)𝑚𝑘𝑙=0𝑠=𝑘+1𝑙𝑟𝐿1𝑎𝐿2𝑙𝑁2(𝑠)+𝑏𝐿2𝑙𝑁1+(𝑠)𝑚𝑙=0𝑘1𝑠=𝑘𝑙𝑟𝐿1𝑎𝐿2𝑙𝑁2(𝑠)+𝑏𝐿2𝑙𝑁1+(𝑠)(2)𝑚𝑘𝑙=0𝑠=𝑘+1𝑙𝑟𝑀2𝑎𝑀1𝑙𝑁1(𝑠)+𝑏𝑀1𝑙𝑁2(𝑠)+𝑐𝑀1𝑙𝐵1𝑁+𝜀2(𝑠)(3)𝑚𝑙=0𝑘1𝑠=𝑘𝑙𝑟𝑀2𝑎𝑀1𝑙𝑁1(𝑠)+𝑏𝑀1𝑙𝑁2(𝑠)+𝑐𝑀1𝑙𝐵1𝑁+𝜀2𝑟(𝑠)}(4)=exp{𝐿1𝑟2(𝑘)𝑟𝑀2𝑟1(𝑘)(5)𝑚𝑙=0𝑟𝐿1𝑎2𝑙(𝑘)𝑎𝐿2𝑙𝑁2𝑏(𝑘𝑙)+2𝑙(𝑘)𝑏𝐿2𝑙𝑁1(𝑘𝑙)(6)𝑚𝑙=0𝑟𝐿1𝑐2𝑙(𝑘)𝑁2(𝑘)𝑁1(𝑘𝑙)(7)𝑚𝑙=0𝑟𝑀2𝑎𝑀1𝑙𝑎1𝑙𝑁(𝑘)1𝑏(𝑘𝑙)+𝑀1𝑙𝑏1𝑙𝑁(𝑘)2+𝑐(𝑘𝑙)(8)𝑀1𝑙𝐵1+𝜀𝑐1𝑙(𝑘)𝑁1𝑁(𝑘)2(𝑘𝑙)(9)𝑚𝑙=0𝑟𝐿1𝑏𝐿2𝑙𝑟𝑀2𝑎𝑀1𝑙𝑁1(𝑘)𝑚𝑙=0𝑟𝐿1𝑎𝐿2𝑙𝑟𝑀2𝑏𝑀1𝑙𝑟𝑀2𝑐𝑀1𝑙𝐵1𝑁+𝜀2(𝑘)}(10)exp{𝑚𝑙=0𝑟𝐿1𝑏𝐿2𝑙𝑟𝑀2𝑎𝑀1𝑙𝑁1(𝑘)𝑚𝑙=0𝑟𝐿1𝑎𝐿2𝑙𝑟𝑀2𝑏𝑀1𝑙𝑟𝑀2𝑐𝑀1𝑙𝐵1𝑁+𝜀2(𝑘)}.(11)expΔ𝜀𝑁1(𝑘)+𝑁2(𝑘)(12)expΔ𝜀𝜎2.(13)(14) it follows from (1.1), (4.13), and (4.14) that𝑘>𝐾2,
That is, for all 𝑢(𝑘)𝑢(𝑘2)expΔ𝜀𝜎2𝑘𝐾2.(4.17)𝑢(𝑘)So from the definition of 𝑁𝑟𝐿12(𝑘)𝑁𝑟𝑀21(𝑘)exp{𝑚𝑙=0𝑘1𝑠=𝑘𝑙𝑟𝐿1𝑎𝐿2𝑙𝑁2(𝑠)+𝑏𝐿2𝑙𝑁1(𝑠)𝑚𝑙=0𝑘1𝑠=𝑘𝑙𝑟𝑀2𝑎𝑀1𝑙𝑁1(𝑠)+𝑏𝑀1𝑙𝑁2(𝑠)+𝑐𝑀1𝑙𝐵1𝑁+𝜀2}(𝑠)×expΔ𝜀𝜎2𝑘𝐾22𝐵1𝑟𝑀2exp{𝑚𝑙=0𝑘1𝑠=𝑘𝑙2𝑟𝐿1𝑎𝐿2𝑙𝐵2+𝑏𝐿2𝑙𝐵1+𝑚𝑙=0𝑘1𝑠=𝑘𝑙2𝑟𝑀2𝑎𝑀1𝑙𝐵1+𝑏𝑀1𝑙𝐵2+𝑐𝑀1𝑙𝐵1𝐵+𝜀2}×expΔ𝜀𝜎2𝑘𝐾20as𝑘+.(4.18) it follows thatlim𝑘𝑁2(𝑘)=0.(4.19)The above analysis shows that𝑁1(𝑘+1)=𝑁1𝑁(𝑘)exp1.42.52+0.02sin(𝑘)1(𝑘)0.5𝑁1(𝑘1)0.55𝑁2(𝑘)0.3𝑁2(𝑘1)0.1𝑁1(𝑘)𝑁2(𝑘)0.09𝑁1(𝑘)𝑁2,𝑁(𝑘1)2(𝑘+1)=𝑁2𝑁(𝑘)exp0.72.62+0.02sin(𝑘)2(𝑘)1.2𝑁2(𝑘1)0.01𝑁1(𝑘)0.01𝑁1(𝑘1)0.09𝑁1(𝑘)𝑁2(𝑘)0.1𝑁2(𝑘)𝑁1.(𝑘1)(5.1)This completes the proof of Theorem 4.2.

5. Examples

The following two examples show the feasibility of our results.

Example 5.1. Consider the following systemΔ11=𝑟𝐿1𝑎𝑀11𝐵11𝑙=0𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1𝐵2Δ0.8271394917>0,21=𝑟𝐿2𝑎𝑀21𝐵21𝑙=0𝑏𝑀2𝑙+𝑐𝑀2𝑙𝐵2𝐵10.3138443044>0.(5.2)One could easily see that𝑎min𝐿10,2𝐵1𝑎𝑀10𝑎𝑀11𝑏+2𝑀2+2𝐵2𝑐𝑀𝑎0.1776281960,min𝐿20,2𝐵2𝑎𝑀20𝑎𝑀21𝑏+2𝑀1+2𝐵1𝑐𝑀0.613080483.(5.3)Clearly, conditions (2.5) are satisfied. From Theorem 2.3, it follows that system (5.1) is permanent. Also, by simple computation, we havelim𝑘𝑁𝑖(𝑘)𝑁𝑖(𝑘)=0,𝑖=1,2.(5.4)The above inequality shows that (H0) is fulfilled. From Theorem 3.1, it follows that(𝑁1(𝑘),𝑁2(𝑘))=(0.42,0.175),(0.41,0.178),Figures 1 and 2 are the numeric simulations of the solution of system (5.1) with initial condition (0.4,0.18),𝑘=1,0. and 𝑁1

310425.fig.001
Figure 1: Dynamic behaviors of the species (𝑁1(𝑝),𝑁2(𝑝))=(0.42,0.175),(0.41,0.178), of system (5.1) with initial conditions (0.4,0.18),𝑝=1,0. and 𝑁2
310425.fig.002
Figure 2: Dynamic behaviors of the species (𝑁1(𝑘),𝑁2(𝑘))=(0.42,0.175),(0.41,0.178), of system (5.1) with initial conditions (0.4,0.18),𝑘=1,0. and 𝑁1(𝑘+1)=𝑁1𝑁(𝑘)exp1.41.5+0.2sin(𝑘)1(𝑘)0.9𝑁1𝑁(𝑘1)0.3+0.2sin(𝑘)2(𝑘)0.5𝑁2𝑁(𝑘1)0.4+0.1cos(𝑘)1(𝑘)𝑁2(𝑘)0.4𝑁1(𝑘)𝑁2,𝑁(𝑘1)2(𝑘+1)=𝑁2𝑁(𝑘)exp0.71.1+0.5sin(𝑘)2(𝑘)0.5𝑁2𝑁(𝑘1)1.1+0.2cos(𝑘)1(𝑘)0.7𝑁1(𝑘1)2.3𝑁1(𝑘)𝑁2(𝑘)0.4𝑁2(𝑘)𝑁1.(𝑘1)(5.5)

Example 5.2. Consider the following system:𝑟𝑀2𝑟𝐿1=0.7𝑏1.4=0.5,𝐿20𝑎𝑀10=0.9𝑎1.70.5294,𝐿20𝑏𝑀10+𝑐𝑀10𝐵10.6𝑏0.5+0.5×1.14760.5588,𝐿21𝑎𝑀11=0.7𝑎0.90.7778,𝐿21𝑏𝑀11+𝑐𝑀11𝐵10.50.5+0.4×1.14760.5214.(5.6)One could easily see that𝑙=0,1,Then, for 𝑟𝑀2𝑟𝐿1𝑏min{𝐿2𝑙𝑎𝑀1𝑙,𝑎𝐿2𝑙𝑏𝑀1𝑙+𝑐𝑀1𝑙𝐵1}<0.(5.7)lim𝑘𝑁2(𝑘)=0.The above inequality shows that (H1*) is fulfilled. From Theorem 4.2, it follows that (𝑁1(𝑘),𝑁2(𝑘))=(0.42,0.6),(𝑘=1,0) Numeric simulation of the dynamic behaviors of system (5.5) with the initial conditions 𝑁2 is presented in Figure 3.

310425.fig.003
Figure 3: Dynamic behaviors of the species (𝑁1(𝑘),𝑁2(𝑘))=(0.42,0.6),(𝑘=1,0), of system (5.5) with initial conditions sin(𝑘),cos(𝑘) respectively.

Remark 5.3. In the above two examples, we can take as the perturbation terms. Our numeric simulations show that if the perturbation terms are large enough, then those terms will greatly influence the dynamic behaviors of the system, and in some cases, may lead to the extinction of the species.

Acknowledgments

The authors are grateful to anonymous referees for their excellent suggestions, which greatly improve the presentation of the paper. Also, this work was supported by the Program for New Century Excellent Talents in Fujian Province University (0330-003383).

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