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

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.

1. Introduction

It is well known that the differential equation𝑥̇₁(𝑡)=𝑥₁(𝑡)[𝑟₁−𝑎₁₁𝑥₁(𝑡)+𝑎₁₂𝑥₂𝑥̇(𝑡)],₂(𝑡)=𝑥₂(𝑡)[𝑟₂+𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡)] (1.1) 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 𝑟_𝑖,𝑎_𝑖𝑗,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 𝑟_𝑖>0,𝑎_𝑖𝑗>0,𝑖,𝑗=1,2, and 𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁>0, then 𝑥₁(𝑡)→𝑥^∗₁,𝑥₂(𝑡)→𝑥^∗₂,as𝑡→∞, (1.2) where 𝑥^∗=(𝑥^∗₁,𝑥^∗₂) is the unique positive equilibrium of system (1.1) and 𝑥^∗₁=𝑟₁𝑎₂₂+𝑟₂𝑎₁₂𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁>0,𝑥^∗₂=𝑟₂𝑎₁₁+𝑟₁𝑎₂₁𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁>0. (1.3) 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 𝑇_𝑐 (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 𝑟_𝑖,𝑖=1,2. Suppose 𝑟_𝑖→𝑟_𝑖+𝜎_𝑖𝐵̇_𝑖(𝑡), where 𝐵_𝑖(𝑡),𝑖=1,2 are mutually independent one dimensional standard Brownian motions with 𝐵_𝑖(0)=0, and 𝜎_𝑖>0,𝑖=1,2 are the intensities of white noises. The stochastic version corresponding to the deterministic system (1.1) takes the following form: 𝑑𝑥₁(𝑡)=𝑥₁(𝑡)[(𝑟₁−𝑎₁₁𝑥₁(𝑡)+𝑎₁₂𝑥₂(𝑡))𝑑𝑡+𝜎₁𝑑𝐵₁(𝑡)],𝑑𝑥₂(𝑡)=𝑥₂(𝑡)[(𝑟₂+𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡))𝑑𝑡+𝜎₂𝑑𝐵₂(𝑡)]. (1.4)

This paper is organized as follows. In Section 2, 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,{ℱ_𝑡}_𝑡≥0,𝑃) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ₀ contains all 𝑃-null sets). Let 𝑅²₊ denote the positive cone of 𝑅², namely 𝑅²₊={𝑥∈𝑅²∶𝑥_𝑖>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 solution 𝑥(𝑡) of system (1.4) for any given initial value 𝑥(0)=𝑥₀∈𝑅²₊ provided 𝑎₁₁𝑎₂₂>𝑎₁₂𝑎₂₁.

Proof. The proof is similar to Theorem 2.1 in [17]. Here we define a 𝐶²-function 𝑉∶𝑅²₊→𝑅₊: 𝑉(𝑥₁,𝑥₂)=𝑎₂₁[𝑥₁−𝑥^∗₁−𝑥^∗₁𝑥log¹𝑥^∗₁]+𝑎₁₂[𝑥₂−𝑥^∗₂−𝑥^∗₂𝑥log²𝑥^∗₂], (2.1) where 𝑥^∗=(𝑥^∗₁,𝑥^∗₂)^⊤ satisfies 𝑟₁−𝑎₁₁𝑥^∗₁+𝑎₁₂𝑥^∗₂𝑟=0,₂+𝑎₂₁𝑥^∗₁−𝑎₂₂𝑥^∗₂=0. (2.2)

Remark 2.2. Theorem 2.1 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:𝑑𝑁(𝑡)=𝑁(𝑡)[(𝑎(𝑡)−𝑏(𝑡)𝑁(𝑡))𝑑𝑡+𝛼(𝑡)𝑑𝐵(𝑡)], (2.3) 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 that 𝑎(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 𝑒𝑁(𝑡)=^{∫𝑡0[𝑎(𝑠)−𝛼2(𝑠)/2]𝑑𝑠+𝛼(𝑠)𝑑𝐵(𝑠)}1/𝑁₀+∫^𝑡₀𝑏(𝑠)𝑒^{∫𝑠0[𝑎(𝜏)−𝛼2(𝜏)/2]𝑑𝜏+𝛼(𝜏)𝑑𝐵(𝜏)}𝑑𝑠,𝑡≥0. (2.4)

Theorem 2.4. Assume that 𝑎₁₁𝑎₂₂>𝑎₁₂𝑎₂₁ and 𝑥(𝑡)∈𝑅²₊ is the solution of system (1.4) with initial value 𝑥₀∈𝑅²₊. Then 𝑥(𝑡) has the property that 𝑥₁(𝑡)≥𝜙₁(𝑡),𝑥₂(𝑡)≥𝜙₂(𝑡), (2.5) where 𝜙₁(𝑡) and 𝜙₂(𝑡) are the solutions of equations: 𝑑𝜙₁(𝑡)=𝜙₁(𝑡)[(𝑟₁−𝑎₁₁𝜙₁(𝑡))𝑑𝑡+𝜎₁𝑑𝐵₁(𝑡)],𝜙₁(0)=𝑥₁(0),𝑑𝜙₂(𝑡)=𝜙₂(𝑡)[(𝑟₂−𝑎₂₂𝜙₂(𝑡))𝑑𝑡+𝜎₂𝑑𝐵₂(𝑡)],𝜙₂(0)=𝑥₂(0). (2.6)(2.7)

The result of Theorem 2.4 follows directly from the classical comparison theorem of stochastic differential equations (see [23]).

Remark 2.5. From Lemma 2.3, we see 𝜙₁𝑒(𝑡)=^{(𝑟1−𝜎21/2)𝑡+𝜎1𝐵1(𝑡)}1/𝑥₁(0)+𝑎₁₁∫^𝑡₀𝑒^{(𝑟1−𝜎21/2)𝑠+𝜎1𝐵1(𝑠)}𝑑𝑠,𝜙₂𝑒(𝑡)=^{(𝑟2−𝜎22/2)𝑡+𝜎2𝐵2(𝑡)}1/𝑥₂(0)+𝑎₂₂∫^𝑡₀𝑒^{(𝑟2−𝜎22/2)𝑠+𝜎2𝐵2(𝑠)}.𝑑𝑠 (2.8) This together with Theorem 2.4 shows that if 𝑟_𝑖>(𝜎²_𝑖/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 if 𝑟_𝑖>0,𝑎_𝑖𝑗>0,𝑖,𝑗=1,2, and 𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁>0, then the unique positive equilibrium (𝑥^∗₁,𝑥^∗₂) 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 domain 𝑈⊂𝐸_𝑙 with regular boundary Γ, having the following properties.(B.1)In the domain 𝑈 and some neighbourhood thereof, the smallest eigenvalue of the diffusion matrix 𝐴(𝑥) is bounded away from zero.(B.2)If 𝑥∈𝐸_𝑙⧵𝑈, the mean time 𝜏 at which a path issuing from 𝑥 reaches the set 𝑈 is finite, and sup_𝑥∈𝐾𝐸_𝑥𝜏<∞ for every compact subset 𝐾⊂𝐸_𝑙.

Lemma 3.1 (see [25]). If (B) holds, then the Markov process 𝑋(𝑡) has a stationary distribution 𝜇(𝐴). Let 𝑓(⋅) be a function integrable with respect to the measure 𝜇. Then 𝑃_𝑥{lim_𝑇→∞1𝑇^𝑇₀𝑓(𝑋(𝑡))𝑑𝑡=_𝐸𝑙𝑓(𝑥)𝜇(𝑑𝑥)}=1 (3.1) for all 𝑥∈𝐸_𝑙.

Theorem 3.2. Assume 𝑎₁₁𝑎₂₂>𝑎₁₂𝑎₂₁, 𝜎₁>0, 𝜎₂>0 and 𝛿<min{𝑚₁(𝑥^∗₁)²,𝑚₂(𝑥^∗₂)²}. Then there is a stationary distribution 𝜇(𝐴) for system (1.4) and it has ergodic property. Here (𝑥^∗₁,𝑥^∗₂) is the solution of (2.2), 𝛿=(𝑎₂₁𝑥^∗₁𝜎²₁+𝑎₁₂𝑥^∗₂𝜎²₂)/2,𝑚₁=𝑎₁₁𝑎₂₁−𝑎₁₂𝑎₂₁/𝜖₀>0 and 𝑚₂=𝑎₁₂𝑎₂₂−𝜖₀𝑎₁₂𝑎₂₁>0, where 𝜖₀>0 satisfies 𝑎₁₂/𝑎₁₁<𝜖₀<𝑎₂₂/𝑎₂₁.

Proof. Define 𝑉∶𝐸_𝑙=𝑅²₊→𝑅₊: 𝑉(𝑥₁,𝑥₂)=𝑎₂₁(𝑥₁−𝑥^∗₁−𝑥^∗₁𝑥log¹𝑥^∗₁)+𝑎₁₂(𝑥₂−𝑥^∗₂−𝑥^∗₂𝑥log²𝑥^∗₂). (3.2) Then 𝑑𝑉=𝑎₂₁𝑥(1−^∗1𝑥₁)𝑑𝑥₁+𝑎₂₁2𝑥^∗₁𝑥²₁(𝑑𝑥₁)²+𝑎₁₂𝑥(1−^∗2𝑥₂)𝑑𝑥₂+𝑎₁₂2𝑥^∗₂𝑥²₂(𝑑𝑥₂)²=𝑎₂₁(𝑥₁−𝑥^∗₁)[(𝑟₁−𝑎₁₁𝑥₁+𝑎₁₂𝑥₂)𝑑𝑡+𝜎₁𝑑𝐵₁1(𝑡)]+2𝑎₂₁𝑥^∗₁𝜎²₁𝑑𝑡+𝑎₁₂(𝑥₂−𝑥^∗₂)[(𝑟₂+𝑎₂₁𝑥₁−𝑎₂₂𝑥₂)𝑑𝑡+𝜎₂𝑑𝐵₂1(𝑡)]+2𝑎₁₂𝑥^∗₂𝜎²₂𝑑𝑡∶=𝐿𝑉𝑑𝑡+𝑎₂₁𝜎₁(𝑥₁−𝑥^∗₁)𝑑𝐵₁(𝑡)+𝑎₁₂𝜎₂(𝑥₂−𝑥^∗₂)𝑑𝐵₂(𝑡), (3.3) where 𝐿𝑉=𝑎₂₁(𝑥₁−𝑥^∗₁)(𝑟₁−𝑎₁₁𝑥₁+𝑎₁₂𝑥₂1)+2𝑎₂₁𝑥^∗₁𝜎²₁+𝑎₁₂(𝑥₂−𝑥^∗₂)(𝑟₂+𝑎₂₁𝑥₁−𝑎₂₂𝑥₂1)+2𝑎₁₂𝑥^∗₂𝜎²₂=𝑎₂₁(𝑥₁−𝑥^∗₁)[−𝑎₁₁(𝑥₁−𝑥^∗₁)+𝑎₁₂(𝑥₂−𝑥^∗₂)]+𝑎₁₂(𝑥₂−𝑥^∗₂)[𝑎₂₁(𝑥₁−𝑥^∗₁)−𝑎₂₂(𝑥₂−𝑥^∗₂)]+𝛿=−𝑎₁₁𝑎₂₁(𝑥₁−𝑥^∗₁)²+2𝑎₁₂𝑎₂₁(𝑥₁−𝑥^∗₁)(𝑥₂−𝑥^∗₂)−𝑎₁₂𝑎₂₂(𝑥₂−𝑥^∗₂)²+𝛿, (3.4) according to the equality (2.2). By Young inequality, we have 2𝑎₁₂𝑎₂₁|𝑥₁−𝑥^∗₁||𝑥₂−𝑥^∗₂|≤𝑎₁₂𝑎₂₁[(𝑥₁−𝑥^∗₁)²𝜖₀+𝜖₀(𝑥₂−𝑥^∗₂)²]. (3.5) Then 𝐿𝑉≤−(𝑎₁₁𝑎₂₁−𝑎₁₂𝑎₂₁𝜖₀)(𝑥₁−𝑥^∗₁)²−(𝑎₁₂𝑎₂₂−𝜖₀𝑎₁₂𝑎₂₁)(𝑥₂−𝑥^∗₂)²+𝛿=−𝑚₁(𝑥₁−𝑥^∗₁)²−𝑚₂(𝑥₂−𝑥^∗₂)²+𝛿. (3.6) Note that 𝛿<min{𝑚₁(𝑥^∗₁)²,𝑚₂(𝑥^∗₂)²}; then the ellipsoid −𝑚₁(𝑥₁−𝑥^∗₁)²−𝑚₂(𝑥₂−𝑥^∗₂)²+𝛿=0 (3.7) lies entirely in 𝑅²₊. We can take 𝑈 to be any neighborhood of the ellipsoid with 𝑈⊆𝐸_𝑙=𝑅²₊, so for 𝑥∈𝑈⧵𝐸_𝑙, 𝐿𝑉≤0, which implies that condition (B.2) in Lemma 3.1 is satisfied. Besides, there is 𝑀>0 such that ^𝑛_𝑖,𝑗=1(^𝑛_𝑘=1𝑔_𝑖𝑘(𝑥)𝑔_𝑗𝑘(𝑥))𝜉_𝑖𝜉_𝑗=𝜎²₁𝑥²₁𝜉²₁+𝜎²₂𝑥²₂𝜉²₂≥𝑀|𝜉²|𝑥∈𝑈,𝜉∈𝑅², (3.8) 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. If 𝑎₁₁𝑎₂₂>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 equation𝑁^⋅(𝑡)=𝑁(𝑡)[𝑎−𝑏𝑁(𝑡)] (3.9) with initial value 𝑁₀>0. It is well known, when 𝑎,𝑏>0, (3.9) has a unique positive solution 𝑒𝑁(𝑡)=^𝑎𝑡1/𝑁₀+(𝑏/𝑎)(𝑒^𝑎𝑡−1),𝑡≥0,lim_𝑡→∞𝑎𝑁(𝑡)=𝑏,lim_𝑡→∞log𝑁(𝑡)𝑡=0. (3.10)(3.11)

Lemma 3.4. Suppose that 𝑟₁>𝜎²₁/2 and 𝜙₁(𝑡) is the solution of (2.6), then one has 𝜓₁(𝑡)𝑒^{−𝜎1(max0≤𝑠≤𝑡𝐵1(𝑠)−𝐵1(𝑡))}≤𝜙₁(𝑡)≤𝜓₁(𝑡)𝑒^{−𝜎1(min0≤𝑠≤𝑡𝐵1(𝑠)−𝐵1(𝑡))}, (3.12) where 𝜓₁(𝑡) is the solution of 𝜓̇₁(𝑡)=𝜓₁(𝑡)[𝑟₁−𝜎²₁2−𝑎₁₁𝜓₁𝜓(𝑡)],₁(0)=𝑥₁(0). (3.13)

Proof. From the representation of the solution 𝜙₁(𝑡), we have 1𝜙₁=1(𝑡)𝑥₁𝑒(0)^{−(𝑟1−𝜎21/2)𝑡−𝜎1𝐵1(𝑡)}+𝑎₁₁^𝑡₀𝑒^{−(𝑟1−𝜎21/2)(𝑡−𝑠)−𝜎(𝐵1(𝑡)−𝐵1(𝑠))}𝑑𝑠=𝑒^{−𝜎1𝐵1(𝑡)}[1𝑥₁𝑒(0)^{−(𝑟1−𝜎21/2)𝑡}+𝑎₁₁^𝑡₀𝑒^{−(𝑟1−𝜎21/2)(𝑡−𝑠)}𝑒^{𝜎1𝐵1(𝑠)}𝑑𝑠]≤𝑒^{−𝜎1𝐵1(𝑡)}[1𝑥₁𝑒(0)^{−(𝑟1−𝜎2/2)𝑡}+𝑎₁₁𝑒^{𝜎1max0≤𝑠≤𝑡𝐵1(𝑠)𝑡}₀𝑒^{−(𝑟1−𝜎21/2)(𝑡−𝑠)}𝑑𝑠]≤𝑒^{𝜎1[max0≤𝑠≤𝑡𝐵1(𝑠)−𝐵1(𝑡)]}[1𝑥₁𝑒(0)^{−(𝑟1−𝜎21/2)𝑡}+𝑎₁₁^𝑡₀𝑒^{−(𝑟1−𝜎21/2)(𝑡−𝑠)}𝑑𝑠], (3.14) where the last inequality is based on the property of Brownian motion that 𝐵(0)=0. Similarly, we have 1𝜙₁(𝑡)≥𝑒^{𝜎1(min0≤𝑠≤𝑡𝐵1(𝑠)−𝐵1(𝑡))}[1𝑥₁𝑒(0)^{−(𝑟1−𝜎21/2)𝑡}+𝑎₁₁^𝑡₀𝑒^{−(𝑟1−𝜎21/2)(𝑡−𝑠)}𝑑𝑠]. (3.15) Therefore 𝑒^{𝜎1(min0≤𝑠≤𝑡𝐵1(𝑠)−𝐵1(𝑡))}1𝜓₁≤1(𝑡)𝜙₁(𝑡)≤𝑒^{𝜎1(max0≤𝑠≤𝑡𝐵1(𝑠)−𝐵1(𝑡))}1𝜓₁,(𝑡) (3.16) which is as required.

Lemma 3.5. Suppose that 𝑟₁>𝜎²₁/2 and 𝜙₁(𝑡) is the solution of (2.6), then one has lim_𝑡→∞log𝜙₁(𝑡)𝑡=0. (3.17)

Proof. It is easy to drive from Lemma 3.4 that 𝜎₁(𝐵₁(𝑡)−max_{0≤𝑠≤𝑡}𝐵₁(𝑠))≤log𝜙₁(𝑡)−log𝜓₁(𝑡)≤𝜎₁(𝐵₁(𝑡)−min_{0≤𝑠≤𝑡}𝐵₁(𝑠)). (3.18) Note that the distribution of max_{0≤𝑠≤𝑡}𝐵₁(𝑠) is the same as |𝐵₁(𝑡)|, and that min_{0≤𝑠≤𝑡}𝐵₁(𝑠) has the same distribution as—max_{0≤𝑠≤𝑡}𝐵₁(𝑠), then by (3.11) and the strong law of large numbers, we get lim_𝑡→∞log𝜙₁(𝑡)𝑡=lim_𝑡→∞log𝜓₁(𝑡)𝑡=0. (3.19) 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 that 𝑟₂>𝜎²₂/2 and 𝜙₂(𝑡) is the solution of (2.7), then one has lim_𝑡→∞log𝜙₂(𝑡)𝑡=0. (3.20)

Lemma 3.7. Let 𝑀(𝑡)=∫^𝑡₀𝑒^𝑠𝑑𝐵(𝑠), where 𝐵(𝑡) is 1-dimensional standard Brownian motion, then limsup_𝑡→∞|𝑒^−𝑡𝑀(𝑡)|log𝑡=1a.s. (3.21)

Proof. The proof can be found on [21, page 70].

Based on these lemmas, now we show the main result in this section.

Let 𝑦₁(𝑡)=log𝑥₁(𝑡), 𝑦₂(𝑡)=log𝑥₂(𝑡); then by Itô's formula we obtain𝑑𝑦₁(𝑡)=(𝑟₁−𝜎²₁2−𝑎₁₁𝑥₁(𝑡)+𝑎₁₂𝑥₂(𝑡))𝑑𝑡+𝜎₁𝑑𝐵₁(𝑡),𝑑𝑦₂(𝑡)=(𝑟₂−𝜎²₂2+𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡))𝑑𝑡+𝜎₂𝑑𝐵₂(𝑡). (3.22) If 𝑎₁₁𝑎₂₂>𝑎₁₂𝑎₂₁ and 2𝑟₁>𝜎²₁,2𝑟₂>𝜎²₂, then the equation 𝑟₁−𝜎²₁2−𝑎₁₁𝑥₁+𝑎₁₂𝑥₂𝑟=0,₂−𝜎²₂2+𝑎₂₁𝑥₁−𝑎₂₂𝑥₂=0 (3.23) has a unique positive solution:𝑥^∗₁=𝑎₂₂(𝑟₁−𝜎²₁/2)+𝑎₁₂(𝑟₂−𝜎²₂/2)𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁,𝑥^∗₂=𝑎₂₁(𝑟₁−𝜎²₁/2)+𝑎₁₁(𝑟₂−𝜎²₂/2)𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁. (3.24)

Lemma 3.8. Assume 𝑎₁₁𝑎₂₂>𝑎₁₂𝑎₂₁ and 2𝑟₁>𝜎²₁,2𝑟₂>𝜎²₂. Then for any initial value 𝑥₀∈𝑅²₊, the solution 𝑥(𝑡) of system (1.4) has the following property: lim_𝑡→∞𝑥₁(𝑡)=𝑥^∗₁,lim_𝑡→∞𝑥₂(𝑡)=𝑥^∗₂, (3.25) where 𝑥_𝑖(𝑡)=(1/𝑡)∫^𝑡₀𝑥_𝑖(𝑠)𝑑𝑠,𝑖=1,2.

Proof. It follows from (3.22) that 𝑦₁(𝑡)𝑡=𝑦₁(0)𝑡+𝑟₁−𝜎²₁2−𝑎₁₁𝑥₁(𝑡)+𝑎₁₂𝑥₂(𝑡)+𝜎₁𝐵₁(𝑡)𝑡,𝑦₂(𝑡)𝑡=𝑦₂(0)𝑡+𝑟₂−𝜎²₂2+𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡)+𝜎₂𝐵₂(𝑡)𝑡. (3.26) Obviously, to prove the result, it is an easy consequence of lim_𝑡→∞𝑦_𝑖(𝑡)𝑡=lim_𝑡→∞log𝑥_𝑖(𝑡)𝑡=0,𝑖=1,2a.s. (3.27) We first show that liminf_𝑡→∞log𝑥_𝑖(𝑡)𝑡≥0,𝑖=1,2a.s. (3.28) In fact, the results of Theorem 2.4 and Lemmas 3.5 and 3.6 imply that (3.28) is true.
Next, we will provelimsup_𝑡→∞log𝑥_𝑖(𝑡)𝑡≤0,𝑖=1,2a.s. (3.29) If 𝑎₁₁𝑎₂₂>𝑎₁₂𝑎₂₁, then there exist positive constants 𝑐₁,𝑐₂,𝑚₁,𝑚₂ such that −𝑎₁₁𝑐₁+𝑎₂₁𝑐₂=−𝑚₁,𝑎₁₂𝑐₁−𝑎₂₂𝑐₂=−𝑚₂. (3.30) From (3.22) we get 𝑑(𝑐₁𝑦₁(𝑡)+𝑐₂𝑦₂(𝑡))=𝑐₁𝑑log𝑥₁(𝑡)+𝑐₂𝑑log𝑥₂(𝑡)=𝑐₁[(𝑟₁−𝜎²₁2−𝑎₁₁𝑥₁(𝑡)+𝑎₁₂𝑥₂(𝑡))𝑑𝑡+𝜎₁𝑑𝐵₁(𝑡)]+𝑐₂[(𝑟₂−𝜎²₂2+𝑎₂₁𝑥₁(𝑡)−𝑎₂₂𝑥₂(𝑡))𝑑𝑡+𝜎₂𝑑𝐵₂(𝑡)]=[(𝑟₁−𝜎²₁2)𝑐₁+(𝑟₂−𝜎²₂2)𝑐₂+(𝑎₂₁𝑐₂−𝑎₁₁𝑐₁)𝑥₁(𝑡)+(𝑎₁₂𝑐₁−𝑎₂₂𝑐₂)𝑥₂(𝑡)]𝑑𝑡+𝑐₁𝜎₁𝑑𝐵₁(𝑡)+𝑐₂𝜎₂𝑑𝐵₂(𝑡)=[(𝑟₁−𝜎²₁2)𝑐₁+(𝑟₂−𝜎²₂2)𝑐₂−𝑚₁𝑥₁(𝑡)−𝑚₂𝑥₂(𝑡)]𝑑𝑡+𝑐₁𝜎₁𝑑𝐵₁(𝑡)+𝑐₂𝜎₂𝑑𝐵₂(𝑡),𝑑[𝑒^𝑡(𝑐₁𝑦₁(𝑡)+𝑐₂𝑦₂(𝑡))]=𝑒^𝑡𝑑(𝑐₁𝑦₁(𝑡)+𝑐₂𝑦₂(𝑡))+𝑒^𝑡(𝑐₁𝑦₁(𝑡)+𝑐₂𝑦₂(𝑡))𝑑𝑡=𝑒^𝑡[(𝑟₁−𝜎²₁2)𝑐₁+(𝑟₂−𝜎²₂2)𝑐₂+𝑐₁𝑦₁(𝑡)−𝑚₁𝑒^𝑦1(𝑡)+𝑐₂𝑦₂(𝑡)−𝑚₂𝑒^𝑦2(𝑡)]𝑑𝑡+𝑒^𝑡[𝑐₁𝜎₁𝑑𝐵₁(𝑡)+𝑐₂𝜎₂𝑑𝐵₂(𝑡)]. (3.31) Note that the function 𝑐₁𝑦₁−𝑚₁𝑒^𝑦1 has its maximum value 𝑐^∗₁=𝑐₁log(𝑐₁/𝑚₁)−𝑐₁ at 𝑦₁=log(𝑐₁/𝑚₁), and the function 𝑐₂𝑦₂−𝑚₂𝑒^𝑦2 has its maximum value 𝑐^∗₂=𝑐₂log(𝑐₂/𝑚₂)−𝑐₂ at 𝑦₂=log(𝑐₂/𝑚₂); then 𝑑[𝑒^𝑡(𝑐₁𝑦₁(𝑡)+𝑐₂𝑦₂(𝑡))]≤𝑒^𝑡[(𝑟₁−𝜎²₁2)𝑐₁+(𝑟₂−𝜎²₂2)𝑐₂+𝑐^∗₁+𝑐^∗₂]𝑑𝑡+𝑒^𝑡[𝑐₁𝜎₁𝑑𝐵₁(𝑡)+𝑐₂𝜎₂𝑑𝐵₂(𝑡)]∶=𝑐^∗𝑒^𝑡𝑑𝑡+𝑒^𝑡[𝑐₁𝜎₁𝑑𝐵₁(𝑡)+𝑐₂𝜎₂𝑑𝐵₂(𝑡)], (3.32) where 𝑐^∗=(𝑟₁−𝜎²₁/2)𝑐₁+(𝑟₂−𝜎²₂/2)𝑐₂+𝑐^∗₁+𝑐^∗₂. Integrating both sides of it from 0 to 𝑡, yields 𝑒^𝑡(𝑐₁𝑦₁(𝑡)+𝑐₂𝑦₂(𝑡))≤𝑐₁𝑦₁(0)+𝑐₂𝑦₂(0)+𝑐^∗(𝑒^𝑡−1)+^𝑡₀𝑒^𝑠[𝑐₁𝜎₁𝑑𝐵₁(𝑠)+𝑐₂𝜎₂𝑑𝐵₂𝑐(𝑠)],₁𝑦₁(𝑡)+𝑐₂𝑦₂(𝑡)≤𝑐^∗+(𝑐₁𝑦₁(0)+𝑐₂𝑦₂(0)−𝑐^∗)𝑒^−𝑡+𝑒^−𝑡𝑡₀𝑒^𝑠[𝑐₁𝜎₁𝑑𝐵₁(𝑠)+𝑐₂𝜎₂𝑑𝐵₂(𝑠)]≤𝑐^∗+𝑐₁𝑦₁(0)+𝑐₂𝑦₂(0)+𝑒^−𝑡𝑡₀𝑒^𝑠[𝑐₁𝜎₁𝑑𝐵₁(𝑠)+𝑐₂𝜎₂𝑑𝐵₂(𝑠)]∶=𝑐+𝑒^−𝑡𝑡₀𝑒^𝑠[𝑐₁𝜎₁𝑑𝐵₁(𝑠)+𝑐₂𝜎₂𝑑𝐵₂(𝑠)], (3.33) where 𝑐=𝑐^∗+𝑐₁𝑦₁(0)+𝑐₂𝑦₂(0). It is easy to drive from Lemma 3.7 that 𝑐₁𝑦₁(𝑡)+𝑐₂𝑦₂(𝑡)≤𝑐+𝑂_a.s.(log𝑡), (3.34) which implies 𝑐₁limsup_𝑡→∞𝑦₁(𝑡)𝑡+𝑐₂limsup_𝑡→∞𝑦₂(𝑡)𝑡≤0,a.s. (3.35) Moreover (3.35) together with (3.28) shows that 𝑐₁limsup_𝑡→∞𝑦₁(𝑡)𝑡≤−𝑐₂limsup_𝑡→∞𝑦₂(𝑡)𝑡≤−𝑐₂liminf_𝑡→∞𝑦₂(𝑡)𝑡≤0,a.s. (3.36) that is, limsup_𝑡→∞𝑦₁(𝑡)𝑡≤0,a.s. (3.37) Similarly, we have limsup_𝑡→∞𝑦₂(𝑡)𝑡≤0,a.s. (3.38) which is as required.

Lemma 3.9. Assume 𝑎₁₁>𝑎₁₂ and 𝑎₂₂>𝑎₂₁. Then for any initial value 𝑥₀∈𝑅²₊, there exists a positive constant 𝐾(𝑝) such that the solution 𝑥(𝑡) of system (1.4) has the following property: 𝐸[𝑐₁𝑥^𝑝₁(𝑡)+𝑐₂𝑥^𝑝₂(𝑡)]≤𝐾(𝑝),∀𝑡∈[0,∞],𝑝>1. (3.39)

Proof. By Itô's formula and Young inequality, we compute 1𝑑(𝑝𝑥^𝑝₁)=𝑥₁^𝑝−1𝑑𝑥₁+𝑝−12𝑥₁^𝑝−2(𝑑𝑥₁)²=[(𝑟₁+𝑝−12𝜎²₁)𝑥^𝑝₁−𝑎₁₁𝑥₁^𝑝+1+𝑎₁₂𝑥^𝑝₁𝑥₂]𝑑𝑡+𝜎₁𝑥^𝑝₁𝑑𝐵₁(𝑡)≤[(𝑟₁+𝑝−12𝜎²₁)𝑥^𝑝₁−𝑎₁₁𝑥₁^𝑝+1+𝑎₁₂𝑝𝑥𝑝+1^1𝑝+1+𝑎₁₂𝑥𝑝+1^2𝑝+1]𝑑𝑡+𝜎₁𝑥^𝑝₁𝑑𝐵₁1(𝑡),𝑑(𝑝𝑥^𝑝₂)=𝑥₂^𝑝−1𝑑𝑥₂+𝑝−12𝑥₂^𝑝−2(𝑑𝑥₂)²=[(𝑟₂+𝑝−12𝜎²₂)𝑥^𝑝₂+𝑎₂₁𝑥₁𝑥^𝑝₂−𝑎₂₂𝑥₂^𝑝+1]𝑑𝑡+𝜎₂𝑥^𝑝₂𝑑𝐵₂(𝑡)≤[(𝑟₂+𝑝−12𝜎²₂)𝑥^𝑝₂+𝑎₂₁𝑥𝑝+1^1𝑝+1+𝑎₂₁𝑝𝑥𝑝+1^2𝑝+1−𝑎₂₂𝑥₂^𝑝+1]𝑑𝑡+𝜎₂𝑥^𝑝₂𝑑𝐵₂(𝑡). (3.40) Hence, for positive constants 𝑐₁ and 𝑐₂, we have 𝑐₁1𝑑(𝑝𝑥^𝑝₁)+𝑐₂1𝑑(𝑝𝑥^𝑝₂)≤[𝑐₁(𝑟₁+𝑝−12𝜎²₁)𝑥^𝑝₁−(𝑐₁𝑎₁₁−𝑐₁𝑎₁₂𝑝−𝑐𝑝+1²𝑎₂₁𝑝+1)𝑥₁^𝑝+1+𝑐₂(𝑟₂+𝑝−12𝜎²₂)𝑥^𝑝₂−(𝑐₂𝑎₂₂−𝑐₁𝑎₁₂−𝑐𝑝+1²𝑎₂₁𝑝𝑝+1)𝑥₂^𝑝+1]𝑑𝑡+𝑐₁𝜎₁𝑥^𝑝₁𝑑𝐵₁(𝑡)+𝑐₂𝜎₂𝑥^𝑝₂𝑑𝐵₂(𝑡). (3.41) Next, we claim that there are 𝑐₁>0,𝑐₂>0 such that 𝑐₁𝑎₁₁−𝑐₁𝑎₁₂𝑝−𝑐𝑝+1²𝑎₂₁𝑝+1>0,𝑐₂𝑎₂₂−𝑐₁𝑎₁₂−𝑐𝑝+1²𝑎₂₁𝑝𝑝+1>0, (3.42) 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.
Let 𝛼₁∶=𝑝(𝑟₁+((𝑝−1)/2)𝜎²₁),𝛼₂∶=𝑝(𝑟₂+((𝑝−1)/2)𝜎²₂), 𝛽₁∶=𝑐₁^{−(𝑝+1)/𝑝}𝑝(𝑐₁𝑎₁₁−𝑐₁𝑎₁₂𝑝/(𝑝+1)−𝑐₂𝑎₂₁/(𝑝+1)),𝛽₂∶=𝑐₂^{−(𝑝+1)/𝑝}𝑝(𝑐₂𝑎₂₂−𝑐₁𝑎₁₂/(𝑝+1)−𝑐₂𝑎₂₁𝑝/(𝑝+1)). Then 𝛼₁>0,𝛼₂>0,𝛽₁>0,𝛽₂>0 and𝑑(𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂)≤[𝑐₁𝛼₁𝑥^𝑝₁+𝑐₂𝛼₂𝑥^𝑝₂−𝑐₁^{(𝑝+1)/𝑝}𝛽₁𝑥₁^𝑝+1−𝑐₂^{(𝑝+1)/𝑝}𝛽₂𝑥₂^𝑝+1+𝑐]𝑑𝑡₁𝑝𝜎₁𝑥^𝑝₁𝑑𝐵₁𝑐(𝑡)+²𝑝𝜎₂𝑥^𝑝₂𝑑𝐵₂(𝑡)≤[max{𝛼₁,𝛼₂}(𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂)−min{𝛽₁,𝛽₂}(𝑐₁^{(𝑝+1)/𝑝}𝑥₁^𝑝+1+𝑐₂^{(𝑝+1)/𝑝}𝑥₂^𝑝+1+𝑐)]𝑑𝑡₁𝑝𝜎₁𝑥^𝑝₁𝑑𝐵₁𝑐(𝑡)+²𝑝𝜎₂𝑥^𝑝₂𝑑𝐵₂(𝑡). (3.43) Hence, 𝑑𝐸[𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂]≤{max{𝛼₁,𝛼₂}𝐸[𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂]−min{𝛽₁,𝛽₂}𝐸[𝑐₁^{(𝑝+1)/𝑝}𝑥₁^𝑝+1+𝑐₂^{(𝑝+1)/𝑝}𝑥₂^𝑝+1]}𝑑𝑡≤{max{𝛼₁,𝛼₂}𝐸[𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂]−2^−𝑝min{𝛽₁,𝛽₂}𝐸[𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂]^{(𝑝+1)/𝑝}}𝑑𝑡. (3.44) By comparison theorem, we can get limsup_𝑡→∞𝐸[𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂2]≤[^𝑝max{𝛼₁,𝛼₂}min{𝛽₁,𝛽₂}]^𝑝∶=𝐶(𝑝), (3.45) which implies that there is a 𝑇>0, such that 𝐸[𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂]≤2𝐶(𝑝),∀𝑡>𝑇. (3.46) Besides, note that 𝐸[𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂] is continuous, then there is a 𝐶(𝑝)>0 such that 𝐸[𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂]≤𝐶(𝑝),∀𝑡∈[0,𝑇]. (3.47) Let 𝐾(𝑝)=max{2𝐶(𝑝),𝐶(𝑝)}, then 𝐸[𝑐₁𝑥^𝑝₁+𝑐₂𝑥^𝑝₂]≤𝐾(𝑝),∀𝑡∈[0,∞]. (3.48)

By the ergodic property, for 𝑚>0, we havelim_𝑡→∞1𝑡^𝑡₀(𝑥^𝑝_𝑖(𝑠)∧𝑚)𝑑𝑠=_𝑅2+(𝑧^𝑝_𝑖∧𝑚)𝜇(𝑑𝑧₁,𝑑𝑧₂),a.s. (3.49) On the other hand, by dominated convergence theorem, we can get 𝐸[lim_𝑡→∞1𝑡^𝑡₀(𝑥^𝑝_𝑖(𝑠)∧𝑚)𝑑𝑠]=lim_𝑡→∞1𝑡^𝑡₀𝐸[𝑥^𝑝_𝑖(𝑠)∧𝑚]𝑑𝑠≤𝐾(𝑝),𝑖=1,2, (3.50) which together with (3.49) implies _𝑅2+(𝑧^𝑝_𝑖∧𝑚)𝜇(𝑑𝑧₁,𝑑𝑧₂)≤𝐾(𝑝),𝑖=1,2. (3.51) Letting 𝑚→∞, we get _𝑅2+𝑧^𝑝_𝑖𝜇(𝑑𝑧₁,𝑑𝑧₂)≤𝐾(𝑝),𝑖=1,2. (3.52) That is to say, functions 𝑓₁(𝑥)=𝑥^𝑝₁ and 𝑓₂(𝑥)=𝑥^𝑝₂ are integrable with respect to the measure 𝜇. Therefore one has the following.

Theorem 3.10. Assume 𝑎₁₁>𝑎₁₂, 𝑎₂₂>𝑎₂₁, 2𝑟₁>𝜎²₁>0,2𝑟₂>𝜎²₂>0 and 𝛿<min{𝑚₁(𝑥^∗₁)²,𝑚₂(𝑥^∗₂)²}, where 𝛿=(𝑎₂₁𝑥^∗₁𝜎²₁+𝑎₁₂𝑥^∗₂𝜎²₂)/2, 𝑚₁=𝑎₁₁𝑎₁₂(𝑎₂₂−𝑎₂₁)/𝑎₁₁𝑎₂₂+𝑎₁₂𝑎₂₁ and 𝑚₂=𝑎₁₂(𝑎₁₁𝑎₂₂−𝑎₁₂𝑎₂₁)/2𝑎₁₁. Then for any initial value 𝑥₀∈𝑅²₊, the solution 𝑥(𝑡) of system (1.4) has the following property: 𝑃{lim_𝑡→∞𝑥₁(𝑡)=_𝑅2+𝑧₁𝜇(𝑑𝑧₁,𝑑𝑧₂)=𝑥^∗₁}=1,𝑃{lim_𝑡→∞𝑥₂(𝑡)=_𝑅2+𝑧₂𝜇(𝑑𝑧₁,𝑑𝑧₂)=𝑥^∗₂}=1. (3.53)

Moreover, we can get the following.

Theorem 3.11. Assume 𝑎₁₁>𝑎₁₂, 𝑎₂₂>𝑎₂₁ and 2𝑟₁>𝜎²₁,2𝑟₂>𝜎²₂. Then for any initial value 𝑥₀∈𝑅²₊, the solution 𝑥(𝑡) has following property lim_𝑡→∞1𝑡^𝑡₀(𝑎₁₁𝑎₂₁𝑥²₁(𝑠)−𝑎₁₂𝑎₂₂𝑥²₂(𝑠))𝑑𝑠=𝑎₂₁𝑟₁𝑥^∗₁−𝑎₁₂𝑟₂𝑥^∗₂, (3.54) where (𝑥^∗₁,𝑥^∗₂) is defined as in (3.24).

Proof. By (3.39), for 𝛿>0, we have 𝑃{𝜔∶sup_{(𝑛−1)𝛿≤𝑡≤𝑛𝛿}𝑥₁(𝑡)𝑡>𝛿}≤𝐸[𝑥^𝑝₁](𝑛−1)^𝑝𝛿≤𝐾(𝑝)(𝑛−1)^𝑝𝛿,𝑝>1. (3.55) In view of the well-known Borel-Cantelli lemma, we see that for almost all 𝜔∈Ω, sup_{(𝑛−1)𝛿≤𝑡≤𝑛𝛿}𝑥₁(𝑡)𝑡≤𝛿 (3.56) 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 𝜔lim_𝑡→∞𝑥₁(𝑡)𝑡=0. (3.57) Similarly, we can obtain lim_𝑡→∞𝑥₂(𝑡)𝑡=0. (3.58) Besides, by (3.39) and its ergodic property, we get lim_𝑡→∞1𝑡^𝑡₀𝑥₁(𝑠)𝑑𝐵₁(𝑠)=0,lim_𝑡→∞1𝑡^𝑡₀𝑥₂(𝑠)𝑑𝐵₂(𝑠)=0. (3.59) On the other hand, we have 𝑑(𝑎₂₁𝑥₁−𝑎₁₂𝑥₂)=[𝑎₂₁𝑥₁(𝑟₁−𝑎₁₁𝑥₁)−𝑎₁₂𝑥₂(𝑟₂−𝑎₂₂𝑥₂)]𝑑𝑡+𝑎₂₁𝜎₁𝑥₁𝑑𝐵₁(𝑡)−𝑎₁₂𝜎₂𝑥₂𝑑𝐵₂(𝑡). (3.60) Then 𝑎₂₁𝑥₁(𝑡)𝑡−𝑎₁₂𝑥₂(𝑡)𝑡=𝑎₂₁𝑥₁(0)𝑡−𝑎₁₂𝑥₂(0)𝑡+𝑎₂₁𝑟₁1𝑡^𝑡₀𝑥₁(𝑠)𝑑𝑠−𝑎₁₂𝑟₂1𝑡^𝑡₀𝑥₂(𝑠)𝑑𝑠−𝑎₁₁𝑎₂₁1𝑡^𝑡₀𝑥²₁(𝑠)𝑑𝑠+𝑎₁₂𝑎₂₂1𝑡^𝑡₀𝑥²₂(𝑠)𝑑𝑠+𝑎₂₁𝜎₁1𝑡^𝑡₀𝑥₁(𝑠)𝑑𝐵₁(𝑠)−𝑎₁₂𝜎₂1𝑡^𝑡₀𝑥₂(𝑠)𝑑𝐵₂(𝑠). (3.61) Therefore, (3.25), (3.57), (3.58), and (3.59) imply lim_𝑡→∞1𝑡^𝑡₀(𝑎₁₁𝑎₂₁𝑥²₁(𝑠)−𝑎₁₂𝑎₂₂𝑥²₂(𝑠))𝑑𝑠=𝑎₂₁𝑟₁𝑥^∗₁−𝑎₁₂𝑟₂𝑥^∗₂. (3.62)

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: 𝑥_1,𝑘+1=𝑥_1,𝑘+𝑥_1,𝑘[(𝑟₁−𝑎₁₁𝑥_1,𝑘+𝑎₁₂𝑥_2,𝑘)Δ𝑡+𝜎₁𝜖_1,𝑘√1Δ𝑡+2𝜎²₁(𝜖²_1,𝑘𝑥Δ𝑡−Δ𝑡)],_2,𝑘+1=𝑥_2,𝑘+𝑥_2,𝑘[(𝑟₂+𝑎₂₁𝑥_1,𝑘−𝑎₂₂𝑥_2,𝑘)Δ𝑡+𝜎₂𝜖_1,𝑘√1Δ𝑡+2𝜎²₁(𝜖²_1,𝑘Δ𝑡−Δ𝑡)], (3.63) given the values of (𝑥_1,0,𝑥_2,0) 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 $(x_1(0),x_2(0))=(\mathrm{1.2,3.5}),\mathrm{\hspace{0.17em}\hspace{0.17em}}{a}_{11}=0.6,\mathrm{\hspace{0.17em}\hspace{0.17em}}{a}_{12}=0.4,\mathrm{\hspace{0.17em}\hspace{0.17em}}{a}_{21}=0.3,\mathrm{\hspace{0.17em}\hspace{0.17em}}{a}_{22}=0.5,\mathrm{\hspace{0.17em}\hspace{0.17em}}{\sigma }_1=0.02,\mathrm{\hspace{0.17em}\hspace{0.17em}}{\sigma }_2=0.02,\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{and}\mathrm{\hspace{0.17em}\hspace{0.17em}}\Delta t=0.001$ such that $a_{11}{a}_{22}>a_{12}{a}_{21}$. The imaginary lines represent the solution of system (1.1), and the real lines represent the solution of system (1.4).

Figure 1 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.

Acknowledgments

The work was supported by the Ministry of Education of China (no. 109051), the Ph.D. Programs Foundation of Ministry of China (no. 200918), Young Teachers of Northeast Normal University (no. 20090104), and the Fundamental Research Funds for the Central Universities (no. 09SSXT117), the Fundamental Research Funds for the Central Universities.