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

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)−^𝑚_𝑙=0𝑎_1𝑙(𝑘)𝑁₁(𝑘−𝑙)−^𝑚_𝑙=0−𝑏_1𝑙(𝑘)𝑁₂(𝑘−𝑙)_𝑚_𝑙=0𝑁𝑐_1𝑙(𝑘)𝑁₁(𝑘)𝑁₂(𝑘−𝑙)],₂(𝑘+1)=𝑁₂(𝑘)exp[𝑟₂(𝑘)−^𝑚_𝑙=0𝑎_2𝑙(𝑘)𝑁₂(𝑘−𝑙)−^𝑚_𝑙=0−𝑏_2𝑙(𝑘)𝑁₁(𝑘−𝑙)_𝑚_𝑙=0𝑐_2𝑙(𝑘)𝑁₂(𝑘)𝑁₁(𝑘−𝑙)](1.1)together with the initial condition𝑁_𝑖(−𝑙)≥0,𝑁_𝑖(0)>0,𝑖=1,2;𝑙=0,1,…,𝑚,(1.2)where 𝑚 is a positive integer, 𝑁_𝑖(𝑘) represent the densities of population 𝑖 at the 𝑘th generation, 𝑟_𝑖(𝑘) are the intrinsic growth rate of population 𝑖 at the 𝑘th generation, 𝑎_𝑖𝑙(𝑘) measure the intraspecific influence of the (𝑘−𝑙)th generation of population 𝑖 on the density of own population, 𝑏_𝑖𝑙(𝑘) stand for the interspecific influence of the (𝑘−𝑙)th generation of population 𝑖 on the density of own population, and 𝑐_𝑖𝑙(𝑘) stand for the effect of toxic inhibition of population 𝑖 by population 𝑗 at the (𝑘−𝑙)th generation, 𝑖,𝑗=1,2 and 𝑖≠𝑗. Also, {𝑟_𝑖(𝑘)},{𝑎_𝑖𝑙(𝑘)},{𝑏_𝑖𝑙(𝑘)} and {𝑐_𝑖𝑙(𝑘)} are all bounded nonnegative sequences defined for 𝑘∈𝑁, denoted by the set of all nonnegative integers, and 𝑙∈{0,1,…,𝑚} such that00<<𝑏𝑟^𝐿𝑖_𝐿𝑖𝑙≤𝑟_𝑖(𝑘)≤𝑟^𝑀𝑖,0<𝑎^𝐿𝑖𝑙≤𝑎_𝑖𝑙(𝑘)≤𝑎^𝑀𝑖𝑙,≤𝑏_𝑖𝑙(𝑘)≤𝑏^𝑀𝑖𝑙,0<𝑐^𝐿𝑖𝑙≤𝑐_𝑖𝑙(𝑘)≤𝑐^𝑀𝑖𝑙,(1.3)here, for any bounded sequence {𝑓(𝑘)}, define𝑓^𝑀=sup_𝑘∈𝑁𝑓(𝑘),𝑓^𝐿=inf_𝑘∈𝑁𝑓(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=𝑥¹𝑑𝑥₂(𝑡)𝑟₁−𝑎₁₁𝑥₁(𝑡)−𝑎₁₂𝑥₂(𝑡)−𝑏₁𝑥₁(𝑡)𝑥₂(𝑡),(𝑡)𝑑𝑡=𝑥²(𝑡)𝑟₂−𝑎₂₁𝑥₁(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 [1–3, 5–24] 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:𝑥𝑥₁(𝑘+1)=𝑥₁(𝑘)exp𝑟₁(𝑘)−𝑎₁₁(𝑘)𝑥₁(𝑘)−𝑎₁₂(𝑘)𝑥₂(𝑘)−𝑏₁(𝑘)𝑥₁(𝑘)𝑥₂(𝑘)₂,(𝑘+1)=𝑥₂(𝑘)exp𝑟₂(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, 25–45].

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, 5–9, 12, 13, 27–30, 33, 34, 37–45] 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), limsup_𝑘→∞𝑁_𝑖(𝑘)≤𝐵_𝑖,𝑖=1,2,(2.1)where 𝐵=_𝑖^defexp𝑟^𝑀𝑖−1𝑎^𝐿𝑖0,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 𝑘∈𝑁, we have𝑁_𝑖(𝑘+1)≤𝑁_𝑖(𝑘)exp𝑟_𝑖(𝑘)−𝑎_𝑖0(𝑘)𝑁_𝑖(𝑘),𝑖=1,2.(2.3)Applying Lemma 2.1 of Yang [44] to (2.3), we can obtainlimsup_𝑘→∞𝑁_𝑖(𝑘)≤exp𝑟^𝑀𝑖−1𝑎^𝐿𝑖0def=𝐵_𝑖,𝑖=1,2.(2.4)This completes the proof of Lemma 2.1.

Lemma 2.2. Assume that Δ=₁₁^def𝑟^𝐿1−^𝑚_𝑙=1𝑎^𝑀1𝑙𝐵₁−^𝑚_𝑙=0𝑏^𝑀1𝑙Δ+𝑐^𝑀1𝑙𝐵₁𝐵²>0,_21def=𝑟^𝐿2−^𝑚_𝑙=1𝑎^𝑀2𝑙𝐵₂−^𝑚_𝑙=0𝑏^𝑀2𝑙+𝑐^𝑀2𝑙𝐵₂𝐵¹>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), liminf_𝑘→∞𝑁_𝑖(𝑘)≥𝐴_𝑖,𝑖=1,2,(2.6)where 𝐴_𝑖=Δ^𝑖1𝑎^𝑀𝑖0Δexp[Δ_𝑖2],𝑖=1,2,₁₂=𝑟^𝐿1−^𝑚_𝑙=0𝑎^𝑀1𝑙𝐵₁−^𝑚_𝑙=0𝑏^𝑀1𝑙Δ₂₂+𝑐^𝑀1𝑙𝐵₁𝐵²,=𝑟^𝐿2−^𝑚_𝑙=0𝑎^𝑀2𝑙𝐵₂−^𝑚_𝑙=0𝑏^𝑀2𝑙+𝑐^𝑀2𝑙𝐵₂𝐵¹.(2.7)

Proof. In view of (2.5), we can choose a constant 𝜀>0 small enough such that𝑟^𝐿1−^𝑚_𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−^𝑚_𝑙=0𝑏^𝑀1𝑙𝑟+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀>0,(2.8)_𝐿2−^𝑚_𝑙=1𝑎^𝑀2𝑙𝐵²+𝜀−^𝑚_𝑙=0𝑏^𝑀2𝑙+𝑐^𝑀2𝑙𝐵²+𝜀𝐵₁+𝜀>0.(2.9)𝜀>0,In view of (2.1), for above 𝑘₀∈𝑁 there exists an integer 𝑁_𝑖(𝑘)≤𝐵_𝑖+𝜀∀𝑘≥𝑘₀,𝑖=1,2.(2.10) such that𝑙₀≥𝑘₀+𝑚We consider the following two cases.
Case (i). We assume that there exists an integer 𝑁₁(𝑙₀+1)≤𝑁₁(𝑙₀). such that 𝑁₁𝑙⁰+1=𝑁₁𝑙⁰exp[𝑟¹𝑙⁰−^𝑚_𝑙=0𝑎_1𝑙𝑙⁰𝑁¹𝑙⁰−𝑙−^𝑚_𝑙=0−𝑏_1𝑙𝑙⁰𝑁²𝑙⁰−𝑙_𝑚_𝑙=0≥𝑁₁𝑐_1𝑙𝑙⁰𝑁¹𝑙⁰𝑁²𝑙⁰−𝑙]𝑙⁰exp{𝑟^𝐿1−^𝑚_𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−^𝑚_𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀−𝑎^𝑀10𝑁₁𝑙⁰}.(2.11) Note that𝑟^𝐿1−^𝑚_𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−^𝑚_𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀−𝑎^𝑀10𝑁₁𝑙⁰≤0.(2.12)So we can obtain𝑁₁𝑙⁰≥𝑟^𝐿1−∑^𝑚𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−∑^𝑚𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀𝑎^𝑀10>0.(2.13)It follows from (2.8) that𝑁₁𝑙⁰+1≥𝑟^𝐿1−∑^𝑚𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−∑^𝑚𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀𝑎^𝑀10×exp{𝑟^𝐿1−^𝑚_𝑙=0𝑎^𝑀1𝑙𝐵¹+𝜀−^𝑚_𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀}.(2.14)Then we have𝑁_1𝜀=𝑟^𝐿1−∑^𝑚𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−∑^𝑚𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀𝑎^𝑀10×exp{𝑟^𝐿1−^𝑚_𝑙=0𝑎^𝑀1𝑙𝐵¹+𝜀−^𝑚_𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀}.(2.15)Let𝐵₁=exp𝑟^𝑀1−1𝑎^𝐿10≥𝑟^𝑀1𝑎^𝐿10≥𝑟^𝐿1𝑎^𝑀10,(2.16)Note that𝑟^𝐿1−𝑎^𝑀10𝐵₁≤0,thus 𝜀>0, and so, for above 𝑟^𝐿1−^𝑚_𝑙=0𝑎^𝑀1𝑙𝐵¹+𝜀−^𝑚_𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀<𝑟^𝐿1−𝑎^𝑀10𝐵¹+𝜀<𝑟^𝐿1−𝑎^𝑀10𝐵₁≤0(2.17)𝑟^𝐿1−∑^𝑚𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−∑^𝑚𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀𝑎^𝑀10≥𝑁_1𝜀.(2.18)or𝑁₁(𝑘)≥𝑁_1𝜀∀𝑘≥𝑙₀.(2.19)
We can claim that𝑝₀≥𝑙₀By way of contradiction, assume that there exists an integer 𝑁₁(𝑝₀)<𝑁_1𝜀. such that 𝑝₀≥𝑙₀+2. Then 𝑝₀≥𝑙₀+2 Let 𝑁₁(𝑝₀)<𝑁_1𝜀. be the smallest integer such that 𝑁₁(𝑝₀−1)>𝑁₁(𝑝₀). Then 𝑁₁(𝑝₀)≥𝑁_1𝜀, The above argument produces that 𝑁₁(𝑘+1)>𝑁₁(𝑘) a contradiction. Thus (2.19) proved.Case (ii). We assume that 𝑘≥𝑘₀+𝑚, for all lim_𝑘→∞𝑁₁(𝑘) then 𝑁₁. exists, denoted by 𝑁₁≥𝑟^𝐿1−∑^𝑚𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−∑^𝑚𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀𝑎^𝑀10.(2.20) We can claim that𝑁₁<𝑟^𝐿1−∑^𝑚𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−∑^𝑚𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀𝑎^𝑀10.(2.21)By the way of contradiction, assume thatlim_𝑘→∞[𝑟₁(𝑘)−^𝑚_𝑙=0𝑎_1𝑙(𝑘)𝑁₁(𝑘−𝑙)−^𝑚_𝑙=0𝑏_1𝑙(𝑘)𝑁₂(𝑘−𝑙)−^𝑚_𝑙=0𝑐_1𝑙(𝑘)𝑁₁(𝑘)𝑁₂(𝑘−𝑙)]=0,(2.22)Taking limit in the first equation of (1.1) giveslim_𝑘→∞[𝑟₁(𝑘)−^𝑚_𝑙=0𝑎_1𝑙(𝑘)𝑁₁(𝑘−𝑙)−^𝑚_𝑙=0𝑏_1𝑙(𝑘)𝑁₂(𝑘−𝑙)−^𝑚_𝑙=0≥𝑟_𝐿1𝑐_1𝑙(𝑘)𝑁₁(𝑘)𝑁₂(𝑘−𝑙)]−^𝑚_𝑙=1𝑎^𝑀1𝑙𝐵¹+𝜀−𝑎^𝑀10𝑁₁−^𝑚_𝑙=0𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂+𝜀>0.(2.23)which is a contradiction since𝑁₁≥𝑁_1𝜀.(2.24)The claim is thus proved.
From (2.20), we see thatliminf_𝑘→∞𝑁₁(𝑘)≥𝑁_1𝜀.(2.25)Combining Cases (i) and (ii), we see that𝜀→0,Setting lim_𝜀→0𝑁_1𝜀=Δ¹¹𝑎^𝑀10expΔ₁₂=^def𝐴₁.(2.26) it follows thatliminf_𝑘→∞𝑁₁(𝑘)≥𝐴₁.(2.27)
So we can easily see thatliminf_𝑘→∞𝑁₂(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 min𝑎^𝐿𝑖0,2𝐵_𝑖−𝑎^𝑀𝑖0−²_𝑗=_{1,𝑗≠𝑖}[^𝑚_𝑙=1𝑎^𝑀𝑖𝑙+(𝑚+1)𝑏^𝑀𝑗+2𝐵_𝑗𝑐^𝑀]>𝜂,𝑖=1,2,(H0) such that 𝑖,𝑗=1,2,𝑖≠𝑗,𝐵_𝑖where, for 𝐵_𝑗 and 𝑏^𝑀𝑗=max𝑏^𝑀𝑗𝑙∶𝑙=0𝑐,1,…,𝑚,𝑐^𝑀𝑖=max𝑐^𝑀𝑖𝑙∶𝑙=0,1,…,𝑚_𝑀,=max𝑐^𝑀𝑖∶𝑖=1,2,(3.1) are defined in (2.2), {(𝑁₁(𝑘),𝑁₂(𝑘))}then for any two positive solutions {(𝑁^∗1(𝑘),𝑁^∗2(𝑘))} and lim_𝑘→∞𝑁^𝑖(𝑘)−𝑁^∗𝑖(𝑘)=0,𝑖=1,2.(3.2) of system (1.1)-(1.2), 𝑉₁₁(𝑘)=|||ln𝑁₁(𝑘)−ln𝑁^∗1(𝑘)|||.(3.3)

Proof. First, let𝑉₁₁(𝑘+1)=≤||||||llnn𝑁𝑁₁₁(𝑘+1)−ln𝑁^∗1(𝑘+1)|||(𝑘)−ln𝑁^∗1(𝑘)−𝑎₁₀(𝑘)𝑁₁(𝑘)−𝑁^∗1(𝑘)|||+^𝑚_𝑙=1+𝑎_1𝑙(𝑘)|||𝑁₁(𝑘−𝑙)−𝑁^∗1(𝑘−𝑙)_𝑚|||_𝑙=0𝑏_1𝑙(𝑘)|||𝑁₂(𝑘−𝑙)−𝑁^∗2(𝑘−𝑙)|||+^𝑚_𝑙=0𝑐_1𝑙(𝑘)|||𝑁₁(𝑘)𝑁₂(𝑘−𝑙)−𝑁^∗1(𝑘)𝑁^∗2(𝑘−𝑙)|||.(3.4)Then from the first equation of (1.1), we have|||ln𝑁₁(𝑘)−ln𝑁^∗1(𝑘)|||=1𝜃₁(𝑘)|||𝑁¹(𝑘)−𝑁^∗1(𝑘)|||,(3.5)Noticing that by mean-value theory0<𝜃₁(𝑘)≤max{𝑁₁(𝑘),𝑁^∗1(𝑘)}.where |||ln𝑁₁=(|||𝑘l)n−𝑁l₁n𝑁^∗1(𝑘)−𝑎₁₀(𝑘)𝑁₁(𝑘)−𝑁^∗1(𝑘)|||+|||ln𝑁(₁𝑘)−ln𝑁^∗1(𝑘)|||−|||ln𝑁₁(𝑘)−ln𝑁^∗1(𝑘)|||=|||ln𝑁₁(𝑘)−ln𝑁^∗1(𝑘)−𝑎₁₀(𝑘)𝑁₁(𝑘)−𝑁^∗1(𝑘)|||(𝑘)−ln𝑁^∗1(𝑘)|||−1𝜃₁(𝑘)−|||𝜃₁1(𝑘)−𝑎¹⁰(𝑘)||||||𝑁₁(𝑘)−𝑁^∗1(𝑘)|||.(3.6) Then𝑉₁₁(𝑘+1)≤|||ln𝑁₁(𝑘)−ln𝑁^∗1(𝑘)|||−1𝜃₁(𝑘)−|||𝜃₁1(𝑘)−𝑎¹⁰+(𝑘)||||||𝑁₁(𝑘)−𝑁^∗1(𝑘)_𝑚|||_𝑙=1𝑎_1𝑙(𝑘)|||𝑁₁(𝑘−𝑙)−𝑁^∗1(𝑘−𝑙)|||+^𝑚_𝑙=0+𝑏_1𝑙(𝑘)|||𝑁₂(𝑘−𝑙)−𝑁^∗2(𝑘−𝑙)_𝑚|||_𝑙=0𝑐_1𝑙(𝑘)|||𝑁₁(𝑘)𝑁₂(𝑘−𝑙)−𝑁^∗1(𝑘)𝑁^∗2(𝑘−𝑙)|||.(3.7)Substituting (3.6) into (3.4) leads toΔ𝑉₁₁≤−1𝜃₁(𝑘)−|||𝜃₁1(𝑘)−𝑎¹⁰+(𝑘)||||||𝑁₁(𝑘)−𝑁^∗1(𝑘)_𝑚|||_𝑙=1𝑎_1𝑙(𝑘)|||𝑁₁(𝑘−𝑙)−𝑁^∗1(𝑘−𝑙)|||+^𝑚_𝑙=0+𝑏_1𝑙(𝑘)|||𝑁₂(𝑘−𝑙)−𝑁^∗2(𝑘−𝑙)_𝑚|||_𝑙=0𝑐_1𝑙(𝑘)|||𝑁₁(𝑘)𝑁₂(𝑘−𝑙)−𝑁^∗1(𝑘)𝑁^∗2(𝑘−𝑙)|||.(3.8)So it follows that𝜀>0,According to (2.1), for any constant 𝑘₀∈𝑁 there exists an integer 𝑁₁(𝑘)≤𝐵₁+𝜀,𝑁₂(𝑘)≤𝐵₂+𝜀∀𝑘≥𝑘₀.(3.9) such that𝑘≥𝑘₀+𝑚,𝑙=0,1,…,𝑚,So for all |||𝑁₁(𝑘=)𝑁|||𝑁₂(𝑘−𝑙)−𝑁^∗1(𝑘)𝑁^∗2(𝑘−𝑙)₁|||=|||𝑁(𝑘)𝑁₂(𝑘−𝑙)−𝑁₁(𝑘)𝑁^∗2(𝑘−𝑙)+𝑁₁(𝑘)𝑁^∗2(𝑘−𝑙)−𝑁^∗1(𝑘)𝑁^∗2(𝑘−𝑙)₁|||≤𝐵(𝑘)𝑁₂(𝑘−𝑙)−𝑁^∗2(𝑘−𝑙)+𝑁^∗2(𝑘−𝑙)𝑁₁(𝑘)−𝑁^∗1(𝑘)₁|||+𝜀|||𝑁₂(𝑘−𝑙)−𝑁^∗2(𝑘−𝑙)|||+𝐵₂+𝜀|||𝑁₁(𝑘)−𝑁^∗1(𝑘)|||.(3.10) it follows that𝑘≥𝑘₀+𝑚,So for all Δ𝑉₁₁≤−(1𝜃₁(𝑘)−|||𝜃₁1(𝑘)−𝑎¹⁰(𝑘)|||−^𝑚_𝑙=0𝐵²++𝜀𝑐_1𝑙(𝑘))|||𝑁₁(𝑘)−𝑁^∗1(𝑘)_𝑚|||_𝑙=1𝑎_1𝑙(𝑘)|||𝑁₁(𝑘−𝑙)−𝑁^∗1(𝑘−𝑙)|||+^𝑚_𝑙=0𝑏^1𝑙(𝑘)+𝐵₁+𝜀𝑐_1𝑙(𝑘)|||𝑁₂(𝑘−𝑙)−𝑁^∗2(𝑘−𝑙)|||.(3.11) it follows from (3.4) that𝑉₁₂(𝑘)=^𝑚_{𝑙=1𝑘−1𝑠=𝑘−𝑙}+𝑎_1𝑙(𝑠+𝑙)|||𝑁₁(𝑠)−𝑁^∗1(𝑠)_𝑚|||_{𝑙=0𝑘−1𝑠=𝑘−𝑙}𝑏^1𝑙(𝑠+𝑙)+𝐵₁+𝜀𝑐_1𝑙(𝑠+𝑙)|||𝑁₂(𝑠)−𝑁^∗2(𝑠)|||,(3.12)Next, letΔ𝑉₁₂==𝑉₁₂(𝑘+1)−𝑉₁₂(𝑘)_𝑚_𝑘𝑙=1_{𝑠=𝑘+1−𝑙}𝑎_1𝑙(𝑠+𝑙)|||𝑁₁(𝑠)−𝑁^∗1(𝑠)|||−^𝑚_{𝑙=1𝑘−1𝑠=𝑘−𝑙}+𝑎_1𝑙(𝑠+𝑙)|||𝑁₁(𝑠)−𝑁^∗1(𝑠)_𝑚|||_𝑘𝑙=0_{𝑠=𝑘+1−𝑙}𝑏^1𝑙−(𝑠+𝑙)+𝐵₁+𝜀𝑐_1𝑙(𝑠+𝑙)|||𝑁₂(𝑠)−𝑁^∗2(𝑠)_𝑚|||_{𝑙=0𝑘−1𝑠=𝑘−𝑙}𝑏^1𝑙=(𝑠+𝑙)+𝐵₁+𝜀𝑐_1𝑙(𝑠+𝑙)|||𝑁₂(𝑠)−𝑁^∗2(𝑠)_𝑚|||_𝑙=1𝑎_1𝑙(𝑘+𝑙)|||𝑁₁(𝑘)−𝑁^∗(𝑘)|||−^𝑚_𝑙=1+𝑎_1𝑙(𝑘)|||𝑁₁(𝑘−𝑙)−𝑁^∗1(𝑘−𝑙)_𝑚|||_𝑙=0𝑏^1𝑙−(𝑘+𝑙)+𝐵₁+𝜀𝑐_1𝑙(𝑘+𝑙)|||𝑁₂(𝑘)−𝑁^∗2(𝑘)_𝑚|||_𝑙=0𝑏^1𝑙(𝑘)+𝐵₁+𝜀𝑐_1𝑙(𝑘)|||𝑁₂(𝑘−𝑙)−𝑁^∗2(𝑘−𝑙)|||.(3.13)and we can obtain𝑉₁Now, we define 𝑉₁(𝑘)=𝑉₁₁(𝑘)+𝑉₁₂(𝑘).(3.14) by𝑘≥𝑘₀+𝑚,So for all Δ𝑉₁=Δ𝑉₁₁+Δ𝑉₁₂≤−(1𝜃₁(𝑘)−|||𝜃₁1(𝑘)−𝑎¹⁰(𝑘)|||−^𝑚_𝑙=0𝐵²+𝜀𝑐_1𝑙(𝑘)−^𝑚_𝑙=1+𝑎_1𝑙(𝑘+𝑙))|||𝑁₁(𝑘)−𝑁^∗1(𝑘)_𝑚|||_𝑙=0𝑏^1𝑙(𝑘+𝑙)+𝐵₁+𝜀𝑐_1𝑙(𝑘+𝑙)|||𝑁₂(𝑘)−𝑁^∗2(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 define𝑉𝑉₂₁(𝑘)=|||ln𝑁₂(𝑘)−ln𝑁^∗2(𝑘)₂₂|||,(𝑘)=^𝑚_{𝑙=1𝑘−1𝑠=𝑘−𝑙}+𝑎_2𝑙(𝑠+𝑙)|||𝑁₂(𝑠)−𝑁^∗2(𝑠)_𝑚|||_{𝑙=0𝑘−1𝑠=𝑘−𝑙}𝑏^2𝑙(𝑠+𝑙)+𝐵₂+𝜀𝑐_2𝑙(𝑠+𝑙)|||𝑁₁(𝑠)−𝑁^∗1(𝑠)|||.(3.17)where𝑘≥𝑘₀+𝑚,Then for all Δ𝑉₂=Δ𝑉₂₁+Δ𝑉₂₂≤−(1𝜃₂(𝑘)−|||𝜃₂1(𝑘)−𝑎²⁰(𝑘)|||−^𝑚_𝑙=0𝐵¹+𝜀𝑐_2𝑙(𝑘)−^𝑚_𝑙=1+𝑎_2𝑙(𝑘+𝑙))|||𝑁₂(𝑘)−𝑁^∗2(𝑘)_𝑚|||_𝑙=0𝑏^2𝑙(𝑘+𝑙)+𝐵₂+𝜀𝑐_2𝑙(𝑘+𝑙)|||𝑁₁(𝑘)−𝑁^∗1(𝑘)|||,(3.18) we can obtain𝜃₂(𝑘)where 𝑁₂(𝑘) lies between 𝑁^∗2(𝑘). and 𝑉
Now, we define 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘).(3.19) by𝑉(𝑘)≥0It is easy to see that 𝑘∈𝑍 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 min𝑎^𝐿𝑖0,2𝐵_𝑖+𝜀−𝑎^𝑀𝑖0−²_{𝑗=1,𝑗≠𝑖}[^𝑚_𝑙=1𝑎_𝑖𝑙(𝑘+𝑙)+(𝑚+1)𝑏^𝑀𝑗+2𝐵_𝑗+𝜀𝑐^𝑀]>𝜂.(3.20)𝑘≥𝑘₀+𝑚,So for all Δ𝑉≤−²_𝑖=1{1𝜃_𝑖(𝑘)−|||𝜃_𝑖1(𝑘)−𝑎^𝑖0−(𝑘)₂|||_𝑗=_{1,𝑗≠𝑖}[^𝑚_𝑙=1𝑎_𝑖𝑙(𝑘+𝑙)+^𝑚_𝑙=0𝑏^𝑗𝑙×|||𝑁(𝑘+𝑙)+𝐵_𝑗+𝜀𝑐_𝑖𝑙(𝑘)+𝑐_𝑗𝑙(𝑘+𝑙)]}_𝑖≤−₂(𝑘)−𝑁^∗𝑖(𝑘)|||_𝑖=1{min𝑎^𝐿𝑖0,2𝐵_𝑖+𝜀−𝑎^𝑀𝑖0−²_{𝑗=1,𝑗≠𝑖}[^𝑚_𝑙=1×|||𝑁𝑎_𝑖𝑙(𝑘+𝑙)+(𝑚+1)𝑏^𝑀𝑗+2𝐵_𝑗+𝜀𝑐^𝑀]}_𝑖≤−𝜂(₂𝑘)−𝑁^∗𝑖(𝑘)|||_𝑖=1|||𝑁_𝑖(𝑘)−𝑁^∗𝑖(𝑘)|||,(3.21) it follows from (3.15) and (3.18) that^𝑘_𝑝_=𝑘0+𝑚𝑉(𝑝+1)−𝑉(𝑝)≤−𝜂^𝑘_{𝑝=𝑘02+𝑚}_𝑖=1|||𝑁_𝑖(𝑝)−𝑁^∗𝑖(𝑝)|||,(3.22)So we have𝑉(𝑘+1)+𝜂^𝑘_𝑝=_𝑘02+𝑚_𝑖=1|||𝑁_𝑖(𝑝)−𝑁^∗𝑖(𝑝)|||≤𝑉𝑘₀+𝑚.(3.23)which implies^𝑘_𝑝_=𝑘02+𝑚_𝑖=1|||𝑁_𝑖(𝑝)−𝑁^∗𝑖(𝑝)|||≤𝑉𝑘⁰+𝑚𝜂.(3.24)It follows that^∞_𝑘_=𝑘02+𝑚_𝑖=1|||𝑁_𝑖(𝑘)−𝑁^∗𝑖(𝑘)|||≤𝑉𝑘⁰+𝑚𝜂<+∞,(3.25)Thenlim_𝑘→∞∑^2𝑖=1|𝑁_𝑖(𝑘)−𝑁^∗𝑖(𝑘)|=0,which implies that lim_𝑘→∞𝑁^𝑖(𝑘)−𝑁^∗𝑖(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 liminf_𝑘→+∞𝑁¹(𝑘)+𝑁₂(𝑘)>𝜎.(4.1) such that 𝐵>0

Proof. By (2.1), there exists a constant 𝑁_𝑖(𝑘)<𝐵∀𝑘>𝑘₀,𝑖=1,2.(4.2) such that𝑘>𝑘₀+𝑚,𝑖=1,2,In view of (1.1) for all 𝑁_𝑖(𝑘)≤𝑁_𝑖(𝑘+1)exp{−𝑟^𝐿𝑖+∑^𝑚_𝑙=0𝑎^𝑀𝑖𝑙+𝑏^𝑀𝑖𝑙+𝑐^𝑀𝑖𝑙𝐵𝐵}.(4.3) it follows that𝑁_𝑖(𝑘−𝑙)≤𝑁_𝑖(𝑘)exp{𝑙[−𝑟^𝐿𝑖+^𝑚_𝑙=0𝑎^𝑀𝑖𝑙+𝑏^𝑀𝑖𝑙+𝑐^𝑀𝑖𝑙𝐵𝐵]},𝑙=0,1,…,𝑚.(4.4)So we have𝐶₁=max{exp{𝑙[−𝑟^𝐿𝑖+^𝑚_𝑙=0𝑎^𝑀𝑖𝑙𝐶+𝑏^𝑀𝑖𝑙+𝑐^𝑀𝑖𝑙𝐵𝐵]}∶𝑖=1,2,𝑙=0,1,…,𝑚},₂=max𝑑^𝑀𝑖𝑙𝐶₁∶𝑖=1,2,𝑙=0,1,…,𝑚,(4.5)Let𝑑^𝑀𝑖𝑙=max{𝑎^𝑀𝑖𝑙,𝑏^𝑀𝑖𝑙+𝑐^𝑀𝑖𝑙𝐵},where 𝑖=1,2,𝑙=0,1,…,𝑚.𝑘>𝑘₀+𝑚,𝑖,𝑗=1,2,𝑖≠𝑗, For all 𝑁_𝑖(𝑘+1)≥𝑁_𝑖(𝑘)exp{𝑟^𝐿𝑖−^𝑚_𝑙=0𝑎^𝑀𝑖𝑙𝐶₁𝑁_𝑖(𝑘)−^𝑚_𝑙=0𝑏^𝑀𝑖𝑙≥𝑁_𝑖+𝑐^𝑀𝑖𝑙𝐵𝐶₁𝑁_𝑗(𝑘)}(𝑘)exp{𝑟^𝐿𝑖−^𝑚_𝑙=0≥𝑁_𝑖𝑑^𝑀𝑖𝑙𝐶₁𝑁^𝑖(𝑘)+𝑁_𝑗(𝑘)}(𝑘)expmin𝑟^𝐿1,𝑟^𝐿2−(𝑚+1)𝐶²𝑁¹(𝑘)+𝑁₂(𝑘),(4.6) it follows from (1.1) and (4.4) that𝑁₁(𝑘+1)+𝑁₂(𝑘+1)≥𝑁₁(𝑘)+𝑁₂(𝑘)expmin𝑟^𝐿1,𝑟^𝐿2−(𝑚+1)𝐶²𝑁¹(𝑘)+𝑁₂(𝑘).(4.7)so we have𝑥(𝑘)=𝑁₁(𝑘)+𝑁₂(𝑘),Let 𝑥(𝑘+1)≥=𝑥𝑥((𝑘𝑘))eexxpp{mminin𝑟𝑟^𝐿1_𝐿1,𝑟^𝐿2−(𝑚+1)𝐶²𝑥(𝑘),𝑟^𝐿2[1−(𝑚+1)𝐶²min𝑟^𝐿1,𝑟^𝐿2𝑥(𝑘)]}_def=𝑥(𝑘)exp𝑟1−𝑎𝑥(𝑘).(4.8) then we have𝑘>𝑘₀,Note that for all 𝑥(𝑘)=𝑁₁(𝑘)+𝑁₂(𝑘)<2𝐵,liminf_𝑘→+∞𝑥(𝑘)≥1𝑎exp𝑟(1−2𝑎𝐵)>0.(4.9) so similar to the proof of Lemma 2.2 of Chen [27], we have𝜎>0Then, there is a positive constant liminf_𝑘→+∞𝑁¹(𝑘)+𝑁₂(𝑘)=liminf_𝑘→+∞𝑥(𝑘)≥1𝑎exp𝑟(1−2𝑎𝐵)>𝜎.(4.10) such that^𝑚_𝑙=0𝑟^𝐿1𝑏^𝐿2𝑙−𝑟^𝑀2𝑎^𝑀1𝑙>0,^𝑚_𝑙=0𝑟^𝐿1𝑎^𝐿2𝑙−𝑟^𝑀2𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵₁>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 𝑟^𝑀2𝑟^𝐿1−min{𝑏^𝐿2𝑙𝑎^𝑀1𝑙,𝑎^𝐿2𝑙𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵₁}<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 ^𝑚_𝑙=0𝑟^𝐿1𝑏^𝐿2𝑙−𝑟^𝑀2𝑎^𝑀1𝑙>0,^𝑚_𝑙=0𝑟^𝐿1𝑎^𝐿2𝑙−𝑟^𝑀2𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀>0.(4.11) small enough such thatΔ^𝜀=min{^𝑚_𝑙=0𝑟^𝐿1𝑏^𝐿2𝑙−𝑟^𝑀2𝑎^𝑀1𝑙,^𝑚_𝑙=0𝑟^𝐿1𝑎^𝐿2𝑙−𝑟^𝑀2𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵¹+𝜀}>0.(4.12)Set𝜀>0For above 𝐾∈𝑁 from (2.1), there is an integer 𝑖=1,2, such that for 𝑁_𝑖(𝑘)≤𝐵_𝑖+𝜀∀𝑘>𝐾.(4.13)𝐾₁>𝐾Lemma 4.1 also implies that there exists 𝑁₁(𝑘)+𝑁₂(𝑘)≥𝜎2∀𝑘≥𝐾¹.(4.14) such that𝑢(𝑘)=𝑁^𝑟𝐿12(𝑘)𝑁^𝑟𝑀2₁(𝑘)exp{−^𝑚_{𝑙=0𝑘−1𝑠=𝑘−𝑙}+𝑟^𝐿1𝑎^𝐿2𝑙𝑁₂(𝑠)+𝑏^𝐿2𝑙𝑁₁(𝑠)_𝑚_{𝑙=0𝑘−1𝑠=𝑘−𝑙}𝑟^𝑀2𝑎^𝑀1𝑙𝑁₁(𝑠)+𝑏^𝑀1𝑙𝑁₂(𝑠)+𝑐^𝑀1𝑙𝐵¹+𝜀𝑁₂(𝑠)}.(4.15)Set𝑘>𝐾₂>𝐾₁+𝑚,So for all 𝑢(𝑘+1)𝑢(𝑘)=exp{𝑟^𝐿1𝑟₂(𝑘)−^𝑚_𝑙=0−𝑟_𝑀2𝑟^𝐿1𝑎^2𝑙(𝑘)𝑁₂(𝑘−𝑙)+𝑏_2𝑙(𝑘)𝑁₁(𝑘−𝑙)+𝑐_2𝑙(𝑘)𝑁₂(𝑘)𝑁₁(𝑘−𝑙)(4.16)𝑟₁(𝑘)+^𝑚_𝑙=0−𝑟^𝑀2𝑎^1𝑙(𝑘)𝑁₁(𝑘−𝑙)+𝑏_1𝑙(𝑘)𝑁₂(𝑘−𝑙)+𝑐_1𝑙(𝑘)𝑁₁(𝑘)𝑁₂(𝑘−𝑙)(1)_𝑚_𝑘𝑙=0_{𝑠=𝑘+1−𝑙}𝑟^𝐿1𝑎^𝐿2𝑙𝑁₂(𝑠)+𝑏^𝐿2𝑙𝑁₁(𝑠)+^𝑚_{𝑙=0𝑘−1𝑠=𝑘−𝑙}+𝑟^𝐿1𝑎^𝐿2𝑙𝑁₂(𝑠)+𝑏^𝐿2𝑙𝑁₁(𝑠)(2)_𝑚_𝑘𝑙=0_{𝑠=𝑘+1−𝑙}−𝑟^𝑀2𝑎^𝑀1𝑙𝑁₁(𝑠)+𝑏^𝑀1𝑙𝑁₂(𝑠)+𝑐^𝑀1𝑙𝐵¹+𝜀𝑁₂(𝑠)(3)_𝑚_{𝑙=0𝑘−1𝑠=𝑘−𝑙}=exp{𝑟_𝐿1𝑟^𝑀2𝑎^𝑀1𝑙𝑁₁(𝑠)+𝑏^𝑀1𝑙𝑁₂(𝑠)+𝑐^𝑀1𝑙𝐵¹+𝜀𝑁₂(𝑠)}(4)−𝑟₂(𝑘)−𝑟^𝑀2𝑟₁(𝑘)(5)_𝑚_𝑙=0−𝑟^𝐿1𝑎^2𝑙(𝑘)−𝑎^𝐿2𝑙𝑁²(𝑘−𝑙)+𝑏_2𝑙(𝑘)−𝑏^𝐿2𝑙𝑁¹(𝑘−𝑙)(6)_𝑚_𝑙=0−𝑟^𝐿1𝑐_2𝑙(𝑘)𝑁₂(𝑘)𝑁₁(𝑘−𝑙)(7)_𝑚_𝑙=0𝑟^𝑀2+𝑎𝑐^𝑀1𝑙−𝑎_1𝑙(𝑘)𝑁₁(𝑘−𝑙)+𝑏^𝑀1𝑙−𝑏_1𝑙(𝑘)𝑁₂(𝑘−𝑙)(8)_𝑀1𝑙−𝐵¹+𝜀−𝑐_1𝑙(𝑘)𝑁₁(𝑘)𝑁₂(𝑘−𝑙)(9)_𝑚_𝑙=0𝑟^𝐿1𝑏^𝐿2𝑙−𝑟^𝑀2𝑎^𝑀1𝑙𝑁¹(𝑘)−^𝑚_𝑙=0𝑟^𝐿1≤exp{−_𝑚𝑎^𝐿2𝑙−𝑟^𝑀2𝑏^𝑀1𝑙−𝑟^𝑀2𝑐^𝑀1𝑙𝐵¹+𝜀𝑁₂(𝑘)}(10)_𝑙=0𝑟^𝐿1𝑏^𝐿2𝑙−𝑟^𝑀2𝑎^𝑀1𝑙𝑁¹(𝑘)−^𝑚_𝑙=0𝑟^𝐿1≤exp−Δ_𝜀𝑎^𝐿2𝑙−𝑟^𝑀2𝑏^𝑀1𝑙−𝑟^𝑀2𝑐^𝑀1𝑙𝐵¹+𝜀𝑁₂(𝑘)}.(11)≤exp−Δ_𝜀𝑁¹(𝑘)+𝑁₂(𝑘)(12)𝜎2.((1134)) it follows from (1.1), (4.13), and (4.14) that𝑘>𝐾₂,
That is, for all 𝑢(𝑘)≤𝑢(𝑘₂)exp−Δ^𝜀𝜎2𝑘−𝐾².(4.17)𝑢(𝑘)So from the definition of 𝑁^𝑟𝐿1₂(𝑘)≤𝑁^𝑟𝑀2₁(𝑘)exp{^𝑚_{𝑙=0𝑘−1𝑠=𝑘−𝑙}−𝑟^𝐿1𝑎^𝐿2𝑙𝑁₂(𝑠)+𝑏^𝐿2𝑙𝑁₁(𝑠)_𝑚_{𝑙=0𝑘−1𝑠=𝑘−𝑙}×exp−Δ_𝜀𝑟^𝑀2𝑎^𝑀1𝑙𝑁₁(𝑠)+𝑏^𝑀1𝑙𝑁₂(𝑠)+𝑐^𝑀1𝑙𝐵¹+𝜀𝑁₂(𝑠)}𝜎≤2𝐵₁2𝑘−𝐾²^𝑟𝑀2exp+{^𝑚_𝑚_{𝑙=0𝑘−1𝑠=𝑘−𝑙}2𝑟^𝐿1𝑎^𝐿2𝑙𝐵₂+𝑏^𝐿2𝑙𝐵₁_{𝑙=0𝑘−1𝑠=𝑘−𝑙}×exp−Δ_𝜀2𝑟^𝑀2𝑎^𝑀1𝑙𝐵₁+𝑏^𝑀1𝑙𝐵₂+𝑐^𝑀1𝑙𝐵¹+𝜀𝐵₂}𝜎2𝑘−𝐾²→0as𝑘→+∞.(4.18) it follows thatlim_𝑘→∞𝑁₂(𝑘)=0.(4.19)The above analysis shows that𝑁₁(𝑘+1)=𝑁₁(𝑘)exp−01..545−𝑁₂2.52+0.02sin(𝑘)𝑁₁(𝑘)−0.5𝑁₁(𝑘−1)−0.1𝑁₁(𝑘)−0.3𝑁₂(𝑘−1)𝑁(𝑘)𝑁₂(𝑘)−0.09𝑁₁(𝑘)𝑁₂(𝑘−1)₂,(𝑘+1)=𝑁₂(𝑘)exp−00..071−𝑁₁2.62+0.02sin(𝑘)𝑁₂(𝑘)−1.2𝑁₂(𝑘−1)−0.09𝑁₁(𝑘)−0.01𝑁₁(𝑘−1)(𝑘)𝑁₂(𝑘)−0.1𝑁₂(𝑘)𝑁₁(𝑘−1).(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=𝑟^𝐿1−𝑎^𝑀11𝐵₁−¹_𝑙=0𝑏^𝑀1𝑙Δ+𝑐^𝑀1𝑙𝐵₁𝐵²≈0.8271394917>0,₂₁=𝑟^𝐿2−𝑎^𝑀21𝐵₂−¹_𝑙=0𝑏^𝑀2𝑙+𝑐^𝑀2𝑙𝐵₂𝐵¹≈0.3138443044>0.(5.2)One could easily see thatmin𝑎^𝐿10,2𝐵₁min𝑎_𝐿20−𝑎^𝑀10−𝑎^𝑀11+2𝑏^𝑀2+2𝐵₂𝑐^𝑀≈0.1776281960,,2𝐵₂−𝑎^𝑀20−𝑎^𝑀21+2𝑏^𝑀1+2𝐵₁𝑐^𝑀≈0.613080483.(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 havelim_𝑘→∞𝑁^𝑖(𝑘)−𝑁^∗𝑖(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

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

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)=𝑁1(𝑘)exp−1.04.3−+01..25s+in0(.𝑘2)sin𝑁(2𝑘)𝑁1(𝑘)−0.9𝑁1(𝑘−1)−0.4+0.1cos(𝑘)𝑁1(𝑘)−0.5𝑁2(𝑘−1)𝑁(𝑘)𝑁2(𝑘)−0.4𝑁1(𝑘)𝑁2(𝑘−1)2,(𝑘+1)=𝑁2(𝑘)exp−0.17.1−+01..21c+os0.(5𝑘)sin𝑁(𝑘1)𝑁2(𝑘)−0.5𝑁2(𝑘−1)−2.3𝑁1(𝑘)−0.7𝑁1(𝑘−1)(𝑘)𝑁2(𝑘)−0.4𝑁2(𝑘)𝑁1(𝑘−1).(5.5)

Example 5.2. Consider the following system:𝑟^𝑀2𝑟^𝐿1=0.7𝑏1.4=0.5,_𝐿20𝑎^𝑀10=0.91.7≈0.5294,𝑏^𝑀10+𝑎𝑐^𝐿20𝑀10𝐵₁≈0.6𝑏0.5+0.5×1.1476≈0.5588,_𝐿21𝑎^𝑀11=0.70.9≈0.7778,𝑏^𝑀11+𝑎𝑐^𝐿21𝑀11𝐵₁≈0.50.5+0.4×1.1476≈0.5214.(5.6)One could easily see that𝑙=0,1,Then, for 𝑟^𝑀2𝑟^𝐿1−min{𝑏^𝐿2𝑙𝑎^𝑀1𝑙,𝑎^𝐿2𝑙𝑏^𝑀1𝑙+𝑐^𝑀1𝑙𝐵₁}<0.(5.7)lim_𝑘→∞𝑁₂(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.

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).