# Dynamics of a General Stochastic Nonautonomous Lotka-Volterra Model with Delays and Impulsive Perturbations

Authors: Qing Wang; Yongguang Yu; Shuo Zhang
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

A stochastic nonautonomous N-species Lotka-Volterra model with delays and impulsive perturbations is investigated. For this model, sufficient conditions for extinction, nonpersistence in the mean, weak persistence and stochastic permanence are given, respectively. The influences of the stochastic noises, and the impulsive perturbations on the properties of the
stochastic model are also discussed. The critical value between weak persistence and extinction is obtained. Finally, numerical simulations are given to support the theoretical analysis results.


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## Body

## 1. Introduction

Population ecology is a major subfield of ecology that deals with the dynamics of species populations and the way these populations interact with the environment. It is concerned with the study of groups of organisms that live together in time and space and compete for the limited resources or in some way inhibit others’ growth. Modelling of dynamic interactions in nature allows us to understand better how these complex interactions and processes work. The well-known model that regards dynamic of population models is the Lotka-Volterra model. The investigation of the Lotka-Volterra model is one of the dominant themes in mathematical ecology due to its importance. The Lotka-Volterra model with delays has received more and more attentions and had lots of nice results [1–5]. More details of the Lotka-Volterra model with delays are discussed in the books by Gopalsamy [6] and Kuang [7].In the real world, the population models are inevitably influenced by the environmental noise which is an important component in an ecosystem [8–10]. Moreover, May [11] has pointed out the fact that, due to environmental noise, the birth rate, carrying capacity, competition coefficient, and other parameters involved with the system exhibit random fluctuation to a greater or lesser extent [12].On the other hand, populations may be affected by a variety of factors both naturally and manly, such as earthquake, drought, flooding, fire, crop-dusting, planting, hunting, and harvesting; the inner discipline of species or environment often suffers some dispersed changes over a relatively short time interval at the fixed times, which makes it unsuitable to be considered continually. In mathematics perspective, such sudden changes could be described by impulses. With the development of the theory of impulsive differential equations [13, 14], we can establish adequate mathematical models of impulsive differential equations to investigate the dynamic behaviors of such ecosystems with impulsive effects. Consequently the dynamical behaviors of impulsive population dynamical models and stochastic population dynamical models have been extensively studied [15–19]. The nonautonomous N-species Lotka-Volterra competitive system with impulsive perturbations was discussed in Hou et al. [15]. N-species nonautonomous Lotka-Volterra competitive system with delays and impulsive perturbations was considered in Zhang and Teng [18]. While these papers did not discuss the persistence and extinction of the stochastic Lotka-Volterra model, from the viewpoint of applications, it is critical to find out when the population will go to extinction or survival.A major problem in population biology is to understand what determines extinction of a population. Population extinction is often a result of habitat destruction and modification which can be widespread. Moreover, dramatic changes in ecosystem structure or function are often caused by the species additions in the form of invasive species. In addition, the extinction of native populations may be caused by the growth of invasive species [20]. Moreover, even large populations may be destroyed by some extraordinary perturbation [21]. When the time is sufficiently large the population of some species may not become extinct, but the size of that population may be close to zero so that the species can be endangered. In other words, there exists a critical number between extinction and survival of population. In this sense, Ma and Hallam [22, 23] proposed the concepts of nonpersistence in the mean and weak persistence for some deterministic models and Lu and Ding [24] applied these concepts to stochastic logistic models instead of the stochastic Lotka-Volterra model.Inspired by the works referred above, in this paper, we will investigate the persistence and extinction of a general stochastic nonautonomous Lotka-Volterra model with delays and impulsive perturbations. To our knowledge, there are few results of this aspect for the stochastic nonautonomous Lotka-Volterra model. Moreover, all the publications have not obtained the persistence-extinction threshold for the general stochastic nonautonomous Lotka-Volterra model with delays and impulsive perturbations. The problems above are explored and some main results are given in this paper. The general stochastic nonautonomous Lotka-Volterra model with delays and impulsive perturbations which has a unique positive global solution is investigated. For this model, sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are established. The influences of the stochastic noises and impulsive perturbations on the properties of the stochastic model are discussed. Comparing with deterministic results [25–27], if the noise is sufficiently small, the property permanence that the related deterministic system possesses is preserved in the stochastic model. However, with the increase of stochastic noise, the properties of the system may be changed greatly. For example, the solution to the associated stochastic model will be extinct with probability one with the increase of stochastic noise being sufficiently large, although the solution to the original deterministic model may be persistent. The properties of the system are not affected by the impulsive perturbations which are bounded; otherwise, the properties may be changed significantly. The critical value between weak persistence and extinction is obtained.The rest of the paper is arranged as follows. The stochastic nonautonomous Lotka-Volterra model with delays and impulsive perturbations is formulated and some notations and preliminaries are given in Section2. Section 3 shows that the general nonautonomous Lotka-Volterra model has a unique positive global solution. Then, sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are established in Section 4. The simulation results in Section 5 are given to illustrate the main results obtained in this paper. Finally, the conclusions are given in Section 6.


## 2. Problem Formulation and Preliminaries

A classical nonautonomous Lotka-Volterra model with time-varying and infinite delays can be expressed as follows:(1)dxitdt=xitrit-∑j=1naijtxjt+∑j=1nbijtxjt-τijt+∑j=1ncijt∫-∞0xjt+θdμijθ,1≤i≤n,1≤j≤n,where x(t)=(x1(t),…,xn(t))T; xi(t) and ri(t) are the population size and intrinsic exponential growth rate for the ith species at time t, respectively; aij(t),  bij(t), and cij(t) represent the effects of interspecific (for i≠j) and intraspecific (for i=j) interaction at time t; τij(t)≥0 represents the time-varying delays; and μij(θ) is the probability measure on (-∞,0], 1≤i,j≤n.In real life, model (1) is affected by environmental noises. The intrinsic growth rate of the ith species ri(t)(1≤i≤n) at time t is estimated by an average value plus an error term. Then the intrinsic growth rate becomes ri(t)→ri(t)+σi(t)B˙i(t). The effects of interspecific (for i≠j) and intraspecific (for i=j) interaction aij(t)(1≤i,j≤n) at time t are estimated by an average value plus an error term. So aij(t) is replaced by -aij(t)→-aij(t)+δij(t)B˙ij(t).  σi(t) and δij(t)(1≤i,j≤n) are continuous nonnegative bounded functions on R¯+=[0,+∞) and the intensities of the white noise at time t. B˙i(t) and B˙ij(t) are the white noises; Bi(t) and Bij(t) are the one-dimensional Brownian motions defined on a complete probability space (Ω,F,P).On the other hand, in practice, model (1) is affected by impulsive perturbations. As a result, model (1) becomes the following stochastic nonautonomous Lotka-Volterra model with impulsive perturbations:(2)dxit=xitrit-∑j=1naijtxjt+∑j=1nbijtxjt-τijt+∑j=1ncijt∫-∞0xjt+θdμijθdt+xitσitdBit+xit∑j=1nδijtxjtdBijt,t≠tk,k∈N,xitk+-xitk=hikxitk,k∈N,1≤i≤n,1≤j≤n, where N denotes the set of positive integers, 0<t1<t2<⋯<limk→+∞⁡tk=+∞. This model will be studied in this paper.Let the initial dataξ(θ)=(ξ1(θ),ξ2(θ),…,ξn(θ))T be positive and belong to the friendly space Cg [28] which is defined by(3)Cg=φ∈C-∞,0;0,+∞n:l<φCg=sup-∞<θ≤0erθφθ<L,where φθ=(φ12(θ)+φ22(θ)+⋯+φn2(θ))1/2, r>0, and l,L are two positive constants, l<L.For model (2) we always assume the following:(H1):
μij(θ)(1≤i,j≤n) is the probability measure on (-∞,0] satisfying μijr=∫-∞0‍e-2rθdμij(θ)<+∞.(H2):
ri(t),aij(t),bij(t) and cij(t)(1≤i,j≤n) are continuous and bounded function on R¯+ and min1≤i,j≤ninft∈R¯+aij(t)>0, bij(t)≥0, cij(t)≥0.(H3):
τij(t)(1≤i,j≤n) are continuously differentiable functions with 0≤τij(t)≤τM and 1-τ˙ij(t)>0 for t∈R, where τM is a constant. Δij-1(t) is inverse function of Δij(t)=t-τij(t).(H4):
we assume that1+hik>0, i=1,2,…,n, k∈N. When hik>0, the perturbation denotes increasing of the species (e.g., planting), while -1<hik<0 represents decreasing (e.g., harvesting).Definition 1 (see [29]).
A stochastic processX(t)=(X1(t),…,Xn(t))T, t∈R¯+, is said to be a solution of ISDE:(4)dXt=Ft,Xtdt+Gt,XtdBt,t≠tk,k∈N,Xtk+-Xtk=BkXtk,k∈N with initial condition X(0), if the following conditions are satisfied:(1)
X(t) is adapted and is continuous on (0,t1) and (tk,tk+1), k∈N;  F(t,X(t))∈L1(R¯+;Rn), G(t,X(t))∈L2(R¯+;Rn), where Lk(R¯+;Rn) is all Rn valued measurable adapted processes f(t) satisfying ∫0T‍|f(t)|kdt<+∞ a.s. (almost surely) for every T>0.(2)
For eachtk,  k∈N,  X(tk+)=limt→tk+⁡X(t) and X(tk-)=limt→tk-⁡X(t) exist and X(tk)=X(tk-) with probability one.(3)
X(t) obeys the equivalent integral equation of (4) for almost every t∈R¯+∖tk and satisfies the impulsive conditions at each t=tk, k∈N with probability one. Then X(t) is said to be a solution of ISDE (4).
For the aim of simplicity, we define the following notations:(5)fu=supt∈R⁡ft,fl=inft∈R⁡ft,xit=1t∫0txisds,xi∗=limsupt→+∞⁡xit,xi∗=liminft→+∞⁡xit,R+=0,+∞,gi∗=limsupt→+∞⁡t-1∑0<tk<tln⁡1+hik+∫0trit-σi2s2ds,1≤i≤n. For any sequence {dij(t)}(1≤i,j≤n), define(6)d¯iju=max1≤i,j≤n⁡supt∈R⁡dijt,d~ijl=min1≤i,j≤n⁡inft∈R⁡dijt. The following definitions are commonly used and we list them here.Definition 2.
(a) The populationxi(t) is said to go toextinction a.s. if limt→+∞xi(t)=0.
(b) The populationxi(t) is said to benonpersistence in the mean a.s. (see, e.g., [22]) if limsupt→+∞〈xi(t)〉=0.
(c) The populationxi(t) is said to beweak persistence a.s. (see, e.g., [23]) if limsupt→+∞xi(t)>0.
(d) The populationx(t)=(x1(t),x2(t),…,xn(t))T is said to bestochastic permanenceif, for arbitrary ɛ>0, there are constants β>0, H>0 such that liminft→+∞P{|x(t)|≥β}≥1-ɛ and liminft→+∞P{|x(t)|≤H}≥1-ɛ, where · denotes the Euclidian norm in R+n.


## 3. Nonexplosion

Theorem 3.
Consider model (2); for any given initial value ξ(θ)=(ξ1(θ),ξ2(θ),…,ξn(θ))T∈Cg, there is a unique solution x(t) on t∈R and the solution remains in R+n with probability 1; in other words, x(t)∈R+n for all t∈R almost surely.Proof.
Consider the following SDE without impulses:(7)dyit=yitrit-∑j=1naijt∏0<tk<t1+hjkyjt+∑j=1nbijt∏0<tk<t-τijt1+hjkyjt-τijt+∑j=1ncijt∫-∞0∏0<tk<t+θ1+hjkyjt+θdμijθdt+yitσitdBit+yit∑j=1nδijt∏0<tk<t1+hjkyjtdBijt,with the same initial condition as model (2).
Now for model (7), there is a unique solution y(t) on t∈R and the solution will remain in R+ with probability 1.
Since the coefficients of model (7) do not fulfil the linear growth condition, the general theorems of existence and uniqueness cannot be implemented for this equation. However, they are locally Lipschitz continuous; hence for any given positive initial condition ξ(θ)=(ξ1(θ),ξ2(θ),…,ξn(θ))T∈Cg, there is a unique local solution y(t) on t∈(-∞,τe), where τe is the explosion time. To show this solution x(t) is global solution, which means that the explosion time τe=+∞, a.s. Let us choose k0>0 which is sufficiently large such that(8)1k0<min-∞<θ≤0⁡ξθ≤max-∞<θ≤0⁡ξθ<k0.For each integer k≥k0, define the stopping time(9)τk=inf⁡t∈-∞,τe:yit≤1koryit≥k,1≤i≤n,where throughout this paper we set inf⁡Ø=+∞ (as usual Ø denotes the empty set). Clearly, τk is increasing as k→+∞. Set τ+∞=limk→+∞τk, whence τ+∞≤τe a.s. and x(t)∈R+n a.s. for all t≥0. In other words, to complete the proof all we need to show is that τ+∞=+∞ a.s. To show this statement, let us define a C2-function V: R+n→R+ by V1(y)=∑i=1n[yi-1-0.5ln⁡(yi)], where y=(y1,…,yn)T. Let k≥k0 and T>0 be arbitrary. Now define(10)Vt,yt=∑i=1n∑j=1n∫t-τijtt∏0<tk<s1+hjk2yj2s1-τ˙ijΔij-1sds+∑i=1n∑j=1n∫-∞0∫t+θt∏0<tk<s1+hjk2yj2sdsdμijθ+V1yt. For 0≤t≤τk∧T, applying Itô’s formula to V(t,y(t)), we can get(11)dVt,yt=Fytdt+∑i=1n12yi0.5t-1σitdBit+∑i=1n12yi0.5t-1∑j=1nδijt∏0<tk<t1+hjkyjtdBijt,where (12)Fyt=∑i=1n∑j=1n∏0<tk<t1+hjk2yj2t1-τ˙ijΔij-1t+n∑j=1n∏0<tk<t1+hjk2yj2t+∑i=1n12rityi0.5t-1-∑i=1n∑j=1n12aijtyi0.5t-1∏0<tk<t1+hjkyjt+∑i=1n∑j=1n116bij2tyi0.5t-12+∑i=1n∑j=1n116cij2tyi0.5t-12-18∑i=1nσi2tyi0.5t+14∑i=1nσi2t-18∑i=1nyi0.5t∑j=1nδijt∏0<tk<t1+hjkyjt2+14∑i=1n∑j=1nδijt∏0<tk<t1+hjkyjt2.With the fact that ∏0<tk<t(1+hik)yi(t)≤∑i=1n∏0<tk<t(1+hik)yi(t), it is easy to see that F(y(t)) is bounded on R+n. In other words, there exists a positive constant K such that F(y(t))≤K. Therefore,(13)dVt,yt≤Kdt+∑i=1n12yi0.5t-1σitdBit+∑i=1n12yi0.5t-1∑j=1nδijt∏0<tk<t1+hjkyjtdBijt. Integrating both sides from 0 to t and then taking expectations, we have(14)EVt,yt≤∑i=1n∑j=1n∫-τij00∏0<tk<t1+hjk2yj2s1-τ˙ijΔij-1sds+∑i=1n∑j=1n∫-∞0∫θ0∏0<tk<s1+hjk2yj2sdsdμijθ+V1y0+Kt. Let t=τk∧T, from τ˙ij(Δij-1(t))<1, we have(15)EV1yτk∧T≤∑i=1n∑j=1n∫-τij00∏0<tk<t1+hjk2yj2s1-τ˙ijΔij-1sds+∑i=1n∑j=1n∫-∞0∫θ0∏0<tk<s1+hjk2yj2sdsdμijθ+V1y0+KT. Note that, for every ω∈{τk≤T}, xi(τk,ω) equals either k or 1/k, and hence V1(y(τk,ω)) is no less than either k-1-0.5log⁡(k) or 1/k-1-0.5log⁡1/k=1/k-1+0.5log⁡(k). Consequently,(16)V1yτk,ω≥k-1-0.5log⁡k∧1k-1+0.5log⁡k. It then follows from (14) that(17)∑i=1n∑j=1n∫-τij00∏0<tk<t1+hjk2yj2s1-τ˙ijΔij-1sds+∑i=1n∑j=1n∫-∞0∫θ0∏0<tk<s1+hjk2yj2sdsdμijθ+V1y0+KT≥EIτk≤TωV1yτk,ω≥Pτk≤Tk-1-0.5log⁡k∧1k-1+0.5log⁡k,where Iτk≤T is the indicator function of {τk≤T}. Letting k→+∞, limk→+∞P{τk≤T}=0 and hence P{τ+∞≤T}=0. Since T>0 is arbitrary, we must have P{τ+∞<+∞}=0, so P{τ+∞=+∞}=1 as required.
Now let(18)xit=∏0<tk<t1+hikyit. Next x(t) is the solution of (2) which will be shown. In fact, x(t) is continuous on (tk,tk+1)⊂(0,+∞), k∈N and for every t≠tk,(19)dxit=d∏0<tk<t1+hikyit=∏0<tk<t1+hikdyit=∏0<tk<t1+hikyitrit-∑j=1naijt∏0<tk<t1+hjkyjt+∑j=1nbijt∏0<tk<t-τijt1+hjkyjt-τijt+∑j=1ncijt∫-∞0∏0<tk<t+θ1+hjkyjt+θdμijθdt+∏0<tk<t1+hikσityitdBit+∏0<tk<t1+hikyit∑j=1nδijt∏0<tk<t1+hjkyjtdBijt=xitrit-∑j=1naijtxjt+∑j=1nbijtxjt-τijt+∑j=1ncijt∫-∞0xjt+θdμijθdt+σitxitdBit+xit∑j=1nδijtxjtdBijt.Moreover, for every k∈N and tk∈[0,+∞),(20)xitk+=limt→tk+⁡∏0<tl<t1+hilyit=∏0<tl≤tk1+hilyitk+=1+hik∏0<tl<tk1+hilyitk=1+hikxitk. In addition,(21)xitk-=limt→tk-⁡∏0<tl<t1+hilyit∏0<tl<tk1+hilyitk-=∏0<tl<tk1+hilyitk=xitk. So the proof is completed.


## 4. The Persistence and Extinction Analysis

In this section, the extinction, nonpersistence in the mean, weak persistence, and stochastic permanence of model (2) are discussed.Theorem 4.
Ifgi∗<0 and min1≤j≤ninft∈R¯+⁡{aij(t)-bij(Δij-1(t))/1-τ˙ij(Δij-1(t))-ciju}≥0, then the ith population xi(t) of model (2) goes to extinction a.s.Proof.
Now applying Itô’s formula to (7), we can have(22)d∑j=1n∫t-τijttbijΔij-1s∏0<tk<s1+hjkyjs1-τ˙ijΔij-1sds+dln⁡yit=∑j=1nbijΔij-1t∏0<tk<t1+hjkyjt1-τ˙ijΔij-1t-bijt∏0<tk<t-τijt1+hjkyjt-τijtdt+rit-σi2t2-∑j=1naijt∏0<tk<t1+hjkyjt+∑j=1nbijt∏0<tk<t-τijt1+hjkyjt-τijt+∑j=1ncijt∫-∞0∏0<tk<t+θ1+hjkyjt+θdμijθ-∑j=1nδijt∏0<tk<t1+hjkyjt22dt+σitdBit+∑j=1nδijt∏0<tk<t1+hjkyjtdBijt.Then we have(23)∑j=1n∫t-τijttbijΔij-1sxjs1-τ˙ijΔij-1sds-∫-τij00bijΔij-1sxjs1-τ˙ijΔij-1sds+ln⁡yit-ln⁡yi0=∫0tris-σi2s2-∑j=1naijs-bijΔij-1s1-τ˙ijΔij-1sxjs+∑j=1ncijs∫-∞0xjs+θdμijθ-∑j=1nδijsxjs22ds+Mi1t+Mi2t,where Mi1(t)=∫0tσi(s)dBi(s) and Mi2(t)=∫0t∑j=1nδij(s)xj(s)dBij(s). By hypothesis (H1),(24)∫0t∑j=1ncijs∫-∞0xjs+θdμijθds=∑j=1n∫0tcijsds∫-∞-sers+θxjs+θe-rs+θdμijθ+∑j=1n∫-t0dμijθ∫-θtxjs+θds≤∑j=1n1rcijuξjCrμijr1-e-rt+∑j=1nciju∫0txjsds.The quadratic form of Mi1(t) is 〈Mi1(t),Mi1(t)〉=∫0t‍σi2(s)ds≤(σiu)2t. Making use of the strong law of large numbers for martingales [30],(25)limt→+∞⁡Mi1tt=0,a.s.The quadratic form of Mi2(t) is 〈Mi2(t),Mi2(t)〉=∫0t[∑j=1nδij(s)xj(s)]2ds. By virtue of the exponential martingale inequality [30], for any positive constants T0,α and β, we have(26)Psup0≤t≤T0⁡Mi2t-α2Mi2t,Mi2t>β≤e-αβ.Choose T0=k, α=1, β=2ln⁡k. Then it follows that(27)Psup0≤t≤k⁡Mi2t-12Mi2t,Mi2t>2ln⁡k≤1k2.Making use of the Borel-Cantelli lemma [30], one gets that, for almost all ω∈Ω, there is a random integer k0=k0(ω) such that, for k≥k0,(28)sup0≤t≤k⁡Mi2t-12Mi2t,Mi2t≤2ln⁡k.This implies that(29)Mi2t≤2ln⁡k+12Mi2t,Mi2t=2ln⁡k+12∫0t∑j=1nδijsxjs2dsfor all 0≤t≤k, k≥k0 a.s. Substituting this inequality, (24), and (25) into (23), we can obtain that(30)∑0<tk<tln⁡1+hik+ln⁡yit-ln⁡yi0≤∑0<tk<tln⁡1+hik+∑j=1n∫-τij00bijΔij-1sxjs1-τ˙ijΔij-1sds+∫0tris-σi2s2-∑j=1naijs-bijΔij-1s1-τ˙ijΔij-1s-cijuxjsds+2ln⁡k+∑j=1n1rcijuξjCrμijr1-e-rt+Mi1t,for all 0≤t≤k, k≥k0 a.s. Therefore,(31)ln⁡xit-ln⁡xi0≤∑0<tk<tln⁡1+hik+∑j=1n∫-τij00bijΔij-1sxjs1-τ˙ijΔij-1sds+∫0tris-σi2s2-∑j=1naijs-bijΔij-1s1-τ˙ijΔij-1s-cijuxjsds+2ln⁡k+∑j=1n1rcijuξjCrμijr1-e-rt+Mi1t,for all 0≤t≤k, k≥k0 a.s. In other words, we can get that for 0<k-1≤t≤k, k≥k0,(32)t-1ln⁡xit-ln⁡xi0≤t-1∑0<tk<tln⁡1+hik+t-1∑j=1n∫-τij00bijΔij-1sxjs1-τ˙ijΔij-1sds+t-1∫0tris-σi2s2ds+2k-1-1ln⁡k+t-1∑j=1n1rcijuξjCrμijr1-e-rt+Mi1tt.
Taking superior limit on both sides of (32) and using (25), we have limsupt→+∞ln⁡xi(t)/t≤gi∗. That is to say, if gi∗<0, one can see that limt→+∞xi(t)=0 a.s. So the proof is completed.Theorem 5.
Ifgi∗=0 and min1≤j≤ninft∈R¯+{aij(t)-bij(Δij-1(t))/1-τ˙ij(Δij-1(t))-ciju}>0, then the ith population xi(t) of model (2) is nonpersistent in the mean a.s.Proof.
Due to (25), for any given ɛ>0, there is a positive constant T, such that(33)t-1∑0<tk<tln⁡1+hik+t-1∑j=1n∫-τij00bijΔij-1sxjs1-τ˙ijΔij-1sds+t-1∑j=1n1rcijuξjCrμijr1-e-rt+2ln⁡kt+Mi1tt<ɛ,for sufficiently large t satisfying t>T. Substituting the above inequality into (31), therefore,(34)ln⁡xit-ln⁡xi0t<ɛ-t-1∫0t∑j=1naijs-bijΔij-1s1-τ˙ijΔij-1s-cijuxjsds,for all T≤k-1≤t, k≥k0 a.s.
Definehi(t)=∫0t‍xi(s)ds, N=min1≤j≤ninfs∈R¯+[aij(s)-bij(Δij-1(s))/1-τ˙ij(Δij-1(s))-ciju], with the fact that xi(t)≤∑i=1n‍xi(t)≤nxt; we have(35)ln⁡dhitdt<ɛt-Nhit+ln⁡xi0;t>T. Consequently,(36)eNhitdhitdt<xi0eɛt;t>T. Integrating this inequality from T to t,(37)N-1eNhit-eNhiT<xi0ε-1eɛt-eɛT. Rewriting this inequality,(38)eNhit<eNhiT+xi0Nε-1eɛt-xi0Nε-1eɛT. Taking the logarithm of both sides, we have(39)hit<N-1ln⁡xi0Nε-1eɛt+eNhiT-xi0Nε-1eɛT. In other words, we can get that(40)t-1∫0txisds≤t-1N-1ln⁡xi0Nε-1eɛt+eNhiT-xi0Nε-1eɛT∗. By applying L’Hospital’s rule, one can obtain(41)xit∗≤N-1t-1lnxi0Nε-1eɛt∗=ɛN. Since ɛ is arbitrary, we have 〈xi(t)〉∗=0, which is the required assertion. So the proof is completed.Theorem 6.
Ifgi∗>0, min1≤j≤ninft∈R¯+{aij(t)-eτij(Δij-1(t))bij(Δij-1(t))/1-τ˙ij(Δij-1(t))-cijuμijr}>0, then the ith population xi(t) of model (2) is weak persistence a.s.Proof.
To begin with, let us claim that(42)limsupt→+∞⁡t-1ln⁡xit≤0a.s. Applying Itô’s formula to (7), we have(43)d∑j=1n∫t-τijttes+τijΔij-1sbijΔij-1s∏0<tk<s1+hjkyjs1-τ˙ijΔij-1sds+etln⁡yit=etln⁡xit-∑0<tk<tln⁡1+hik+rit-σi2t2-∑j=1naijt-eτijΔij-1tbijΔij-1t1-τ˙ijΔij-1txjt+∑j=1ncijt∫-∞0xjt+θdμijθ-∑j=1nδijtxjt22dt+etσitdBit+et∑j=1nδijtxjtdBijt. Thus,(44)∑j=1n∫t-τijttes+τijΔij-1sbijΔij-1sxjs1-τ˙ijΔij-1sds-∑j=1n∫-τij00es+τijΔij-1sbijΔij-1sxjs1-τ˙ijΔij-1sds+etln⁡yit-ln⁡yi0=∫0tesln⁡xis-∑0<tk<sln⁡1+hik+ris-σi2s2-∑j=1naijs-eτijΔij-1sbijΔij-1s1-τ˙ijΔij-1sxjs+∑j=1ncijs∫-∞0xjs+θdμijθ-∑j=1nδijsxjs22ds+Ni1t+Ni2t, where  Ni1(t)=∫0t‍esσi(s)dBi(s), Ni2(t)=∫0t‍es∑j=1n‍δij(s)xj(s)dBij(s). Now we have(45)∫0tes∑j=1ncijs∫-∞0xjs+θdμijθds=∑j=1n∫0tcijsesds∫-∞-sers+θxjs+θe-rs+θdμijθ+∑j=1n∫-t0dμijθ∫0t+θcijs-θes-θxjsds≤∑j=1ncijuμijrξjCrt+∑j=1ncijuμijr∫0tesxjsds.Note that Ni1(t)(1≤i≤n) is a local martingale with the quadratic form 〈Ni1(t),Ni1(t)〉=∫0t‍e2sσi2(s)ds. Ni2(t)(1≤i≤n) is also a local martingale with the quadratic form 〈Ni2(t),Ni2(t)〉=∫0t‍e2s[∑j=1n‍δij(s)xj(s)]2ds. Following from the exponential martingale inequality (26) by choosing T0=μk, α=e-μk, β=ρeμkln⁡k,(46)Psup0≤t≤μk⁡Niλt-0.5e-μkNiλt,Niλt>ρeμkln⁡k≤k-ρ, where ρ>1 and μ>1, λ=1,2. In view of the Borel-Cantelli lemma [30], for almost all ω∈Ω, there exists a k0(ω) such that, for every k≥k0(ω),(47)Niλt≤0.5e-μkNiλt,Niλt+ρeμkln⁡k,0≤t≤μk.By hypotheses (H2) and (H3), it is easy to see that there exists a constant C independent of k such that(48)ln⁡xis-∑0<tk<sln1+hik+ris-σi2s2-∑j=1naijs-eτijΔij-1sbijΔij-1s1-τ˙ijΔij-1s-cijuμijrxjs+es-μkσi2s2-∑j=1nδijsxjs21-es-μk2≤C,for any 0≤s≤μk and xi(s)>0. Substituting the above inequalities (45), (47), and (48) into (44), for any 0≤t≤μk, we have(49)∑j=1n∫t-τijttes+τijΔij-1sbijΔij-1sxjs1-τ˙ijΔij-1sds-∑j=1n∫-τij00es+τijΔij-1sbijΔij-1sxjs1-τ˙ijΔij-1sds+etln⁡yit-ln⁡yi0≤Cet-1+∑j=1ncijuμijrξjCrt+2ρeμkln⁡k. On the other hand, we can obtain that(50)et∑0<tk<tln⁡1+hik+etln⁡yit-ln⁡yi0≤∑j=1n∫-τij00es+τijΔij-1sbijΔij-1sxjs1-τ˙ijΔij-1sds+et∑0<tk<tln⁡1+hik+Cet-1+∑j=1ncijuμijrξjCrt+2ρeμkln⁡k. That is to say,(51)etln⁡xit-ln⁡xi0≤∑j=1n∫-τij00es+τijΔij-1sbijΔij-1sxjs1-τ˙ijΔij-1sds+et∑0<tk<tln⁡1+hik+Cet-1+∑j=1ncijuμijrξjCrt+2ρeμkln⁡k. Therefore,(52)ln⁡xit≤e-tln⁡xi0+∑0<tk<tln⁡1+hik+C1-e-t+2ρeμk-tln⁡k+e-t∑j=1ncijuμijrξjCrt+∑j=1n∫-τij00es-t+τijΔij-1sbijΔij-1sxjs1-τ˙ijΔij-1sds. Consequently, if μ(k-1)≤t≤μk and k≥k0(ω), one can observe that(53)t-1ln⁡xit≤t-1e-tln⁡xi0+t-1∑0<tk<tln⁡1+hik+t-1C1-e-t+2t-1ρeμk-tln⁡k+t-1e-t∑j=1ncijuμijrξjCrt+t-1∑j=1n∫-τij00es-t+τijΔij-1sbijΔij-1sxjs1-τ˙ijΔij-1sds,which becomes the desired assertion (42) by letting t→+∞.
Now suppose thatgi∗>0; we will prove limsupt→+∞xi(t)>0 a.s. If this assertion is not true, let F={limsupt→+∞xi(t)=0} and suppose P(F)>0. In the light of (23), we can get that(54)t-1∑j=1n∫t-τijttbijΔij-1sxjs1-τ˙ijΔij-1sds-t-1∑j=1n∫-τij00bijΔij-1sxjs1-τ˙ijΔij-1sds+t-1ln⁡xit-t-1ln⁡xi0=t-1∑0<tk<tln⁡1+hik+t-1∫0tris-σi2s2-∑j=1naijs-bijΔij-1s1-τ˙ijΔij-1sxjs+∑j=1ncijs∫-∞0xjs+θdμijθ-∑j=1nδijsxjs22ds+Mi1tt+Mi2tt. On the other hand, for ∀ω∈F, we have limt→+∞xi(t,ω)=0 and the fact that xi(t)≤∑i=1n‍xi(t)≤nxt. The law of large numbers for local martingales [30] indicates that limt→+∞Mi2(t)/t=0. Substituting this equality and (25) into (68),(55)limsupt→+∞⁡t-1ln⁡xit,ω≥limsupt→+∞⁡t-1∑0<tk<tln⁡1+hik+∫0tris-σi2s2ds=gi∗>0. Then P(limsupt→+∞[t-1lnxi(t)]>0)>0, which contradicts (42). So the proof is completed.When it comes to the study of population model, the role of stochastic permanence indicating the eternal existence of the population can never be ignorant with its theoretical and practical significance. So now let us show that the populationxi(t) is stochastic permanence in some cases.(H5):
there are two positive constantsm and M such that m≤∏0<tk<t‍(1+hik)≤M   (i=1,2,…,n) for all t>0.Theorem 7.
Under assumption (H5), ifr~il≥2(σ¯iu)2, r≥1, and there exists ε2∈(0,2r) such that min1≤i,j≤ninft∈R¯+{eε2(t+τij(t))-eε2(Δij-1(t))/1-τ˙ij(t)}≥0, then the population x(t) of model (2) is stochastic permanence.Proof.
We prove that, for arbitraryɛ∈(0,1), there is a positive constant H=H(ɛ) such that liminft→+∞P{xt≤H}≥1-ɛ. Consider 0<p<1; by Itô’s formula, we have(56)d∑i=1nyipt≤Fxtdt-∑i=1nε2xipt∏0<tk<t1+hikp+∑i=1n∑j=1neε2τijtxj2t-∑i=1n∑j=1nxj2t-τijt-∑i=1n∑j=1n∫-∞0xj2t+θdμijθ+∑i=1n∑j=1nμijrxj2tdt+∑i=1npxiptσit∏0<tk<t1+hikpdBit+∑i=1npxipt∏0<tk<t1+hikp∑j=1nδijtxjtdBijt, where(57)Fxt=∑i=1n∑j=1neε2τijtxj2t+∑i=1nε2+ritpxipt∏0<tk<t1+hikp+∑i=1n∑j=1np2bij2txi2pt4∏0<tk<t1+hik2p+∑i=1n∑j=1nμijrxj2t+∑i=1n∑j=1nxj2t-τijt+∑i=1n∑j=1np2cij2txi2pt4∏0<tk<t1+hik2p-∑i=1n12p1-pσi2txipt∏0<tk<t1+hikp-∑i=1n12p1-pxipt∏0<tk<t1+hikp∑j=1nδijtxjt2.With the fact that xi(t)≤∑i=1n‍xi(t)≤n|x(t)|, it is easy to see that F(x(t)) is bounded in R+n; in other words, M1=supx(t)∈R+nF(x(t))<+∞. Therefore,(58)d∑i=1nyipt≤M1-∑i=1nε2xipt∏0<tk<t1+hikp-∑i=1n∑j=1neε2τijtxj2t+∑i=1n∑j=1nxj2t-τijtdt+∑i=1n∑j=1n∫-∞0xj2t+θdμijθdt-∑i=1n∑j=1nμijrxj2tdt+pxiptσitdBit+pxipt∑j=1nδijtxjtdBijt. Once again by Itô’s formula, we have(59)deε2t∑i=1nyipt=eε2t∑i=1nε2yiptdt+d∑i=1nyipt≤eε2tM1-∑i=1n∑j=1neε2τijtxj2t+∑i=1n∑j=1nxj2t-τijt+∑i=1n∑j=1n∫-∞0xj2t+θdμijθ-∑i=1n∑j=1nμijrxj2tdt+eε2tpxiptσitdBit+eε2tpxipt∑j=1nδijtxjtdBijt. We hence derive that(60)eε2tE∑i=1nyipt≤∑i=1nξip0+eε2tM1ε2-M1ε2-E∑i=1n∑j=1n∫0teε2s+ε2τijsxj2sds+E∑i=1n∑j=1n∫-τij0t-τijteε2Δij-1sxj2sds1-τ˙ijsds+E∑i=1n∑j=1n∫0teε2s∫-∞0xj2s+θdμijθds-E∑i=1n∑j=1n∫0tμijreε2sxj2sds≤∑i=1nξip0+eε2tM1ε2-M1ε2+E∑i=1n∑j=1n∫-τij00eε2s+ε2τijsxj2sds+E∑i=1n∑j=1n∫0teε2s∫-∞0xj2s+θdμijθds-E∑i=1n∑j=1nμijr∫0teε2sxj2sds. From hypothesis (H1), we have(61)∫0teε2s∫-∞0xj2s+θdμijθds=∫0teε2sds∫-∞-se2rs+θxj2s+θe-2rs+θdμijθ+∫-t0dμijθ∫0t+θeε2s-θxj2sds≤ξjCr2μijrt+μijr∫0teε2sxj2sds. This implies that limsupt→+∞⁡E∑i=1nyipt≤M1/ε2.Therefore,(62)limsupt→+∞⁡E∑i=1n∏0<tk<t1+hikpyipt≤MpM1ε2. So we can get that(63)limsupt→+∞⁡E∑i=1nxipt≤MpM1ε2. On the other hand, we have xt2≤nmax1≤i≤nsupt∈R⁡xit2, so(64)xtp≤np/2max1≤i≤nsupt∈R⁡xitp≤np/2∑i=1nxipt. Therefore, finally we have (65)limsupt→+∞⁡Extp≤np/2MpM1ε2. Setting K=np/2MpM1/ε2 and choosing p=1/2,(66)limsupt→+∞⁡Ext≤K. Now for any ɛ>0, let H=K2/ε2. Then, by Chebyshev’s inequality,(67)Pxt>H=Pxt>H≤ExtH. Hence limsupt→+∞⁡Pxt>H≤ɛ. This implies(68)liminft→+∞⁡Pxt≤H≥1-ɛ.Next, we claim that, for arbitrary ɛ>0, there is a constant β>0 such that liminft→+∞P{|x(t)|≥β}≥1-ɛ. Denote V(y(t))=∑i=1n‍yi(t). Applying Itô’s formula, we get(69)dVyt=∑i=1nyitrit-∑j=1naijt∏0<tk<t1+hjkyjt+∑j=1nbijt∏0<tk<t-τijt1+hjkyjt-τijt+∑j=1ncijt∫-∞0∏0<tk<t+θ1+hjkyjt+θdμijθdt+∑i=1nyitσitdBit+∑i=1nyit∑j=1nδijt∏0<tk<t1+hjkyjtdBijt. Define U(y(t))=1/V(y(t)) on t≥0. Then we have(70)dU=-U2∑i=1nyitrit-∑j=1naijt∏0<tk<t1+hjkyjt+∑j=1nbijt∏0<tk<t-τijt1+hjkyjt-τijt+∑j=1ncijt∫-∞0∏0<tk<t+θ1+hjkyjt+θdμijθ+U3∑i=1nyitσit2+∑i=1nyit∑j=1nδijt∏0<tk<t1+hjkyjt2dt-U2∑i=1nyitσitdBit-U2∑i=1nyit∑j=1nδijt∏0<tk<t1+hjkyjtdBijt,dropping y(t) from U(y(t)). Define the function V¯(y(t))=U2+p(y(t)); then by Itô’s formula, we have(71)LV¯yt=2+pUp-U3∑i=1nyitrit-∑j=1naijt∏0<tk<t1+hjkyjt+∑j=1nbijt∏0<tk<t-τijt1+hjkyjt-τijt+∑j=1ncijt∫-∞0∏0<tk<t+θ1+hjkyjt+θdμijθ+U4∑i=1nyitσit2+∑i=1nyit∑j=1nδijt∏0<tk<t1+hjkyjt2+p+12U4∑i=1nyitσit2+∑i=1nyit∑j=1nδijt∏0<tk<t1+hjkyjt2. Since we know that(72)U∑i=1nyitrit≥r~il,U2∑i=1nyit∑j=1naijt∏0<tk<t1+hjkyjt≤Ma¯iju,U2∑i=1nyitσit2≤σ¯iu2,U4∑i=1nyit∑j=1nδijt∏0<tk<t1+hjkyjt2≤M2δ¯iju2. It follows from (71) that(73)LV¯yt≤2+p-r~il-p+32σ¯iu2U2+p+Ma¯ijuU1+p+p+32M2δ¯iju2Up. Now choose a constant k>0 sufficiently small such that it satisfies k-(2+p)(r~il-p+3/2(σ¯iu)2)<0. Therefore,(74)LektV¯yt=kektV¯yt+ektLV¯yt≤kektU2+pyt+ekt2+p-r~il-p+32σ¯iu2U2+p+Ma¯ijuU1+p+p+32M2δ¯iju2Up=ektk-2+pr~il-p+32σ¯iu2U2+p+2+pMa¯ijuU1+p+2+pp+32M2δ¯iju2Up≤Kekt.This implies that(75)limsupt→+∞⁡EU2+pyt≤K. For y(t)∈R+n, note that ∑i=1nyit2+p≤n2+pyt2+p. Consequently(76)limsupt→+∞⁡E1xt2+p≤n-2-plimsupt→+∞⁡E1∑i=1nxit2+p=n-2-plimsupt→+∞⁡E1∑i=1n∏0<tk<t1+hikyit2+p≤mn-2-pK≕d. So for any ɛ>0, setting β=(ɛ/d)1/2+p, by Chebyshev’s inequality, gets that(77)Pxt<β=Pxt2+p<β2+p=P1xt2+p>1β2+p≤E1/xt2+p1/β2+p=β2+pE1xt2+p,which means that limsupt→+∞P{xt<β}≤β2+pd=ɛ. We can get that(78)liminft→+∞⁡Pxt≥β≥1-ɛ. So the whole proof is completed.Remark 8.
Ifgi∗>0 and min1≤j≤ninft∈R¯+{aij(t)-eτij(Δij-1(t))bij(Δij-1(t))/1-τ˙ij(Δij-1(t))-cijuμijr}>0, then the population xi(t) will be weak persistence. If gi∗<0 and min1≤j≤ninft∈R¯+{aij(t)-bij(Δij-1(t))/1-τ˙ij(Δij-1(t))-ciju}≥0, then the population xi(t) will go to extinction. That is to say, if min1≤j≤ninft∈R¯+{aij(t)-eτij(Δij-1(t))bij(Δij-1(t))/1-τ˙ij(Δij-1(t))-cijuμijr}>0 and min1≤j≤ninft∈R¯+{aij(t)-bij(Δij-1(t))/1-τ˙ij(Δij-1(t))-ciju}≥0 hold, then gi∗ is the critical number between weak persistence and extinction for the population xi(t).Remark 9.
Generally speaking, as the biology has implied, in Theorem4, on one hand, if the species in the process of planting, that is, hik>0, or harvesting, that is, hik<0, is affected by stochastic environmental noises which plays a dominant role, then the species will be extinct a.s. In a word, population probably will come to an end in the worst cases which is revealed in Theorem 4, while if the growth rate and the influences of the stochastic noises and impulsive perturbations cancel each other out, then the effects of interspecific (for i≠j) and intraspecific (for i=j) interaction at time t, that is, aij(t), are the dominant factor. So the living chances are considerably rare which is shown in Theorem 5. In Theorem 6, even though the growth rate is larger than the influences of the stochastic noises and impulsive perturbations, aij(t) plays the dominant role; then the population size is limited to zero with the time permitted; however, the opportunity of the survival of it still exists. In Theorem 7, if the growth rate is large enough, then the species will be stochastic permanence. This can well explain why the conditions are gradually stronger from Theorems 4–6.Remark 10.
According togi∗=limsupt→+∞t-1[∑0<tk<t‍ln(1+hik)+∫0t‍(ri(s)-σi2(s)/2)ds], on one hand, we are conscious of the fact that the stochastic noise on ri(t) is detrimental to the survival of the population but the stochastic noise on aij(t) has hardly impressed on the persistence or extinction of the population. Thus, in true ecological modelling, the stochastic noise on ri(t) should be realized but the stochastic noise on aij(t) could be overlooked in some cases.Remark 11.
In view ofgi∗=limsupt→+∞t-1[∑0<tk<t‍ln(1+hik)+∫0t‍(ri(s)-σi2(s)/2)ds], we can find that the properties including extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are not affected by the impulsive perturbations which are bounded and if the impulsive perturbations are unbounded the properties will change significantly.Remark 12.
Assumption (H5) is easy to be satisfied. For example, for theith population, if hik=e(-1)k+1/k2-1, then e0.75<∏0<tk<t‍(1+hik)<e for all t>0. Thus 1≤∏0<tk<t‍(1+hik)≤e for all t>0.


## 5. Examples and Numerical Simulations

In this section, we explore system behavior numerical solutions of model (2). For convenience, consider the case n=2 and let the probability measure μij(θ)=eθ(i=1,2,j=1,2) on (-∞,0] satisfying μijr=∫-∞0‍e-2rθdμij(θ)<+∞. Thus the nonautonomous stochastic model (2) will be written as(79)dxt=xtr1t-a11txt-a12tyt+b11txt-τ11t+b12tyt-τ12t+retc11t∫-∞0erθξ1θdθ+retc12t∫-∞0erθξ2θdθ+retc11t∫0terθxθdθ+retc12t∫0terθyθdθdt+σ1ttxtdB1t+δ11tx2tdB11t+δ12txtytdB12t,dyt=ytr2t-a21txt-a22tyt+b21txt-τ21t+b22tyt-τ22t+retc21t∫-∞0erθξ1θdθ+retc22t∫-∞0erθξ2θdθ+retc21t∫0terθxθdθ+retc22t∫0terθyθdθdt+σ2ttytdB2t+δ21txtytdB21t+δ22ty2tdB22t,xθ=ξ1θ,yθ=ξ2θ,ξθ=ξ1θ,ξ2θT∈Cg. By employing the Milstein method mentioned in Higham [31] to discretize the two equations, where the integral term is approximated by using the composite θ-rule as a quadrature [32] and taking ξ1(θ)=e-0.5θ,  ξ2(θ)=2e-0.5θ,  τij(t)≡0.3(i=1,2,j=1,2), we can obtain the discrete approximate solution with respect to (79)(80)xk+1=xk+xkr1kΔt-a11kΔtxk-a12kΔtyk+b11kΔtxk-300+b12kΔtyk-300+c11kΔtre-kΔt2+c12kΔtre-kΔt+c11kΔtre-kΔt∑j=1ke-rjΔtxjΔt+c12kΔtre-kΔt∑j=1ke-rjΔtxjΔtΔt+σ1kΔtxkΔtζ1k+12σ12kΔtxkζ1k2-1Δt+δ11kΔtxk2Δtζ11k+12δ112kΔtxk2ζ11k2-1Δt+δ12kΔtxkykΔtζ12k+12δ122kΔtxkykζ12k2-1Δt,yk+1=yk+ykr2kΔt-a21kΔtxk-a22kΔtyk+b21kΔtxk-300+b22kΔtyk-300+c21kΔtre-kΔt2+c22kΔtre-kΔt+c21kΔtre-kΔt∑j=1ke-rjΔtxjΔt+c22kΔtre-kΔt∑j=1ke-rjΔtyjΔtΔt+σ2kΔtykΔtζ2k+12σ22kΔtykζ2k2-1Δt+δ21kΔtxkykΔtζ21k+12δ212kΔtxkykζ21k2-1Δt+δ22kΔtyk2Δtζ22k+12δ222kΔtyk2ζ22k2-1Δt,where ζik and ζijk(i=1,2,j=1,2,k=1,2,…,n) are the Gaussian random variables which follow N(0,1).Here, we choosetk=100k,aij(t)=1.2+0.01cost(i=1,2,j=1,2), bij(t)=0.3+0.05sint,cij(t)=0.006+0.02cost, σi(t)=1.5,δij(t)=0.1,hik=e-0.005-1 for all k∈N and step size Δt=0.001. The only difference between conditions of Figures 1(a)–1(d) is that the representations of ri(t) are different.Figure 1
Persistence and extinction of model (68). (a) Extinction a.s. (b) Nonpersistence in the mean a.s. (c) Weak persistence a.s. (d) Stochastic permanence.
(a)
(b)
(c)
(d)Because of-1<hik=e-0.005-1<0, both species x(t) and y(t) are in the process of harvesting. Moreover, in Figure 1(a), we choose ri(t)=0.1+0.4sint; stochastic environmental noises play a dominant role; then the conditions of Theorem 4 are satisfied. In view of Theorem 4, the population x(t) and the population y(t) will go to extinction a.s. In Figure 1(b), we consider ri(t)=1.12505+0.4sint; the effects of interspecific (for i≠j) and intraspecific (for i=j) interaction at time t, that is, aij(t), are the dominant factor; then the conditions of Theorem 5 hold. By virtue of Theorem 5, the population x(t) and the population y(t) are nonpersistent in the mean a.s. In Figure 1(c), we choose ri(t)=1.6+0.4sint; aij(t) plays the dominant role; then the conditions of Theorem 6 are satisfied. That is to say, the population x(t) and the population y(t) are weak persistence a.s. This means that even though the species x(t) and y(t) are in the process of harvesting, aij(t) plays the dominant role; then the opportunity of the survival of it still exists. In Figure 1(d), we consider ri(t)=4.9; the growth rate is large enough; then the conditions of Theorem 7 hold. Making use of Theorem 7, the population x(t) and the population y(t) are stochastic permanence. By the numerical simulations, we can find that stochastic noise on ri(t)(1≤i≤n) can change the properties of the population models significantly.


## 6. Conclusions

In this paper, the persistence and extinction of a general stochastic nonautonomous N-species Lotka-Volterra model with time-varying, infinite delays and impulsive perturbations are investigated. Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are established in Theorems4–7. The influences of the stochastic noises and impulsive perturbations on the properties of the stochastic model are discussed. On one hand, if the noise is small enough, the property permanence that the related deterministic system possesses is preserved in the stochastic model. On the other hand, with the increase of noise, the solution of the considered model (2) that will become extinct with probability one, nonpersistent in the mean, or weakly persistent has also been shown in this paper. According to gi∗=limsupt→+∞t-1[∑0<tk<t‍ln(1+hik)+∫0t‍(ri(s)-σi2(s)/2)ds], we can obtain the result that if the impulsive perturbations are bounded, the properties including extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are not affected by the impulsive perturbations and if the impulsive perturbations are unbounded the properties will be affected by the impulsive perturbations and change greatly. Moreover, the critical number between extinction and weak persistence is obtained. Through the observation of Theorems 4–7, there is a very interesting phenomenon that the stochastic noise on ri(t) is detrimental to the survival of the population but the stochastic noise on aij(t) has hardly impressed on the persistence or extinction of the population. Finally, the numerical simulations are given to confirm the theoretical analysis results.



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