﻿ 一类非自治随机互惠系统的渐近性态 Asymptotic Behavior of a Non-Autonomous Stochastic Mutualism System

Pure Mathematics
Vol. 09  No. 04 ( 2019 ), Article ID: 30964 , 13 pages
10.12677/PM.2019.94068

Asymptotic Behavior of a Non-Autonomous Stochastic Mutualism System

Ao Guo, Xiaoquan Ding*

School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang Henan

Received: May 31st, 2019; accepted: Jun. 10th, 2019; published: Jun. 26th, 2019

ABSTRACT

This paper is devoted to the asymptotic behavior of a non-autonomous stochastic mutualism system. Firstly, the existence and uniqueness of global positive solution to the system is established for any positive initial value. Then by using the comparison theorem for stochastic differential equations and Lyapunov functions, the sufficient conditions for the permanence, extinction, global attractivity, and existence of periodic solutions to the system are derived respectively. Finally, some numerical simulations are given to illustrate our theoretical results.

Keywords:Stochastic Mutualism System, Global Attractivity, Permanence, Extinction, Periodic Solution

1. 引言

2006年，Gravesa等人 [3] 提出以下互惠模型

$\left\{\begin{array}{l}\frac{\text{d}{x}_{1}\left(t\right)}{\text{d}t}={x}_{1}\left(t\right)\left[{r}_{1}-{b}_{1}{\text{e}}^{-{k}_{1}{x}_{2}\left(t\right)}-{c}_{1}{x}_{1}\left(t\right)\right],\hfill \\ \frac{\text{d}{x}_{2}\left(t\right)}{\text{d}t}={x}_{2}\left(t\right)\left[{r}_{2}-{b}_{2}{\text{e}}^{-{k}_{2}{x}_{1}\left(t\right)}-{c}_{2}{x}_{2}\left(t\right)\right],\hfill \end{array}$ (1.1)

$\left\{\begin{array}{l}\frac{\text{d}{x}_{1}\left(t\right)}{\text{d}t}={x}_{1}\left(t\right)\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){x}_{2}\left(t\right)}-{c}_{1}\left(t\right){x}_{1}\left(t\right)\right],\hfill \\ \frac{\text{d}{x}_{2}\left(t\right)}{\text{d}t}={x}_{2}\left(t\right)\left[{r}_{2}\left(t\right)-{b}_{2}\left(t\right){\text{e}}^{-{k}_{2}\left(t\right){x}_{1}\left(t\right)}-{c}_{2}\left(t\right){x}_{2}\left(t\right)\right],\hfill \end{array}$ (1.2)

$\left\{\begin{array}{c}\text{d}{x}_{1}\left(t\right)={x}_{1}\left(t\right)\left[{r}_{1}-{b}_{1}{\text{e}}^{-{k}_{1}{x}_{2}\left(t\right)}-{c}_{1}{x}_{1}\left(t\right)\right]\text{d}t+{\sigma }_{1}{x}_{1}\left(t\right)\text{d}{B}_{1}\left(t\right),\\ \text{d}{x}_{2}\left(t\right)={x}_{2}\left(t\right)\left[{r}_{2}-{b}_{2}{\text{e}}^{-{k}_{2}{x}_{1}\left(t\right)}-{c}_{2}{x}_{2}\left(t\right)\right]\text{d}t+{\sigma }_{2}{x}_{2}\left(t\right)\text{d}{B}_{2}\left(t\right),\end{array}$ (1.3)

$\left\{\begin{array}{l}\text{d}{x}_{1}\left(t\right)={x}_{1}\left(t\right)\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){x}_{2}\left(t\right)}-{c}_{1}\left(t\right){x}_{1}\left(t\right)\right]\text{d}t+{\sigma }_{1}\left(t\right){x}_{1}\left(t\right)\text{d}{B}_{1}\left(t\right),\\ \text{d}{x}_{2}\left(t\right)={x}_{2}\left(t\right)\left[{r}_{2}\left(t\right)-{b}_{2}\left(t\right){\text{e}}^{-{k}_{2}\left(t\right){x}_{1}\left(t\right)}-{c}_{2}\left(t\right){x}_{2}\left(t\right)\right]\text{d}t+{\sigma }_{2}\left(t\right){x}_{2}\left(t\right)\text{d}{B}_{2}\left(t\right),\end{array}$ (1.4)

1) ${R}_{+}^{2}=\left\{\left({x}_{1},{x}_{2}\right):{x}_{1}>0,{x}_{2}>0\right\}$

2) 对 $\left[0,\infty \right)$ 上的有界函数f，定义 $\stackrel{⌣}{f}=\underset{t\in \left[0,\infty \right)}{\mathrm{sup}}f\left(t\right),\stackrel{⌢}{f}=\underset{t\in \left[0,\infty \right)}{\mathrm{inf}}f\left(t\right)$

3) 给定 $\theta >0$，对 $\left[0,\theta \right]$ 上的可积函数g，定义 ${〈g〉}_{\theta }=\frac{1}{\theta }{\int }_{0}^{\theta }g\left(s\right)\text{d}s$

2. 正解的存在唯一性

${\tau }_{m}=\mathrm{inf}\left\{t\in \left[0,{\tau }_{e}\right):{x}_{1}\left(t\right)\notin \left(\frac{1}{m},m\right)或者{x}_{2}\left(t\right)\notin \left(\frac{1}{m},m\right)\right\}.$

$ℙ\left\{{\tau }_{\infty }\le T\right\}>\epsilon$ .

$ℙ\left\{{\tau }_{m}\le T\right\}\ge \epsilon$ . (2.1)

$V\left({x}_{1},{x}_{2}\right)=\left({x}_{1}-1-\mathrm{ln}{x}_{1}\right)+\left({x}_{2}-1-\mathrm{ln}{x}_{2}\right).$

$\text{d}V\left({x}_{1},{x}_{2}\right)=LV\left({x}_{1},{x}_{2}\right)+\left({x}_{1}-1\right){\sigma }_{1}\left(t\right)\text{d}{B}_{1}\left(t\right)+\left({x}_{2}-1\right){\sigma }_{2}\left(t\right)\text{d}{B}_{2}\left(t\right),$ (2.2)

$\begin{array}{c}LV\left({x}_{1},{x}_{2}\right)=\left({x}_{1}-1\right)\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){x}_{2}\left(t\right)}-{c}_{1}\left(t\right){x}_{1}\left(t\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left({x}_{2}-1\right)\left[{r}_{2}\left(t\right)-{b}_{2}\left(t\right){\text{e}}^{-{k}_{2}\left(t\right){x}_{1}\left(t\right)}-{c}_{2}\left(t\right){x}_{2}\left(t\right)\right]+\text{0}\text{.5}\left({\sigma }_{1}^{2}\left(t\right)+{\sigma }_{2}^{2}\left(t\right)\right)\\ \le {x}_{1}\left({\stackrel{⌣}{r}}_{1}+{\stackrel{⌣}{c}}_{1}-{\stackrel{⌢}{c}}_{1}{x}_{1}\right)+{x}_{2}\left({\stackrel{⌣}{r}}_{2}+{\stackrel{⌣}{c}}_{2}-{\stackrel{⌢}{c}}_{2}{x}_{2}\right)+{\stackrel{⌣}{b}}_{1}+{\stackrel{⌣}{b}}_{2}+0.5\left({\stackrel{⌣}{\sigma }}_{1}^{2}+{\stackrel{⌣}{\sigma }}_{2}^{2}\right)\\ =:G\left({x}_{1},{x}_{2}\right).\end{array}$ (2.3)

$\mathbb{E}V\left({x}_{1}\left({\tau }_{m}\wedge T\right),{x}_{2}\left({\tau }_{m}\wedge T\right)\right)=V\left({x}_{1}\left(0\right),{x}_{2}\left(0\right)\right)+\mathbb{E}{\int }_{0}^{{\tau }_{m}\wedge T}LV\text{d}t\le V\left({x}_{1}\left(0\right),{x}_{2}\left(0\right)\right)+MT.$ (2.4)

$m\ge {m}_{1}$，记 ${\Omega }_{m}=\left\{{\tau }_{m}\le T\right\}$。由 ${\tau }_{m}$ 的定义可知，对每个 $\omega \in {\Omega }_{m}$，存在 $x\left({\tau }_{m},\omega \right)$ 的某个分量 ${x}_{i}\left({\tau }_{m},\omega \right)\left(i=1或2\right)$ 等于m或者1/m。结合(2.1)，有

$\mathbb{E}V\left({x}_{1}\left({\tau }_{m}\wedge T\right),{x}_{2}\left({\tau }_{m}\wedge T\right)\right)\ge \mathbb{E}\left[{I}_{{\Omega }_{m}}V\left({x}_{1}\left({\tau }_{m}\wedge T\right),{x}_{2}\left({\tau }_{m}\wedge T\right)\right)\right]\ge \epsilon \left[m-1-\mathrm{ln}m\right]\wedge \left[\mathrm{ln}m-1+\frac{1}{m}\right],$ (2.5)

$V\left({x}_{1}\left(0\right),{x}_{2}\left(0\right)\right)+MT\ge \epsilon \left[m-1-\mathrm{ln}m\right]\wedge \left[\mathrm{ln}m-1+\frac{1}{m}\right].$

$m\to \infty$，有

$V\left({x}_{1}\left(0\right),{x}_{2}\left(0\right)\right)+MT\ge \infty ,$

3. 全局吸引性

$\mathbb{E}{|x\left(t\right)-x\left(s\right)|}^{\alpha }\le c{|t-s|}^{1+\beta },\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\le s,t<+\infty ,$

$P\left\{\omega :\underset{0<|t-s|

$\begin{array}{l}\text{d}\left({\text{e}}^{t}{x}_{1}^{p}\right)={\text{e}}^{t}{x}_{1}^{p}\left[1+p\left({r}_{1}\left(t\right)-{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){x}_{2}}-{c}_{1}\left(t\right){x}_{1}\right)+0.5p\left(p-1\right){\sigma }_{1}^{2}\left(t\right)\right]\text{d}t+p{\text{e}}^{t}{x}_{1}^{p}{\sigma }_{1}\text{d}{B}_{1}\left(t\right)\\ \text{}\le {\text{e}}^{t}{x}_{1}^{p}\left[1+p{\stackrel{⌣}{r}}_{1}+0.5p\left(p-1\right){\stackrel{⌣}{\sigma }}_{1}^{2}-p{\stackrel{⌢}{c}}_{1}{x}_{1}\right]\text{d}t+p{\text{e}}^{t}{x}_{1}^{p}{\sigma }_{1}\left(t\right)\text{d}{B}_{1}\left(t\right)\end{array}$ (3.1)

$H\left(x\right)={x}^{p}\left[1+p{\stackrel{⌣}{r}}_{1}+0.5p\left(p-1\right){\stackrel{⌣}{\sigma }}_{1}^{2}-p{\stackrel{⌢}{c}}_{1}x\right]$。易知 $H\left(x\right)$ 在上 $\left(0,\infty \right)$ 有正上界，记为 $K\left(p\right)$。对(3.1)两边从0到t积分，并取数学期望得

${\text{e}}^{t}\mathbb{E}{x}_{1}^{p}\left(t\right)-\mathbb{E}{x}_{1}^{p}\left(0\right)\le {\int }_{0}^{t}{\text{e}}^{s}H\left({x}_{1}\left(s\right)\right)\text{d}s\le K\left(p\right){\text{e}}^{t}.$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\mathbb{E}{x}_{1}^{p}\left(t\right)\le K\left(p\right).$

$\mathbb{E}{x}_{1}^{p}\left(t\right)\le \stackrel{˜}{K}\left(p,{x}_{1}\left(0\right)\right).$ (3.2)

${x}_{1}\left(s\right)={x}_{1}\left(0\right)+{\int }_{0}^{t}{f}_{1}\left(s\right)\text{d}s+{\int }_{0}^{t}{g}_{1}\left(s\right)\text{d}{B}_{1}\left(s\right),$ (3.3)

${f}_{1}\left(s\right)={x}_{1}\left(s\right)\left[{r}_{1}\left(s\right)-{b}_{1}\left(s\right){\text{e}}^{-{k}_{1}\left(s\right){x}_{2}\left(s\right)}-{c}_{1}\left(s\right){x}_{1}\left(s\right)\right],\text{\hspace{0.17em}}{g}_{1}\left(s\right)={\sigma }_{1}\left(s\right){x}_{1}\left(s\right).$

$\begin{array}{c}\mathbb{E}{|{f}_{1}\left(s\right)|}^{p}=\mathbb{E}\left({x}_{1}^{p}\left(s\right){|{r}_{1}\left(s\right)-{b}_{1}\left(s\right){\text{e}}^{-{k}_{1}\left(s\right){x}_{2}\left(s\right)}-{c}_{1}\left(s\right){x}_{1}\left(s\right)|}^{p}\right)\\ \le 0.5\mathbb{E}{x}_{1}^{2p}\left(s\right)+0.5\mathbb{E}{|{r}_{1}\left(s\right)-{b}_{1}\left(s\right){\text{e}}^{-{k}_{1}\left(s\right){x}_{2}\left(s\right)}-{c}_{1}\left(s\right){x}_{1}\left(s\right)|}^{2p}\\ \le 0.5\mathbb{E}{x}_{1}^{2p}\left(s\right)+{2}^{2p-2}{\left({\stackrel{⌣}{r}}_{1}+{\stackrel{⌣}{b}}_{1}\right)}^{2p}+{2}^{2p-2}{\stackrel{⌣}{c}}_{1}\mathbb{E}{x}_{1}^{2p}\left(s\right)\\ \le \left(0.5+{2}^{2p-2}{\stackrel{⌣}{c}}_{1}\right)\stackrel{˜}{K}\left(2p,{x}_{1}\left(0\right)\right)+{2}^{2p-2}{\left({\stackrel{⌣}{r}}_{1}+{\stackrel{⌣}{b}}_{1}\right)}^{2p},\end{array}$ (3.4)

$\mathbb{E}{|{g}_{1}\left(s\right)|}^{p}=\mathbb{E}\left({\sigma }_{1}^{p}\left(s\right){x}_{1}^{p}\left(s\right)\right)\le {\stackrel{⌣}{\sigma }}_{1}{}^{p}\mathbb{E}{x}_{1}^{p}\left(s\right)\le {\stackrel{⌣}{\sigma }}_{1}^{p}\stackrel{˜}{K}\left(p,{x}_{1}\left(0\right)\right).$ (3.5)

$\begin{array}{c}\mathbb{E}{|{\int }_{{t}_{1}}^{{t}_{2}}{g}_{1}\left(s\right)\text{d}{B}_{1}\left(s\right)|}^{p}\le {\left[\frac{p\left(p-1\right)}{2}\right]}^{\frac{p}{2}}{\left({t}_{2}-{t}_{1}\right)}^{\frac{p-2}{2}}{\int }_{{t}_{1}}^{{t}_{2}}\mathbb{E}{|{g}_{1}\left(s\right)|}^{p}\text{d}s\\ \le {\stackrel{⌣}{\sigma }}_{1}^{p}\stackrel{˜}{K}\left(p,{x}_{1}\left(0\right)\right){\left[\frac{p\left(p-1\right)}{2}\right]}^{\frac{p}{2}}{\left({t}_{2}-{t}_{1}\right)}^{\frac{p}{2}}.\end{array}$ (3.6)

$\frac{1}{p}+\frac{1}{q}=1,$

$\begin{array}{c}\mathbb{E}{\left({\int }_{{t}_{1}}^{{t}_{2}}|{f}_{1}\left(s\right)|\text{d}s\right)}^{p}\le \mathbb{E}{\left\{{\left({\int }_{{t}_{1}}^{{t}_{2}}{1}^{q}\text{d}s\right)}^{\frac{1}{q}}\cdot {\left({\int }_{{t}_{1}}^{{t}_{2}}{|{f}_{1}\left(s\right)|}^{p}\text{d}s\right)}^{\frac{1}{p}}\right\}}^{p}={\left({t}_{2}-{t}_{1}\right)}^{\frac{p}{q}}\cdot {\int }_{{t}_{1}}^{{t}_{2}}\mathbb{E}{|{f}_{1}\left(s\right)|}^{p}\text{d}s\\ \le \left[\left(0.5+{2}^{2p-2}{\stackrel{⌣}{c}}_{1}\right)\stackrel{˜}{K}\left(2p,{x}_{1}\left(0\right)\right)+{2}^{2p-2}{\left({\stackrel{⌣}{r}}_{1}+{\stackrel{⌣}{b}}_{1}\right)}^{2p}\right]{\left({t}_{2}-{t}_{1}\right)}^{p}.\end{array}$ (3.7)

$\begin{array}{l}\mathbb{E}{|{x}_{1}\left({t}_{2}\right)-{x}_{1}\left({t}_{1}\right)|}^{p}\\ \le {2}^{p-1}\mathbb{E}{\left({\int }_{{t}_{1}}^{{t}_{2}}|{f}_{1}\left(s\right)|\text{d}s\right)}^{p}+{2}^{p-1}\mathbb{E}{|{\int }_{{t}_{1}}^{{t}_{2}}{g}_{1}\left(s\right)\text{d}{B}_{1}\left(s\right)|}^{p}\\ \le {2}^{p-1}\left\{\left(0.5+{2}^{2p-2}{\stackrel{⌣}{c}}_{1}\right)\stackrel{˜}{K}\left(2p,{x}_{1}\left(0\right)\right)+{2}^{2p-2}{\left({\stackrel{⌣}{r}}_{1}+{\stackrel{⌣}{b}}_{1}\right)}^{2p}+{\stackrel{⌣}{\sigma }}_{1}^{p}\stackrel{˜}{K}\left(p,{x}_{1}\left(0\right)\right){\left[\frac{p\left(p-1\right)}{2}\right]}^{\frac{p}{2}}\right\}{\left({t}_{2}-{t}_{1}\right)}^{\frac{p}{2}}.\end{array}$

$\mu =\mathrm{min}\left\{\underset{t\ge 0}{\mathrm{inf}}\left[{c}_{1}\left(t\right)-{b}_{2}\left(t\right){k}_{2}\left(t\right)\right],\underset{t\ge 0}{\mathrm{inf}}\left[{c}_{2}\left(t\right)-{b}_{1}\left(t\right){k}_{1}\left(t\right)\right]\right\},$

$\mu >0$。设 $x\left(t\right)=\left({x}_{1}\left(t\right),{x}_{2}\left(t\right)\right)$$\stackrel{¯}{x}\left(t\right)=\left({\stackrel{¯}{x}}_{1}\left(t\right),{\stackrel{¯}{x}}_{2}\left(t\right)\right)$ 分别系统(1.4)的初值为 $x\left(0\right)$$\stackrel{¯}{x}\left(0\right)$ 的两个解，由Itô公式和Lagrange中值定理，可得

$\begin{array}{c}d\left(\mathrm{ln}{x}_{1}\left(t\right)-\mathrm{ln}{\stackrel{¯}{x}}_{1}\left(t\right)\right)=\left\{-{b}_{1}\left(t\right)\left[{\text{e}}^{-{k}_{1}\left(t\right){x}_{2}\left(t\right)}-{\text{e}}^{-{k}_{1}\left(t\right){\stackrel{¯}{x}}_{2}\left(t\right)}\right]-{c}_{1}\left(t\right)\left[{x}_{1}\left(t\right)-{\stackrel{¯}{x}}_{1}\left(t\right)\right]\right\}\text{d}t\\ =\left\{{b}_{1}\left(t\right){k}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){\xi }_{1}\left(t\right)}\left[{x}_{2}\left(t\right)-{\stackrel{¯}{x}}_{2}\left(t\right)\right]-{c}_{1}\left(t\right)\left[{x}_{1}\left(t\right)-{\stackrel{¯}{x}}_{1}\left(t\right)\right]\right\}\text{d}t,\end{array}$ (3.8)

$\begin{array}{c}d\left(\mathrm{ln}{x}_{2}\left(t\right)-\mathrm{ln}{\stackrel{¯}{x}}_{2}\left(t\right)\right)=\left\{-{b}_{2}\left(t\right)\left[{\text{e}}^{-{k}_{2}\left(t\right){x}_{1}\left(t\right)}-{\text{e}}^{-{k}_{2}\left(t\right){\stackrel{¯}{x}}_{1}\left(t\right)}\right]-{c}_{2}\left(t\right)\left[{x}_{2}\left(t\right)-{\stackrel{¯}{x}}_{2}\left(t\right)\right]\right\}\text{d}t\\ =\left\{{b}_{2}\left(t\right){k}_{2}\left(t\right){\text{e}}^{-{k}_{2}\left(t\right){\xi }_{2}\left(t\right)}\left[{x}_{1}\left(t\right)-{\stackrel{¯}{x}}_{1}\left(t\right)\right]-{c}_{2}\left(t\right)\left[{x}_{2}\left(t\right)-{\stackrel{¯}{x}}_{2}\left(t\right)\right]\right\}\text{d}t,\end{array}$ (3.9)

${V}_{2}\left(t\right)=|\mathrm{ln}{x}_{1}\left(t\right)-\mathrm{ln}{\stackrel{¯}{x}}_{1}\left(t\right)|+|\mathrm{ln}{x}_{2}\left(t\right)-\mathrm{ln}{\stackrel{¯}{x}}_{2}\left(t\right)|,$

$\begin{array}{l}{V}_{2}\left(t\right)-{V}_{2}\left(0\right)\\ ={\int }_{0}^{t}\mathrm{sgn}\left(\mathrm{ln}{x}_{1}\left(s\right)-\mathrm{ln}{\stackrel{¯}{x}}_{1}\left(s\right)\right)\text{d}\left(\mathrm{ln}{x}_{1}\left(s\right)-\mathrm{ln}{\stackrel{¯}{x}}_{1}\left(s\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{0}^{t}\mathrm{sgn}\left(\mathrm{ln}{x}_{2}\left(s\right)-\mathrm{ln}{\stackrel{¯}{x}}_{2}\left(s\right)\right)\text{d}\left(\mathrm{ln}{x}_{2}\left(s\right)-\mathrm{ln}{\stackrel{¯}{x}}_{2}\left(s\right)\right)\\ ={\int }_{0}^{t}\mathrm{sgn}\left({x}_{1}\left(s\right)-{\stackrel{¯}{x}}_{1}\left(s\right)\right)\left\{{b}_{1}\left(s\right){k}_{1}\left(s\right){\text{e}}^{-{k}_{1}\left(s\right){\xi }_{1}\left(s\right)}\left[{x}_{2}\left(s\right)-{\stackrel{¯}{x}}_{2}\left(s\right)\right]-{c}_{1}\left(s\right)\left[{x}_{1}\left(s\right)-{\stackrel{¯}{x}}_{1}\left(s\right)\right]\right\}\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{0}^{t}\mathrm{sgn}\left({x}_{2}\left(s\right)-{\stackrel{¯}{x}}_{2}\left(s\right)\right)\left\{{b}_{2}\left(s\right){k}_{2}\left(s\right){\text{e}}^{-{k}_{2}\left(s\right){\xi }_{2}\left(s\right)}\left[{x}_{1}\left(s\right)-{\stackrel{¯}{x}}_{1}\left(s\right)\right]-{c}_{2}\left(s\right)\left[{x}_{2}\left(s\right)-{\stackrel{¯}{x}}_{2}\left(s\right)\right]\right\}\text{d}s\end{array}$

$\begin{array}{l}\le {\int }_{0}^{t}\left[{b}_{1}\left(s\right){k}_{1}\left(s\right)|{x}_{2}\left(s\right)-{\stackrel{¯}{x}}_{2}\left(s\right)|-{c}_{1}\left(s\right)|{x}_{1}\left(s\right)-{\stackrel{¯}{x}}_{1}\left(s\right)|\right]\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }+{\int }_{0}^{t}\left[{b}_{2}\left(s\right){k}_{2}\left(s\right)|{x}_{1}\left(s\right)-{\stackrel{¯}{x}}_{1}\left(s\right)|-{c}_{2}\left(s\right)|{x}_{2}\left(s\right)-{\stackrel{¯}{x}}_{2}\left(s\right)|\right]\text{d}s\\ \le -{\int }_{0}^{t}\left\{\left[{c}_{1}\left(s\right)-{b}_{2}\left(s\right){k}_{2}\left(s\right)\right]|{x}_{1}\left(s\right)-{\stackrel{¯}{x}}_{1}\left(s\right)|+\left[{c}_{2}\left(s\right)-{b}_{1}\left(s\right){k}_{1}\left(s\right)\right]|{x}_{2}\left(t\right)-{\stackrel{¯}{x}}_{2}\left(t\right)|\right\}\text{d}s\\ \le -\mu {\int }_{0}^{t}\left[|{x}_{1}\left(s\right)-{\stackrel{¯}{x}}_{1}\left(s\right)|+|{x}_{2}\left(s\right)-{\stackrel{¯}{x}}_{2}\left(s\right)|\right]\text{d}s.\end{array}$

${V}_{2}\left(t\right)+\mu {\int }_{0}^{t}\left[|{x}_{1}\left(s\right)-{\stackrel{¯}{x}}_{1}\left(s\right)|+|{x}_{2}\left(s\right)-{\stackrel{¯}{x}}_{2}\left(s\right)|\right]\text{d}t\le {V}_{2}\left(0\right).$

$\underset{t\to \infty }{\mathrm{lim}}\left[|{x}_{1}\left(t\right)-{\stackrel{¯}{x}}_{1}\left(t\right)|+|{x}_{2}\left(t\right)-{\stackrel{¯}{x}}_{2}\left(t\right)|\right]=0.$

4. 持续性与灭绝性

$\text{d}u\left(t\right)=u\left(t\right)\left[r\left(t\right)-a\left(t\right)u\left(t\right)\right]\text{d}t+\sigma \left(t\right)u\left(t\right)\text{d}B\left(t\right),$ (4.1)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}ℙ\left\{u\left(t\right)\le \alpha \right\}\ge 1-\epsilon ,\text{}\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}ℙ\left\{u\left(t\right)\ge \beta \right\}\ge 1-\epsilon .$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{1}{t}{\int }_{0}^{t}u\left(s\right)\text{d}s>0.$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{{x}_{i}\left(t\right)\le \alpha \right\}\ge 1-\epsilon ,\text{}\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{{x}_{i}\left(t\right)\ge \beta \right\}\ge 1-\epsilon ,\text{}i=1,2.$

$\begin{array}{c}\text{d}{x}_{1}\left(t\right)={x}_{1}\left(t\right)\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){x}_{2}\left(t\right)}-{c}_{1}\left(t\right){x}_{1}\left(t\right)\right]\text{d}t+{\sigma }_{1}\left(t\right){x}_{1}\left(t\right)\text{d}{B}_{1}\left(t\right)\\ \le {x}_{1}\left(t\right)\left[{r}_{1}\left(t\right)-{c}_{1}\left(t\right){x}_{1}\left(t\right)\right]\text{d}t+{\sigma }_{1}\left(t\right){x}_{1}\left(t\right)\text{d}{B}_{1}\left(t\right),\end{array}$ (4.2)

$\begin{array}{c}\text{d}{x}_{1}\left(t\right)={x}_{1}\left(t\right)\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){x}_{2}\left(t\right)}-{c}_{1}\left(t\right){x}_{1}\left(t\right)\right]\text{d}t+{\sigma }_{1}\left(t\right){x}_{1}\left(t\right)\text{d}{B}_{1}\left(t\right)\\ \ge {x}_{1}\left(t\right)\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right)-{c}_{1}\left(t\right){x}_{1}\left(t\right)\right]\text{d}t+{\sigma }_{1}\left(t\right){x}_{1}\left(t\right)\text{d}{B}_{1}\left(t\right).\end{array}$ (4.3)

$\text{d}\Lambda \left(t\right)=\Lambda \left(t\right)\left[{r}_{1}\left(t\right)-{c}_{1}\left(t\right)\Lambda \left(t\right)\right]\text{d}t+{\sigma }_{1}\left(t\right)\Lambda \left(t\right)\text{d}{B}_{1}\left(t\right),$ (4.4)

$\text{d}\lambda \left(t\right)=\lambda \left(t\right)\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right)-{c}_{1}\left(t\right)\lambda \left(t\right)\right]\text{d}t+{\sigma }_{1}\left(t\right)\lambda \left(t\right)\text{d}{B}_{1}\left(t\right).$ (4.5)

$\Lambda \left(t\right)$$\lambda \left(t\right)$ 分别是方程(4.4)和(4.5)的初值为 $\Lambda \left(0\right)={x}_{1}\left(0\right)$$\lambda \left(0\right)={x}_{1}\left(0\right)$ 的解。根据随机微分方程的比较定理 [22] ，由(4.2)和(4.3)得

$\lambda \left(t\right)\le {x}_{1}\left(t\right)\le \Lambda \left(t\right),\text{\hspace{0.17em}}\text{a}\text{.s}\text{.}$ (4.6)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{\Lambda \left(t\right)\le {\alpha }_{1}\right\}\ge 1-\epsilon$$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{\lambda \left(t\right)\ge {\beta }_{1}\right\}\ge 1-\epsilon$ (4.7)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{{x}_{1}\left(t\right)\le {\alpha }_{1}\right\}\ge 1-\epsilon$$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{{x}_{1}\left(t\right)\ge {\beta }_{1}\right\}\ge 1-\epsilon$ (4.8)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{{x}_{2}\left(t\right)\le {\alpha }_{2}\right\}\ge 1-\epsilon$$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{{x}_{2}\left(t\right)\ge {\beta }_{2}\right\}\ge 1-\epsilon$ (4.9)

$\alpha =\mathrm{max}\left\{{\alpha }_{1},{\alpha }_{2}\right\}$$\beta =\mathrm{min}\left\{{\beta }_{1},{\beta }_{2}\right\}$，由(4.8)和(4.9)可知，对任意 $i=1,2$，有

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{{x}_{i}\left(t\right)\le \alpha \right\}\ge 1-\epsilon ,\text{\hspace{0.17em}}\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}P\left\{{x}_{i}\left(t\right)\ge \beta \right\}\ge 1-\epsilon .$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{1}{t}{\int }_{0}^{t}{x}_{i}\left(t\right)\text{d}t>0,\text{a}\text{.s}\text{.,}\text{\hspace{0.17em}}i=1,2.$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{1}{t}{\int }_{0}^{t}\left[{r}_{1}\left(s\right)-{b}_{1}\left(s\right)-0.5{\sigma }_{1}^{2}\left(s\right)\right]\text{d}s>0$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{1}{t}{\int }_{0}^{t}\lambda \left(t\right)\text{d}t>0.$ (4.10)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{1}{t}{\int }_{0}^{t}{x}_{1}\left(s\right)\text{d}s\ge \underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{1}{t}{\int }_{0}^{t}\lambda \left(t\right)\text{d}t>0.$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{1}{t}{\int }_{0}^{t}{x}_{2}\left(s\right)\text{d}s>0.$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\mathrm{ln}{x}_{i}\left(t\right)}{t}<0,\text{}i=1,2.$

$\text{d}\mathrm{ln}{x}_{1}\left(t\right)=\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){x}_{2}}-{c}_{1}\left(t\right){x}_{1}-0.5{\sigma }_{1}^{2}\left(t\right)\right]\text{d}t+{\sigma }_{1}\left(t\right)\text{d}{B}_{1}\left(t\right).$ (4.11)

$\mathrm{ln}{x}_{1}\left(t\right)\le \mathrm{ln}{x}_{1}\left(0\right)+{\int }_{0}^{t}\left[{r}_{1}\left(s\right)-0.5{\sigma }_{1}^{2}\left(s\right)\right]\text{d}s+M\left(t\right),$ (4.12)

${〈M,M〉}_{t}={\int }_{0}^{t}{\sigma }_{1}^{2}\left(s\right)\text{d}s\le {\stackrel{⌣}{\sigma }}_{1}^{2}t.$

$\underset{t\to \infty }{\mathrm{lim}}\frac{M\left(t\right)}{t}=0,\text{\hspace{0.17em}}\text{a}\text{.s}.$ (4.13)

$\underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{\mathrm{ln}{x}_{1}\left(t\right)}{t}\le \underset{t\to \infty }{\mathrm{lim}\mathrm{sup}}\frac{1}{t}{\int }_{0}^{t}\left[{r}_{1}\left(s\right)-\frac{1}{2}{\sigma }_{1}^{2}\left(s\right)\right]\text{d}s<0,\text{\hspace{0.17em}}\text{a}\text{.s}.$

5. 周期解

$\text{d}x\left(t\right)=f\left(t,x\left(t\right)\right)\text{d}t+g\left(t,x\left(t\right)\right)\text{d}{B}_{t},$ (5.1)

$LV\left(t,x\right)={V}_{t}\left(t,x\right)+{V}_{x}\left(t,x\right)f\left(t,x\right)+0.5\text{trace}\left[{g}^{\text{T}}\left(t,x\right){V}_{xx}\left(t,x\right)g\left(t,x\right)\right].$

1) $\underset{k\to \infty }{\mathrm{lim}}\underset{|x|>k}{\mathrm{inf}}V\left(t,x\right)\to \infty$

2) $\underset{k\to \infty }{\mathrm{lim}}\underset{|x|>k}{\mathrm{sup}}LV\left(t,x\right)=-\infty$

${u}_{1}\left(t\right)=\mathrm{ln}{x}_{1}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}{u}_{2}\left(t\right)=\mathrm{ln}{x}_{2}\left(t\right).$

$\left\{\begin{array}{l}\text{d}{u}_{1}\left(t\right)=\left[{r}_{1}\left(t\right)-0.5{\sigma }_{1}^{2}\left(t\right)-{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){\text{e}}^{{u}_{2}\left(t\right)}}-{c}_{1}\left(t\right){\text{e}}^{{u}_{1}\left(t\right)}\right]\text{d}t+{\sigma }_{1}\left(t\right)\text{d}{B}_{1}\left(t\right),\\ \text{d}{u}_{2}\left(t\right)=\left[{r}_{2}\left(t\right)-0.5{\sigma }_{2}^{2}\left(t\right)-{b}_{2}\left(t\right){\text{e}}^{-{k}_{2}\left(t\right){\text{e}}^{{u}_{1}\left(t\right)}}-{c}_{2}\left(t\right){\text{e}}^{{u}_{2}\left(t\right)}\right]\text{d}t+{\sigma }_{2}\left(t\right)\text{d}{B}_{2}\left(t\right).\end{array}$ (5.2)

${V}_{3}\left(t,{u}_{1},{u}_{2}\right):\left[0,+\infty \right)×{R}^{2}\to {R}_{+}$ , ${V}_{3}\left(t,{u}_{1},{u}_{2}\right)=\underset{i=1}{\overset{2}{\sum }}\left\{{\text{e}}^{{u}_{i}}+{\text{e}}^{q\left[{w}_{i}\left(t\right)-{u}_{i}\right]}\right\}$ ,

${{w}^{\prime }}_{i}\left(t\right)={r}_{i}\left(t\right)-{b}_{i}\left(t\right)-0.5{\sigma }_{i}^{2}\left(t\right)-{〈{r}_{i}-{b}_{i}-0.5{\sigma }_{i}^{2}〉}_{\theta }$ (5.3)

${〈{r}_{i}-{b}_{i}-0.5{\sigma }_{i}^{2}〉}_{\theta }-0.5q{\stackrel{⌣}{\sigma }}_{i}^{2}>0$ (5.4)

${w}_{i}\left(t+\theta \right)-{w}_{i}\left(t\right)={\int }_{t}^{t+\theta }\left[{r}_{i}\left(s\right)-{b}_{i}\left(s\right)-0.5{\sigma }_{i}^{2}\left(s\right)-{〈{r}_{i}-{b}_{i}-0.5{\sigma }_{i}^{2}〉}_{\theta }\right]\text{d}s=0.$

$\underset{k\to \infty }{\mathrm{lim}}\underset{|u|>k}{\mathrm{inf}}{V}_{3}\left(t,{u}_{1},{u}_{2}\right)\to \infty ,$

${V}_{3}$ 满足引理5.1的条件(1)。由(5.3)和(5.4)得

$\begin{array}{c}L\left[{\text{e}}^{{u}_{1}}+{\text{e}}^{q\left[{w}_{1}\left(t\right)-{u}_{1}\right]}\right]={\text{e}}^{{u}_{1}}\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){e}^{{u}_{2}}}-{c}_{1}\left(t\right){\text{e}}^{{u}_{1}}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+q{e}^{q\left[{w}_{1}\left(t\right)-{u}_{1}\right]}\left[{{w}^{\prime }}_{1}{}_{1}\left(t\right)-{r}_{1}\left(t\right)+0.5{\sigma }_{1}^{2}\left(t\right)+{b}_{1}\left(t\right){\text{e}}^{-{k}_{1}\left(t\right){\text{e}}^{{u}_{2}}}+{c}_{1}\left(t\right){\text{e}}^{{u}_{1}}+0.5q{\sigma }_{1}^{2}\left(t\right)\right]\\ \le {\text{e}}^{{u}_{1}}\left[{\stackrel{⌣}{r}}_{1}-{\stackrel{⌢}{c}}_{1}{\text{e}}^{{u}_{1}}\right]+q{\text{e}}^{q\left[{w}_{1}\left(t\right)-{u}_{1}\right]}\left[{{w}^{\prime }}_{1}\left(t\right)-{r}_{1}\left(t\right)+0.5{\sigma }_{1}^{2}\left(t\right)+{b}_{1}\left(t\right)+{\stackrel{⌣}{c}}_{1}{\text{e}}^{{u}_{1}}+0.5q{\stackrel{⌣}{\sigma }}_{1}^{2}\right]\\ ={\text{e}}^{{u}_{1}}\left[{\stackrel{⌣}{r}}_{1}-{\stackrel{⌢}{c}}_{1}{\text{e}}^{{u}_{1}}\right]+q{\text{e}}^{q\left[{w}_{1}\left(t\right)-{u}_{1}\right]}\left[{\stackrel{⌣}{c}}_{1}{\text{e}}^{{u}_{1}}-{〈{r}_{1}-{b}_{1}-0.5{\sigma }_{1}^{2}〉}_{\theta }+0.5q{\stackrel{⌣}{\sigma }}_{1}^{2}\right]\\ \le {\text{e}}^{{u}_{1}}\left[{\stackrel{⌣}{r}}_{1}+q{\stackrel{⌣}{c}}_{1}{\text{e}}^{q\left[{\stackrel{⌣}{w}}_{1}-{u}_{1}\right]}-{\stackrel{⌢}{c}}_{1}{\text{e}}^{{u}_{1}}\right]-q\left[{〈{r}_{1}-{b}_{1}-0.5{\sigma }_{1}^{2}〉}_{\theta }-0.5q{\stackrel{⌣}{\sigma }}_{1}^{2}\right]{\text{e}}^{q\left[{\stackrel{⌢}{w}}_{1}-u\right]}.\end{array}$

$L\left[{\text{e}}^{{u}_{2}}+{\text{e}}^{q\left[{w}_{2}\left(t\right)-{u}_{2}\right]}\right]\le {\text{e}}^{{u}_{2}}\left[{\stackrel{⌣}{r}}_{2}+q{\stackrel{⌣}{c}}_{2}{\text{e}}^{q\left[{\stackrel{⌣}{w}}_{2}-{u}_{1}\right]}-{\stackrel{⌢}{c}}_{2}{\text{e}}^{{u}_{2}}\right]-q\left[{〈{r}_{2}-{b}_{2}-0.5{\sigma }_{2}^{2}〉}_{\theta }-0.5q{\stackrel{⌣}{\sigma }}_{2}^{2}\right]{\text{e}}^{q\left[{\stackrel{⌢}{w}}_{2}-{u}_{2}\right]}.$

$L{V}_{3}\le \underset{i=1}{\overset{2}{\sum }}\left\{{\text{e}}^{{u}_{i}}\left[{\stackrel{⌣}{r}}_{i}+q{\stackrel{⌣}{c}}_{i}{\text{e}}^{q\left[{\stackrel{⌣}{w}}_{i}-{u}_{i}\right]}-{\stackrel{⌢}{c}}_{i}{\text{e}}^{{u}_{i}}\right]-q\left[{〈{r}_{i}-{b}_{i}-0.5{\sigma }_{i}^{2}〉}_{\theta }-0.5q{\stackrel{⌣}{\sigma }}_{i}^{2}\right]{\text{e}}^{q\left[{\stackrel{⌢}{w}}_{i}-{u}_{i}\right]}\right\}.$ (5.5)

$\underset{k\to \infty }{\mathrm{lim}}\underset{|u|>k}{\mathrm{inf}}L{V}_{3}\left(t,{u}_{1},{u}_{2}\right)\to -\infty$ ,

${V}_{3}$ 满足引理5.1的条件(2)。于是，系统(5.2)存在一个 $\theta$ -周期解。从而，系统(1.4)存在一个正的 $\theta$ -周期解。证毕。

6. 数值模拟

$\mathrm{min}\left\{\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right)-0.5{\sigma }_{1}^{2}\left(t\right)\right],\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\left[{r}_{2}\left(t\right)-{b}_{2}\left(t\right)-0.5{\sigma }_{2}^{2}\left(t\right)\right]\right\}>0,$

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\left[{r}_{1}\left(t\right)-{b}_{1}\left(t\right)-0.5{\sigma }_{1}^{2}\left(t\right)\right]>0,\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\left[{r}_{2}\left(t\right)-{b}_{2}\left(t\right)-0.5{\sigma }_{2}^{2}\left(t\right)\right]<0,$

$\begin{array}{l}\mathrm{min}\left\{{〈{r}_{1}-{b}_{1}-0.5{\sigma }_{1}^{2}〉}_{\text{2π}},{〈{r}_{2}-{b}_{2}-0.5{\sigma }_{2}^{2}〉}_{\text{2π}}\right\}>0,\\ \mathrm{min}\left\{\underset{t\in \left[0,2\text{π}\right]}{\mathrm{min}}\left[{c}_{1}\left(t\right)-{b}_{2}\left(t\right){k}_{2}\left(t\right)\right],\underset{t\in \left[0,2\text{π}\right]}{\mathrm{min}}\left[{c}_{2}\left(t\right)-{b}_{1}\left(t\right){k}_{1}\left(t\right)\right]\right\}>0,\end{array}$

Figure 1. A solution of example 6.1 with initial value 

Figure 2. A solution of example 6.2 with initial value $x\left(0\right)={\left(3,5\right)}^{\text{T}}$

Figure 3. Solutions of example 6.3 and its determined counterpart with initial value $x\left(0\right)={\left(3,5\right)}^{\text{T}}$

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27. NOTES

*通讯作者。