JAM Journal of Applied Mathematics 1687-0042 1110-757X Hindawi Publishing Corporation 10.1155/2014/963712 963712 Research Article Stochastically Ultimate Boundedness and Global Attraction of Positive Solution for a Stochastic Competitive System Guo Shengliang 1,2 Liu Zhijun 1,2 http://orcid.org/0000-0003-3713-0813 Xiang Huili 1,2 Fan Meng 1 Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province Hubei University for Nationalities Enshi, Hubei 445000 China hubu.edu.cn 2 Department of Mathematics Hubei University for Nationalities Enshi, Hubei 445000 China hubu.edu.cn 2014 292014 2014 12 06 2014 25 08 2014 25 08 2014 2 9 2014 2014 Copyright © 2014 Shengliang Guo 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.

A stochastic competitive system is investigated. We first show that the positive solution of the above system does not explode to infinity in a finite time, and the existence and uniqueness of positive solution are discussed. Later, sufficient conditions for the stochastically ultimate boundedness of positive solution are derived. Also, with the help of Lyapunov function, sufficient conditions for the global attraction of positive solution are established. Finally, numerical simulations are presented to justify our theoretical results.

1. Introduction

In recent years, many researches have been done on the dynamics of many types of Lotka-Volterra competitive systems. Owing to their theoretical and practical significance, these competitive systems have been investigated extensively and there exists a large volume of literature relevant to many good results (see ). Particularly, in , Gopalsamy introduced the following autonomous two-species competitive system: (1)y1(t)=y1(t)[a1-b1y1(t)-c1y2(t)-d1y12(t)],y2(t)=y2(t)[a2-b2y2(t)-c2y1(t)-d2y22(t)], where y1(t) and y2(t) can be interpreted as the population size of two competing species at time t, respectively. All parameters involved with the above model are positive constants and can be interpreted in more detail; a1 and a2 are the intrinsic growth rates of two species; b1, d1, b2, and d2 represent the effects of intraspecific competition; c1 and c2 are the effects of interspecific competition. Recently, Tan et al.  have considered the effect of impulsive perturbations and discussed the uniformly asymptotic stability of almost periodic solutions for a corresponding nonautonomous impulsive version of (1). It has also been noticed that the ecological systems, in the real world, are often perturbed by various types of environmental noise. May  also pointed out that, due to environmental fluctuation, the birth rate, the death rate, and other parameters usually show random fluctuation to a certain extent. To accurately describe such systems, it is necessary to use stochastic differential equations. Recently, various stochastic dynamical models have been introduced extensively in  and many interesting and valuable results including extinction, persistence, and stability can be found in .

Motivated by the above works, in this contribution, we assume that the environmental noise affects mainly the intrinsic growth rate ai with (2)aiai+σiw˙i(t),i=1,2, where w˙i(t) are independent white noises, wi(t) are standard Brownian motions defined on the complete probability space (Ω,F,{Ft}t0,P) with a filtration {Ft}t0 satisfying the usual conditions, and σi2 represent the intensities of the white noises. Then the stochastically perturbed sytem (1) can be Itô’s equations (3)dy1(t)=y1(t)[a1-b1y1(t)-c1y2(t)-d1y12(t)]dt+σ1y1(t)dw1(t),dy2(t)=y2(t)[a2-b2y2(t)-c2y1(t)-d2y22(t)]dt+σ2y2(t)dw2(t) with the initial values yi(0)>0.

In this paper, we will focus on the stochastically ultimate boundedness and global attraction of positive solutions of system (3). To the best of our knowledge, there are few published papers concerning system (3). The rest of this paper is organized as follows. In Section 2, we present some assumptions, definitions, and lemmas. In Section 3, we investigate the existence and uniqueness of positive solution, and then, we discuss the stochastically ultimate boundedness of positive solutions. In Section 4, we discuss the global attraction of positive solutions. Finally, we conclude and present a specific example to justify the analytical results.

2. Preliminaries

Throughout this paper, we give the notation R+2={y1>0,y2>0} and assumptions.

b1>d1, b2>d2.

For any initial value (y1(0),y2(0))R+2, there exists p>1 such that (4)yi(0)ai+(1/2)(p-1)σi2bi,i=1,2.

b1>c2, b2>c1.

In the following, let us briefly review several basic definitions and lemmas which will be useful for establishing our main results.

Definition 1.

The solution (y1(t),y2(t)) of system (3) is stochastically ultimately bounded a.s. if for arbitrary εi(0,1), there exists a positive constant φi=φ(εi) such that (5)limsupt+P{|yi(t)|>φi}<εi,i=1,2.

Definition 2.

Let (y1(t),y2(t)) be a positive solution of system (3). Then (y1(t),y2(t)) is said to be globally attractive provided that any other solution (y1*(t),y2*(t)) of system (3) satisfies (6)limt+|y1(t)-y1*(t)|=0,limt+|y2(t)-y2*(t)|=0a.s.

Lemma 3 (see [<xref ref-type="bibr" rid="B21">21</xref>]).

Let p>2,gMp([0,T];Rm) such that (7)E0T|g(s)|pds<, where Mp([0,T];Rm) is the family of processes {h(t)}0tT such that (8)E0T|h(t)|pdt<. Then (9)E|0Tg(s)dw(s)|p(p(p-1)2)p/2T(p-2)/2E0T|g(s)|pds.

Lemma 4 (see [<xref ref-type="bibr" rid="B22">22</xref>]).

Suppose that a1,a2,,an are real numbers; then (10)|a1+a2++an|pCp(|a1|p+|a2|p++|an|p), where p>0 and (11)Cp={1,0<p1;np-1,p>1.

Lemma 5 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Assume that an n-dimensional stochastic process X(t) on t0 satisfies the condition (12)E|X(t)-X(s)|αλ|t-s|1+β,0s,t<, for positive constants α, β, and λ. Then there exists a continuous version X~(t) of X(t) which has the property that, for every ϑ(0,β/α), there is a positive random variable ψ(ω) such that (13)P{ω:sup0<|t-s|<ψ(ω),0s,t<|X~(t,ω)-X(t,ω)||t-s|ϑ21-2-ϑ}=1. In other words, almost every sample path of X~(t) is locally but uniformly Hölder continuous with exponent ϑ.

Lemma 6 (see [<xref ref-type="bibr" rid="B24">24</xref>]).

Let f(t) be a nonnegative function on t0 such that f(t) is integrable on t0 and is uniformly continuous on t0. Then limt+f(t)=0.

3. Stochastically Ultimate Boundedness

In this section, we first show, under the assumption (S1), that the positive solution of system (3) will not explode to infinity at any finite time.

Lemma 7.

Let (S1) hold and the initial value (y1(0),y2(0))R+2. Then system (3) has a unique solution (y1(t),y2(t)) on t0, which will remain in R+2 with probability one.

Proof.

It is easy to see that the coefficients of system (3) satisfy the local Lipschitz condition. Then for any given initial value (y1(0),y2(0))R+2, there exists a unique local solution (y1(t),y2(t)) on [0,τe), where τe is the explosion time. To show that the positive solution is global, we only need to show that τe=+  a.s. Let n0 be sufficiently large for every component of (y1(0),y2(0)) which remains in the interval [1/n0,n0]. For each integer nn0, one can define the stopping time (14)τn=inf{t[0,τe):y1(t)(1n,n)ory2(t)(1n,n)}. Clearly, τn is increasing as n+. Assign τ+=limn+τn, whence τ+τe  a.s. If we can show that τ+=+  a.s., then τe=+  a.s. and (y1(t),y2(t))R+2  a.s. for all t0. In other words, to complete the proof, we just need to show that τ+=+  a.s.

By reduction to absurdity, we suppose that τ++; then there exists a pair of constants T>0 and ɛ(0,1) such that (15)P{τ+T}>ɛ. As a result, there exists an integer n1n0 such that for all nn1(16)P{τnT}ɛ. Define a C2-function V:R+2R+ as (17)V(y1,y2)=y1-1-lny1+y2-1-lny2. Obviously, V(y1,y2) is a nonnegative function. If (y1(t),y2(t))R+2, then using Itô’s formula, one can derive that (18)dV(y1,y2)=(1-1y1)dy1+0.51y12(dy1)2+(1-1y2)dy2+0.51y22(dy2)2={-d1y13(t)-b1y12(t)+d1y12(t)hhhhh+a1y1(t)+b1y1(t)+c2y1(t)hhhhh-c1y1(t)y2(t)-c2y1(t)y2(t)-d2y23(t)hhhhh-b2y22(t)+d2y22(t)+a2y2(t)hhhhh+b2y2(t)+c1y2(t)-a1-a2hhhhh+0.5σ12+0.5σ22}dt+(y1(t)-1)σ1dw1(t)+(y2(t)-1)σ2dw2(t)=F(y1,y2)dt+(y1(t)-1)σ1dw1(t)+(y2(t)-1)σ2dw2(t), where (19)F(y1,y2)=-d1y13(t)-b1y12(t)+d1y12(t)+a1y1(t)+b1y1(t)+c2y1(t)-c1y1(t)y2(t)-c2y1(t)y2(t)-d2y23(t)-b2y22(t)+d2y22(t)+a2y2(t)+b2y2(t)+c1y2(t)-a1-a2+0.5σ12+0.5σ22. A simple calculation shows that (20)F(y1,y2)-(b1-d1)y12(t)+(a1+b1+c2)y1(t)-(b2-d2)y22(t)+(a2+b2+c1)y2(t)-a1-a2+0.5σ12+0.5σ22. It then follows from (S1) that the upper bound of F(y1,y2), noted by K, exists. We therefore have (21)dV(y1,y2)Kdt+(y1(t)-1)σ1dw1(t)+(y2(t)-1)σ2dw2(t). Integrating both sides from 0 to τkT yields that (22)0τkTdV(y1,y2)0τkTKdt+0τkT(y1(t)-1)σ1dw1(t)+0τkT(y2(t)-1)σ2dw2(t), whence taking expectations leads to (23)EV(y1(τkT),y2(τkT))V(y1(0),y2(0))+KE(τkT)V(y1(0),y2(0))+KT. Set Ωn={τnT} for nn1, and then P(Ωn)ɛ. Note that arbitrary ωΩn, there exist some i such that yi(τn,ω) equals either n or 1/n. Then V(y1(τn,ω),y2(τn,ω)) is not less than (24)min{(n-1-lnn),(1n-1+lnn)}. As a consequence, (25)V(y1(0),y2(0))+KTE[1Ωn(ω)V(y1(τn,ω),y2(τn,ω))]ɛmin{(n-1-lnn),(1n-1+lnn)}, where 1Ωn is the indicator function of Ωn. Let n+ lead to the contradiction (26)+>V(y1(0),y2(0))+KT=+. So we must have τ+=+  a.s. This completes the proof of Lemma 7.

Lemma 7 is fundamental in this paper. In the following, we will show that the pth moment of the positive solution of system (3) is upper bounded and then discuss the stochastically ultimate boundedness.

Theorem 8.

Let (S1) and (S2) hold; then the positive solution (y1(t),y2(t)) of system (3) with initial value (y1(0),y2(0))R+2 is stochastically ultimately bounded.

Proof.

From Lemma 7, we can see that system (3) has a unique positive solution under assumption (S1). Assign p>1 arbitrarily; then by Itô’s formula we can show that (27)dyip(t)=pyip-1(t)dyi(t)+12p(p-1)yip-2(dyi(t))2=pyip(t)(12ai-biyi(t)-ciyj(t)-diyi2(t)hhhhhhhhh+12(p-1)σi2)dt+pσiyipdwi(t). Integrating and taking expectations on both sides yield that (28)E(yip(t))-E(yip(0))=0tpE{yip(s)(p-12ai-biyi(s)-ciyj(s)hhhhhhhhhhhhhihh-diyi2(s)+p-12σi2)}ds. We then derive that (29)dE(yip(t))dt=pE{yip(t)(p-12ai-biyi(t)-ciyj(t)hhhhhhhhhhhi-diyi2(t)+p-12σi2)}paiE(yip(t))-pbiE(yip+1(t))+12p(p-1)σi2E(yip(t))=pE(yip(t))[ai+p-12σi2]-pbiE(yip+1(t)). By Hölder inequality one has (30)E(yip+1(t))(E(yip(t)))(p+1)/p, and moreover (31)dE(yip(t))dtpE(yip(t)){ai+p-12σi2-bi[E(yip(t))]1/p}. Denote Xi(t)=E(yip(t)), Xi(0)=yip(0); then (31) can be rewritten as (32)dXi(t)dtpXi(t)[ai+p-12σi2-biXi1/p(t)]. It follows from (S2) that (33)0<biXi1/p(0)=byi(0)ai+12(p-1)σi2; that is, (34)0<Xi(0)[ai+(1/2)(p-1)σi2bi]pHi(p). Meanwhile, it is easy to see that (35)dXi(t)dt|Xi(t)=Hi(p)0. By the standard comparison theorem, we therefore derive that (36)E(yip(t))Hi(p), which implies that the pth moment of positive solution is upper bounded.

Let us now proceed to discuss the stochastically ultimate boundedness of system (3). Setting φi=[Hi(p)/εi]1/p, then by the Chebyshev inequality, we obtain that (37)P{|yi(t)|>φi}<E(yip(t))φipHi(p)Hi(p)/εi=εi. This gives that (38)limsupt+P{|yi(t)|>φi}<εi,i=1,2. The proof of Theorem 8 is complete.

4. Global Attraction

In this section, we will establish sufficient conditions for global attraction of system (3).

Lemma 9.

Let (S2) hold and let (y1(t),y2(t)) be a solution of (3) with initial value (y1(0),y2(0))R+2; then almost every sample path of (y1(t),y2(t)) is uniformly continuous for t0.

Proof.

We first prove  y1(t). Let us consider the following integral equation: (39)y1(t)=y1(0)+0tf1(s,y1(s),y2(s))ds+0tg1(s,y1(s),y2(s))dw1(s), where (40)f1(s,y1(s),y2(s))=y1(s)[a1-b1y1(s)-c1y2(s)-d1y12(s)],g1(s,y1(s),y2(s))=σ1y1(s). Recalling (S2), (32), and the standard comparison theorem, we can know that (41)E(yip(t))Hi(p),i=1,2. Then from Lemma 4 and (41) one derives that (42)E(|f1(s,y1(s),y2(s))|p)=E(y1p(s)|a1-b1y1(s)-c1y2(s)-d1y12(s)|p)0.5E(y12p(s))+0.5E(-d1y12(s)|2p|y12a1-b1y1(s)-c1y2(s)hhhhhhhhhhhhhhhhhhhhh-d1y12(s)|2p)0.5E(y12p(s))+0.5E(42p-1(|d1y12(s)|2p|a1|2p+|b1y1(s)|2phhhhhhhhhhhhhhh+|c1y2(s)|2p+|d1y12(s)|2p))0.5E(y12p(s))+0.5·42p-1|a1|2p+0.5·42p-1|b1|2pE(y12p(s))+0.5·42p-1|c1|2pE(y22p(s))+0.5·42p-1|d1|2pE(y14p(s))0.5H1(2p)+0.5·42p-1|a1|2p+0.5·42p-1|b1|2pH1(2p)+0.5·42p-1|c1|2pH2(2p)+0.5·42p-1|d1|2pH1(4p)L(p),E(|g1(s,y1(s),y2(s))|p)=E(σ1py1p(s))σ1pE(y1p(s))σ1pH1(p)R(p). Meanwhile, by Lemma 3, we obtain that, for 0t1<t2<+ and p>2, (43)E|t1t2g1(s,y1(s),y2(s))dw1(s)|p[p(p-1)2]p/2(t2-t1)(p-2)/2×t1t2E|g1(s,y1(s),y2(s))|pds. Let (44)t2-t11,1p+1q=1, and then from (42), (43), and Lemma 4, one can derive that (45)E|y1(t2)-y1(t1)|p=E|t1t2f1(s,y1(s),y2(s))dshhhhhh+t1t2g1(s,y1(s),y2(s))dw1(s)|p2p-1E(t1t2|f1(s,y1(s),y2(s))|ds)p+2p-1E|t1t2g1(s,y1(s),y2(s))dw1(s)|p2p-1(t1t21qds)p/qE(t1t2|f1(s,y1(s),y2(s))|pds)+2p-1[p(p-1)2]p/2(t2-t1)(p-2)/2×t1t2E|g1(s,y1(s),y2(s))|pds2p-1(t1t21qds)p/qt1t2L(p)ds+2p-1[p(p-1)2]p/2(t2-t1)(p-2)/2t1t2R(p)ds2p-1(t2-t1)p/q+1L(p)+2p-1[p(p-1)2]p/2(t2-t1)p/2R(p)2p-1(t2-t1)pL(p)+2p-1[p(p-1)2]p/2(t2-t1)p/2R(p)2p-1(t2-t1)p/2×{(t2-t1)p/2L(p)+[p(p-1)2]p/2R(p)}2p-1(t2-t1)p/2{L(p)+[p(p-1)2]p/2R(p)}. Thus, it follows from Lemma 5 that almost every sample path of y1(t) is locally but uniformly Hölder-continuous with exponent ϑ for ϑ(0,(p-2)/2p) and therefore almost every sample path of y1(t) is uniformly continuous on t0.

By a similar procedure as above, y2(t) can be proven. Thus, (y1(t),y2(t)) is uniformly continuous on t0. The proof of Lemma 9 is complete.

Theorem 10.

Let (S1), (S2), and (S3) hold; then the unique positive solution of system (3) is globally attractive for initial data (y1(0),y2(0))R+2.

Proof.

It follows from (S1) that, for initial data (y1(0),y2(0))R+2, system (3) has a unique solution (y1(t),y2(t))R+2 (see Lemma 7). Assume that (y1*(t),y2*(t)) is another solution of system (3) with initial values y1*(0),y2*(0)>0.

Define a Lyapunov function V(t) as (46)V(t)=|lny1(t)-lny1*(t)|+|lny2(t)-lny2*(t)|. Using Itô’s formula, a calculation of the right differential D+V(t) along (3) shows that (47)D+V(t)=sgn(y1(t)-y1*(t))d(lny1(t)-lny1*(t))+sgn(y2(t)-y2*(t))d(lny2(t)-lny2*(t))=sgn(y1(t)-y1*(t))×{[dy1(t)y1(t)-(dy1(t))22y12(t)]-[dy1*(t)y1*(t)-(dy1*(t))22(y1*(t))2]}+sgn(y2(t)-y2*(t))×{[dy2(t)y2(t)-(dy2(t))22y22(t)]-[dy2*(t)y2*(t)-(dy2*(t))22(y2*(t))2]}=sgn(y1(t)-y1*(t))×{[(a1-b1y1(t)-c1y2(t)-d1y12(t)-σ122)dthhhhhhhh+σ1dw1(a1-b1y1(t)-c1y2(t)-d1y12(t)-σ122)]-[(a1-b1y1*(t)-c1y2*(t)-d1y1*(t)2-σ122)dthhhhhhhh+σ1dw1(a1-b1y1*(t)-c1y2*(t)-d1y1*(t)2-σ122)]}+sgn(y2(t)-y2*(t))×{[(a2-b2y2(t)-c2y1(t)-d2y22(t)-σ222)dthhhhhhh+σ2dw2(a2-b2y2(t)-c2y1(t)-d2y22(t)-σ222)]-[(a2-b2y2*(t)-c2y1*(t)-d2y2*(t)2-σ222)dtHHHHHH+σ2dw2(a2-b2y2*(t)-c2y1*(t)-d2y2*(t)2-σ222)]}=sgn(y1(t)-y1*(t))×{-b1(y1(t)-y1*(t))-c1(y2(t)-y2*(t))hhhhhh-d1(y1(t)+y1*(t))(y1(t)-y1*(t))}dt+sgn(y2(t)-y2*(t))×{-b2(y2(t)-y2*(t))-c2(y1(t)-y1*(t))hhhhhh-d2(y2(t)+y2*(t))(y2(t)-y2*(t))}dt{-b1-d1(y1(t)+y1*(t))+c2}|y1(t)-y1*(t)|dthhihh+{-b2-d2(y2(t)+y2*(t))+c1}|y2(t)-y2*(t)|dt-(b1-c2)|y1(t)-y1*(t)|dt-(b2-c1)|y2(t)-y2*(t)|dt. Integrating both sides yields that (48)V(t)-V(0)-(b1-c2)0t|y1(s)-y1*(s)|ds-(b2-c1)0t|y2(s)-y2*(s)|ds; that is, (49)V(t)+(b1-c2)0t|y1(t)-y1*(t)|ds+(b2-c1)0t|y2(t)-y2*(t)|dsV(0)<. In view of (S3) and V(t)0, then it follows from (49) that (50)|y1(t)-y1*(t)|L1[0,),|y2(t)-y2*(t)|L1[0,). So recalling Lemmas 9 and 6, we can show that (51)limt+|y1(t)-y1*(t)|=0,limt+|y2(t)-y2*(t)|=0a.s. This completes the proof of Theorem 10.

5. Numerical Simulations

In this paper, we derived the sufficient conditions for the existence, uniqueness, stochastically ultimate boundness, and global attraction of positive solutions of system (3). In order to illustrate the above theoretical results, we will perform a specific example. Motivated by the Milsten method mentioned in Higham , we can obtain the following discrete version of system (3): (52)y1(k+1)=y1(k)+y1(k)[d1y12a1-b1y1(k)-c1y2(k)hhhhhhhhhhhh-d1y12(k)]Δt+σ1y1(k)Δtξ1(k)+0.5σ12y1(k)(ξ12(k)-1)Δt,y2(k+1)=y2(k)+y2(k)[d2y22a2-b2y2(k)-c2y1(k)hhhhhhhhhhhh-d2y22(k)]Δt+σ2y2(k)Δtξ2(k)+0.5σ22y2(k)(ξ22(k)-1)Δt, where ξ1(k) and ξ2(k) are Gaussian random variables that follow N(0,1). Let a1=0.6, b1=0.5, c1=0.2, d1=0.3, and σ1=0.1; a2=0.7, b2=0.4, c2=0.3, d2=0.2, and σ2=0.1; Δt=0.001; and the initial value (y1(0),y2(0))=(0.3,0.2), (y1*(0),y2*(0))=(0.6,0.5). After a calculation, we can see that the assumptions of Theorems 8 and 10 hold. Figures 1 and 2 show that the positive solution of system (52) is stochastically ultimately bounded and globally attractive.

The sample path of (y1(t);y1*(t)) in the same coordinate system.

The sample path of (y2(t);y2*(t)) in the same coordinate system.

It follows from Theorem 8 that a preliminary result on the stochastically ultimate boundness of system (3) is obtained. We would like to mention here that an interesting but challenging problem associated with the investigation of system (3) should be the stochastic permanence; we leave this for future work.

Conflict of Interests

The authors declare that they have no conflict of interests.

Acknowledgments

The authors thank the anonymous referees for their valuable comments. The work is supported by the National Natural Science Foundation of China (no. 11261017), the Foundation of Hubei University for Nationalities (PY201401), the Key Laboratory of Biological Resources Protection and Utilization of Hubei Province (PKLHB1329), and the Key Subject of Hubei Province (Forestry).

Fan M. Wang K. Jiang D. Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments Mathematical Biosciences 1999 160 1 47 61 10.1016/S0025-5564(99)00022-X MR1704338 Liu Z. Fan M. Chen L. Globally asymptotic stability in two periodic delayed competitive systems Applied Mathematics and Computation 2008 197 1 271 287 10.1016/j.amc.2007.07.086 MR2396310 ZBL1148.34046 Liu B. Chen L. The periodic competitive Lotka-Volterra model with impulsive effect Mathematical Medicine and Biology 2004 21 2 129 145 Song Y. Han M. Peng Y. Stability and hopf bifurcations in a competitive Lotka-Volterra system with two delays Chaos, Solitons &Fractals 2004 22 5 1139 1148 10.1016/j.chaos.2004.03.026 MR2078839 Tang X. H. Zou X. Global attractivity of non-autonomous Lotka-Volterra competition system without instantaneous negative feedback Journal of Differential Equations 2003 192 2 502 535 10.1016/S0022-0396(03)00042-1 MR1990850 ZBL1035.34085 Jin Z. Zhien M. Maoan H. The existence of periodic solutions of the n-species Lotka-Volterra competition systems with impulsive Chaos, Solitons & Fractals 2004 22 1 181 188 10.1016/j.chaos.2004.01.007 MR2058760 Ahmad S. Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations Journal of Mathematical Analysis and Applications 1987 127 2 377 387 10.1016/0022-247X(87)90116-8 MR915064 ZBL0648.34037 Gopalsamy K. Stability and Oscillations in Delay Differential Equation of Population Dynamics 1992 Dordrecht, The Netherlands Kluwer Academic MR1163190 Tan R. Liu W. Wang Q. Liu Z. Uniformly asymptotic stability of almost periodic solutions for a competitive system with impulsive perturbations Advances in Difference Equations 2014 2014, article 2 10.1186/1687-1847-2014-2 May R. M. Stability and Complexity in Model Ecosystems 2001 New York, NY, USA Princeton University Press Li Y. Gao H. Existence, uniqueness and global asymptotic stability of positive solutions of a predator-prey system with Holling II functional response with random perturbation Nonlinear Analysis. Theory, Methods & Applications 2008 68 6 1694 1705 10.1016/j.na.2007.01.008 MR2388843 Li X. Mao X. Population dynamical behavior of non-autonomous Lotka-Volterra competitive system with random perturbation Discrete and Continuous Dynamical Systems A 2009 24 2 523 545 10.3934/dcds.2009.24.523 MR2486589 Lv J. Wang K. Asymptotic properties of a stochastic predator-prey system with Holling II functional response Communications in Nonlinear Science and Numerical Simulation 2011 16 10 4037 4048 10.1016/j.cnsns.2011.01.015 MR2802710 Jiang D. Ji C. Li X. O'Regan D. Analysis of autonomous Lotka-Volterra competition systems with random perturbation Journal of Mathematical Analysis and Applications 2012 390 2 582 595 10.1016/j.jmaa.2011.12.049 MR2890539 ZBL1258.34099 Liu M. Wang K. Dynamics and simulations of a logistic model with impulsive perturbations in a random environment Mathematics and Computers in Simulation 2013 92 53 75 10.1016/j.matcom.2013.04.011 MR3082356 Mao X. Yuan C. Zou J. Stochastic differential delay equations of population dynamics Journal of Mathematical Analysis and Applications 2005 304 1 296 320 10.1016/j.jmaa.2004.09.027 MR2124664 ZBL1062.92055 Jiang D. Yu J. Ji C. Shi N. Asymptotic behavior of global positive solution to a stochastic SIR model Mathematical and Computer Modelling 2011 54 1-2 221 232 10.1016/j.mcm.2011.02.004 MR2801881 Liu M. Wang K. Persistence, extinction and global asymptotical stability of a non-autonomous predator-prey model with random perturbation Applied Mathematical Modelling 2012 36 11 5344 5353 10.1016/j.apm.2011.12.057 MR2956748 Li X. Gray A. Jiang D. Mao X. Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching Journal of Mathematical Analysis and Applications 2011 376 1 11 28 10.1016/j.jmaa.2010.10.053 MR2745384 ZBL1205.92058 Xing Z. Peng J. Boundedness, persistence and extinction of a stochastic non-autonomous logistic system with time delays Applied Mathematical Modelling 2012 36 8 3379 3386 10.1016/j.apm.2011.10.022 MR2920898 Mao X. Stochastic Differential Equations and Applications 2008 Chichester, UK Horwood Publishing Hardy G. H. Littlewood J. E. Pólya G. Inequalities 1952 Cambridge University Press MR0046395 Karatzas I. Shreve S. E. Brownian Motion and Stochastic Calculus 1991 Berlin , Germany Springer MR1121940 Barbalat I. Systems d'equations differentielle d'oscillations nonlineaires Revue Roumaine de Mathématiques Pures et Appliquées 1959 4 2 267 270 Higham D. J. An algorithmic introduction to numerical simulation of stochastic differential equations Society for Industrial and Applied Mathematics 2001 43 3 525 546 10.1137/S0036144500378302 MR1872387