Processing math: 5%

数学物理学报, 2018, 38(5): 984-1000 doi:

论文

具有Holling Ⅲ功能性反应的随机捕食食饵模型的平稳分布和周期解

蓝桂杰, 付盈洁, 魏春金, 张树文,

Stationary Distribution and Periodic Solution for Stochastic Predator-Prey Systems with Holling-Type Ⅲ Functional Response

Lan Guijie, Fu Yingjie, Wei Chunjin, Zhang Shuwen,

通讯作者: 张树文, E-mail: zhangsw_123@126.com

收稿日期: 2017-07-25  

基金资助: 福建省自然科学基金.  2016J05012
福建省自然科学基金.  2016J01667

Received: 2017-07-25  

Fund supported: the Fujian Provincial Natural Science Foundation.  2016J05012
the Fujian Provincial Natural Science Foundation.  2016J01667

摘要

该文研究了一类具有Holling Ⅲ功能性反应的随机捕食-食饵系统的动力学行为.对于自治系统,首先获得,对于任意的正初始值,系统都存在唯一的全局正解;第二,利用随机微分方程比较定理,得到系统的平均持续生存与灭绝的充分条件;第三,通过构造Lyapunov函数,证明了系统存在唯一的平稳分布且具有遍历性;而对于非自治系统,通过应用Has'minskii定理证明了,系统至少存在一个非平凡的正周期解;最后,给出数值模拟来验证主要结果

关键词: 捕食-食饵系统 ; 随机干扰 ; 平稳分布 ; 周期解

Abstract

In this paper, we investigate the dynamics of stochastic predator-prey systems with Holling-type Ⅲ functional response. For the autonomous system, we firstly obtain that the system admits unique positive global solution starting from the positive initial value. Then, by comparison theorem for stochastic differential equation, sufficient conditions for extinction and persistence in mean are obtained. Thirdly, by constructing some suitable Lyapunov function, we prove that there are unique stationary distribution and they are ergodic. On the other hand, for the non-autonomous periodic system, we prove that there exists at least one nontrivial positive periodic solution according to the theory of Has'minskii. Finally, some numerical simulations are introduced to illustrate our theoretical result.

Keywords: Predator-prey system ; Random perturbation ; Stationary distribution and ergodicity ; Periodic solution

PDF (1105KB) 元数据 多维度评价 相关文章 导出 EndNote| Ris| Bibtex  收藏本文

本文引用格式

蓝桂杰, 付盈洁, 魏春金, 张树文. 具有Holling Ⅲ功能性反应的随机捕食食饵模型的平稳分布和周期解. 数学物理学报[J], 2018, 38(5): 984-1000 doi:

Lan Guijie, Fu Yingjie, Wei Chunjin, Zhang Shuwen. Stationary Distribution and Periodic Solution for Stochastic Predator-Prey Systems with Holling-Type Ⅲ Functional Response. Acta Mathematica Scientia[J], 2018, 38(5): 984-1000 doi:

1 引言

近年来,许多学者对捕食食饵系统进行了深入研究,获得丰富的研究成果.而实际的生态系统中,各种随机干扰无处不在[1-5],所以随机捕食-食饵模型已经受到国内外数学家与生物学家的极大关注[5-14].本文考虑下列具有随机干扰的捕食-食饵模型

{dx=x(r1a1xb1xy1+x2)dt+σ1xdB1(t),dy=y(r2b2yk2+x)dt+σ2ydB2(t),
(1.1)

其中x(t), y(t)分别表示食饵和捕食者在t时刻的种群密度; r1>0, r2>0分别表示食饵和捕食者的内禀增长率; a1>0表示食饵种群的密度制约系数; b1x21+x2是Holling Ⅲ功能反应函数且b1>0; b2yk2+x表示具有Leslies形式的捕食者数量反应,系数b2, k2均为正常数; σ2i(i=1,2)表示白噪声强度; B1(t),B2(t)是互相独立的标准布朗运动,定义在带有滤子Ft并且满足通常条件(即{Ft}t0是右连续单调递增,且F0包含所有零测集)的完备概率空间(Ω,F,{Ft}t0,P)上.

另一方面,在生态系统中周期现象也是普遍存在的,例如白昼黑夜的变化,四季的更替,食物的供应,个体生命周期等其他原因,使得种群的出生率、死亡率和其他参数不会永久保持不变,而会表现出或多或少的周期性[4, 8].所以,研究周期因素影响下的生态系统的动力学行为显得尤为重要.所以相应系统(1.1)的非自治系统如下

{dx=x(r1(t)a1(t)xb1(t)xy1+x2)dt+σ1(t)xdB1(t),dy=y(r2(t)b2(t)yk2(t)+x)dt+σ2(t)ydB2(t),
(1.2)

其中r1(t), r2(t), a1(t), b1(t), b2(t), k2(t), σ1(t), σ2(t)均为正的、有界的、连续的正ω周期函数.

2 预备知识

下面给出一些基本定义、引理及定理.为方便起见给出以下记号

(1)  Rn+={(a1,a2,,an)Rn|ai>0,i=1,2,,n}.

(2)  对于一个在[0,)上有界的函数h(t),定义hu=sup.

(3)  如果f[0, \infty)上是可积函数,定义\langle f\rangle_w=\frac{1}{w}\int_0^{w}f(s){\rm d}s (w>0).

假设x(t)=(x_1(t), x_2(t), \cdots, x_n(t))\, (t\geq0)是随机微分方程

{\rm d}x(t)=f(x(t), t){\rm d}t+g(x(t), t){\rm d}B(t)
(2.1)

的解,其中f\in{\cal L}^1({\mathbb{R}}^n\times {\mathbb{R}}_+, {\mathbb{R}}^n), g\in{\cal L}^2({\mathbb{R}}^n\times {\mathbb{R}}_+, {\mathbb{R}}^{n\times m}), B(t)n维布朗运动.

定理2.1[15](存在唯一性定理)  假设f(x(t), t)g(x(t), t)关于x(t)满足下列条件:

(1)  局部Lipschitz条件:存在c_k>0(k=1, 2, \cdots), 使得\forall x, y\in {\mathbb{R}}^n\mid x\mid\vee\mid y\mid\leq k有不等式

\mid f(x, t)-f(y, t)\mid\vee\mid g(x, t)-g(y, t)\mid\leq c_k\mid x-y\mid,

成立;

(2)  线性增长条件:存在c>0,使得

\mid f(x, t)\mid\vee\mid g(x, t)\mid\leq c\mid 1+\mid x\mid\, \mid, \forall(x, t)\in {\mathbb{R}}^n\times {\mathbb{R}}_+,

则初始条件为x(0)=x_0\in {\mathbb{R}}^n的系统(2.1)存在唯一连续的局部解x(t)\big(t\in[0, \tau_e)\big), \, \tau_e是爆破时间.

定理2.2[15] ( {\rm{It\hat o}} 公式)  设x(t)\, (t\geq 0) {\rm{It\hat o}} 过程,其随机微分为

{\rm d}x(t)=f(t){\rm d}t+g(t){\rm d}B(t),

其中f\in{\cal L}^1({\mathbb{R}}_+, {\mathbb{R}}^n), g\in{\cal L}^2({\mathbb{R}}_+, {\mathbb{R}}^{n\times m}).V(x(t), t)\in C^{2, 1}({\mathbb{R}}^n\times {\mathbb{R}}_+;{\mathbb{R}}),则V(x(t), t)仍然是 {\rm{It\hat o}} 过程,具有如下随机微分

{\rm d}V(x(t), t)=V_t(x(t), t){\rm d}t+V_x(x(t), t){\rm d}x(t)+\frac{1}{2}{\rm d}x^T(t)V_{xx}(x(t), t){\rm d}x(t).

定理2.3[15](随机微分方程比较定理)  设x_i(t)(i=1, 2)分别是随机微分方程

{\rm d}x_i(t)=f_i(x_i(t), t){\rm d}t+g(x_i(t), t){\rm d}B(t)

的解,其中f_i(x, t)\in C([0, +\infty)\times {\mathbb{R}}), g(x, t)\in C([0, +\infty)\times {\mathbb{R}}).若满足:

(1)  存在定义在[0, +\infty)上满足\rho(0)=0\int_{0^+}^{+\infty}\rho(s){\rm d}s=\infty的函数\rho(s),使得

|g(x, t)-g(y, t)|\leq \rho(|x-y|), x, y\in {\mathbb{R}}, t\geq 0;

(2)  f_1(x, t)\leq f_2(x, t), x\in{\mathbb{R}}, t\geq 0;

(3)  x_1(0)\leq x_2(0),

则有x_1(t)\leq x_2(t) a.s., t\geq 0.

定义2.1[9]  设x(t)是系统(1.1)的任意解,则

(1)  若\lim\limits_{t\rightarrow+\infty}x(t)=0,则称种群x(t)为灭绝的;

(2)  若\lim\limits_{t\rightarrow+\infty}\inf\frac{1}{t}\int_0^tx(s){\rm d}s>0,则称种群x(t)为平均持续生存的.

考虑下列随机微分方程

{\rm d}X(t)=\mu(X(t), t){\rm d}t+\sigma(X(t), t){\rm d}B(t).
(2.2)

引理2.1[9]  设X(t)是方程(2.2)的解,若S(-\infty)>-\inftyS(+\infty)=+\infty成立,则\lim\limits_{t\rightarrow\infty}X(t)=-\infty,其中

S(u)=\int_0^u{e^{-\int_0^v{\frac{2\mu(y)}{\sigma^2(y)}}{\rm d}y}}{\rm d}v.

考虑下列随机微分方程

{\rm d}X(t)= X(t)[r_1-a_1X(t)]{\rm d}t+\sigma X(t){\rm d}B(t),
(2.3)

其中r_1, a_1, \sigma都是正常数, B(t)是标准的布朗运动.

引理2.2[9]  X(t)是方程(2.3)的解,当r_1>\frac{\sigma^2}{2},并且满足初始值X(0)>0,则

\lim\limits_{t\rightarrow \infty} \frac{\ln X(t)}{t}=0 \; \; \; {\rm a.s.}

\lim\limits_{t\rightarrow \infty} \frac{1}{t} \int_0^t {X(s)}{\rm d}s=\frac{r_1-\frac{\sigma^2}{2}}{a_1} \; \; \; {\rm a.s.. }

X(t)E_l(l维欧几里得空间)中的一个Markov自治过程,可表示为如下随机微分方程

{\rm d}X(t)=b(X){\rm d}t+\sum\limits_{s=1}^k g_s^i(X){\rm d}B_s(t).

其扩散矩阵为

A(X)=a_{ij}(X), \ \ a_{ij}(X)=\sum\limits_{s=1}^k g_s^i(X)g_s^j(X).

作如下假设:

(A)  存在具有正则边界\Gamma的有界区域U\subset E_l,具有如下性质:

(A1)  在U和它的一些邻域,扩散矩阵A(X)的最小特征值是非零的.

(A2)  当X\in E_l \backslash U时,从X出发的轨道到达集合U的平均时间\tau是有限的,且对每个紧子集K\subset E_l\sup\limits_{X\in K}E_X\tau<\infty.

引理2.3[1, 16]  如果假设(A)成立,则Markov过程X(t)存在平稳分布\mu(\cdot).f(\cdot)是关于测度\mu可积的函数,则有如下结论成立

P_x\bigg \{\lim\limits_{T\rightarrow \infty} \frac{1}{T} \int_0^T{f(X(t))}{\rm d}t=\int_{E_l}f(x)\mu({\rm d}x)\bigg\}=1, \; \; \; x\in E_l.

为了验证(A1)成立,我们只需证明LU中是一致椭圆的,其中LV=b(X)V_X+\frac{trA(X)V_{XX}}{2},即证明存在正数M满足

\sum\limits_{i, j=1}^k a_{ij}(X)\xi_i\xi_j\geq M|\xi|^2, \ \ X\in U, \ \ \xi \in {\mathbb{R}}^k.

为了验证(A2)成立,只要证明存在非负的C^2 -函数及邻域U,使得对于任意的X\in E_l \backslash U, LV(X)是负的.

下面,考虑如下方程

X(t)=X(t_0)+\int_{t_0}^t {b(s, X(s))}{\rm d}s+\sum\limits_{r=1}^k \int_{t_0}^t{\sigma_r(s, X(s))}{\rm d}B_r(s), \ \ X\in {\mathbb{R}}^l.
(2.4)

假设方程(2.4)的系数b(s, X(s)), \sigma_i(s, X(s))(i=1, 2, \cdots, k)满足下列条件

\begin{array}{ll} |b(s, X)-b(s, Y)|+\sum\limits_{r=1}^k{|\sigma_r(s, X)-\sigma_r(s, Y)|}\leq B|X-Y|, \\[4mm] |b(s, X)|+\sum\limits_{r=1}^k{|\sigma_r(s, X)|}\leq B(1+|X|), \end{array}
(2.5)

其中B是一个常数.

引理2.4[4, 16]  设方程(2.4)的系数关于tw周期的,且在每个柱形I\times U中条件(2.5)成立,并且假设存在一个C^2 -函数V(t, x)关于tw周期的,且下列条件在某个紧集外成立

\begin{array}{ll} (1)\;\;\;\; \liminf\limits_{|x|\rightarrow+\infty}V(t, x)=+\infty, \\\; (2)\;\;\;\; LV(t, x)\leq -1, \end{array}
(2.6)

则方程(2.4)存在一个w周期解,即该解是一个w周期的Markov过程.

3 主要结果

定理3.1  对任意给定的初值(x(0), y(0))\in {\mathbb{R}}^2_+,系统(1.1)存在唯一解(x(t), y(t))\in {\mathbb{R}}_+^2,并且以概率1存在于{\mathbb{R}}^2_+中.

  首先考虑如下方程

\left\{ \begin{array}{llc} \left. \begin{array}{llc} \displaystyle {\rm d}u=\Big(r_1-\frac{1}{2}\sigma_1^2-a_1e^u-\frac{b_1e^ue^v}{1+e^{2u}}\Big){\rm d}t+\sigma_1{\rm d}B_1, \\[3mm] \displaystyle {\rm d}v=\Big(r_2-\frac{1}{2}\sigma_2^2-\frac{b_2e^v}{k_2+e^u}\Big){\rm d}t+\sigma_2{\rm d}B_2, \\[3mm] \end{array}\right. \end{array} \right.
(3.1)

显然,系统(3.1)满足局部Lipschitz条件,则系统存在唯一的局部解(u(t), v(t)), (t\in[0, \tau_{e}))其中\tau_{e}是爆破时间.由 {\rm{It\hat o}} 公式可得(x(t), y(t))=(e^{u(t)}, e^{v(t)})是系统(1.1)满足初值(x(0), y(0))\in {\mathbb{R}}^2_+的唯一解.为了证明这个解是全局的,只需证明\tau_{e}=+\infty.

k_{0}>0足够大,使得(x(t), y(t))\in[\frac{1}{k_0}, k_{0}]\times[\frac{1}{k_0}, k_{0}],对于任意的正数k\geq k_{0},定义一个停时序列

\tau_{_{k}}= \inf \Big\{t\in[0, \tau_{e}):x(t)\notin(\frac{1}{k}, k)\ \mbox{或} \ y(t)\notin(\frac{1}{k}, k)\Big\},

定义\inf\Phi=+\infty (\Phi代表一个空集).显然,当k\rightarrow\infty时, \tau_{k}是单调递增的,且\tau_{k}<\tau_{e},因此有\tau_{\infty}=\lim\limits_{k\rightarrow+\infty}\tau_{k},其中\tau_{\infty}\leq\tau_{e} a.s.,因此,只需证明\tau_{\infty}\rightarrow\infty a.s..

假设\tau_{\infty}\nrightarrow\infty,则存在常数T\geq0, \epsilon\in(0, 1)和一个整数k_{1}\geq k_{0},有

P\{\tau_{k} \leq T\}\geq\epsilon, \;\;\;\; \forall k\geq k_{1}.

定义一个C^{2} -函数V: {\mathbb{R}}_{+}^{2}\rightarrow{\mathbb{R}}_{+}.

\begin{eqnarray*}V(x, y)&=&\bigg[x^{\frac{1}{2}}-1-\frac{\ln x}{2}+y^\frac{1}{2}-1-\frac{\ln y}{2}\bigg]+(k_2+x)y\\ &=&V_1+V_2+V_3, \end{eqnarray*}

其中V_1=x^{\frac{1}{2}}-1-\frac{\ln x}{2}, V_2=y^{\frac{1}{2}}-1-\frac{\ln y}{2}, V_3=(k_2+x)y. {\rm{It\hat o}} 公式可得

\begin{eqnarray*}LV_1&=&\frac{1}{2}\big(x^\frac{1}{2}-1\big)\Big(r_1-a_1x-\frac{b_1xy}{1+x^2}\Big)+\frac{1}{2}\Big(-\frac{1}{4}x^{-\frac{3}{2}}+\frac{1}{2}x^{-2}\Big)\sigma_1^2x^2\\ &=&-\frac{a_1}{2}x^\frac{3}{2}+\frac{a_1}{2}x+\Big(\frac{r_1}{2}-\frac{1}{8}\sigma_1^2\Big)x^\frac{1}{2}-\frac{b_1x^\frac{3}{2}y}{2(1+x^2)}+\frac{b_1xy}{2(1+x^2)}-\frac{1}{2}r_1+\frac{1}{4}\sigma_1^2.\end{eqnarray*}

同理可得

\begin{eqnarray*}LV_2&=&\frac{1}{2}\big(y^\frac{1}{2}-1\big)\Big(r_2-\frac{b_2y}{k_2+x}\Big)+\frac{1}{2}\Big(-\frac{1}{4}y^{-\frac{3}{2}}+\frac{1}{2}y^{-2}\Big)\sigma_2^2y^2\\&\leq& \frac{1}{2}\Big(r_2-\frac{1}{4}\sigma_2^2\Big)y^\frac{1}{2}+\frac{b_2y}{2k_2}-\frac{r_2}{2}+\frac{1}{4}\sigma_2^2, \\LV_3&=&xy\Big(r_1-a_1x-\frac{b_1xy}{1+x^2}\Big)+(k_2+x)y*\Big(r_2-\frac{b_2y}{k_2+x}\Big)\\&=&r_1xy-a_1x^2y-\frac{b_1x^2y^2}{1+x^2}+r_2(k_2+x)y-b_2y^2, \end{eqnarray*}

\begin{eqnarray*}LV&=&LV_1+LV_2+LV_3\\&\leq&-\frac{a_1}{2}x^\frac{3}{2}+\frac{a_1}{2}x+\Big(\frac{r_1}{2}-\frac{1}{8}\sigma_1^2\Big)x^\frac{1}{2}-\frac{b_1x^\frac{3}{2}y}{2(1+x^2)}+\frac{b_1xy}{2(1+x^2)}-\frac{1}{2}r_1+\frac{1}{4}\sigma_1^2\\&&+\frac{1}{2}\Big(r_2-\frac{1}{4}\sigma_2^2\Big)y^\frac{1}{2}+\frac{b_2y}{2k_2}-\frac{r_2}{2}+\frac{1}{4}\sigma_2^2+r_1xy-a_1x^2y-\frac{b_1x^2y^2}{1+x^2}\\&&+r_2(k_2+x)y-b_2y^2\\&\leq& -\frac{a_1}{2}x^\frac{3}{2}+\frac{a_1}{2}x+\Big(\frac{r_1}{2}-\frac{1}{8}\sigma_1^2\Big)x^\frac{1}{2}+\frac{b_1y}{4}-\frac{1}{2}r_1+\frac{1}{4}\sigma_1^2-b_2y^2\\&&+\frac{1}{2}\Big(r_2-\frac{1}{4}\sigma_2^2\Big)y^\frac{1}{2}+\frac{b_2y}{2k_2}-\frac{r_2}{2}+\frac{1}{4}\sigma_2^2+r_1xy-a_1x^2y+r_2(k_2+x)y\\&\leq& N.\end{eqnarray*}

事实上,由于

\begin{eqnarray*}&&-\frac{a_1}{2}x^\frac{3}{2}+\frac{a_1}{2}x+\Big(\frac{r_1}{2}-\frac{1}{8}\sigma_1^2\Big)x^\frac{1}{2}+\frac{b_1y}{4}-\frac{1}{2}r_1+\frac{1}{4}\sigma_1^2\\&&-b_2y^2+\frac{1}{2}\Big(r_2-\frac{1}{4}\sigma_2^2\Big)y^\frac{1}{2}+\frac{b_2y}{2k_2}-\frac{r_2}{2}+\frac{1}{4}\sigma_2^2+r_1xy-a_1x^2y+r_2(k_2+x)y\end{eqnarray*}

关于x y的最高次项的系数都小于零,所以可以找到一个常数N,使得

\begin{eqnarray*}&&-\frac{a_1}{2}x^\frac{3}{2}+\frac{a_1}{2}x+\Big(\frac{r_1}{2}-\frac{1}{8}\sigma_1^2\Big)x^\frac{1}{2}+\frac{b_1y}{4}-\frac{1}{2}r_1+\frac{1}{4}\sigma_1^2\\&&-b_2y^2+\frac{1}{2}\Big(r_2-\frac{1}{4}\sigma_2^2\Big)y^\frac{1}{2}+\frac{b_2y}{2k_2}-\frac{r_2}{2}+\frac{1}{4}\sigma_2^2+r_1xy-a_1x^2y+r_2(k_2+x)y<N.\end{eqnarray*}

下面的证明与文献[17]的相似,所以省略.

定理3.2  设x(t), y(t)是系统(1.1)满足初始条件x(0)>0, y(0)>0的任意解,若系统(1.1)满足r_1-\frac{1}{2}\sigma_1^2-\ell_1 (r_2-\frac{\sigma_2^2}{2})>0, r_2>\frac{\sigma_2^2}{2}b_1^2k_2^2+4\ell_1 b_2(b_1-\ell_1 b_2)<0,其中\ell_1>0,则系统(1.1)是平均持续生存的.

  由文献[7]可以得到,种群y是平均持续生存的且有\lim\limits_{t\rightarrow\infty}\frac{1}{t}\int_0^t \frac{y(s)}{k_2+x(s)}{\rm d}s=\frac{r_2-\frac{1}{2}\sigma_2^2}{b_2}成立.下面证明\frac{b_1xy}{1+x^2}<\frac{\ell_1 b_2y}{k_2+x}.事实上

\begin{eqnarray*}\frac{b_1xy}{1+x^2}-\frac{\ell_1 b_2y}{k_2+x}&=&\frac{b_1xy(k_2+x)-\ell_1 b_2y(1+x^2)}{(1+x^2)(k_2+x)}\\&=&\frac{b_1k_2xy+b_1x^2y-\ell_1 b_2y-\ell_1 b_2x^2y}{(1+x^2)(k_2+x)}\\&=&\frac{[(b_1-\ell_1 b_2)x^2+b_1k_2x-\ell_1 b_2]y}{(1+x^2)(k_2+x)}.\end{eqnarray*}

因为b_1^2k_2^2+4\ell_1 b_2(b_1-\ell_1 b_2)<0,可得b_1-\ell_1 b_2<0 [(b_1-\ell_1 b_2)x^2+b_1k_2x-\ell_1 b_2]<0,即\frac{b_1xy}{1+x^2}<\frac{\ell_1 b_2y}{k_2+x}恒成立.

现在证明种群x也是平均持续生存的.由 {\rm{It\hat o}} 公式可得

\begin{eqnarray*}d\ln x&=&\frac{1}{x}{\rm d}t-\frac{1}{2x^2}({\rm d}x)^2\\& =&\Big(r_1-a_1x-\frac{b_1xy}{1+x^2}\Big){\rm d}t+\sigma_1{\rm d}B_1-\frac{1}{2}\sigma_1^2{\rm d}t\\ &=&\Big(r_1-a_1x-\frac{b_1xy}{1+x^2}-\frac{1}{2}\sigma_1^2\Big){\rm d}t+\sigma_1{\rm d}B_1\\ &\geq&\Big (r_1-a_1x-\frac{\ell_1 b_2y}{k_2+x}-\frac{1}{2}\sigma_1^2\Big){\rm d}t+\sigma_1{\rm d}B_1, \end{eqnarray*}

对上述不等式两边同时从0到t积分可得

\ln x(t)-\ln x(0)\geq\Big(r_1-\frac{1}{2}\sigma_1^2\Big)t-\int_0^t{a_1x}{\rm d}s-\int_0^t{\frac{\ell_1 b_2y}{k_2+x}}{\rm d}s+\sigma_1B_1,

两边同时除以t并取极限有

\lim\limits_{t\rightarrow\infty}\frac{\ln x(t)}{t}\geq r_1-\frac{1}{2}\sigma_1^2-\lim\limits_{t\rightarrow\infty}\frac{1}{t}\int_0^t{a_1x}{\rm d}s-\lim\limits_{t\rightarrow\infty}\frac{1}{t}\int_0^t{\frac{\ell_1 b_2y}{k_2+x}}{\rm d}s.

根据引理2.2和随机微分方程的比较定理,可得

\lim\limits_{t\rightarrow\infty}\inf\frac{1}{t}\int_0^t{x}{\rm d}s\geq\frac{r_1-\frac{1}{2}\sigma_1^2-\ell_1(r_2-\frac{\sigma_2^2}{2})}{a_1} \;\;\;\;\mbox{a.s..}

所以,当r_1-\frac{1}{2}\sigma_1^2-\ell_1 (r_2-\frac{\sigma_2^2}{2})>0时,种群x是平均持续生存的.另一方面

\lim\limits_{t\rightarrow\infty}\inf\frac{1}{t}\int_0^t{y}{\rm d}s\geq\lim\limits_{t\rightarrow\infty}\inf\frac{1}{t}\int_0^t \frac{k_2y(s)}{k_2+x(s)}{\rm d}s=\frac{k_2(r_2-\frac{1}{2}\sigma_2^2)}{b_2} \;\;\;\;\mbox{a.s..}

所以,当r_1-\frac{1}{2}\sigma_1^2-\ell_1 (r_2-\frac{\sigma_2^2}{2})>0r_2>\frac{\sigma_2^2}{2}时,系统(1.1)是平均持续生存的.

定理3.3  设x(t), y(t)是系统(1.1)满足初始条件x(0)>0, y(0)>0的任意解,

(i)  若r_1<\frac{\sigma_1^2}{2}, r_2>\frac{\sigma_2^2}{2},则种群x是灭绝的,种群y是平均持续生存的.

(ii)  若r_2<\frac{\sigma_2^2}{2}, r_1>\frac{\sigma_1^2}{2},则种群y是灭绝的,种群x是平均持续生存的.

(iii)  若r_1<\frac{\sigma_1^2}{2}, r_2<\frac{\sigma_2^2}{2},则种群x和种群y均灭绝.

  (i)由系统(3.1)可得

{\rm d}u\leq \Big(r_1-\frac{1}{2}\sigma_1^2\Big){\rm d}t+\sigma_1{\rm d}B_1.

由条件(i)和引理2.1有

\lim\limits_{t\rightarrow\infty} u(t)=-\infty \;\;\;\;\mbox{a.s., }

\lim\limits_{t\rightarrow\infty}x(t)=0 \;\;\;\;\mbox{a.s..}
(3.2)

即对于任意的\varepsilon>0,存在t_0\Omega_\varepsilon,当t>t_0\omega\in\Omega_\varepsilon,都有P(\Omega_\varepsilon)\geq 1-\varepsilon\frac{x}{k_2+x}\leq \varepsilon成立.又因为

\begin{eqnarray*}{\rm d}y&=&y\Big(r_2-\frac{b_2y}{k_2+x(t)}\Big){\rm d}t+\sigma_2y{\rm d}B_2\\ &=&y\Big(r_2-\frac{b_2y}{k_2}+\frac{b_2xy}{(k_2+x)k_2}\Big){\rm d}t+\sigma_2y{\rm d}B_2. \end{eqnarray*}

可得

y\Big(r_2-\frac{b_2y}{k_2}\Big){\rm d}t+\sigma_2y{\rm d}B_2\leq {\rm d}y\leq y\Big(r_2-\frac{b_2y(1-\varepsilon)}{k_2}\Big){\rm d}t+\sigma_2y{\rm d}B_2.

又由于r_2>\frac{\sigma_2^2}{2},根据引理2.2和随机微分方程的比较定理,则有

\liminf\limits_{t\rightarrow\infty}\frac{1}{t}\int_0^t y(s){\rm d}s\geq \frac{k_2(r_2-\frac{\sigma_2^2}{2})}{b_2} \;\;\;\;\mbox{a.s., }

\limsup\limits_{t\rightarrow\infty}\frac{1}{t}\int_0^t y(s){\rm d}s\leq\frac{k_2(r_2-\frac{\sigma_2^2}{2})}{(1-\varepsilon)b_2} \;\;\;\;\mbox{a.s., }

由于\varepsilon的任意性,则有

\lim\limits_{t\rightarrow\infty}\frac{1}{t}\int_0^t y(s){\rm d}s=\frac{k_2(r_2-\frac{\sigma_2^2}{2})}{b_2} \;\;\;\;\mbox{a.s..}

所以,当r_1<\frac{\sigma_1^2}{2}, r_2>\frac{\sigma_2^2}{2}时, x是灭绝的,并且y是平均持续生存的.

(ii)  从系统(3.1)可得

{\rm d}v=\Big(r_2-\frac{1}{2}\sigma_2^2-\frac{b_2e^v}{k_2+e^u}\Big){\rm d}t+\sigma_2{\rm d}B_2 \leq\Big(r_2-\frac{\sigma_2^2}{2}\Big){\rm d}t+\sigma_2{\rm d}B_2,

与方程(3.2)的证明类似,当r_2<\frac{\sigma_2^2}{2}时,可得

\lim\limits_{t\rightarrow\infty}y(t)=0 \;\;\;\;\mbox{a.s..}

类似的,对于任意的\varepsilon>0,存在T_0\Omega_\varepsilon,当t>t_0\omega\in\Omega_\varepsilon,都有P(\Omega_\varepsilon)\geq 1-\varepsilon\frac{b_1xy}{1+x^2}\leq\varepsilon.所以有

x(r_1-a_1x-\varepsilon){\rm d}t+\sigma_1x{\rm d}B_1\leq {\rm d}x\leq x(r_1-a_1x){\rm d}t+\sigma_1x{\rm d}B_1.

r_1>\frac{\sigma_1^2}{2},根据引理2.2和随机微分方程比较定理可得

\frac{r_1-\varepsilon-\frac{\sigma_1^2}{2}}{a_1}\leq\liminf\limits_{t\rightarrow\infty}\frac{1}{t}\int_0^t{x(s)}{\rm d}s\leq\frac{r_1-\frac{\sigma_1^2}{2}}{a_1}\;\;\;\;{\rm a.s., }

由于\varepsilon的任意性可得

\liminf\limits_{t\rightarrow\infty}\frac{1}{t}\int_0^t{x(s)}{\rm d}s=\frac{r_1-\frac{\sigma_1^2}{2}}{a_1} \;\;\;\;\mbox{a.s..}

所以,当r_2<\frac{\sigma_2^2}{2}, r_1>\frac{\sigma_1^2}{2}时,种群y是灭绝的, x是平均持续生存的.

(iii)  当r_1<\frac{\sigma_1^2}{2}, r_2<\frac{\sigma_2^2}{2}成立时,从前面两种情况讨论容易得到

\lim\limits_{t\rightarrow\infty}x(t)=0 \;\;\;\; \mbox{a.s., } \lim\limits_{t\rightarrow\infty}y(t)=0 \;\;\;\;\mbox{a.s..}

即,当r_1<\frac{\sigma_1^2}{2}, r_2<\frac{\sigma_2^2}{2}时,种群x和种群y均灭绝.

定理3.4  设系统(1.1)满足0<\sigma_1^2<r_1, 0<\sigma_2^2<r_2,则对于任意的正初始值x(0), y(0),系统(1.1)存在唯一的平稳分布\mu(\cdot),而且是遍历的.

  定义C^2 -函数V(x, y):{\mathbb{R}}^2_+\rightarrow {\mathbb{R}}_+:

V(x, y)=\frac{1}{x}+\ln x+\ell_2\Big(\frac{1}{y}+\ln y\Big),

其中\ell_2>0并且满足b_1^2+4\ell_2 b_2(b_1k_2-\ell_2 b_2)<0. {\rm{It\hat o}} 公式可得

\begin{eqnarray*} {\rm d}V&=&\Big(-\frac{1}{x^2}+\frac{1}{x}\Big){\rm d}x+\ell_2\Big(-\frac{1}{y^2}+\frac{1}{y}\Big){\rm d}y+\frac{1}{2}(2x^{-3}-x^{-2})({\rm d}x)^2+\frac{\ell_2}{2}(2y^{-3}-y^{-2})({\rm d}y)^2\\ &=&\Big(-\frac{1}{x}+1\Big)\bigg[\Big(r_1-a_1x-\frac{b_1xy}{1+x^2}\Big){\rm d}t+\sigma_1{\rm d}B_1\bigg]\\&&+\ell_2\Big(-\frac{1}{y}+1\Big)\bigg[\Big(r_2-\frac{b_2y}{k_2+x}\Big){\rm d}t+\sigma_2{\rm d}B_2\bigg]+\frac{(2x^{-1}-1)\sigma_1^2+\ell_2(2y^{-1}-1)\sigma_2^2}{2}{\rm d}t\\& =&LV{\rm d}t+\Big(-\frac{1}{x}+1\Big)\sigma_1{\rm d}B_1+\ell_2\Big(-\frac{1}{y}+1\Big)\sigma_2{\rm d}B_2, \end{eqnarray*}

其中

\begin{eqnarray*} LV&=&-\frac{r_1}{x}+a_1+\frac{b_1y}{1+x^2}+r_1-a_1x-\frac{b_1xy}{1+x^2}+\ell_2\Big(-\frac{r_2}{y}+\frac{b_2}{k_2+x}+r_2-\frac{b_2y}{k_2+x}\Big)\\ &&+\frac{\sigma_1^2}{x}+\ell_2\frac{\sigma_2^2}{y}-\frac{\sigma_1^2+\ell_2\sigma_2^2}{2}\\ &\leq& -a_1x-\frac{r_1}{x}+a_1+\frac{b_1y}{1+x^2}+r_1-\frac{\ell_2 r_2}{y}+\frac{\ell_2 b_2}{k_2+x}+\ell_2 r_2-\frac{\ell_2 b_2y}{k_2+x}\\&&+\frac{\sigma_1^2}{x}+\ell_2\frac{\sigma_2^2}{y}-\frac{\sigma_1^2+\ell_2\sigma_2^2}{2}\\ &\leq& -a_1x-\frac{r_1-\sigma_1^2}{x}+\Big(\frac{b_1}{1+x^2}-\frac{\ell_2 b_2}{k_2+x}\Big)y-\frac{\ell_2(r_2-\sigma_2^2)}{y}+a_1+r_1\\&&+\frac{\ell_2 b_2}{k_2}+\ell_2 r_2-\frac{\sigma_1^2+\ell_2 \sigma_2^2}{2}\\& \leq& -a_1x-\frac{r_1-\sigma_1^2}{x}+\frac{-\ell_2 b_2x^2+b_1x+b_1k_2-\ell_2 b_2}{(1+x^2)(k_2+x)}y-\frac{\ell_2 (r_2-\sigma_2^2)}{y}+\lambda_0, \end{eqnarray*}

其中\lambda_0=\max\{a_1+r_1+\frac{\ell_2 b_2}{k_2}+\ell_2 r_2-\frac{\sigma_1^2+\ell_2 \sigma_2^2}{2}, 0\}.\ell_2的定义可知

\Delta=b_1^2+4\ell_2 b_2(b_1k_2-\ell_2 b_2)<0,

所以\ell_2 b_2x^2-b_1x-b_1k_2+\ell_2 b_2有最小值且大于零不妨设其为m_1.

LV\leq -a_1x-\frac{r_1-\sigma_1^2}{x}-\frac{m_1y}{(1+x^2)(k_2+x)}-\frac{\ell_2 (r_2-\sigma_2^2)}{y}+\lambda_0.

选择足够小的\varepsilon_1, \varepsilon_2使得

0<\varepsilon_1<\min\Big\{\frac{r_1-\sigma_1^2}{\lambda_0+1}, \frac{a_1}{\lambda_0+1}\Big\},
(3.3)

0<\varepsilon_2<\frac{\ell_2(r_2-\sigma_2^2)}{\lambda_0+1}, \ \ \ \frac{m_1}{(1+\frac{1}{\varepsilon_1^2})(k_2+\frac{1}{\varepsilon_1})\varepsilon_2}>\lambda_0+1.
(3.4)

下面,考虑有界集

D_{\varepsilon_{1, 2}}=\Big\{(x, y)\in {\mathbb{R}}_+^2| \varepsilon_1<x<\frac{1}{\varepsilon_1}\ \ \ \varepsilon_2<y<\frac{1}{\varepsilon_2}\Big\},

假设

D_{\varepsilon_{1, 2}}^1=\Big\{(x, y)\in {\mathbb{R}}_+^2| 0<y\leq \varepsilon_2 \Big\}, D_{\varepsilon_{1, 2}}^2=\Big\{(x, y)\in {\mathbb{R}}_+^2| 0<x\leq \varepsilon_1 \Big\},

D_{\varepsilon_{1, 2}}^3=\Big\{(x, y)\in {\mathbb{R}}_+^2| x\geq \frac{1}{\varepsilon_1} \Big\}, D_{\varepsilon_{1, 2}}^4=\Big\{(x, y)\in {\mathbb{R}}_+^2| \varepsilon_1 \leq x \leq \frac{1}{\varepsilon_1}, y \geq \frac{1}{\varepsilon_2} \Big\}.

显然, D_{\varepsilon_{1, 2}}^C=D_{\varepsilon_{1, 2}}^1 \cup D_{\varepsilon_{1, 2}}^2 \cup D_{\varepsilon_{1, 2}}^3 \cup D_{\varepsilon_{1, 2}}^4.

下面证明,对于任意的(x, y)\in D_{\varepsilon_{1, 2}}^C,都有LV<-1成立.

(x, y)\in D_{\varepsilon_{1, 2}}^1,根据不等式(3.4),可得

LV\leq-\frac{\ell_2(r_2-\sigma_2^2)}{\varepsilon_2}+\lambda_0<-1.

(x, y)\in D_{\varepsilon_{1, 2}}^2,根据不等式(3.3),可得

LV\leq-\frac{r_1-\sigma_1^2}{\varepsilon_1}+\lambda_0<-1.

(x, y)\in D_{\varepsilon_{1, 2}}^3,根据不等式(3.3),可得

LV\leq-\frac{a_1}{\varepsilon_1}+\lambda_0<-1.

(x, y)\in D_{\varepsilon_{1, 2}}^4,根据不等式(3.4),可得

LV\leq -\frac{m_1}{(1+\frac{1}{\varepsilon_1^2})(k_2+\frac{1}{\varepsilon_1})\varepsilon_2}+\lambda_0<-1.

从上述的讨论可知:对任意的(x, y)\in D_{\varepsilon_{1, 2}}^C,有LV<-1, D_{\varepsilon_{1, 2}}的一个邻域U, 并且\bar{U}\subseteq {\mathbb{R}}_+^2.显然对任意的(x, y)\in {\mathbb{R}}_+^2/ULV<-1.因此条件(A2)得到满足.

此外,可以找到一个常数M=\min\limits_{(x, y)\in\bar{U}}\{\sigma_1^2x^2, \sigma_2^2y^2\}>0,使得对于任意的(x, y)\in \bar{U}, \xi=(\xi_1, \xi_2)\in {\mathbb{R}}_+^2,有

\sum\limits_{i, j=1}^2{a_{ij}(x, y)\xi_i\xi_j}=\sigma_1^2x^2\xi_1^2+\sigma_2^2y^2\xi_2^2\geq M\|\xi\|^2,

因此条件(A1)也满足.所以系统(1.1)存在平稳分布,且具有遍历性.

定理3.5  若\langle r_1(t)-\sigma_1^2(t)\rangle_{\omega}>0, \langle r_2(t)-\sigma_2^2(t)\rangle_{\omega}>0,则系统(1.2)存在正的\omega周期解.

  构造V(t, x, y)函数

V(t, x, y)=\Big(\frac{e^{\omega_1(t)}}{x}+\ln x\Big)+\ell_3\Big(\frac{e^{\omega_2(t)}}{y}+e^{|\omega_1^u|}\ln y\Big),

其中\ell_3>0且满足

(b_1^u)^2+4\ell_3 b_2^l(b_1^uk_2^u-\ell_3 b_2^2)<0,

\omega_1'(t)=r_1(t)-\sigma_1(t)^2-\langle r_1(t)-\sigma_1^2(t)\rangle_\omega,

\omega_2'(t)=r_2(t)-\sigma_2(t)^2-\langle r_2(t)-\sigma_2^2(t)\rangle_\omega.

\omega_i的定义可得

\begin{eqnarray*}\omega_i(t+\omega)-\omega_i(t)&=&\int_t^{t+\omega}{\omega_i'(s)}{\rm d}s\\ &=&\int_t^{t+\omega}{r_i(s)-\sigma_i^2(s)}{\rm d}s-\int_t^{t+\omega}{\langle r_i(s)-\sigma_i^2(s)\rangle_\omega}{\rm d}s\\ &=&\int_t^{t+\omega}{r_i(s)-\sigma_i^2(s)}{\rm d}s-\int_t^{t+w}{\int_0^\omega{\frac{r_i(u)-\sigma_i^2(u)}{\omega}}{\rm d}u}{\rm d}s\\ &=&\int_t^{t+\omega}{r_i(s)-\sigma_i^2(s)}{\rm d}s-\int_0^{\omega}{r_i(s)-\sigma_i^2(s)}{\rm d}s\\ &=&0. \end{eqnarray*}

所以, \omega_1(t), \omega_2(t)都是\omega周期函数.假设

U_k=\Big(\frac{1}{k}, k\Big)\times\Big(\frac{1}{k}, k\Big),

显然,当k\rightarrow\infty

\inf\limits_{(t, x, y)\in[0, +\infty)\times {\mathbb{R}}^2_+\setminus U_k}V(t, x, y)\rightarrow \infty,

这就验证了方程(2.6)的条件(1).下面只需验证方程(2.6)条件(2).假设

V_1=\frac{e^{\omega_1(t)}}{x}+\ln x, \;\;\;\; V_2=\frac{e^{\omega_2(t)}}{y}+e^{|\omega_1^u|}\ln y,

V_1应用 {\rm{It\hat o}} 公式可得

\begin{eqnarray*} {\rm d}V_1&=&\frac{e^{\omega_1(t)}\omega_1'(t)}{x}{\rm d}t+\Big(-\frac{e^{\omega_1(t)}}{x^2}+\frac{1}{x}\Big){\rm d}x+\frac{1}{2}(2x^{-3}e^{\omega_1(t)}-x^{-2})\sigma_1^2(t)x^2{\rm d}t\\ &=&\frac{e^{\omega_1(t)}\omega_1'(t)}{x}{\rm d}t+\Big(-\frac{e^{\omega_1(t)}}{x}+1\Big)\Big[(r_1(t)-a_1(t)x-\frac{b_1(t)xy}{1+x^2}){\rm d}t+\sigma_1(t){\rm d}B_1\Big]\\ &&+\Big(\frac{e^{\omega_1(t)}\sigma_1^2(t)}{x}-\frac{1}{2}\sigma_1^2(t)\Big){\rm d}t\\ &=&LV_1{\rm d}t+\Big(-\frac{e^{\omega_1(t)}}{x}+1\Big)\sigma_1(t){\rm d}B_1, \end{eqnarray*}

其中

\begin{eqnarray*}LV_1&=&\frac{e^{\omega_1(t)}\omega_1'(t)}{x}+\Big(-\frac{e^{\omega_1(t)}}{x}+1\Big)\Big(r_1(t)-a_1(t)x-\frac{b_1(t)xy}{1+x^2}\Big)+\frac{e^{\omega_1(t)}\sigma_1^2}{x}-\frac{1}{2}\sigma_1^2(t)\\ &=&\frac{e^{\omega_1(t)}}{x}(\omega_1'(t)+\sigma_1^2(t)-r_1(t))+a_1(t)e^{\omega_1}+e^{\omega_1}\frac{b_1(t)y}{1+x^2}+r_1(t)-a_1(t)x\\&&-\frac{b_1(t)xy}{1+x^2}-\frac{1}{2}\sigma_1^2(t)\\ &=&-\frac{e^{\omega_1(t)}}{x}\langle r_1(t)-\sigma_1^2(t)\rangle_{\omega}+a_1(t)e^{\omega_1}+e^{\omega_1}\frac{b_1(t)y}{1+x^2}+r_1(t)-a_1(t)x\\&&-\frac{b_1(t)xy}{1+x^2}-\frac{1}{2}\sigma_1^2(t).\end{eqnarray*}

同理可得

\begin{eqnarray*}{\rm d}V_2&=&\frac{e^{\omega_2(t)}\omega_2'(t)}{y}{\rm d}t+\Big(-\frac{e^{\omega_2(t)}}{y^2}+\frac{e^{|\omega_1^u|}}{y}\Big)\Big[y\Big(r_2(t)-\frac{b_2(t)y}{k_2(t)+x}\Big)+\sigma_2(t)y{\rm d}B_2\Big]\\ &&+\frac{1}{2}(2y^{-3}e^{\omega_2(t)}-e^{|\omega_1^u|}y^{-2})\sigma_2^2y^2{\rm d}t\\& =&LV_2{\rm d}t+\Big(-\frac{e^{\omega_2(t)}}{y^2}+\frac{e^{|\omega_1^u|}}{y}\Big)\sigma_2(t)y{\rm d}B_2, \end{eqnarray*}

其中

\begin{eqnarray*}LV_2&=&\frac{e^{\omega_2(t)}\omega_2'(t)}{y}+\Big(-\frac{e^{\omega_2(t)}}{y}+e^{|\omega_1^u|}\Big)\Big(r_2(t)-\frac{b_2(t)y}{k_2(t)+x}\Big)+\frac{e^{\omega_2}\sigma_2^2(t)}{y}-\frac{1}{2}e^{|\omega_1^u|}\sigma_2^2(t)\\ &=&\frac{e^{\omega_2(t)}}{y}(\omega_2'(t)+\sigma_2^2(t)-r_2(t))+\frac{b_2(t)e^{\omega_2}}{k_2(t)+x}+r_2(t)e^{|\omega_1^u|}-\frac{b_2(t)e^{|\omega_1^u|}y}{k_2(t)+x}-\frac{1}{2}e^{|\omega_1^u|}\sigma_2^2\\ &=&-\frac{e^{\omega_2(t)}}{y}\langle r_2(t)-\sigma_2^2(t)\rangle_{\omega}+\frac{b_2(t)e^{\omega_2}}{k_2(t)+x}+r_2(t)e^{|\omega_1^u|}-\frac{b_2(t)e^{|\omega_1^u|}y}{k_2(t)+x}-\frac{1}{2}e^{|\omega_1^u|}\sigma_2^2. \end{eqnarray*}

所以有

\begin{eqnarray*}LV&=&LV_1+\ell_3 LV_2\\&=& -\frac{e^{\omega_1(t)}}{x}\langle r_1(t)-\sigma_1^2(t)\rangle_{\omega}-\frac{\ell_3 e^{\omega_2(t)}}{y}\langle r_2(t)-\sigma_2^2(t)\rangle_{\omega}-a_1(t)x+e^{\omega_1}\frac{b_1(t)y}{1+x^2}\\&&-\ell_3\frac{b_2(t)e^{|\omega_1^u|}y}{k_2(t)+x}+a_1(t)e^{\omega_1}+r_1(t)-\frac{b_1(t)xy}{1+x^2}-\frac{1}{2}\sigma_1^2(t)+\ell_3\frac{b_2(t)e^{\omega_2}}{k_2(t)+x}\\&&+\ell_3 r_2(t)e^{|\omega_1^u|}-\frac{1}{2}\ell_3 e^{|\omega_1^u|}\sigma_2^2(t)\\&\leq& -\frac{e^{\omega_1^l}}{x}\langle r_1(t)-\sigma_1^2(t)\rangle_{\omega}-\frac{\ell_3e^{\omega_2^l}}{y}\langle r_2(t)-\sigma_2^2(t)\rangle_{\omega}-a_1^lx+e^{|\omega_1^u|}\frac{b_1^uy}{1+x^2}\\&&-\ell_3\frac{b_2^le^{|\omega_1^u|}y}{k_2^u+x}+a_1^ue^{|\omega_1^u|}+r_1^u-\frac{1}{2}|\sigma_1^l|^2+\ell_3\frac{b_2^ue^{|\omega_2^u|}}{k_2^l}+\ell_3 r_2^ue^{|\omega_1^u|}-\frac{1}{2}\ell_3 e^{|\omega_1^u|}|\sigma_2^l|^2.\end{eqnarray*}

因为

\begin{eqnarray*}e^{|\omega_1^u|}\frac{b_1^uy}{1+x^2}-\ell_3\frac{b_2^le^{|\omega_1^u|}y}{k_2^u+x}&=&e^{|\omega_1^u|}y\Big(\frac{b_1^u}{1+x^2}-\frac{\ell_3 b_2^l}{k_2^u+x}\Big)\\&=&(e^{|\omega_1^u|}y)\frac{b_1^u(k_2^u+x)-\ell_3 b_2^l(1+x^2)}{(1+x^2)(k_2^u+x)}\\&=&e^{|\omega_1^u|}y\frac{b_1^uk_2^u+b_1^ux-\ell_3 b_2^l-\ell_3 b_2^lx^2}{(1+x^2)(k_2^u+x)}, \end{eqnarray*}

又因为

\Delta=(b_1^u)^2+4\ell_3 b_2^l(b_1^uk_2^u-\ell_3 b_2^2)<0.

所以-b_1^uk_2^u-b_1^ux+\ell_3 b_2^l+\ell_3 b_2^lx^2有最小值且大于零不妨设其为m_2.

LV\leq-\frac{e^{\omega_1^l}}{x}\langle r_1(t)-\sigma_1^2(t)\rangle_{\omega}-\ell_3\frac{e^{\omega_2^l}}{y}\langle r_2(t)-\sigma_2^2(t)\rangle_{\omega}-a_1^lx -\frac{m_2e^{|\omega_1^u|}y}{(1+x^2)(k_2^u+x)}+\lambda^*,

其中

\lambda^*=\max\Big\{a_1^ue^{|\omega_1^u|}+r_1^u-\frac{1}{2}|\sigma_1^l|^2+\ell_3\frac{b_2^ue^{|\omega_2^u|}}{k_2^l}+\ell_3 r_2^ue^{|\omega_1^u|}-\frac{1}{2}\ell_3 e^{|\omega_1^u|}|\sigma_2^l|^2, 0\Big\}.

选择足够小的\varepsilon_1, \varepsilon_2使得

0<\varepsilon_1<\min\Big\{\frac{e^{w_1^l\langle r_1-\sigma_1^2 \rangle_\omega}}{\lambda^*+1}, \frac{a_1^l}{\lambda^*+1}\Big\},
(3.5)

0<\varepsilon_2<\frac{\ell_3 e^{w_2^l}\langle r_2-\sigma_2^2 \rangle_\omega}{\lambda^*+1}, \ \ \ \frac{m_2e^{|w_1^u|}}{(1+\frac{1}{\varepsilon_1^2})(k_2^u+\frac{1}{\varepsilon_1})\varepsilon_2}>\lambda^*+1.
(3.6)

考虑有界集

D_{\varepsilon_{1, 2}}=\Big\{(x, y)\in {\mathbb{R}}_+^2| \varepsilon_1<x<\frac{1}{\varepsilon_1}\ \ \ \varepsilon_2<y<\frac{1}{\varepsilon_2}\Big\},

假设

D_{\varepsilon_{1, 2}}^1=\{(x, y)\in {\mathbb{R}}_+^2| 0<y\leq \varepsilon_2 \},

D_{\varepsilon_{1, 2}}^2=\{(x, y)\in {\mathbb{R}}_+^2| 0<x\leq \varepsilon_1 \},

D_{\varepsilon_{1, 2}}^3=\Big\{(x, y)\in {\mathbb{R}}_+^2| x\geq \frac{1}{\varepsilon_1} \Big\},

D_{\varepsilon_{1, 2}}^4=\Big\{(x, y)\in {\mathbb{R}}_+^2| \varepsilon_1 \leq x \leq \frac{1}{\varepsilon_1}, y \geq \frac{1}{\varepsilon_2} \Big\}.

显然, D_{\varepsilon_{1, 2}}^C=D_{\varepsilon_{1, 2}}^1 \cup D_{\varepsilon_{1, 2}}^2 \cup D_{\varepsilon_{1, 2}}^3 \cup D_{\varepsilon_{1, 2}}^4.

下面的证明与定理3.4的证明类似,所以省略.

4 数值模拟与结论

文章研究了白噪声扰动下的随机捕食-食饵系统.利用随机分析的方法,证明了系统(1.1)对于任意的正初始值,存在唯一的全局正解;平均意义下的持久性;使用Has'minskii的平稳分布理论及周期性理论得到了系统(1.1)满足一定条件,存在平稳分布并且是遍历的,这些条件反映了大幅度环境噪声可能会使系统变得不稳定;进而证明了对于任意的正初始值,系统(1.2)存在正周期解.

为了验证理论结果,采用Milstein高阶方法[18]对随机系统(1.1)和(1.2)进行数值模拟.

对于系统(1.1),取r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \ell_1=3; \sigma_1=0.01; \sigma_2=0.01可得\frac{r_1-\frac{1}{2}\sigma_1^2}{r_2-\frac{\sigma_2^2}{2}}\approx3.3344>\ell_1, b_1k_2^2+4\ell_1b_2(b_1-\ell_1 b_2)=-4,显然满足定理3.2和3.4所需的条件.从图 1可知系统(1.1)的解围绕其确定性方程的解在小邻域内波动且存在平稳分布.

图 1

图 1   左侧是系统(1.1)的密度函数图,右侧是系统(1.1)的解与其确定性系统的解.这里r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \sigma_1=0.01; \sigma_2=0.01.


对于系统(1.1),取r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \ell_1=3; \sigma_1=0.1; \sigma_2=0.1可得\frac{r_1-\frac{1}{2}\sigma_1^2}{r_2-\frac{\sigma_2^2}{2}}\approx4.4222>\ell_1, b_1k_2^2+4\ell_1b_2(b_1-\ell_1 b_2)=-4,显然满足定理3.2和3.4所需的条件.从图 2可知系统(1.1)的解围绕其确定性方程的解在小邻域内波动且存在平稳分布.相比图 1系统(1.1)的解的波动区域变大.

图 2

图 2   左侧是系统(1.1)的密度函数图,右侧是系统(1.1)的解与其确定性系统的解.这里r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \sigma_1=0.01; \sigma_2=0.01.


对于系统(1.1),取r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \sigma_1=1.5; \sigma_2=0.1,显然满足定理3.3 (i)条件.从图 3可知食饵x灭绝,捕食者y平均持续生存.

图 3

图 3   系统(1.1)的解与其确定性系统的解.这里r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \sigma_1=1.5; \sigma_2=0.1.


对于系统(1.1),取r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \sigma_1=0.1; \sigma_2=0.85,显然满足定理3.3 (ii)条件.从图 4可知捕食者y灭绝,食饵x平均持续生存.

图 4

图 4   系统(1.1)的解与其确定性系统的解.这里r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \sigma_1=0.1; \sigma_2=0.85.


对于系统(1.1),取r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \sigma_1=1.5; \sigma_2=0.85,显然满足定理3.3 (iii)条件.从图 5可知食饵x与捕食者y均灭绝.

图 5

图 5   系统(1.1)的解与其确定性系统的解.这里r_1=1; r_2=0.3; a_1=0.5; b_1=0.5; b_2=0.5; k_2=2; x(0)=1; y(0)=0.2; \sigma_1=1.5; \sigma_2=0.85.


图 4图 5对比可知,在y灭绝的情况下,小的白噪声可促进x的生长;过大的白噪声可使x灭绝.

对于系统(1.2),取r_1=1+0.5\sin t; r_2=0.3+0.05\sin t; a_1=0.05+0.002\sin t; b_1=1+0.5\sin t; b_2=1+0.5\sin t; k_2=5+0.5\sin t; x(0)=10; y(0)=5; \sigma_1=0.01+0.002\sin t; \sigma_2=0.01+0.0006\sin t.显然满足定理3.5的条件.从图 7可以看出,经过一段时间后,确定系统的解会进入周期轨道;当噪声强度较小时,随机系统(1.2)的解会在周期轨道的小邻域内震动.

图 6

图 6   左侧是系统(1.2)的解,右侧是相应的确定系统的解,这里r_1=1+0.5\sin t; r_2=0.3+0.05\sin t; a_1=0.05+0.002\sin t; b_1=1+0.5\sin t; b_2=1+0.5\sin t; k_2=5+0.5\sin t; x(0)=10; y(0)=5; \sigma_1=0.01+0.002\sin t; \sigma_2=0.01+0.0006\sin t.


图 7

图 7   左侧为系统(1.2)的相轨线,右侧为相应的确定系统的相轨线.这里r_1=1+0.5\sin t; r_2=0.3+0.05\sin t; a_1=0.05+0.002\sin t; b_1=1+0.5\sin t; b_2=1+0.5\sin t; k_2=5+0.5\sin t; x(0)=10; y(0)=5; \sigma_1=0.01+0.002\sin t; \sigma_2=0.01+0.0006\sin t.


参考文献

Cai Y L , Kang Y , Wang W M .

A stochastic SIRS epidemic model with nonlinear incidence rate

Appl Math Comput, 2017, 305: 221- 240

URL     [本文引用: 2]

Liu Q , Jiang D Q , Shi N Z , et al.

Dynamical behavior of a stochastic HBV infection model with logistic hepatocyte growth

Acta Math Sci, 2017, 37B (4): 927- 940

URL    

魏凤英, 林青腾.

一类具有校正隔离率随机SIQS模型的绝灭性与分布

数学物理学报, 2017, 37A (6): 1148- 1161

URL    

Wei F Y , Lin Q T .

Extinction and distribution for an SIQS epidemi model with quarantined-adjusted incidence

Acta Math Sci, 2017, 37A (6): 1148- 1161

URL    

Wang W M , Cai Y L , Li J L , Gui Z J .

Periodic behavior in a FIV model with seasonality as well as environment fluctuations

J Franklin I, 2017, 354: 7410- 7428

DOI:10.1016/j.jfranklin.2017.08.034      [本文引用: 2]

Liu M , Mandal P S .

Dynamical behavior of a one-prey two-predator model with random perturbations

Commun Nonlinear Sci, 2015, 28 (1/3): 123- 137

URL     [本文引用: 2]

Ji C Y , Jiang D Q , Shi N Z .

A note on a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation

J Math Anal Appl, 2011, 377 (1): 435- 440

DOI:10.1016/j.jmaa.2010.11.008     

Mandal P S , Banerjee M .

Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model

Physica A, 2012, 391 (4): 1216- 1233

DOI:10.1016/j.physa.2011.10.019      [本文引用: 1]

Jiang D Q , Zuo W J , Hayat T , Alsaedi A .

Stationary distribution and periodic solutions for stochastic Holling-Leslie predator-prey systems

Physica A, 2016, 460: 16- 28

DOI:10.1016/j.physa.2016.04.037      [本文引用: 1]

Ji C Y , Jiang D Q , Shi N Z .

Analysis of a predator-prey model with modified Leslie-Gower and Holling-type Ⅱ schemes with stochastic perturbation

J Math Anal Appl, 2009, 359 (2): 482- 498

DOI:10.1016/j.jmaa.2009.05.039      [本文引用: 3]

Zhao D L , Yuan S L .

Dynamics of the stochastic Leslie-Gower predator-prey system with randomized intrinsic growth rate

Physica A, 2016, 461: 419- 428

DOI:10.1016/j.physa.2016.06.010     

Xu Y , Liu M , Yang Y .

Analysis of a stochastic two-predators one-prey system with modified Leslie-Gower and Holling-type Ⅱ schemes

J Appl Anal Comput, 2017, 7 (2): 713- 727

URL    

Liu M , Wang K .

Dynamics of a two-prey one-predator system in random environments

J Nonlinear Sci, 2013, 23 (5): 751- 775

DOI:10.1007/s00332-013-9167-4     

Qiu H , Liu M , Wang K , Wang Y .

Dynamics of a stochastic predator-prey system with BeddingtonDeAngelis functional response

Appl Math Comput, 2012, 219 (4): 2303- 2312

Liu M , Wang K .

Persistence, extinction and global asymptotical stability of a non-autonomous predatorprey model with random perturbation

Appl Math Model, 2012, 36 (11): 5344- 5353

DOI:10.1016/j.apm.2011.12.057      [本文引用: 1]

王克. 随机生物数学模型. 北京: 科学出版社, 2010

[本文引用: 3]

Wang K . Random Biological Mathematical Model. Beijing: Science Press, 2010

[本文引用: 3]

Khasminskii R . Stochastic Stability of Differential Equations. Alphen aan den Rijn: Sijthoff & Noordhoff, 1980

[本文引用: 2]

Mao X R , Marion G , Renshaw E .

Environmental Brownian noise suppresses explosions in population dynamics

Stoch Proc Appl, 2002, 97 (1): 95- 110

DOI:10.1016/S0304-4149(01)00126-0      [本文引用: 1]

Kloeden P , Platen E . Numerical Solution of Stochastic Differential Equations. Berlin: Springer, 1999

[本文引用: 1]

/