# Persistence and Nonpersistence of a Nonautonomous Stochastic Mutualism System

Authors: Peiyan Xia; Xiaokun Zheng; Daqing Jiang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

In this paper, a two-species nonautonomous stochastic mutualism system is investigated. The intrinsic growth rates of the two species at timet are estimated by rit+σitB˙i(t),i=1,2, respectively. Viewing the different intensities of the noises σi(t), i=1,2 as two parameters at time t, we conclude that there exists a global positive solution and the pth moment of the solution is bounded. We also show that the system is permanent, including stochastic permanence, persistence in mean, and asymptotic boundedness in time average. Besides, we show that the large white noise will make the system nonpersistent. Finally, we establish sufficient criteria for the global attractivity of the system.


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## Body

## 1. Introduction

For more than three decades, mutualism of multispecies has attracted the attention of both mathematicians and ecologists. By definition, in a mutualism of multispecies, the interaction is beneficial for the growth of other species. Lotka-Volterra mutualism systems have long been used as standard models to mathematically address questions related to this interaction. Among these, nonautonomous Lotka-Volterra mutualism models are studied by many authors, see [1–7] and references therein. The classical nonautonomous Lotka-Volterra mutualism system can be expressed as follows:
(1)x˙i(t)=xi(t)[ri(t)-aii(t)xi(t)+∑j=1j≠inaij(t)xj(t)],i=1,2,…,n,
where xi(t),i=1,2,…,n is the density of the ith population at time t, ri(t)>0,i=1,2,…,n is the intrinsic growth rate of the ith population at time t,ri(t)/aii(t)>0,i=1,2,…,n is the carrying capacity at time t, and coefficient aij(t)>0,i,j=1,2,…,n describes the influence of the jth population upon the ith population at time t.It is shown in [1] that if different conditions hold (see conditions (a)–(e) in [1]), then the solution of system (1) is bounded, permanent, extinct, and global attractive, respectively. However, when the intrinsic growth rate and coefficient aij(t) are periodic, it is shown in [3] that there exists positive periodic solution and almost periodic solutions are obtained.From another point of view, environmental noise always exists in real life. It is an interesting problem, both mathematically and biologically, to determine how the structure of the model changes under the effect of a fluctuating environment. Many authors studied the biological models with stochastic perturbation, see [8–12] and references therein. In [8] Ji et al. discussed the following two-species stochastic mutualism system
(2)dx1(t)=x1(t)[(r1-a11x1(t)+a12x2(t))dt+σ1dB1(t)],dx2(t)=x2(t)[(r2+a21x1(t)-a22x2(t))dt+σ2dB2(t)],
where Bi(t),i=1,2 are mutually independent one dimensional standard Brownian motions with Bi(0)=0,i=1,2, and σi,i=1,2 are the intensities of white noise. It is shown in [8] that if a11a22>a12a21 then there is a unique nonnegative solution of system (2). For small white noise there is a stationary distribution of (2) and it has ergodic property. Biologically, this implies that with small perturbation of environment, the stability of the two species varies with the intensity of white noise, and both species will survive.However, almost all known stochastic models assume that the growth rate and the carrying capacity of the population are independent of timet. In contrast, the natural growth rates of many populations vary with t in real situation, for example, due to the seasonality. As a matter of fact, nonautonomous stochastic population systems have recently been studied by many authors, for example, [13–17].In this paper we consider the system(3)dx1(t)=x1(t)[(r1(t)-a11(t)x1(t)+a12(t)x2(t))dt+σ1(t)dB1(t)(r1(t)-a11(t)x1(t)+a12(t)x2(t))],dx2(t)=x2(t)[(r2(t)+a21(t)x1(t)-a22(t)x2(t))dt+σ2(t)dB2(t)(r1(t)-a11(t)x1(t)+a12(t)x2(t))],
where ri(t), aij(t), σi(t), i,j=1,2 are all continuous bounded nonnegative functions on [0,+∞). The objective of our study is to investigate the long-time behavior of system (3). As in [8], we mainly discuss when the system is persistent and when it is not under a fewer conditions. More specifically, we show that there is a positive solution of system (3) and its pth moment bounded in Section 2. In Section 3, we deduce the persistence of the system. If the white noise is not large such that ril-((σiu)2/2)>0,i=1,2, we will prove that the solution of system (3) is a stochastic persistence. In addition, we show that every component of the solution is persistent in mean. We further deduce that every component of the solution of system (3) is an asymptotic boundedness in mean. In Section 4, we show that larger white noise will make system (3) nonpersistent. Finally, we study the global attractivity of system (3).Throughout this paper, unless otherwise specified, let(Ω,{ℱt}t≥0,P) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all P-null sets). Let R+2 be the positive cone of R2, namely, R+2={x∈R2:xi>0,i=1,2}. If x∈Rn, its norm is denoted by |x|=(∑i=1n‍xi2)1/2. If f(t) is a continuous bounded function on [0,+∞), we use the notationsup(4)fu=supt∈[0,+∞)f(t),fl=mint∈[0,+∞)f(t).


## 2. Existence and Uniqueness of the Positive Solution

In population dynamics, the first concern is that the solution should be nonnegative. In order to do that a stochastic differential equation can have a unique global (i.e., no explosion at any finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (Mao [18]). However, the coefficients of system (3) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of system (3) may explode at a finite time. Following the way developed by Mao et al. [19], we show that there is a unique positive solution of (3).Theorem 1.
Assume thata11la22l>a12ua21u. Then, there is a unique positive solution x(t)=(x1(t),x2(t)) of system (3) on t≥0 for any given initial value x(0)∈R+2, and the solution will remain in R+2 with probability 1, namely, x(t)∈R+2 for all t≥0 almost surely.The proof of Theorem1 is similar to [8]. But it is skilled in taking the value of ϵ. We show it here.Proof.
Since the coefficients of the equation are locally Lipschitz continuous, for any given initial valuex(0)∈R+2 there is an unique local solution x(t)=(x1(t),x2(t)) on t∈[0,τe), where τe is the explosion time. To show that this solution is global, we need to show that τe=∞ a.s. Let m0>1 be sufficiently large for every component of x(0) lying within the interval [1/m0,m0]. For each integer m≥m0, define the stopping time
(5)τm=inf{1mt∈[0,τe):min{x1(t),x2(t)}≤1mormax{x1(t),x2(t)}≥m},
where throughout this paper we set inf∅=∞ (as usual ∅ denotes the empty set). Clearly, τm is increasing as m→∞. Set τ∞=limm→∞τm, whence τ∞≤τe a.s. If we can show that τ∞=∞ a.s., then τe=∞ a.s. and x(t)∈R+2 a.s. for all t≥0. In other words, to complete the proof, all we need to show is that τ∞=∞ a.s. If this statement is false, there is a pair of constant T>0 and ε∈(0,1) such that
(6)P{τ∞≤T}>ε.
Hence, there is an integer m1≥m0 such that
(7)P{τm≤T}≥ε∀m≥m1.
We define
(8)V(x)=a21u(x1-1-logx1)+a12u(x2-1-logx2).
By Itô's formula, we have
(9)dV(x)={a21u(1-1x1)x1[r1(t)-a11(t)x1+a12(t)x2]+a12u(1-1x2)x2[r2(t)+a21(t)x1-a22(t)x2]+12[a21uσ12(t)+a12uσ22(t)](1-1x1)}dt+a21uσ1(t)(x1-1)dB1(t)+a12uσ2(t)(x2-1)dB2(t)≔LVdt+a21uσ1(t)(x1-1)dB1(t)+a12uσ2(t)(x2-1)dB2(t),
where
(10)LV=a21u(1-1x1)x1[r1(t)-a11(t)x1+a12(t)x2]+a12u(1-1x2)x2[r2(t)+a21(t)x1-a22(t)x2]+12[a21uσ12(t)+a12uσ22(t)]≤a21u[12(r1u+a11u)x1-a12lx2-a11lx12+a12ux1x2-r1l+12(σ1u)2]+a12u[12(r2u+a22u)x2-a21lx1-a22lx22+a21ux1x2-r2l+12(σ2u)2].
According to Young inequality, note that x1x2≤ϵx12+(1/4ϵ)x22, where a21u/2a22l<ϵ<a11l/2a12u, then,
(11)LV≤a21u[12(r1u+a11u)x1-a12lx2-a11lx12+a12u(ϵx12+14ϵx22)-r1l+12(σ1u)2]+a12u[12(r2u+a22u)x2-a21lx1-a22lx22+a21u(ϵx12+14ϵx22)-r2l+12(σ2u)2]=-(a21ua11l-2ϵa21ua12u)x12+[a21u(r1u+a11u)-a12ua21l]x1-a21ur1l+12a21u(σ1u)2-(a12ua22l-12ϵa21ua12u)x22+[a12u(r2u+a22u)-a21ua12l]x2-a12ur2l+12a12u(σ2u)2≤K.
Since a21u/2a22l<ϵ<a11l/2a12u, we obtain -(a21ua11l-2ϵa12u)<0 and -(a12ua22l-(1/2ϵ)a21ua12u)<0. Hence, K is a positive constant. Integrating both sides of (9) from 0 to τm∧T, we therefore obtain
(12)V(x(τm∧T))-V(x(0))V≤∫0τm∧TKdt+∫0τm∧Ta21uσ1(t)(x1(t)-1)dB1(t)+∫0τm∧Ta12uσ2(t)(x2(t)-1)dB2(t).
Whence, taking expectations yields
(13)E[V(x(τm∧T))]≤V(x(0))+KE(τm∧T)≤V(x(0))+KT.
Set Ωm={τm≤T} for m≥m1 and by (7), P(Ωm)≥ε. Note that for every ω∈Ωm, there is x1(τm,ω) or x2(τm,ω) equals either m or 1/m, and therefore
(14)V(x(τm,ω))≥min{a21u,a12u}(m-1-logm)∧(1m-1-log1m):=h(m),
where limm→∞h(m)=∞. It then follows from (13) that
(15)E[V(x(0))]+KT≥E[1Ωm·V(x(τm,ω))]≥ϵh(m),
where 1Ωm is the indicator function of Ωm. Letting m→∞ leads to the contradiction
(16)∞>V(x(0))+KT=∞,
so we must have τ∞=∞ a.s. This completes the proof of Theorem 1.Remark 2.
By Theorem1, we observe that for any given initial value x(0)∈R+2, there is a unique solution x(t)=(x1(t),x2(t)) of system (3) on t≥0 and the solution will remain in R+2 with probability 1, no matter how large the intensities of white noise are. So, under the same assumption there is an global unique positive solution of the corresponding deterministic system of system (3).Next, we show that thepth moment of the solution of system (3) is bounded in time average.Theorem 3.
Assume thata11la22l>a12ua21u. Then there exists a positive constant K(p) such that the solution x(t) of system (3) has the following property:
(17)E[c1x1p(t)+c2x2p(t)]≤K(p),∀t∈[0,∞),p>1,
where c1,c2 satisfy
(18)(a21u)p+1a11l(a22l)p<c1c2<a22l(a11l)p(a12u)p+1.Proof.
By Itô's formula, we have(19)dx1p(t)=px1p(t)[(r1(t)-a11(t)x1(t)+a12(t)x2(t))dt+σ1(t)dB1(t)(r1(t)-a11(t)x1(t)+a12(t)x2(t))]+12p(p-1)x1p(t)σ12(t)dt=p[(r1(t)+p-12σ12(t))x1p(t)-a11(t)x1p+1(t)+a12(t)x1p(t)x2(t)(r1(t)+p-12σ12(t))]dt+σ1(t)px1p(t)dB1(t)=p[α1(t)x1p(t)-a11(t)x1p+1(t)+a12(t)x1p(t)x2(t)]dt+σ1(t)px1p(t)dB1(t)≤p[α1ux1p(t)-a11lx1p+1(t)+a12ux1p(t)x2(t)]dt+pσ1ux1p(t)dB1(t),
where α1(t)=r1(t)+((p-1)/2)σ12(t), and
(20)dx2p(t)=px2p(t)[(r2(t)-a22(t)x2(t)+a21(t)x1(t))dt+σ2(t)dB2(t)(r2(t)-a22(t)x2(t)+a21(t)x1(t))]+12p(p-1)x2p(t)σ22(t)dt=p[(r2(t)+p-12σ22(t))x2p(t)-a22(t)x2p+1(t)+a21(t)x2p(t)x1(t)(r2(t)+p-12σ22(t))]dt+σ2(t)px2p(t)dB2(t)=p[α2(t)x2p(t)-a22(t)x2p+1(t)+a21(t)x2p(t)x1(t)]dt+σ2(t)px2p(t)dB2(t)≤p[α2ux2p(t)-a22lx2p+1(t)+a21ux2p(t)x1(t)]dt+pσ2ux2p(t)dB2(t),
where α2(t)=r2(t)+((p-1)/2)σ22(t). According to Young inequality, we obtain
(21)x1p(t)x2(t)≤ϵ1x1p+1(t)+1p+1(pp+1)p(1ϵ1)px2p+1(t),ϵ1=pa11l(p+1)a12u,x2p(t)x1(t)≤ϵ2x2p+1(t)+1p+1(pp+1)p(1ϵ2)px1p+1(t),ϵ2=pa22l(p+1)a21u.
Thus, we have
(22)dx1p(t)≤p[(pp+1)p(1ϵ1)pα1ux1p(t)-a11lx1p+1(t)+a12uϵ1x1p+1(t)+a12u1p+1(pp+1)p(1ϵ1)px2p+1(t)]dt+pσ1ux1p(t)dB1(t),dx2p(t)≤p[(pp+1)p(1ϵ2)pα2ux2p(t)-a22lx2p+1(t)+a21uϵ2x2p+1(t)+a21u1p+1(pp+1)p(1ϵ2)px1p+1(t)]dt+pσ2ux2p(t)dB2(t).
Since a11la22l>a12ua21u, there exist two positive constants c1,c2 which satisfy
(23)(a21u)p+1a11l(a22l)p<c1c2<a22l(a11l)p(a12u)p+1.
Therefore,
(24)d(c1x1p(t)+c2x2p(t))≤-p[(c1a11l-c1a12uϵ1-c2a21upp(p+1)p+1ϵ2p)x1p+1(t)+(c2a22l-c1a21uϵ2-c1a12upp(p+1)p+1ϵ1p)x2p+1(t)-∑i=12ciαiuxip(t)]dt+∑i=12cipσiuxip(t)dBi(t).
From (23) and the values of ϵ1,ϵ2, we obtain
(25)a21u(pp/(p+1)p+1ϵ2p)a11l-a12uϵ1<c1c2<a22l-a21uϵ2a12u(pp/(p+1)p+1ϵ1p),
which implies that c1a11l-c1a12uϵ1-c2a21u(pp/(p+1)p+1ϵ2p)>0 and c2a22l-c1a21uϵ2-c1a12u(pp/((p+1)p+1ϵ1p))>0. Let
(26)α=max{α1u,α2u},β=min{c1-(p+1)/p[c1a11l-c1a12uϵ1-c2a21upp(p+1)p+1ϵ2p],β=minjjc2-(p+1)/p[c2a22l-c1a21uϵ2-c1a12upp(p+1)p+1ϵ1p]},
then we have
(27)d(c1x1p(t)+c2x2p(t))≤p[α(∑i=12cixip(t))-β(∑i=12ci1+(1/p)xip+1)]dt+∑i=12‍cipσiuxip(t)dBi(t).
Hence, we get
(28)dE[c1x1p(t)+c2x2p(t)]dt≤pαE[c1x1p(t)+c2x2p(t)]-pβE[c11+(1/p)x1p+1(t)+c21+(1/p)x2p+1(t)]≤pαE[c1x1p(t)+c2x2p(t)]-pβ{[E(c1x1p(t))]1+(1/p)+[E(c2x2p(t))]1+(1/p)}≤pαE[c1x1p(t)+c2x2p(t)]-pβ·2-1/p[E(c1x1p(t)+c2x2p(t))]1+(1/p).
By the comparison theorem, we get
(29)limsupt→∞E[c1x1p(t)+c2x2p(t)]≤2(αβ)p:=M(p),
which implies that there is a T0>0, such that
(30)E[c1x1p(t)+c2x2p(t)]≤2M(p),∀t>T0.
Besides, note that E[c1x1p(t)+c2x2p(t)] is continuous, then there is a M~(p)>0 such that
(31)E[c1x1p(t)+c2x2p(t)]≤M~(p),∀t∈[0,T0].
Let K(p)=max{2M(p),M~(p)}, then
(32)E[c1x1p(t)+c2x2p(t)]≤K(p),∀t∈[0,∞).


## 3. Persistence

Theorem1 shows that the solution of system (3) will remain in the positive cone R+2 if a11la22l>a12ua21u. Studying a population system, we pay more attention on whether the system is persistent. In this section, we first show that the solution is a stochastic permanence. Next we show that the solution is persistent in time average. Moreover, we show that the solution x(t) of system (3) is an asymptotic boundedness in time average.


### 3.1. Stochastic Permanence

Lety(t) be the solution of a randomized nonautonomous competitive equation:
(33)dyi(t)=yi(t)[(bi(t)-∑j=1n‍aij(t)yj(t))dt+σi(t)dBi(t)],i=1,2,…,n,
where Bi(t),i=1,2,…,n, are independent standard Brownian motions, y(0)=y0>0 while y0 is independent of B(t), and bi(t), aij(t), σi(t) are all continuous bounded nonnegative functions on [0,+∞).Lemma 4 (see [15]).
Assume thatbil-((σiu)2/2)>0,i=1,2,…,n, then for any given initial value y(0)∈R+n, the solution y(t) of (36) has the properties
(34)limsupt→∞E(1|y(t)|θ)≤H,
where H is a constant, θ is an arbitrary positive constant satisfying
(35)θmax1≤i≤n(σiu)2<2min1≤i≤n(bi(t)-σi2(t)2)l.LetN(t) be the solution of a randomized nonautonomous logistic equation
(36)dN(t)=N(t)[(a(t)-b(t)N(t))dt+α(t)dB(t)],
where B(t) is a 1-dimensional standard Brownian motion, N(0)=N0>0, and N0 is independent of B(t).Lemma 5 (see [13]).
Assume thata(t),b(t), and α(t) are bounded continuous functions defined on [0,∞), a(t)>0 and b(t)>0. Then there exists a unique continuous positive solution of (36) for any initial value N(0)=N0>0, which is global and represented by
(37)N(t)=exp{∫0t[a(s)-(α2(s)2)]ds+α(s)dB(s)}×((1N0)+∫0tb(s)exp{∫0s[a(τ)-(α2(τ)2)]dτ+α(τ)dB(τ)∫0s[a(τ)-(α2(τ)/2)]dτ}ds∫0tb(s)exp)-1,t≥0.From Lemma4 we have the following.Lemma 6.
Assume thatal-((αu)2/2)>0, then for any given initial value N(0)∈R+, the solution N(t) of (36) has the properties
(38)limsupt→∞E(1Nθ(t))≤H,
where H is a constant, θ is positive constant satisfying
(39)θ(αu)2<2[al-(αu)22].Letϕ(t)=(ϕ1(t),ϕ2(t))T be the solution of
(40)dϕi(t)=ϕi(t)[(ri(t)-aii(t)ϕi(t))dt+σi(t)dBi(t)],i=1,2,
where Bi(t),i=1,2, are independent standard Brownian motions, ϕ(0)=ϕ0>0, and ϕ0∈R+2, ri(t),aii(t),σi(t),i=1,2 are all continuous bounded nonnegative functions on [0,+∞). From Lemma 4 it is easy to know the following.Lemma 7.
Assume thatr-il=ril-((σiu)2/2)>0,i=1,2, then for any given initial value ϕ(0)∈R+2, the solution ϕ(t) of (40) has the properties
(41)limsupt→∞E(1ϕiθ(t))≤Hi,i=1,2,
where Hi, i=1,2 are two constants, θ is positive constant satisfying
(42)θ(σiu)2<2r-il,i=1,2.Lemma 8.
Assume thatr-il>0,i=1,2, then for any given initial value x0∈R+2, the solution x(t) of system (3) has the properties
(43)xi(t)≥ϕi(t),i=1,2,(44)limsupt→∞E(1xiθ(t))≤Hi,i=1,2,
where Hi,i=1,2 are two constants, θ is positive constant satisfying
(45)θ(σiu)2<2r-il,i=1,2.Proof.
Equation (43) follows directly from the classical comparison theorem of stochastic differential equations (see [20]). Thus, we obtain
(46)limsupt→∞E(1xiθ(t))≤limsupt→∞E(1ϕiθ(t))≤Hi,i=1,2.Definition 9.
System (3) is said to be stochastically permanent if for any ε∈(0,1), there exists a pair of positive constants δ=δ(ϵ) and M=M(ϵ) such that for any initial value x0∈R+2, the solution obeys
(47)liminft→∞P{xi(t)≤M(ϵ)}≥1-ϵ,liminft→∞P{xi(t)≥δ(ϵ)}≥1-ϵ,i=1,2.Theorem 10.
Assume thata11la22l>a12ua21u,r-il>0,i=1,2, then system (3) is stochastically permanent.The proof is a simple application of the Chebyshev inequality, we omit it.


### 3.2. Persistence in Time Average

Theorem10 shows that if the white noise is not large, the solution of system (3) is survive with large probability. In this part, we show x(t) is persistence in mean.Lemma 11.
Assume thatr-il>0,i=1,2, then for any given initial value ϕ(0)∈R+2, the solution ϕ(t) of (40) has the properties
(48)zi(t)e-[max0≤s≤t∫0sσi(τ)dBi(τ)-∫0tσi(τ)dBi(τ)]≤ϕi(t)≤zi(t)e-[min0≤s≤t∫0sσi(τ)dBi(τ)-∫0tσi(τ)dBi(τ)],i=1,2,
where z(t)=(z1(t),z2(t)) is the solution of
(49)dzi(t)=zi(t)[ri(t)-σi2(t)2-aiiuzi(t)]dt,zi(0)=ϕi(0),i=1,2.Proof.
From Lemma5, we know
(50)1ϕi(t)=1ϕi(0)e-∫0t[ri(s)-(σi2(s)/2)]ds+σi(s)dBi(s)+aiiu∫0te-∫0t[ri(s)-(σi2(s)/2)]ds+σi(s)dBi(s)×e+∫0s[ri(τ)-(σi2(τ)/2)]dτ+σi(τ)dBi(τ)ds=e-∫0tσi(s)dBi(s)[∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτe∫0sσi(τ)dBi(τ)ds∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτ1ϕi(0)e-∫0t[ri(s)-(σi2(s)/2)]ds+aiiu∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτ×e∫0sσi(τ)dBi(τ)ds∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτ1ϕi(0)]≤e-∫0tσi(s)dBi(s)[∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτe∫0sσi(τ)dBi(τ)ds1ϕi(0)e-∫0t[ri(s)-(σi2(s)/2)]ds+aiiuemax0≤s≤t(∫0sσi(τ)dBi(τ))×∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτds]≤emax0≤s≤t[∫0sσi(τ)dBi(τ)]-∫0tσi(s)dBi(s)zi(t).
Similarly, we have
(51)1ϕi(t)≥e-∫0tσi(s)dBi(s)[∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτ1ϕi(0)e-∫0t[ri(s)-(σi2(s)/2)]ds+aiiuemin0≤s≤t(∫0sσi(τ)dBi(τ))×∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτds]≥emin0≤s≤t[∫0sσi(τ)dBi(τ)]-∫0tσi(s)dBi(s)zi(t).Lemma 12.
Assume thatr-il>0,i=1,2, then for any given initial value z(0)∈R+2, the solution z(t) of (49) has the following properties
(52)z~i(t)≤zi(t)≤z^i(t),limt→∞z~i(t)=r-ilaiiu,limt→∞z^i(t)=r-iuaiiu,
where z~(t)=(z~1(t),z~2(t)),z^(t)=(z^1(t),z^2(t)) are the solutions of the two equations, respectively,
(53)dz~i(t)=z~i(t)[r-il-aiiuz~i(t)]dt,z~i(0)=zi(0),i=1,2,(54)dz^i(t)=z^i(t)[r-iu-aiiuz^i(t)]dt,z^i(0)=zi(0),i=1,2.Proof.
Letz~(t)=(z~1(t),z~2(t)), z^(t)=(z^1(t),z^2(t)) are the solutions of SDE (53) and (54), respectively, with the positive initial value z(0). By Lemma 5, we know
(55)z~i(t)=er-ilt1/z~i(0)+(aiiu/r-il)(er-ilt-1),z^i(t)=er-iut1/z^i(0)+(aiiu/r-iu)(er-iut-1).
Thus,
(56)limt→∞z~i(t)=r-ilaiiu,limt→∞z^i(t)=r-iuaiiu.
By the classical comparison theorem of ordinary differential equations, we know
(57)z~i(t)≤zi(t)≤z^i(t).Lemma 13.
Assume thatr-il>0,i=1,2, then for any given initial value ϕ(0)∈R+2, the solution ϕ(t) of (40) has the properties
(58)limt→∞logϕi(t)t=0,a.s.Proof.
By Lemma12, we know
(59)e-[max0≤s≤t∫0sσi(τ)dBi(τ)-∫0tσi(τ)dBi(τ)]≤ϕi(t)zi(t)≤e-[min0≤s≤t∫0sσi(τ)dBi(τ)-∫0tσi(τ)dBi(τ)].
So, we have
(60)∫0tσi(τ)dBi(τ)-max0≤s≤t∫0sσi(τ)dBi(τ)≤logϕi(t)-logzi(t)≤∫0tσi(τ)dBi(τ)-min0≤s≤t∫0sσi(τ)dBi(τ).
Let Mi(t)=∫0tσi(τ)dBi(τ), then
(61)〈Mi,Mi〉t=∫0tσi2(τ)dτ.
Since σi(t),i=1,2 are bounded, then
(62)limt→∞〈Mi,Mi〉tt=limt→∞1t∫0tσi2(τ)dτ<∞,a.s.
By the strong law of large numbers, we know
(63)limt→∞Mi(t)t=limt→∞∫0tσi(τ)dBi(τ)t=0,a.s.
Thus,
(64)limt→∞max0≤s≤t|Mi(s)t|=0,a.s.
Then from (60) we obtain
(65)limt→∞logϕi(t)t=0,a.s.Lemma 14.
Assume thatr-il>0,i=1,2, then for any given initial value ϕ(0)∈R+2, the solution ϕ(t) of (40) has the properties
(66)liminft→∞1t∫0tϕi(s)ds≥r-ilaiiu,a.s.Proof.
By Itô's formula, we have(67)dlogϕi(t)=[ri(t)-σi2(t)2-aiiuϕi(t)]dt+σi(t)dBi(t).
Integrating both sides of this equation from 0 to t yields
(68)logϕi(t)t-logϕi(0)t=∫0t[ri(s)-(σi2(s)/2)]dst-aiiu∫0tϕi(s)dst+∫0tσi(s)dBi(s)t.
By Lemma 13, we know that
(69)limt→∞∫0tσi(s)dBi(s)t=limt→∞logϕi(t)t=0,a.s.
Hence,
(70)liminft→∞1t∫0tϕi(s)ds=1aiiuliminft→∞1t∫0t[ri(s)-σi2(s)2]ds≥r-ilaiiu,a.s.Definition 15.
System (3) is said to be persistent in time average if
(71)liminft→∞1t∫0txi(s)ds>0,i=1,2.Theorem 16.
Assume thata11la22l>a12ua21u and r-il>0,i=1,2, then the solution x(t) of system (3) with any initial value x(0)∈R+2 has the following property:
(72)liminft→∞1t∫0txi(s)ds≥r-ilaiiu,liminft→∞logxi(t)t≥0,a.s.,
and so system (3) is persistent in time average.Proof.
By Lemma8, we know that
(73)xi(t)≥ϕi(t)i=1,2,
where ϕ(t)=(ϕ1(t),ϕ2(t)) is the solution of system (40). Moreover, by Lemma 14 we know that
(74)liminft→∞1t∫0txi(s)ds≥liminft→∞1t∫0tϕi(s)ds≥r-ilaiiu,a.s.
Hence, by Lemma 13 we know that
(75)liminft→∞logxi(t)t≥liminft→∞logϕi(t)t=0,a.s.


### 3.3. Asymptotic Boundedness of Integral Average

Theorem16 shows that every component of the solution x(t) of system (3) will survive forever in time average, if the white noise is not large. In this part, we further deduce that every component of x(t) of system (3) will be an asymptotic boundedness in time average. Before we give the result, we do some preparation work.Lemma 17.
Letf∈C[[0,∞)×Ω,(0,∞)], F(t)∈((0,∞)×Ω,R). If there exist positive constants λ0 and λ such that
(76)logf(t)≥λt-λ0∫0tf(s)ds+F(t),t≥0,a.s.,
and limt→∞(F(t)/t)=0 a.s., then
(77)liminft→∞1t∫0tf(s)ds≥λλ0,a.s.Proof.
The proof is similar to the proof of Lemma in [21]. Let
(78)φ(t)=∫0tf(s)ds.
Since f∈C[[0,∞)×Ω,(0,∞)],φ(t) is differentiable on [0,∞) and
(79)dφ(t)dt=f(t)>0,fort≥0.
Substituting dφ(t)/dt and φ(t) into (76), we obtain the following:
(80)logdφ(t)dt≥λt-λ0φ(t)+F(t),
thus
(81)eλ0φ(t)dφ(t)dt≥eλt+F(t),fort≥0.
Note that limt→∞(F(t)/t)=0a.s., then for 0<ε<min{1,λ}, ∃T=T(ω)>0 and Ωε⊂Ω such that P(Ωε)>1-ε and F(t)≥-εt,t≥T,ω∈Ωε. Then we have
(82)eλ0φ(t)dφ(t)dt≥e(λ-ε)t,fort≥T,ω∈Ωε.
Integrating inequality (82) from 0 to t results in the following:
(83)λ0-1[eλ0φ(t)-eλ0φ(T)]≥(λ-ε)-1[e(λ-ε)t-e(λ-ε)T].
This inequality can be rewritten into
(84)eλ0φ(t)≥eλ0φ(T)+λ0(λ-ε)-1[e(λ-ε)t-e(λ-ε)T].
Taking the logarithm of both sides and dividing both sides by t(>0) yields
(85)φ(t)t≥λ0-1log{eλ0φ(T)+λ0(λ-ε)-1[e(λ-ε)t-e(λ-ε)T]}t.
Then,
(86)liminft→∞φ(t)t≥λ-ελ0,ω∈Ωε.
Letting ε→∞ yields
(87)liminft→∞1t∫0tf(s)ds≥λλ0,a.s.
This finishes the proof of the Lemma.Theorem 18.
Assume thata11la22l>a12ua21u and r-il>0,i=1,2, then the solution x(t) of system (3) with any initial value x(0)∈R+2 has the property
(88)x~i*≤limt→∞1t∫0txi(s)ds≤x^i*,i=1,2,a.s.,
where
(89)x~1*=a22ur-1l+a12lr-2la11ua22u-a12la21l,x~2*=a11ur-2l+a21lr-1la11ua22u-a12la21l,x^1*=a22lr-1u+a12ur-2ua11la22l-a12ua21u,x^2*=a11lr-2u+a21ur-1ua11la22l-a12ua21u.Proof.
To prove the results, we only need to prove(90)liminft→∞1t∫0txi(s)ds≥x~i*,i=1,2,a.s.(91)limsupt→∞1t∫0txi(s)ds≤x^i*,i=1,2,a.s.
By Itô's formula, we have
(92)dlogx1(t)=[r1(t)-12σ12(t)-a11(t)x1(t)+a12(t)x2(t)]dt+σ1(t)dB1(t),dlogx2(t)=[r2(t)-12σ22(t)+a21(t)x1(t)-a22(t)x2(t)]dt+σ2(t)dB2(t).
First, we prove (91). Integrating both sides of (92) from 0 to t yields
(93)logx1(t)=logx1(0)+∫0tr-1(s)ds-∫0ta11(s)x1(s)ds+∫0ta12(s)x2(s)ds+∫0tσ1(s)dB1(s),logx2(t)=logx2(0)+∫0tr-2(s)ds-∫0ta22(s)x2(s)ds+∫0ta21(s)x1(s)ds+∫0tσ2(s)dB2(s),
where r-i(s)=ri(s)-(1/2)σi2(s),i=1,2. Since xi(t)>0,i=1,2, hence
(94)logx1(t)≤logx1(0)+r-1ut-a11l∫0tx1(s)ds+a12u∫0tx2(s)ds+∫0tσ1(s)dB1(s),logx2(t)≤logx2(0)+r-2ut-a22l∫0tx2(s)ds+a21u∫0tx1(s)ds+∫0tσ2(s)dB2(s).
So we have
(95)a22llogx1(t)+a12ulogx2(t)≤a22l[logx1(0)+∫0tσ1(s)dB1(s)]+a22lr-1ut+a12u[logx2(0)+∫0tσ2(s)dB2(s)]+a12ur-2ut-(a11la22l-a21ua12u)∫0tx1(s)ds.
By Theorem 16, we know that
(96)liminft→∞logxi(t)t≥0,i=1,2,a.s.
Obviously,
(97)limt→∞logxi(0)+∫0tσi(s)dBi(s)t=0,i=1,2,a.s.
Hence, we have
(98)limsupt→∞1t∫0tx1(s)ds≤a22lr-1u+a12ur-2ua11la22l-a21ua12u≜x^1*,a.s.
Similarly, we have
(99)limsupt→∞1t∫0tx2(s)ds≤a11lr-2u+a12ur-1ua11la22l-a21ua12u≜x^2*,a.s.
Next, we prove that (90) is true. Taking integration both sides of (92) from 0 to t, we have
(100)logx1(t)≥logx1(0)+r-1lt-a11u∫0tx1(s)ds+a12l∫0tx2(s)ds+∫0tσ1(s)dB1(s),logx2(t)≥logx2(0)+r-2lt-a22u∫0tx2(s)ds+a21l∫0tx1(s)ds+∫0tσ2(s)dB2(s).
By Theorem 16 we know that
(101)liminft→∞1t∫0tx1(s)ds≥r-1la11u≜M1,a.s.,liminft→∞1t∫0tx2(s)ds≥r-2la22u≜N1,a.s.,
then for any ε>0, there is a T(ω)>0 such that
(102)1t∫0tx2(s)ds≥N1-ε,
for t>T(ω). It follows from (100) that, for t>T(ω),
(103)logx1(t)≥logx1(0)+r-1lt-a11u∫0tx1(s)ds+a12l(N1-ε)t+∫0tσ1(s)dB1(s)=logx1(0)-a11u∫0tx1(s)ds+[r-1l+a12l(N1-ε)]t+∫0tσ1(s)dB1(s).
From Lemma 17, we have
(104)liminft→∞1t∫0tx1(s)ds≥r-1l+a12l(N1-ε)a11u:=M2>M1.
Similarly, we have
(105)liminft→∞1t∫0tx2(s)ds≥r-2l+a21l(M1-ε)a22u:=N2>N1.
Continuing this process, we obtain two sequences Mn,Nn(n=1,2,…) such that
(106)Mn=r-1l+a12l(Nn-1-ε)a11u,(107)Nn=r-2l+a21l(Mn-1-ε)a22u.
By induction, we can easily show that Mn+1>Mn,Nn+1>Nn,n=1,2,…, that is, sequences {Mn,n=1,2,…} and {Nn,n=1,2,…} are nondecreasing. Moreover, note that (98) and (99), then the sequences {Mn,n=1,2,…} and {Nn,n=1,2,…}, have upper bounds. Therefore, there are two positive M,N such that
(108)limn→∞Mn=M,limn→∞Nn=N,liminft→∞1t∫0tx1(s)ds≥M,liminft→∞1t∫0tx2(s)ds≥N,
which together with (106) implies
(109)a11uM-a12lN=r-1l-εa12l,a22uN-a21lM=r-2l-εa21l.
Letting ε→0 yields
(110)M=a22ur-1l+a12lr-2la11ua22u-a12la21l≜x~1*,N=a11ur-1l+a21lr-2la11ua22u-a12la21l≜x~1*.
Hence,
(111)liminft→∞1t∫0txi(s)ds≥x~i*,i=1,2,a.s.,
which is as required.


## 3.1. Stochastic Permanence

Lety(t) be the solution of a randomized nonautonomous competitive equation:
(33)dyi(t)=yi(t)[(bi(t)-∑j=1n‍aij(t)yj(t))dt+σi(t)dBi(t)],i=1,2,…,n,
where Bi(t),i=1,2,…,n, are independent standard Brownian motions, y(0)=y0>0 while y0 is independent of B(t), and bi(t), aij(t), σi(t) are all continuous bounded nonnegative functions on [0,+∞).Lemma 4 (see [15]).
Assume thatbil-((σiu)2/2)>0,i=1,2,…,n, then for any given initial value y(0)∈R+n, the solution y(t) of (36) has the properties
(34)limsupt→∞E(1|y(t)|θ)≤H,
where H is a constant, θ is an arbitrary positive constant satisfying
(35)θmax1≤i≤n(σiu)2<2min1≤i≤n(bi(t)-σi2(t)2)l.LetN(t) be the solution of a randomized nonautonomous logistic equation
(36)dN(t)=N(t)[(a(t)-b(t)N(t))dt+α(t)dB(t)],
where B(t) is a 1-dimensional standard Brownian motion, N(0)=N0>0, and N0 is independent of B(t).Lemma 5 (see [13]).
Assume thata(t),b(t), and α(t) are bounded continuous functions defined on [0,∞), a(t)>0 and b(t)>0. Then there exists a unique continuous positive solution of (36) for any initial value N(0)=N0>0, which is global and represented by
(37)N(t)=exp{∫0t[a(s)-(α2(s)2)]ds+α(s)dB(s)}×((1N0)+∫0tb(s)exp{∫0s[a(τ)-(α2(τ)2)]dτ+α(τ)dB(τ)∫0s[a(τ)-(α2(τ)/2)]dτ}ds∫0tb(s)exp)-1,t≥0.From Lemma4 we have the following.Lemma 6.
Assume thatal-((αu)2/2)>0, then for any given initial value N(0)∈R+, the solution N(t) of (36) has the properties
(38)limsupt→∞E(1Nθ(t))≤H,
where H is a constant, θ is positive constant satisfying
(39)θ(αu)2<2[al-(αu)22].Letϕ(t)=(ϕ1(t),ϕ2(t))T be the solution of
(40)dϕi(t)=ϕi(t)[(ri(t)-aii(t)ϕi(t))dt+σi(t)dBi(t)],i=1,2,
where Bi(t),i=1,2, are independent standard Brownian motions, ϕ(0)=ϕ0>0, and ϕ0∈R+2, ri(t),aii(t),σi(t),i=1,2 are all continuous bounded nonnegative functions on [0,+∞). From Lemma 4 it is easy to know the following.Lemma 7.
Assume thatr-il=ril-((σiu)2/2)>0,i=1,2, then for any given initial value ϕ(0)∈R+2, the solution ϕ(t) of (40) has the properties
(41)limsupt→∞E(1ϕiθ(t))≤Hi,i=1,2,
where Hi, i=1,2 are two constants, θ is positive constant satisfying
(42)θ(σiu)2<2r-il,i=1,2.Lemma 8.
Assume thatr-il>0,i=1,2, then for any given initial value x0∈R+2, the solution x(t) of system (3) has the properties
(43)xi(t)≥ϕi(t),i=1,2,(44)limsupt→∞E(1xiθ(t))≤Hi,i=1,2,
where Hi,i=1,2 are two constants, θ is positive constant satisfying
(45)θ(σiu)2<2r-il,i=1,2.Proof.
Equation (43) follows directly from the classical comparison theorem of stochastic differential equations (see [20]). Thus, we obtain
(46)limsupt→∞E(1xiθ(t))≤limsupt→∞E(1ϕiθ(t))≤Hi,i=1,2.Definition 9.
System (3) is said to be stochastically permanent if for any ε∈(0,1), there exists a pair of positive constants δ=δ(ϵ) and M=M(ϵ) such that for any initial value x0∈R+2, the solution obeys
(47)liminft→∞P{xi(t)≤M(ϵ)}≥1-ϵ,liminft→∞P{xi(t)≥δ(ϵ)}≥1-ϵ,i=1,2.Theorem 10.
Assume thata11la22l>a12ua21u,r-il>0,i=1,2, then system (3) is stochastically permanent.The proof is a simple application of the Chebyshev inequality, we omit it.


## 3.2. Persistence in Time Average

Theorem10 shows that if the white noise is not large, the solution of system (3) is survive with large probability. In this part, we show x(t) is persistence in mean.Lemma 11.
Assume thatr-il>0,i=1,2, then for any given initial value ϕ(0)∈R+2, the solution ϕ(t) of (40) has the properties
(48)zi(t)e-[max0≤s≤t∫0sσi(τ)dBi(τ)-∫0tσi(τ)dBi(τ)]≤ϕi(t)≤zi(t)e-[min0≤s≤t∫0sσi(τ)dBi(τ)-∫0tσi(τ)dBi(τ)],i=1,2,
where z(t)=(z1(t),z2(t)) is the solution of
(49)dzi(t)=zi(t)[ri(t)-σi2(t)2-aiiuzi(t)]dt,zi(0)=ϕi(0),i=1,2.Proof.
From Lemma5, we know
(50)1ϕi(t)=1ϕi(0)e-∫0t[ri(s)-(σi2(s)/2)]ds+σi(s)dBi(s)+aiiu∫0te-∫0t[ri(s)-(σi2(s)/2)]ds+σi(s)dBi(s)×e+∫0s[ri(τ)-(σi2(τ)/2)]dτ+σi(τ)dBi(τ)ds=e-∫0tσi(s)dBi(s)[∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτe∫0sσi(τ)dBi(τ)ds∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτ1ϕi(0)e-∫0t[ri(s)-(σi2(s)/2)]ds+aiiu∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτ×e∫0sσi(τ)dBi(τ)ds∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτ1ϕi(0)]≤e-∫0tσi(s)dBi(s)[∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτe∫0sσi(τ)dBi(τ)ds1ϕi(0)e-∫0t[ri(s)-(σi2(s)/2)]ds+aiiuemax0≤s≤t(∫0sσi(τ)dBi(τ))×∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτds]≤emax0≤s≤t[∫0sσi(τ)dBi(τ)]-∫0tσi(s)dBi(s)zi(t).
Similarly, we have
(51)1ϕi(t)≥e-∫0tσi(s)dBi(s)[∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτ1ϕi(0)e-∫0t[ri(s)-(σi2(s)/2)]ds+aiiuemin0≤s≤t(∫0sσi(τ)dBi(τ))×∫0te-∫st[ri(τ)-(σi2(τ)/2)]dτds]≥emin0≤s≤t[∫0sσi(τ)dBi(τ)]-∫0tσi(s)dBi(s)zi(t).Lemma 12.
Assume thatr-il>0,i=1,2, then for any given initial value z(0)∈R+2, the solution z(t) of (49) has the following properties
(52)z~i(t)≤zi(t)≤z^i(t),limt→∞z~i(t)=r-ilaiiu,limt→∞z^i(t)=r-iuaiiu,
where z~(t)=(z~1(t),z~2(t)),z^(t)=(z^1(t),z^2(t)) are the solutions of the two equations, respectively,
(53)dz~i(t)=z~i(t)[r-il-aiiuz~i(t)]dt,z~i(0)=zi(0),i=1,2,(54)dz^i(t)=z^i(t)[r-iu-aiiuz^i(t)]dt,z^i(0)=zi(0),i=1,2.Proof.
Letz~(t)=(z~1(t),z~2(t)), z^(t)=(z^1(t),z^2(t)) are the solutions of SDE (53) and (54), respectively, with the positive initial value z(0). By Lemma 5, we know
(55)z~i(t)=er-ilt1/z~i(0)+(aiiu/r-il)(er-ilt-1),z^i(t)=er-iut1/z^i(0)+(aiiu/r-iu)(er-iut-1).
Thus,
(56)limt→∞z~i(t)=r-ilaiiu,limt→∞z^i(t)=r-iuaiiu.
By the classical comparison theorem of ordinary differential equations, we know
(57)z~i(t)≤zi(t)≤z^i(t).Lemma 13.
Assume thatr-il>0,i=1,2, then for any given initial value ϕ(0)∈R+2, the solution ϕ(t) of (40) has the properties
(58)limt→∞logϕi(t)t=0,a.s.Proof.
By Lemma12, we know
(59)e-[max0≤s≤t∫0sσi(τ)dBi(τ)-∫0tσi(τ)dBi(τ)]≤ϕi(t)zi(t)≤e-[min0≤s≤t∫0sσi(τ)dBi(τ)-∫0tσi(τ)dBi(τ)].
So, we have
(60)∫0tσi(τ)dBi(τ)-max0≤s≤t∫0sσi(τ)dBi(τ)≤logϕi(t)-logzi(t)≤∫0tσi(τ)dBi(τ)-min0≤s≤t∫0sσi(τ)dBi(τ).
Let Mi(t)=∫0tσi(τ)dBi(τ), then
(61)〈Mi,Mi〉t=∫0tσi2(τ)dτ.
Since σi(t),i=1,2 are bounded, then
(62)limt→∞〈Mi,Mi〉tt=limt→∞1t∫0tσi2(τ)dτ<∞,a.s.
By the strong law of large numbers, we know
(63)limt→∞Mi(t)t=limt→∞∫0tσi(τ)dBi(τ)t=0,a.s.
Thus,
(64)limt→∞max0≤s≤t|Mi(s)t|=0,a.s.
Then from (60) we obtain
(65)limt→∞logϕi(t)t=0,a.s.Lemma 14.
Assume thatr-il>0,i=1,2, then for any given initial value ϕ(0)∈R+2, the solution ϕ(t) of (40) has the properties
(66)liminft→∞1t∫0tϕi(s)ds≥r-ilaiiu,a.s.Proof.
By Itô's formula, we have(67)dlogϕi(t)=[ri(t)-σi2(t)2-aiiuϕi(t)]dt+σi(t)dBi(t).
Integrating both sides of this equation from 0 to t yields
(68)logϕi(t)t-logϕi(0)t=∫0t[ri(s)-(σi2(s)/2)]dst-aiiu∫0tϕi(s)dst+∫0tσi(s)dBi(s)t.
By Lemma 13, we know that
(69)limt→∞∫0tσi(s)dBi(s)t=limt→∞logϕi(t)t=0,a.s.
Hence,
(70)liminft→∞1t∫0tϕi(s)ds=1aiiuliminft→∞1t∫0t[ri(s)-σi2(s)2]ds≥r-ilaiiu,a.s.Definition 15.
System (3) is said to be persistent in time average if
(71)liminft→∞1t∫0txi(s)ds>0,i=1,2.Theorem 16.
Assume thata11la22l>a12ua21u and r-il>0,i=1,2, then the solution x(t) of system (3) with any initial value x(0)∈R+2 has the following property:
(72)liminft→∞1t∫0txi(s)ds≥r-ilaiiu,liminft→∞logxi(t)t≥0,a.s.,
and so system (3) is persistent in time average.Proof.
By Lemma8, we know that
(73)xi(t)≥ϕi(t)i=1,2,
where ϕ(t)=(ϕ1(t),ϕ2(t)) is the solution of system (40). Moreover, by Lemma 14 we know that
(74)liminft→∞1t∫0txi(s)ds≥liminft→∞1t∫0tϕi(s)ds≥r-ilaiiu,a.s.
Hence, by Lemma 13 we know that
(75)liminft→∞logxi(t)t≥liminft→∞logϕi(t)t=0,a.s.


## 3.3. Asymptotic Boundedness of Integral Average

Theorem16 shows that every component of the solution x(t) of system (3) will survive forever in time average, if the white noise is not large. In this part, we further deduce that every component of x(t) of system (3) will be an asymptotic boundedness in time average. Before we give the result, we do some preparation work.Lemma 17.
Letf∈C[[0,∞)×Ω,(0,∞)], F(t)∈((0,∞)×Ω,R). If there exist positive constants λ0 and λ such that
(76)logf(t)≥λt-λ0∫0tf(s)ds+F(t),t≥0,a.s.,
and limt→∞(F(t)/t)=0 a.s., then
(77)liminft→∞1t∫0tf(s)ds≥λλ0,a.s.Proof.
The proof is similar to the proof of Lemma in [21]. Let
(78)φ(t)=∫0tf(s)ds.
Since f∈C[[0,∞)×Ω,(0,∞)],φ(t) is differentiable on [0,∞) and
(79)dφ(t)dt=f(t)>0,fort≥0.
Substituting dφ(t)/dt and φ(t) into (76), we obtain the following:
(80)logdφ(t)dt≥λt-λ0φ(t)+F(t),
thus
(81)eλ0φ(t)dφ(t)dt≥eλt+F(t),fort≥0.
Note that limt→∞(F(t)/t)=0a.s., then for 0<ε<min{1,λ}, ∃T=T(ω)>0 and Ωε⊂Ω such that P(Ωε)>1-ε and F(t)≥-εt,t≥T,ω∈Ωε. Then we have
(82)eλ0φ(t)dφ(t)dt≥e(λ-ε)t,fort≥T,ω∈Ωε.
Integrating inequality (82) from 0 to t results in the following:
(83)λ0-1[eλ0φ(t)-eλ0φ(T)]≥(λ-ε)-1[e(λ-ε)t-e(λ-ε)T].
This inequality can be rewritten into
(84)eλ0φ(t)≥eλ0φ(T)+λ0(λ-ε)-1[e(λ-ε)t-e(λ-ε)T].
Taking the logarithm of both sides and dividing both sides by t(>0) yields
(85)φ(t)t≥λ0-1log{eλ0φ(T)+λ0(λ-ε)-1[e(λ-ε)t-e(λ-ε)T]}t.
Then,
(86)liminft→∞φ(t)t≥λ-ελ0,ω∈Ωε.
Letting ε→∞ yields
(87)liminft→∞1t∫0tf(s)ds≥λλ0,a.s.
This finishes the proof of the Lemma.Theorem 18.
Assume thata11la22l>a12ua21u and r-il>0,i=1,2, then the solution x(t) of system (3) with any initial value x(0)∈R+2 has the property
(88)x~i*≤limt→∞1t∫0txi(s)ds≤x^i*,i=1,2,a.s.,
where
(89)x~1*=a22ur-1l+a12lr-2la11ua22u-a12la21l,x~2*=a11ur-2l+a21lr-1la11ua22u-a12la21l,x^1*=a22lr-1u+a12ur-2ua11la22l-a12ua21u,x^2*=a11lr-2u+a21ur-1ua11la22l-a12ua21u.Proof.
To prove the results, we only need to prove(90)liminft→∞1t∫0txi(s)ds≥x~i*,i=1,2,a.s.(91)limsupt→∞1t∫0txi(s)ds≤x^i*,i=1,2,a.s.
By Itô's formula, we have
(92)dlogx1(t)=[r1(t)-12σ12(t)-a11(t)x1(t)+a12(t)x2(t)]dt+σ1(t)dB1(t),dlogx2(t)=[r2(t)-12σ22(t)+a21(t)x1(t)-a22(t)x2(t)]dt+σ2(t)dB2(t).
First, we prove (91). Integrating both sides of (92) from 0 to t yields
(93)logx1(t)=logx1(0)+∫0tr-1(s)ds-∫0ta11(s)x1(s)ds+∫0ta12(s)x2(s)ds+∫0tσ1(s)dB1(s),logx2(t)=logx2(0)+∫0tr-2(s)ds-∫0ta22(s)x2(s)ds+∫0ta21(s)x1(s)ds+∫0tσ2(s)dB2(s),
where r-i(s)=ri(s)-(1/2)σi2(s),i=1,2. Since xi(t)>0,i=1,2, hence
(94)logx1(t)≤logx1(0)+r-1ut-a11l∫0tx1(s)ds+a12u∫0tx2(s)ds+∫0tσ1(s)dB1(s),logx2(t)≤logx2(0)+r-2ut-a22l∫0tx2(s)ds+a21u∫0tx1(s)ds+∫0tσ2(s)dB2(s).
So we have
(95)a22llogx1(t)+a12ulogx2(t)≤a22l[logx1(0)+∫0tσ1(s)dB1(s)]+a22lr-1ut+a12u[logx2(0)+∫0tσ2(s)dB2(s)]+a12ur-2ut-(a11la22l-a21ua12u)∫0tx1(s)ds.
By Theorem 16, we know that
(96)liminft→∞logxi(t)t≥0,i=1,2,a.s.
Obviously,
(97)limt→∞logxi(0)+∫0tσi(s)dBi(s)t=0,i=1,2,a.s.
Hence, we have
(98)limsupt→∞1t∫0tx1(s)ds≤a22lr-1u+a12ur-2ua11la22l-a21ua12u≜x^1*,a.s.
Similarly, we have
(99)limsupt→∞1t∫0tx2(s)ds≤a11lr-2u+a12ur-1ua11la22l-a21ua12u≜x^2*,a.s.
Next, we prove that (90) is true. Taking integration both sides of (92) from 0 to t, we have
(100)logx1(t)≥logx1(0)+r-1lt-a11u∫0tx1(s)ds+a12l∫0tx2(s)ds+∫0tσ1(s)dB1(s),logx2(t)≥logx2(0)+r-2lt-a22u∫0tx2(s)ds+a21l∫0tx1(s)ds+∫0tσ2(s)dB2(s).
By Theorem 16 we know that
(101)liminft→∞1t∫0tx1(s)ds≥r-1la11u≜M1,a.s.,liminft→∞1t∫0tx2(s)ds≥r-2la22u≜N1,a.s.,
then for any ε>0, there is a T(ω)>0 such that
(102)1t∫0tx2(s)ds≥N1-ε,
for t>T(ω). It follows from (100) that, for t>T(ω),
(103)logx1(t)≥logx1(0)+r-1lt-a11u∫0tx1(s)ds+a12l(N1-ε)t+∫0tσ1(s)dB1(s)=logx1(0)-a11u∫0tx1(s)ds+[r-1l+a12l(N1-ε)]t+∫0tσ1(s)dB1(s).
From Lemma 17, we have
(104)liminft→∞1t∫0tx1(s)ds≥r-1l+a12l(N1-ε)a11u:=M2>M1.
Similarly, we have
(105)liminft→∞1t∫0tx2(s)ds≥r-2l+a21l(M1-ε)a22u:=N2>N1.
Continuing this process, we obtain two sequences Mn,Nn(n=1,2,…) such that
(106)Mn=r-1l+a12l(Nn-1-ε)a11u,(107)Nn=r-2l+a21l(Mn-1-ε)a22u.
By induction, we can easily show that Mn+1>Mn,Nn+1>Nn,n=1,2,…, that is, sequences {Mn,n=1,2,…} and {Nn,n=1,2,…} are nondecreasing. Moreover, note that (98) and (99), then the sequences {Mn,n=1,2,…} and {Nn,n=1,2,…}, have upper bounds. Therefore, there are two positive M,N such that
(108)limn→∞Mn=M,limn→∞Nn=N,liminft→∞1t∫0tx1(s)ds≥M,liminft→∞1t∫0tx2(s)ds≥N,
which together with (106) implies
(109)a11uM-a12lN=r-1l-εa12l,a22uN-a21lM=r-2l-εa21l.
Letting ε→0 yields
(110)M=a22ur-1l+a12lr-2la11ua22u-a12la21l≜x~1*,N=a11ur-1l+a21lr-2la11ua22u-a12la21l≜x~1*.
Hence,
(111)liminft→∞1t∫0txi(s)ds≥x~i*,i=1,2,a.s.,
which is as required.


## 4. Nonpersistence

In this section, we discuss the dynamics of system (3) as the white noise is getting larger. We show that system (3) will be nonpersistent if the white noise is large, which does not happen in the deterministic system.Definition 19.
System (3) is said to be nonpersistent, if there are positive constants q1,q2 such that
(112)limt→∞∏i=12xiqi(t)=0a.s.Theorem 20.
Assume thata11la22l>a12ua21u and a22lr-1u+a12ur-2u<0, then system (3) is nonpersistent, where r-i(s)=ri(s)-(σi2(s)/2),i=1,2.Proof.
Sincexi(t)>0,i=1,2 and a11la22l>a12ua21u, from (93) we have
(113)a22llogx1(t)+a12ulogx2(t)≤{a22lr-1u+a12ur-2u}t-(a11la22l-a21ua12u)∫0tx1(s)ds≤+a22l[logx1(0)+∫0tσ1(s)dB1(s)]+a12u[logx2(0)+∫0tσ2(s)dB2(s)]≤K1t+a22l[logx1(0)+∫0tσ1(s)dB1(s)]+a12u[logx2(0)+∫0tσ2(s)dB2(s)],
where K1=a22lr-1u+a12ur-2u which together with
(114)limt→∞a22l[logx1(0)+∫0tσ1(s)dB1(s)]t=limt→∞a12u[logx2(0)+∫0tσ2(s)dB2(s)]t=0,a.s.,
implies
(115)limt→∞1t[a22llogx1(t)+a12ulogx2(t)]≤K1,a.s.
If K1<0, then there must be
(116)limt→∞x1a22l(t)x2a12u(t)=0,a.s.
Hence, system (3) is nonpersistent.Theorem 21.
Assume thata11la22l>a12ua21u and (a21ur-1u+a11lr-2u)<0, then system (3) is nonpersistent, where r-i(s)=ri(s)-(σi2(s)/2),i=1,2.Here we omit the proof of Theorem21 which is similar to the proof of Theorem 20.Remark 22.
If(σil)2>2riu,i=1,2, then the conditions in Theorems 20 and 21 are obviously satisfied, respectively. That is to say, the large white noise will lead to the population system being non-persistent.


## 5. Global Attractivity

In this section, we turn to establishing sufficient criteria for the global attractivity of stochastic system (3).Definition 23.
Letx(t), y(t) be two arbitrary solutions of system (3) with initial values x(0),y(0)∈R+2, respectively. If
(117)limt→∞|x(t)-y(t)|=0,a.s.,
then we say system (3) is globally attractive.Theorem 24.
Assume thata11la22l>a12ua21u, then system (3) is globally attractive.Proof.
Letx(t), y(t) be two arbitrary solutions of system (3) with initial values x(0),y(0)∈R+2. By the Itô's formula, we have
(118)dlogxi(t)=[ri(t)-12σi2(t)-aii(t)xi(t)+aij(t)xj(t)]dt+σi(t)dBi(t),i,j=1,2,j≠i,dlogyi(t)=[ri(t)-12σi2(t)-aii(t)yi(t)+aij(t)yj(t)]dt+σi(t)dBi(t),i,j=1,2,j≠i.
Then,
(119)d(logxi(t)-logyi(t))={-aii(t)[xi(t)-yi(t)]+aij(t)[xi(t)-yi(t)]}dt,i,j=1,2,j≠i.
Since a11la22l>a12ua21u, there exist two positive constants c1, c2 which satisfy
(120)a21ua11l<c1c2<a22la12u.
Thus, c1a11l-c2a21u>0, c2a22l-c1a12u>0.
Consider a Lyapunov functionV(t) defined by
(121)V(t)=c1|logx1(t)-logy1(t)|+c2|logx2(t)-logy2(t)|,t≥0.
A direct calculation of the right differential d+V(t) of V(t) along the ordinary differential equation (119) leads to
(122)d+V(t)=c1sgn(x1(t)-y1(t))d[logx1(t)-logy1(t)]+c2sgn(x2(t)-y2(t))d[logx2(t)-logy2(t)]=c1sgn(x1(t)-y1(t))×[-a11(t)(x1(t)-y1(t))dt+a12(t)(x2(t)-y2(t))dt]+c2sgn(x2(t)-y2(t))×[a21(t)(x1(t)-y1(t))dt-a22(t)(x2(t)-y2(t))dt]≤-c1a11l|x1(t)-y1(t)|dt+c1a12u|x2(t)-y2(t)|dt-c2a22l|x2(t)-y2(t)|dt+c2a21u|x1(t)-y1(t)|dt=-(c1a11l-c2a21u)|x1(t)-y1(t)|dt-(c2a22l-c1a12u)|x2(t)-y2(t)|dt≤-γ∑i=12|xi(t)-yi(t)|dt,
where γ=min{c1a11l-c2a21u,c2a22l-c1a12u}. Integrating both sides of (122) form 0 to t, we have
(123)V(t)+γ∫0t∑i=12|xi(s)-yi(s)|ds≤V(0)<∞.
Let t→∞, we obtain
(124)∫0∞|x(s)-y(s)|ds≤∫0∞∑i=12|xi(s)-yi(s)|ds≤V(0)γ<∞a.s.
Note that u(t)=x(t)-y(t). Clearly, u(t)∈C(R+,R2)a.s. It is straightforward to see from (124) that
(125)liminft→∞|u(t)|=0a.s.
Next, we prove that
(126)limt→∞|u(t)|=0a.s.
By Theorem 3 we obtain that the pth moment of the solution of system (3) is bounded, the following proof is similar to the proof of Theorem 6.2 in [15] and hence is omitted.



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