# Existence, Uniqueness and Ergodicity of Positive Solution of Mutualism System with Stochastic Perturbation

Authors: Chunyan Ji; Daqing Jiang; Hong Liu; Qingshan Yang
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

We discuss a two-species Lotka-Volterra mutualism system with stochastic perturbation. We show that there is a unique nonnegative solution of this system. Furthermore, we investigate that there exists a stationary distribution for this system, and it has ergodic property.


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## Body

## 1. Introduction

It is well known that the differential equation(1.1)ẋ1(t)=x1(t)[r1-a11x1(t)+a12x2(t)],ẋ2(t)=x2(t)[r2+a21x1(t)-a22x2(t)]
denotes the population growth of mutualism system for the two species. x1(t) and x2(t) represent the densities of the two species at time t, respectively, and the parameters ri,aij,i,j=1,2 are all positive. Goh [1] showed that the asymptotic stability equilibrium state of (1.1) in local must be asymptotic stability in global. That is, if ri>0,aij>0,i,j=1,2, and a11a22-a12a21>0, then (1.2)x1(t)→x1*,x2(t)→x2*,ast→∞,
where x*=(x1*,x2*) is the unique positive equilibrium of system (1.1) and (1.3)x1*=r1a22+r2a12a11a22-a12a21>0,x2*=r2a11+r1a21a11a22-a12a21>0.
While if a11a22-a12a21<0, then the population of both species increase to infinite. There are extensive literature concerned with mutualism system; see [2–7].The papers mentioned above are all deterministic models, which do not incorporate the effect of fluctuating environment. In fact, environmental fluctuations are important components in the population system. Most of natural phenomena do not follow strictly deterministic laws, but rather oscillate randomly around some average values, hence the deterministic equilibrium is no longer an absolutely fixed state [8]. Therefore stochastic differential equation models play a significant role in various branches of applied sciences including the population system, as they provide some additional degree of realism compared to their deterministic counterpart [9–13]. Recently, many authors have paid attention to how population systems are affected by random fluctuations from environment (see, e.g., [14–18]). However, as far as we known, there is few paper consider how environmental noises affect the dynamical behaviors of the mutualism system, Zeng et al. [19] discussed the effects of noise and time delay on C(s) (the normalized correlation function) and Tc (the associated relaxation time) of a mutualism system, in which they considered the intraspecies interaction parameters were stochastically perturbed. Motivated by this, the main aim of this paper is to study the dynamical behaviors of the mutualism system with stochastic perturbation.In this paper, considering the effect of randomly fluctuating environment, we incorporate white noise in each equation of system (1.1). Here we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in the natural growth rates ri,i=1,2. Suppose ri→ri+σiḂi(t), where Bi(t),i=1,2 are mutually independent one dimensional standard Brownian motions with Bi(0)=0, and σi>0,i=1,2 are the intensities of white noises. The stochastic version corresponding to the deterministic system (1.1) takes the following form: (1.4)dx1(t)=x1(t)[(r1-a11x1(t)+a12x2(t))dt+σ1dB1(t)],dx2(t)=x2(t)[(r2+a21x1(t)-a22x2(t))dt+σ2dB2(t)].This paper is organized as follows. In Section2, we show there is a unique positive solution of (1.4) if a11a22-a12a21>0, and give out the estimation of the solution. The stability of system (1.4) is investigated in Section 3. Since (1.4) does not have interior equilibrium, we cannot discuss the stability as the deterministic system. First, we show there is a stationary distribution of (1.4) and it has ergodic property. Next, by estimating the p moment, we explore some properties of the solution.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 denote the positive cone of R2, namely R+2={x∈R2:xi>0,i=1,2}. If A is a vector, its transpose is denoted by A⊤. For x∈R2,|x|=|x1|+|x2|.


## 2. The Existence and Estimation of the Solution

To investigate the dynamical behavior, the first concern thing is whether the solution is global existence. Moreover, for a population model, whether the value is nonnegative is also considered. Hence in this section we first show the solution of (1.4) is global and nonnegative. As we have known, for a stochastic differential equation to have a unique global solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition (cf. Arnold [20], Mao [21]). However, the coefficients of (1.4) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of (1.4) may explode at a finite time. In this section, following the way developed by Mao et al. [17], we show there is a unique positive solution of (1.4).Theorem 2.1.
There is a unique positive solutionx(t) of system (1.4) for any given initial value x(0)=x0∈R+2 provided a11a22>a12a21.Proof.
The proof is similar to Theorem2.1 in [17]. Here we define a C2-function V:R+2→R+:
(2.1)V(x1,x2)=a21[x1-x1*-x1*logx1x1*]+a12[x2-x2*-x2*logx2x2*],
where x*=(x1*,x2*)⊤ satisfies
(2.2)r1-a11x1*+a12x2*=0,r2+a21x1*-a22x2*=0.Remark 2.2.
Theorem2.1 shows stochastic equation (1.4) also has a global positive solution under the same condition of the corresponding deterministic system (1.1). That is to say, the white noise does not affect the existence of the unique global positive solution.In the remaining of this section, we give the estimation of the solution of system (1.4).Jiang and Shi [22] discussed a randomized nonautonomous logistic equation:(2.3)dN(t)=N(t)[(a(t)-b(t)N(t))dt+α(t)dB(t)],
where B(t) is 1-dimensional standard Brownian motion, N(0)=N0 and N0 is independent of B(t). They showed the following.Lemma 2.3 (see [22]).
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 (2.3) for any initial value N(0)=N0>0, which is global and represented by
(2.4)N(t)=e∫0t[a(s)-α2(s)/2]ds+α(s)dB(s)1/N0+∫0tb(s)e∫0s[a(τ)-α2(τ)/2]dτ+α(τ)dB(τ)ds,t≥0.Theorem 2.4.
Assume thata11a22>a12a21 and x(t)∈R+2 is the solution of system (1.4) with initial value x0∈R+2. Then x(t) has the property that
(2.5)x1(t)≥ϕ1(t),x2(t)≥ϕ2(t),
where ϕ1(t) and ϕ2(t) are the solutions of equations:
(2.6)dϕ1(t)=ϕ1(t)[(r1-a11ϕ1(t))dt+σ1dB1(t)],ϕ1(0)=x1(0),(2.7)dϕ2(t)=ϕ2(t)[(r2-a22ϕ2(t))dt+σ2dB2(t)],ϕ2(0)=x2(0).The result of Theorem2.4 follows directly from the classical comparison theorem of stochastic differential equations (see [23]).Remark 2.5.
From Lemma2.3, we see
(2.8)ϕ1(t)=e(r1-σ12/2)t+σ1B1(t)1/x1(0)+a11∫0te(r1-σ12/2)s+σ1B1(s)ds,ϕ2(t)=e(r2-σ22/2)t+σ2B2(t)1/x2(0)+a22∫0te(r2-σ22/2)s+σ2B2(s)ds.
This together with Theorem 2.4 shows that if ri>(σi2/2)(i=1,2), then both species will not extinct.


## 3. Stationary Distribution and Ergodicity for System (1.4)

In the introduction, we have mentioned that ifri>0,aij>0,i,j=1,2, and a11a22-a12a21>0, then the unique positive equilibrium (x1*,x2*) of (1.1) is globally stable. But there is none positive equilibrium for (1.4). We investigate there is a stationary distribution for system (1.4) instead of asymptotically stable equilibria [24]. Before giving the main theorem, we first give a lemma (see [25]).Assumption B.
There exists a bounded domainU⊂El with regular boundary Γ, having the following properties.(B.1)
In the domainU and some neighbourhood thereof, the smallest eigenvalue of the diffusion matrix A(x) is bounded away from zero.(B.2)
Ifx∈El∖U, the mean time τ at which a path issuing from x reaches the set U is finite, and supx∈KExτ<∞ for every compact subset K⊂El.Lemma 3.1 (see [25]).
If (B) holds, then the Markov processX(t) has a stationary distribution μ(A). Let f(·) be a function integrable with respect to the measure μ. Then
(3.1)Px{limT→∞1T∫0Tf(X(t))dt=∫Elf(x)μ(dx)}=1
for all x∈El.Theorem 3.2.
Assumea11a22>a12a21, σ1>0, σ2>0 and δ<min{m1(x1*)2,m2(x2*)2}. Then there is a stationary distribution μ(A) for system (1.4) and it has ergodic property. Here (x1*,x2*) is the solution of (2.2), δ=(a21x1*σ12+a12x2*σ22)/2,m1=a11a21-a12a21/ϵ0>0 and m2=a12a22-ϵ0a12a21>0, where ϵ0>0 satisfies a12/a11<ϵ0<a22/a21.Proof.
DefineV:El=R+2→R+:
(3.2)V(x1,x2)=a21(x1-x1*-x1*logx1x1*)+a12(x2-x2*-x2*logx2x2*).
Then
(3.3)dV=a21(1-x1*x1)dx1+a212x1*x12(dx1)2+a12(1-x2*x2)dx2+a122x2*x22(dx2)2=a21(x1-x1*)[(r1-a11x1+a12x2)dt+σ1dB1(t)]+12a21x1*σ12dt+a12(x2-x2*)[(r2+a21x1-a22x2)dt+σ2dB2(t)]+12a12x2*σ22dt:=LVdt+a21σ1(x1-x1*)dB1(t)+a12σ2(x2-x2*)dB2(t),
where
(3.4)LV=a21(x1-x1*)(r1-a11x1+a12x2)+12a21x1*σ12+a12(x2-x2*)(r2+a21x1-a22x2)+12a12x2*σ22=a21(x1-x1*)[-a11(x1-x1*)+a12(x2-x2*)]+a12(x2-x2*)[a21(x1-x1*)-a22(x2-x2*)]+δ=-a11a21(x1-x1*)2+2a12a21(x1-x1*)(x2-x2*)-a12a22(x2-x2*)2+δ,
according to the equality (2.2). By Young inequality, we have
(3.5)2a12a21|x1-x1*||x2-x2*|≤a12a21[(x1-x1*)2ϵ0+ϵ0(x2-x2*)2].
Then
(3.6)LV≤-(a11a21-a12a21ϵ0)(x1-x1*)2-(a12a22-ϵ0a12a21)(x2-x2*)2+δ=-m1(x1-x1*)2-m2(x2-x2*)2+δ.
Note that δ<min{m1(x1*)2,m2(x2*)2}; then the ellipsoid
(3.7)-m1(x1-x1*)2-m2(x2-x2*)2+δ=0
lies entirely in R+2. We can take U to be any neighborhood of the ellipsoid with U¯⊆El=R+2, so for x∈U∖El, LV≤0, which implies that condition (B.2) in Lemma 3.1 is satisfied. Besides, there is M>0 such that
(3.8)∑i,j=1n(∑k=1ngik(x)gjk(x))ξiξj=σ12x12ξ12+σ22x22ξ22≥M|ξ2|x∈U¯,ξ∈R2,
which  implies condition (B.1) is also satisfied. Therefore, the stochastic system (1.4) has a stable a stationary distribution μ(A) and it is ergodic.Remark 3.3.
Ifa11a22>a12a21 and a22>a21, we can choose ϵ0=(1/2)(a12/a11+a22/a21), then m1=a11a12(a22-a21)/(a11a22+a12a21) and m2=a12(a11a22-a12a21)/2a11.Since system (1.4) is ergodic, next we explore some properties of the solution.Consider the equation(3.9)N·(t)=N(t)[a-bN(t)]
with initial value N0>0. It is well known, when a,b>0, (3.9) has a unique positive solution (3.10)N(t)=eat1/Ñ0+(b/a)(eat-1),t≥0,(3.11)limt→∞N(t)=ab,limt→∞logN(t)t=0.Lemma 3.4.
Suppose thatr1>σ12/2 and ϕ1(t) is the solution of (2.6), then one has
(3.12)ψ1(t)e-σ1(max0≤s≤tB1(s)-B1(t))≤ϕ1(t)≤ψ1(t)e-σ1(min0≤s≤tB1(s)-B1(t)),
where ψ1(t) is the solution of
(3.13)ψ̇1(t)=ψ1(t)[r1-σ122-a11ψ1(t)],ψ1(0)=x1(0).Proof.
From the representation of the solutionϕ1(t), we have
(3.14)1ϕ1(t)=1x1(0)e-(r1-σ12/2)t-σ1B1(t)+a11∫0te-(r1-σ12/2)(t-s)-σ(B1(t)-B1(s))ds=e-σ1B1(t)[1x1(0)e-(r1-σ12/2)t+a11∫0te-(r1-σ12/2)(t-s)eσ1B1(s)ds]≤e-σ1B1(t)[1x1(0)e-(r1-σ2/2)t+a11eσ1max0≤s≤tB1(s)∫0te-(r1-σ12/2)(t-s)ds]≤eσ1[max0≤s≤tB1(s)-B1(t)][1x1(0)e-(r1-σ12/2)t+a11∫0te-(r1-σ12/2)(t-s)ds],
where the last inequality is based on the property of Brownian motion that B(0)=0. Similarly, we have
(3.15)1ϕ1(t)≥eσ1(min0≤s≤tB1(s)-B1(t))[1x1(0)e-(r1-σ12/2)t+a11∫0te-(r1-σ12/2)(t-s)ds].
Therefore
(3.16)eσ1(min0≤s≤tB1(s)-B1(t))1ψ1(t)≤1ϕ1(t)≤eσ1(max0≤s≤tB1(s)-B1(t))1ψ1(t),
which is as required.Lemma 3.5.
Suppose thatr1>σ12/2 and ϕ1(t) is the solution of (2.6), then one has
(3.17)limt→∞logϕ1(t)t=0.Proof.
It is easy to drive from Lemma3.4 that
(3.18)σ1(B1(t)-max0≤s≤tB1(s))≤logϕ1(t)-logψ1(t)≤σ1(B1(t)-min0≤s≤tB1(s)).
Note that the distribution of max0≤s≤tB1(s) is the same as |B1(t)|, and that min0≤s≤tB1(s) has the same distribution as—max0≤s≤tB1(s), then by (3.11) and the strong law of large numbers, we get
(3.19)limt→∞logϕ1(t)t=limt→∞logψ1(t)t=0.
This completes the proof of Lemma 3.5.Now consider the solution of (2.7), by the same reasons as Lemmas 3.4 and 3.5, we have the following.Lemma 3.6.
Suppose thatr2>σ22/2 and ϕ2(t) is the solution of (2.7), then one has
(3.20)limt→∞logϕ2(t)t=0.Lemma 3.7.
LetM(t)=∫0tesdB(s), where B(t) is 1-dimensional standard Brownian motion, then
(3.21)lim supt→∞|e-tM(t)|logt=1a.s.Proof.
The proof can be found on [21, page 70].Based on these lemmas, now we show the main result in this section.Lety1(t)=logx1(t), y2(t)=logx2(t); then by Itô's formula we obtain(3.22)dy1(t)=(r1-σ122-a11x1(t)+a12x2(t))dt+σ1dB1(t),dy2(t)=(r2-σ222+a21x1(t)-a22x2(t))dt+σ2dB2(t).
If a11a22>a12a21 and 2r1>σ12,2r2>σ22, then the equation (3.23)r1-σ122-a11x1+a12x2=0,r2-σ222+a21x1-a22x2=0
has a unique positive solution:(3.24)x̃1*=a22(r1-σ12/2)+a12(r2-σ22/2)a11a22-a12a21,x̃2*=a21(r1-σ12/2)+a11(r2-σ22/2)a11a22-a12a21.Lemma 3.8.
Assumea11a22>a12a21 and 2r1>σ12,2r2>σ22. Then for any initial value x0∈R+2, the solution x(t) of system (1.4) has the following property:
(3.25)limt→∞x¯1(t)=x̃1*,limt→∞x¯2(t)=x̃2*,
where x¯i(t)=(1/t)∫0txi(s)ds,i=1,2.Proof.
It follows from (3.22) that
(3.26)y1(t)t=y1(0)t+r1-σ122-a11x¯1(t)+a12x¯2(t)+σ1B1(t)t,y2(t)t=y2(0)t+r2-σ222+a21x¯1(t)-a22x¯2(t)+σ2B2(t)t.
Obviously, to prove the result, it is an easy consequence of
(3.27)limt→∞yi(t)t=limt→∞logxi(t)t=0,i=1,2a.s.
We first show that
(3.28)lim inft→∞logxi(t)t≥0,i=1,2a.s.
In fact, the results of Theorem 2.4 and Lemmas 3.5 and 3.6 imply that (3.28) is true.
Next, we will prove(3.29)lim supt→∞logxi(t)t≤0,i=1,2a.s.
If a11a22>a12a21, then there exist positive constants c1,c2,m1,m2 such that
(3.30)-a11c1+a21c2=-m1,a12c1-a22c2=-m2.
From (3.22) we get
(3.31)d(c1y1(t)+c2y2(t))=c1dlogx1(t)+c2dlogx2(t)=c1[(r1-σ122-a11x1(t)+a12x2(t))dt+σ1dB1(t)]+c2[(r2-σ222+a21x1(t)-a22x2(t))dt+σ2dB2(t)]=[(r1-σ122)c1+(r2-σ222)c2+(a21c2-a11c1)x1(t)+(a12c1-a22c2)x2(t)]dt+c1σ1dB1(t)+c2σ2dB2(t)=[(r1-σ122)c1+(r2-σ222)c2-m1x1(t)-m2x2(t)]dt+c1σ1dB1(t)+c2σ2dB2(t),d[et(c1y1(t)+c2y2(t))]=etd(c1y1(t)+c2y2(t))+et(c1y1(t)+c2y2(t))dt=et[(r1-σ122)c1+(r2-σ222)c2+c1y1(t)-m1ey1(t)+c2y2(t)-m2ey2(t)]dt+et[c1σ1dB1(t)+c2σ2dB2(t)].
Note that the function c1y1-m1ey1 has its maximum value c1*=c1log(c1/m1)-c1 at y1=log(c1/m1), and the function c2y2-m2ey2 has its maximum value c2*=c2log(c2/m2)-c2 at y2=log(c2/m2); then
(3.32)d[et(c1y1(t)+c2y2(t))]≤et[(r1-σ122)c1+(r2-σ222)c2+c1*+c2*]dt+et[c1σ1dB1(t)+c2σ2dB2(t)]:=c*etdt+et[c1σ1dB1(t)+c2σ2dB2(t)],
where c*=(r1-σ12/2)c1+(r2-σ22/2)c2+c1*+c2*. Integrating both sides of it from 0 to t, yields
(3.33)et(c1y1(t)+c2y2(t))≤c1y1(0)+c2y2(0)+c*(et-1)+∫0tes[c1σ1dB1(s)+c2σ2dB2(s)],c1y1(t)+c2y2(t)≤c*+(c1y1(0)+c2y2(0)-c*)e-t+e-t∫0tes[c1σ1dB1(s)+c2σ2dB2(s)]≤c*+c1y1(0)+c2y2(0)+e-t∫0tes[c1σ1dB1(s)+c2σ2dB2(s)]:=c+e-t∫0tes[c1σ1dB1(s)+c2σ2dB2(s)],
where c=c*+c1y1(0)+c2y2(0). It is easy to drive from Lemma 3.7 that
(3.34)c1y1(t)+c2y2(t)≤c+Oa.s.(logt),
which implies
(3.35)c1lim supt→∞y1(t)t+c2lim supt→∞y2(t)t≤0,a.s.
Moreover (3.35) together with (3.28) shows that
(3.36)c1lim supt→∞y1(t)t≤-c2lim supt→∞y2(t)t≤-c2lim inft→∞y2(t)t≤0,a.s.
that is,
(3.37)lim supt→∞y1(t)t≤0,a.s.
Similarly, we have
(3.38)lim supt→∞y2(t)t≤0,a.s.
which is as required.Lemma 3.9.
Assumea11>a12 and a22>a21. Then for any initial value x0∈R+2, there exists a positive constant K(p) such that the solution x(t) of system (1.4) has the following property:
(3.39)E[c1x1p(t)+c2x2p(t)]≤K(p),∀t∈[0,∞],p>1.Proof.
By Itô's formula and Young inequality, we compute(3.40)d(1px1p)=x1p-1dx1+p-12x1p-2(dx1)2=[(r1+p-12σ12)x1p-a11x1p+1+a12x1px2]dt+σ1x1pdB1(t)≤[(r1+p-12σ12)x1p-a11x1p+1+a12pp+1x1p+1+a12p+1x2p+1]dt+σ1x1pdB1(t),d(1px2p)=x2p-1dx2+p-12x2p-2(dx2)2=[(r2+p-12σ22)x2p+a21x1x2p-a22x2p+1]dt+σ2x2pdB2(t)≤[(r2+p-12σ22)x2p+a21p+1x1p+1+a21pp+1x2p+1-a22x2p+1]dt+σ2x2pdB2(t).
Hence, for positive constants c1 and c2, we have
(3.41)c1d(1px1p)+c2d(1px2p)≤[c1(r1+p-12σ12)x1p-(c1a11-c1a12pp+1-c2a21p+1)x1p+1+c2(r2+p-12σ22)x2p-(c2a22-c1a12p+1-c2a21pp+1)x2p+1]dt+c1σ1x1pdB1(t)+c2σ2x2pdB2(t).
Next, we claim that there are c1>0,c2>0 such that
(3.42)c1a11-c1a12pp+1-c2a21p+1>0,c2a22-c1a12p+1-c2a21pp+1>0,
if a11>a12 and a22>a21. In fact, we only need a21/(a11(p+1)-a12p)<c1/c2<(a22(p+1)-a21p)/a12, which can be simplified to a12a21<[a11(p+1)-a12p][a22(p+1)-a21p]. It is obviously true, if a11>a12 and a22>a21.
Letα1:=p(r1+((p-1)/2)σ12),α2:=p(r2+((p-1)/2)σ22), β1:=c1-(p+1)/pp(c1a11-c1a12p/(p+1)-c2a21/(p+1)),β2:=c2-(p+1)/pp(c2a22-c1a12/(p+1)-c2a21p/(p+1)). Then α1>0,α2>0,β1>0,β2>0 and(3.43)d(c1x1p+c2x2p)≤[c1α1x1p+c2α2x2p-c1(p+1)/pβ1x1p+1-c2(p+1)/pβ2x2p+1]dt+c1pσ1x1pdB1(t)+c2pσ2x2pdB2(t)≤[max{α1,α2}(c1x1p+c2x2p)-min{β1,β2}(c1(p+1)/px1p+1+c2(p+1)/px2p+1)]dt+c1pσ1x1pdB1(t)+c2pσ2x2pdB2(t).
Hence,
(3.44)dE[c1x1p+c2x2p]≤{max{α1,α2}E[c1x1p+c2x2p]-min{β1,β2}E[c1(p+1)/px1p+1+c2(p+1)/px2p+1]}dt≤{max{α1,α2}E[c1x1p+c2x2p]-2-pmin{β1,β2}E[c1x1p+c2x2p](p+1)/p}dt.
By comparison theorem, we can get
(3.45)lim supt→∞E[c1x1p+c2x2p]≤[2pmax{α1,α2}min{β1,β2}]p:=C(p),
which implies that there is a T>0, such that
(3.46)E[c1x1p+c2x2p]≤2C(p),∀t>T.
Besides, note that E[c1x1p+c2x2p] is continuous, then there is a C̃(p)>0 such that
(3.47)E[c1x1p+c2x2p]≤C̃(p),∀t∈[0,T].
Let K(p)=max{2C(p),C̃(p)}, then
(3.48)E[c1x1p+c2x2p]≤K(p),∀t∈[0,∞].By the ergodic property, form>0, we have(3.49)limt→∞1t∫0t(xip(s)∧m)ds=∫R+2(zip∧m)μ(dz1,dz2),a.s.
On the other hand, by dominated convergence theorem, we can get (3.50)E[limt→∞1t∫0t(xip(s)∧m)ds]=limt→∞1t∫0tE[xip(s)∧m]ds≤K(p),i=1,2,
which together with (3.49) implies (3.51)∫R+2(zip∧m)μ(dz1,dz2)≤K(p),i=1,2.
Letting m→∞, we get (3.52)∫R+2zipμ(dz1,dz2)≤K(p),i=1,2.
That is to say, functions f1(x)=x1p and f2(x)=x2p are integrable with respect to the measure μ. Therefore one has the following.Theorem 3.10.
Assumea11>a12, a22>a21, 2r1>σ12>0,2r2>σ22>0 and δ<min{m1(x1*)2,m2(x2*)2}, where δ=(a21x1*σ12+a12x2*σ22)/2, m1=a11a12(a22-a21)/a11a22+a12a21 and m2=a12(a11a22-a12a21)/2a11. Then for any initial value x0∈R+2, the solution x(t) of system (1.4) has the following property:
(3.53)P{limt→∞x¯1(t)=∫R+2z1μ(dz1,dz2)=x̃1*}=1,P{limt→∞x¯2(t)=∫R+2z2μ(dz1,dz2)=x̃2*}=1.Moreover, we can get the following.Theorem 3.11.
Assumea11>a12, a22>a21 and 2r1>σ12,2r2>σ22. Then for any initial value x0∈R+2, the solution x(t) has following property
(3.54)limt→∞1t∫0t(a11a21x12(s)-a12a22x22(s))ds=a21r1x̃1*-a12r2x̃2*,
where (x̃1*,x̃2*) is defined as in (3.24).Proof.
By (3.39), for δ>0, we have
(3.55)P{ω:sup(n-1)δ≤t≤nδx1(t)t>δ}≤E[x1p](n-1)pδ≤K(p)(n-1)pδ,p>1.
In view of the well-known Borel-Cantelli lemma, we see that for almost all ω∈Ω,
(3.56)sup(n-1)δ≤t≤nδx1(t)t≤δ
holds for all but finitely many n. Hence there exists a n0(ω), for all ω∈Ω excluding a P-null set, for which (3.56) holds whenever n≥n0. Consequently, letting δ→0, we have, for almost all ω(3.57)limt→∞x1(t)t=0.
Similarly, we can obtain
(3.58)limt→∞x2(t)t=0.
Besides, by (3.39) and its ergodic property, we get
(3.59)limt→∞1t∫0tx1(s)dB1(s)=0,limt→∞1t∫0tx2(s)dB2(s)=0.
On the other hand, we have
(3.60)d(a21x1-a12x2)=[a21x1(r1-a11x1)-a12x2(r2-a22x2)]dt+a21σ1x1dB1(t)-a12σ2x2dB2(t).
Then
(3.61)a21x1(t)t-a12x2(t)t=a21x1(0)t-a12x2(0)t+a21r11t∫0tx1(s)ds-a12r21t∫0tx2(s)ds-a11a211t∫0tx12(s)ds+a12a221t∫0tx22(s)ds+a21σ11t∫0tx1(s)dB1(s)-a12σ21t∫0tx2(s)dB2(s).
Therefore, (3.25), (3.57), (3.58), and (3.59) imply
(3.62)limt→∞1t∫0t(a11a21x12(s)-a12a22x22(s))ds=a21r1x̃1*-a12r2x̃2*.At the end of this section, to conform the results above, we numerically simulate the solution of (1.4). By the method mentioned in [26], we consider the discretized equation: (3.63)x1,k+1=x1,k+x1,k[(r1-a11x1,k+a12x2,k)Δt+σ1ϵ1,kΔt+12σ12(ϵ1,k2Δt-Δt)],x2,k+1=x2,k+x2,k[(r2+a21x1,k-a22x2,k)Δt+σ2ϵ1,kΔt+12σ12(ϵ1,k2Δt-Δt)],
given the values of (x1,0,x2,0) and parameters in the system, by Matlab software we get Figure 1.Figure 1
The solution of system (1.1) and system (1.4) with (x1(0),x2(0))=(1.2,3.5),a11=0.6,a12=0.4,a21=0.3,a22=0.5,σ1=0.02,σ2=0.02,andΔt=0.001 such that a11a22>a12a21. The imaginary lines represent the solution of system (1.1), and the real lines represent the solution of system (1.4).Figure1 gives the solutions of (1.1) and (1.4), and the real lines and the imaginary lines represent the deterministic and the stochastic, respectively. In this figure, we choose parameters such that the conditions said in theorems are satisfied. Hence, although there is no equilibrium of the stochastic system (1.4) as the deterministic system (1.1), but the solution of (1.4) is ergodicity. From the figure, we can see that the solution of system (1.4) is fluctuating around a constant.



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*Source: 684926-2010-07-21.xml*