在现实世界中,生态系统经常经历人类的开发活动与自然因素的影响,如收获、种植、洪水、深林砍伐、狩猎等. 这些现象不能被认为是连续的. 用脉冲微分方程描述这样的现象更符合实际的. 随着脉冲微分方程理论的发展,用脉冲微分方程描述的种群动力系统被提出并得到广泛地研究^{[1, 2, 3, 4, 5, 6, 7, 8, 9]}. 文献[8, 9, 10, 11]给出了 脉冲扰动系统具有一些重要且有趣的动力学行为,包括持久性、灭绝、正解的全局吸引及复杂动力学行为等.

另一方面,自然界中种群的增长不可避免地受到随机噪声的影响^{[12, 13, 15, 16, 17, 18, 19, 20, 21, 22, 23]}. May^[14]指出生态种群的增长率应该是随机的. 通常说环境噪声具有两类,一类是白噪声,另一类是颜色噪声也叫做电报噪声^[24]. 可以把颜色噪声看做两个或更多个环境状态间转换^[25]. 已经发现种群的增长率和环境容纳量也会受到 颜色噪声的影响. 然而从目前掌握的文献可知研究脉冲效应和颜色噪声扰动的种群动力系统的文献较少.

通过上面叙述和著名的时滞捕食-食饵系统

$\left\{ \begin{array}{lll} \left. \begin{array}{lll} {\rm d}u_1(t)=u_1(r_{1}-b_{11}u_1(t)-b_{12}u_2(t-\tau_1(t))){\rm d}t,\\ {\rm d}u_2(t)=u_2(-r_{2}-b_{22}u_2(t)+b_{21}u_1(t-\tau_2(t))){\rm d}t,\\ \end{array} \right. \end{array} \right.$

(1.1)

第2节提出具有环境噪声和脉冲扰动的捕食-食饵模型. 第3节我们给出一些定义和引理. 第4节将证明系统存在全局正解,给出系统解的均值上确界的极限估计且证明系统的解 是最终随机有界的. 第五节研究系统的渐近性质. 第六节确定种群灭绝、非平均持续生存的条件. 最后一节给出简单结论.

令 $(\Omega,F,\{F_t\}_{t\geq 0},p)$ 是滤子$\{F_t\}_{t\geq 0}$满足通常条件的完备概率空间. $B(t)=\{B_1(t),B_2(t)\}$为完备概率空间$(\Omega,F,\{F_t\}_{t\geq 0},p)$上的2维独立的布朗运动. $\gamma(t)$是取值在有限状态空间$S=\{1,2,\cdots,L\}$ 具有生成元 $Q=(q_{ij})_{L\times L}$右连续的Markov链. 而$Q=(q_{ij})_{L\times L}$由 $$ P=\{\gamma(t+\Delta t)=j|\gamma(t)=i\}= \left\{\begin{array}{lll}\left.\begin{array}{lll}q_{ij}\Delta t+o(\Delta t),& \mbox{如果} \ \ j\neq i;\\1+q_{ii}\Delta t+o(\Delta t),~~&\mbox{如果} \ \ j=i \end{array} \right. \end{array} \right. $$ 确定,其中$\Delta t\geq 0$. 当$i\neq j$时,$q_{ij}\geq 0$ 是状态$i$到状态$j$转移率且满足$\sum\limits_{j=1}^{L}q_{ij}=0,i=1,2,\cdots,L$. 假设Markov链$\gamma(t)$是不可约的,即系统能在任意状态转换到其他状态. 从Markov链$\gamma(t)$的不可约性知道此链有唯一的平稳分布 $\pi=(\pi_1,\pi_2,\cdots,\pi_L)$ 满足 $\pi Q=0,\sum\limits_{i=1}^L\pi_i=1,$ $\pi_i>0$,$\forall i\in S.$

在现实世界种群系统不可避免地经历环境噪声的干扰,种群系统会发生明显的变化. 因此考虑随机扰动的生态系统是非常重要的. 首先考虑白噪声. 假设食饵种群的增长率$r_1(t)$和捕食者的死亡率$r_2(t)$承受白噪声的影响,令$r_i\rightarrow r_i+\sigma_i{\rm d}B_i(t)$,其中${\rm d}B_i(t)$是白噪声,$\sigma_i^2$是白噪声的强度. 其次考虑著名的电报噪声. 由于在不同环境状态之间转换是无记忆和下一个转换的等待时间具有 指数分布,文献[26]给出电报噪声可以用具有有限状态空间$S$的有连续Markov链$\gamma(t)$来描述. 因此方程(1.1)改写为

$\left\{\begin{array}{lll}\left.\begin{array}{lll}{\rm d}u_1=u_1(r_{1}(\gamma(t))-b_{11}(\gamma(t))u_1-b_{12}(\gamma(t))u_2(t-\tau_1(t))){\rm d}t+\sigma_1(\gamma(t))u_1{\rm d}B_1(t),\\{\rm d}u_2=u_2(-r_{2}(\gamma(t))-b_{22}(\gamma(t))u_2+b_{21}(\gamma(t))u_1(t-\tau_2(t))){\rm d}t-\sigma_2(\gamma(t))u_2{\rm d}B_2(t).\\ \end{array} \right. \end{array} \right.$

(2.1)

方程(2.1)描述的生态系统解释如下. 如果初始状态$\gamma(0)=i\in S$,则 方程(2.1)在$\gamma(t)$转换到一个新的状态$j\in S$前满足 $$\left\{ \begin{array}{l} \left. \begin{array}{l} {\rm d}u_1=u_1(r_{1}(i)-b_{11}(i)u_1-b_{12}(i)u_2(t-\tau_1(t))){\rm d}t+\sigma_1(i)u_1{\rm d}B_1(t),\\ {\rm d}u_2=u_2(-r_{2}(i)-b_{22}(i)u_2+b_{21}(i)u_1(t-\tau_2(t))){\rm d}t-\sigma_2(i)u_2{\rm d}B_2(t),\\ \end{array} \right. \end{array} \right. $$直到下一个转换系统(2.1)满足方程 $$\left\{\begin{array}{lll}\left.\begin{array}{lll}{\rm d}u_1=u_1(r_{1}(j)-b_{11}(j)u_1-b_{12}(j)u_2(t-\tau_1(t))){\rm d}t+\sigma_1(j)u_1{\rm d}B_1(t),\\{\rm d}u_2=u_2(-r_{2}(j)-b_{22}(j)u_2+b_{21}(j)u_1(t-\tau_2(t))){\rm d}t-\sigma_2(j)u_2{\rm d}B_2(t).\\ \end{array} \right. \end{array} \right. $$

本文研究下列具有Markov链转换和脉冲扰动的随机时滞捕食-食饵系统

$\left\{ \begin{array}{lll} \left. \begin{array}{lll} {\rm d}u_1=u_1(r_{1}(\gamma(t))-b_{11}(\gamma(t))u_1-b_{12}(\gamma(t))u_2(t-\tau_1(t))){\rm d}t+\sigma_1(\gamma(t))u_1{\rm d}B_1(t)\\{\rm d}u_2=u_2(-r_{2}(\gamma(t))-b_{22}(\gamma(t))u_2+b_{21}(\gamma(t))u_1(t-\tau_2(t))){\rm d}t-\sigma_2(\gamma(t))u_2{\rm d}B_2(t)\\\end{array} \right\}\ t\neq t_k,\\[3mm]\left. \begin{array}{lll} u_1(t^+)=(1+a^1_k)u_1(t)\\ u_2(t^+)=(1+a^2_k)u_2(t)\\ \end{array} \right \} \ \ \ \ \ \ t=t_k,\\ \end{array} \right.$

(2.2)

其中$u_1(t),u_2(t)$ 表示 $t$时刻食饵种群与捕食者种群的密度,$k\in N$,$N$是正整数的集合.

论文做如下假设:

($H_1$) $\check{r_i}>0,\check{b_{ij}}>0,\check{ \sigma_i}>0,i,j=1,2,$ 其中 $\check{f}=\min\limits_{i\in S}f(i),\hat{f}=\max\limits_{i\in S}f(i).$

($H_2$) 脉冲时刻$t_k$ 满足$0=t_0<t_1<t_2<\cdots<t_k<t_{k+1}<\cdots$ 且$\lim\limits_{k\rightarrow +\infty}t_k=+\infty,k\in N$.

($H_3$) 当 $a_k^i>0,i=1,2$时,表示对种群进行放养; 当 $a_k^i<0,i=1,2,$时,表示对种群进行收获. $\{a_k^i\}_{k=1}^{+\infty}$ 是满足$a^i_k\in (-1,1),i=1,2$的实数列.

($H_4$) 对 $0\leq s<t$,有 $m_i\leq\prod\limits_{s\leq t_k<t}(1+a_k^i)\leq M_i$,其中 $m_i,M_i,i=1,2$ 是正常数.

($H_5$) $\tau_i(t),t\in[0,+\infty)$是非负、有界的连续可微函数. $\tau_i' (t)$是满足 $\tau^{'u}=\sup\limits_{t\in[0,+\infty)}\tau' (t)<1$的有界函数.

($H_6$) 令 $\tau=\max\limits_{i=1,2}\sup\limits_{t\geq0}\tau_i(t).$ $C=C([-\tau,0];{\Bbb R}_+)$ 表示定义在$[-\tau,0]$ 的连续函数的集合. 对于任意给定$\varphi_i(\theta)\in C$,系统 (2.2)的初始值满足

$u_i(\theta)=\varphi_i(\theta)\geq 0,-\tau\leq \theta\leq 0; \sup\limits_{-\tau\leq \theta\leq 0}\varphi_i(\theta)<\infty,i=1,2.$

(2.3)

本节给出将要用到的一些定义和引理. 考虑下列具有Markov链转换的脉冲微分方程

$\left\{\begin{array}{lll}{\rm d}Y(t)=F(t,\gamma(t),Y(t)){\rm d}t+G(t,\gamma(t),Y(t)){\rm d}B(t),\ & t\neq t_k,k\in N,\\ Y(t^+)-Y(t)=B_kY(t),& t=t_k. \end{array} \right.$

(3.1)

定义3.1 设函数 $Y(t)=(y_1(t),y_2(t),\cdots,y_n(t))^T,t\in {\Bbb R}_+=[0,+\infty)$ 满足

(1) $Y(t)$是$F_t$适应的且在$(0,t_1]$和每个区间$(t_k,t_{k+1}],k\in N$上连续的; $F(t,\gamma (t),Y(t))\in {{L}^{1}}([0,+\infty ),{{\mathbb{R}}^{n}}),G(t,\gamma (t),Y(t))\in {{L}^{2}}([0,+\infty );{{\mathbb{R}}^{n}})$,其中$L^k(([0,+\infty);{\Bbb R}^n)$是满足对$T>0$,$\int_0^T|g(s)|^k{\rm d}s<\infty$的${\Bbb R}^n$值可测的$F_t$适应过程$g(t)$的全体;

(2) 对每个$t_k$,$Y(t_{k}^{-})=\underset{t\to t_{k}^{-}}{\mathop{\lim }}\,Y(t)$和$Y(t_{k}^{+})=\underset{t\to t_{k}^{+}}{\mathop{\lim }}\,Y(t)$ 存在且$Y(t_k)=Y(t_k^-),$ a.s.;

(3) 对$t\in {\Bbb R}_{+}-\{t_k\}$,$Y(t)$满足与系统(3.1)等价的积分方程,在$t=t_k,k\in N$ 以概率1满足脉冲条件; \\则称$Y(t)$是系统(3.1)具初始条件$Y(0),\gamma(0)$的解.

考虑下列具有Markov转换无脉冲的捕食-食饵系统

$\left\{\begin{array}{lll}\left.\begin{array}{lll}{\rm d}x_1(t)=x_1\left(r_1(\gamma(t))-a_{11}(\gamma(t))x_1(t)-a_{12}(\gamma(t))x_2(t-\tau_1(t))\right){\rm d}t+\sigma_1(\gamma(t))x_1{\rm d}B_1(t),\\{\rm d}x_2(t)=x_2\left(-r_2(\gamma(t))-a_{22}(\gamma(t))x_2(t)+a_{21}(\gamma(t))x_1(t-\tau_2(t))\right){\rm d}t-\sigma_2(\gamma(t))x_2{\rm d}B_2(t),\\ \end{array} \right. \end{array} \right.$

(3.2)

初始条件为(2.3)式且

$a_{11}(\gamma(t))=b_{11}(\gamma(t))\prod\limits_{0<t_k<t}(1+a_k^1),a_{12}(\gamma(t))=b_{12}(\gamma(t))\prod\limits_{0<t_k<t-\tau_1(t)}(1+a_k^2),$

(3.3)

$a_{21}(\gamma(t))=b_{21}(\gamma(t))\prod\limits_{0<t_k<t-\tau_2(t)}(1+a_k^1),a_{22}(\gamma(t))=b_{22}(\gamma(t))\prod\limits_{0<t_k<t}(1+a_k^2).$

(3.4)

引理3.1 如果假设($H_1$)-($H_3$),($H_6$)成立,则有

(1) 若$X(t)=(x_1(t),x_2(t)),t\in [-\tau,+\infty)$是系统(3.2)的解,则$U(t)=(\prod\limits_{0<t_k<t}(1+a_k^1)x_1(t),\prod\limits_{0<t_k<t}(1+a_k^2)x_2(t)),t\in [-\tau,+\infty)$是系统(2.2)的解.

(2) 若$U(t)=(u_1(t),u_2(t)),t\in [-\tau,+\infty)$是系统(2.2)的解,则$X(t)=(\prod\limits_{0<t_k<t}(1+a_k^1)^{-1}u_1(t),\prod\limits_{0<t_k<t}(1+a_k^2)^{-1}u_2(t)),t\in [-\tau,+\infty)$为系统(3.2)的解.

证 (1) 易见$U(t)=(\prod\limits_{0<t_k<t}(1+a_k^1)x_1(t),\prod\limits_{0<t_k<t}(1+a_k^2)x_2(t))$ 在$(t_k,t_{k+1}]$是绝对连续的,对任意$t\neq t_k$有 \begin{eqnarray*}{\rm d}u_1(t)&=&\prod\limits_{0<t_k<t}(1+a_k^1){\rm d}x_1(t)\\&=&\prod\limits_{0<t_k<t}(1+a_k^1)\big(x_1\left(r_1(\gamma(t))-a_{11}(\gamma(t))x_1(t)-a_{12}(\gamma(t))x_2(t)\right){\rm d}t\\&&+\sigma_1(\gamma(t))x_1{\rm d}B_1(t)\big)\\&=&u_1(t)\left(r_1(\gamma(t))-b_{11}(\gamma(t))u_1(t)-b_{12}(\gamma(t))u_2(t)\right){\rm d}t+\sigma_1(\gamma(t))u_1{\rm d}B_1(t),\\[2mm]{\rm d}u_2(t)&=&\prod\limits_{0<t_k<t}(1+a_k^2){\rm d}x_2(t)\\&=&\prod\limits_{0<t_k<t}(1+a_k^2)\big(x_2\left(r_2(\gamma(t))-a_{22}(\gamma(t))x_2(t)+a_{21}(\gamma(t))x_1(t)\right){\rm d}t\\&&-\sigma_2(\gamma(t))x_2{\rm d}B_2(t)\big)\\&=&u_2(t)\left(-r_2(\gamma(t))-b_{22}(\gamma(t))u_2(t)+b_{21}(\gamma(t))u_1(t)\right){\rm d}t-\sigma_2(\gamma(t))u_2{\rm d}B_2(t).\end{eqnarray*}

另一方面,对每个$t=t_k$可得 $$u_1(t_k^+)=\lim\limits_{t\rightarrow t_k^+}\prod\limits_{0<t_j<t}(1+a_j^1)x_1(t)=\prod\limits_{0<t_j\leq t_k}(1+a_j^1)x_1(t_k), $$ $$u_2(t_k^+)=\lim\limits_{t\rightarrow t_k^+}\prod\limits_{0<t_j<t}(1+a_j^2)x_2(t)=\prod\limits_{0<t_j\leq t_k}(1+a_j^2)x_2(t_k), $$ $$u_1(t_k)=\prod\limits_{0<t_j<t_k}(1+a_j^1)x_1(t_k),u_2(t_k)=\prod\limits_{0<t_j<t_k}(1+a_j^1)x_2(t_k), $$所以对$k=1,2,\cdots,$有$u_1(t_k^+)=(1+a_k^1)u_1(t_k),u_2(t_k^+)=(1+a_k^2)u_2(t_k).$ 同理可证(2).

定义3.2 对任意的$\varepsilon\in(0,1)$,存在整数$H=H(\varepsilon)$,使得系统(2.2)具初始值$u_i(\theta)=\varphi_i(\theta)\geq 0,-\tau\leq\theta\leq 0$的解$U(t)=(u_1(t),u_2(t))$满足$\lim\limits_{t\rightarrow\infty}P\{|U(t)|>H\}<\varepsilon$,则称该解是随机最终有界的.

定义 3.3 对系统(2.2)的任意一个解$U(t)=(u_1(t),u_2(t))$,则

1) 如果 $\lim\limits_{t\rightarrow +\infty}u_i(t)=0,i=1,2,$ 则称种群$u_i(t)(i=1,2)$灭绝.

2) 如果$ \lim\limits_{t\rightarrow+\infty}\sup\frac{1}{t}\int_0^tu_i(s){\rm d}s=0,i=1,2,$则称种群$u_i(t)(i=1,2)$是平均非持续生存.

引理3.2 随机微分方程

${\rm d}x_i(t)=f_i(t,x_i(t)){\rm d}t+g(t,x_i(t){\rm d}B(t),i=1,2,$

(3.5)

其中 $f_i(t,x_i)\in C([0,+\infty),{\Bbb R})(i=1,2)$ 和 $g(t,x_i)\in C([0,+\infty),{\Bbb R})$.若满足

(1) 方程(3.5) 有初始值为$x_i(t_0)\in {\Bbb R}$的唯一全局连续解$\overline{x_i}(t,t_0,x_i(t_0)),i=1,2$,$t\geq t_0\geq 0$;

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

(3) 存在 $\rho\in C([0,+\infty),{\Bbb R})$ 满足 $\rho(0)=0$ 以及 $\int_{0^+}^{+\infty}\rho(s){\rm d}s=\infty$的函数 $\rho(s),$ 使得 $$|g(x,t)-g(y,t)|\leq \rho(|x-y|),x,y\in {\Bbb R},t\geq 0; $$如果$x_1(0)\leq x_2(0),$ 则有$P\{\overline{x}_1(t,0,x_1(0))\leq \overline{x}_2(t,0,x_2(0)),t\geq 0\}=1.$

因为系统(3.2)描述一个种群动力系统,所以保证系统解的正性和不在有限时间内爆破是至关重要的. 为了使模型有意义需要证明系统解是全局的和非负的. 然而系统(3.2)不满足随机微分方程解的存在唯一性定理^{[27, 28]}. 所以使用文献[29]确定的方法,证明系统(3.2)的全局正解的存在性与唯一性.

定理4.1 令假设($H_1-H_3$,$H_5-H_6$)成立. 对任意给定初始值(2.3),系统(3.2)存在唯一解$X(t)=(x_1(t),x_2(t))$ $(t>-\tau)$ 且以概率1停留在${\Bbb R}_+^2$中.

证 因为系统(3.2)的系数满足局部李氏且连续,对任意给定初始值$x_i(\theta)=\varphi_i(\theta)\geq 0,-\tau\leq \theta\leq 0$,系统(3.2)存在唯一局部正解$X(t)$,$t\in [0,\tau_e)$,$\tau_e$是爆破时刻. 为了证明该解是全局的,只需证明$\tau_e=\infty$ a.s.即可.

设$k_0>0$充分大,使得$\varphi_i(t)(i=1,2)$均落在区间$[k_0^{-1},k_0]$内. 对任意正整数$k>k_0$,定义停时 $$\tau_k=\inf \{t \in [0,\tau_e): x_1(t)\bar{\in}(k^{-1},k)\ \ \mbox{或} \ \ x_2(t)\bar{\in}(k^{-1},k)\}. $$

显然$\tau_k$随 $k\rightarrow \infty$单调增加. 令$\tau_{\infty}=\lim\limits_{k\rightarrow \infty}\tau_k,$ 有$\tau_{\infty}<\tau_e$ a.s. 如果能证明$\tau_{\infty}=\infty$ a.s.,那么 $\tau_e=\infty$ a.s. 且当$t\geq 0$时$(x_1(t),x_2(t)\in {\Bbb R}_+^2,$ a.s. 现在只需证明$\tau_{\infty}=\infty$ a.s.成立,否则存在常数$T>0$ 和$\varepsilon\in (0,1)$,有$p\{\tau_\infty\leq T\}>\varepsilon.$成立. 所以存在$k_1>k_0$,对所有的$k>k_1$有$p\{\tau_k\leq T\}\geq\varepsilon$成立. 定义李氏函数$V:{\Bbb R}_+^2\rightarrow {\Bbb R}_+$ 为 $$V(X)=\sum\limits_{i=1}^2(x_i+1-\ln x_i)+\frac{1}{2(1-\tau_2^{'u})}\check{a}_{11}\int_{t-\tau_2(t)}^tx_1^2(s){\rm d}s+\frac{1}{2(1-\tau_1^{'u})}\check{a}_{22}\int_{t-\tau_1(t)}^tx_2^2(s){\rm d}s. $$

当$y>0$时,$y+1-\ln y>0$. 如果 $(x_1(t),x_2(t))\in {\Bbb R}_+^2,$则 $V(x(t))$ 正定的. 根据广义 Itô 公式 $$\begin{array}{lll}{\rm d}V(t)&=& LV(t){\rm d}t+ \sigma_{1}(\gamma(t))x_1(t){\rm d}B_1(t)-\sigma_{2}(\gamma(t))x_2(t){\rm d}B_2(t),\\\end{array} $$其中\begin{eqnarray*}LV(t)&\leq& (x_1(t)-1)(r_1(\gamma(t))-a_{11}(\gamma(t))x_1(t)-a_{12}(\gamma(t))x_2(t-\tau_1(t)))\\ &&+ (x_2(t)-1)(r_2(\gamma(t))+a_{21}(\gamma(t))x_1(t-\tau_2(t))-a_{22}(\gamma(t))x_2(t))\\ &&+ \frac{\check{a}_{11}}{2(1-\tau_2^{'u})}x_1^2(t)+ \frac{\check{a}_{22}}{2(1-\tau_2^{'u})}x_2^2(t) -\frac{\breve{a}_{11}}{2}x_1^2(t-\tau_2(t))-\frac{\breve{a}_{11}}{2}x_2^2(t-\tau_1(t))\\ &&+ \frac{1}{2}\sigma_1^2(\gamma(t))+\frac{1}{2}\sigma_2^2(\gamma(t)),\end{eqnarray*}一定存在正数 $M$ 有 $LV(t)\leq M$,所以

$\text{d}V(t)\le M\text{d}t+{{\sigma }_{1}}(\gamma (t)){{x}_{1}}(t)\text{d}{{B}_{1}}(t)-{{\sigma }_{2}}(\gamma (t)){{x}_{2}}(t)\text{d}{{B}_{2}}(t).$

(4.1)

(4.1)式两端从$0$到$\tau_k\wedge T$积分并取均值可得

$EV(X(\tau_k\wedge T))\leq V(X(0))+MT.$

(4.2)

对每个 $\omega\in \{\tau_k\leq T\}$,则存在某个 $i$使得 $x_i(\tau_k,\omega)\bar{\in}(k^{-1},k),i=1,2$成立. 这样有 $$V(X(\tau_k))\geq x_i(\tau_k)+1-\ln x_i(\tau_k)\geq (\frac{1}{k}+1-\ln \frac{1}{k})\wedge (k+1-\ln k), $$从而可得

$\begin{align} & \infty >V(X(0)+KT\ge EV(X({{\tau }_{k}}\wedge T))= \\ & P({{\tau }_{k}}\le T)V(X({{\tau }_{k}}))+P({{\tau }_{k}}>T)V(X(T))\ge \\ & p({{\tau }_{k}}\le T)(\frac{1}{k}+1-\ln \frac{1}{k})\wedge (k+1-\ln k). \\ \end{align}$

令$k\rightarrow \infty$,得到$\infty> V(X(0)+KT=\infty$矛盾. 所以一定有$\tau_\infty=\infty$ a.s.

从定理4.1和引理3.1,下列结论成立.

定理4.2 定理4.1的条件成立. 对任意给定初始值(2.3),系统(2.2)存在唯一解$U(t)=(u_1(t),u_2(t))$ $(t>-\tau)$ 且以概率1停留在${\Bbb R}_+^2$中.

引理4.3 定理4.1的条件成立且满足$\frac{\hat{a}_{21}}{(p+1)(1-\tau_2^{'u})}-a_{11}(\gamma(t))<0,\frac{a_{21}(\gamma(t))}{p+1}-a_{22}(\gamma(t))<0$,则系统(3.2)的解$X(t)=(x_1(t),x_2(t))$满足 $$\lim\limits_{t\rightarrow +\infty}\sup E|X(t)|^p\leq C,p\in (1,2], $$其中$C$是不依赖初值$x_i(\theta)=\varphi_i(\theta)\geq 0,-\tau\leq \theta\leq 0,i=1,2$的常数.

证 令$V(X(t))=x_1^p(t)+x_2^p(t),$ 对${\rm e}^tV(X(t))$使用Itô公式得

$\begin{align} & \text{d}({{\text{e}}^{t}}V(X(t)))\le P{{\text{e}}^{t}}\{\frac{1}{p}x_{1}^{p}+\frac{1}{p}x_{2}^{p}+{{r}_{1}}(\gamma (t))x_{1}^{p}-{{a}_{11}}(\gamma (t))x_{1}^{p+1}- \\ & {{r}_{2}}(\gamma (t))x_{2}^{p}+{{a}_{21}}(\gamma (t)){{x}_{1}}(t-{{\tau }_{2}}(t))x_{2}^{p}-{{a}_{22}}(\gamma (t))x_{2}^{p+1} \\ & +\frac{p-1}{2}\sigma _{1}^{2}(\gamma (t))x_{1}^{p}-\frac{p-1}{2}\sigma _{2}^{2}(\gamma (t))x_{2}^{p}\}\text{d}t \\ & +p{{\text{e}}^{t}}({{\sigma }_{1}}(\gamma (t))x_{1}^{p}\text{d}{{B}_{1}}(t)-{{\sigma }_{2}}(\gamma (t))x_{2}^{p}\text{d}{{B}_{2}}(t)). \\ \end{align}$

(4.3)

令$p\in (1,2)$,利用不等式$x_1x_2^p\leq\frac{1}{1+p}x_1^{p+1}+\frac{p}{1+p}x_2^{p+1}$和(4.3)式推得\begin{eqnarray*}{\rm d}({\rm e}^tV(X(t)))&\leq&P{\rm e}^t\bigg\{\frac{1}{p}x_1^p+\frac{1}{p}x_2^p+r_1(\gamma(t))x_1^p-a_{11}(\gamma(t))x_1^{p+1}-r_2(\gamma(t))x_2^p\\ &&+\frac{a_{21}(\gamma(t))}{p+1}x_1^{p+1}(t-\tau_2(t))+\frac{a_{21}(\gamma(t))}{p+1}x_2^{p+1}-a_{22}(\gamma(t))x_2^{p+1}\\ &&+\frac{p-1}{2}\sigma_1^2(\gamma(t))x_1^p -\frac{p-1}{2}\sigma_2^2(\gamma(t))x_2^p\bigg\}{\rm d}t\\ &&+p{\rm e}^t(\sigma_1(\gamma(t))x_1^p{\rm d}B_1(t)-\sigma_2(\gamma(t))x_2^p{\rm d}B_2(t)).\end{eqnarray*}

为了估计时滞项引进非负函数$V_1(X(t))=\frac{p \hat{a}_{21}}{(p+1)(1-\tau_2^{'u})}\int_{t-\tau_2(t)}^t{\rm e}^sx_1^{p+1}(s){\rm d}s,$ 所以 $$\begin{align} & \text{d}({{\text{e}}^{t}}V(X(t))+{{V}_{1}}(X(t)))\le \\ & {{\text{e}}^{t}}F(X(t))\text{d}t+p{{\text{e}}^{t}}({{\sigma }_{1}}(\gamma (t))x_{1}^{p}\text{d} \\ & {{B}_{1}}(t)-{{\sigma }_{2}}(\gamma (t))x_{2}^{p}\text{d}{{B}_{2}}(t)),\\ \end{align} $$其中\begin{eqnarray*}F(X(t))&=&P{\rm e}^t\bigg\{\frac{1}{p}x_1^p+\frac{1}{p}x_2^p+r_1(\gamma(t))x_1^p-a_{11}(\gamma(t))x_1^{p+1}-r_2(\gamma(t))x_2^p\\ &&+ \frac{\hat{a}_{21}x_1^{p+1}}{(p+1)(1-\tau_2^{'u})}+\frac{a_{21}(\gamma(t))}{p+1}x_2^{p+1}-a_{22} (\gamma(t))x_2^{p+1}\\ &&-\frac{p-1}{2}\sigma_1^2(\gamma(t))x_1^p+\frac{p-1}{2}\sigma_2^2(\gamma(t))x_2^p\bigg\}.\end{eqnarray*}

根据定理的假设,存在不依赖$p$的正常数 $K$使得$F(X(t))\leq K$且有 $${\rm d}({\rm e}^tV(X(t))+V_1(X(t)))\leq {\rm e}^tK{\rm d}t+p{\rm e}^t(\sigma_1(\gamma(t))x_1^p{\rm d}B_1(t)-\sigma_2(\gamma(t))x_2^p{\rm d}B_2(t)). $$

对上式从$0$到$t$积分且取均值得${\rm e}^tEV(X(t))\leq V(X(0))+V_1(X(0))+K({\rm e}^t-1),$进而有$EV(X(t))\leq {\rm e}^{-t}(V(X(0))+V_1(X(0)))+K.$

由于$|X|^2\leq 2\max\limits_{1\leq i\leq 2}x_i^2$,可得$|X|^p\leq 2^{\frac{p}{2}}\max\limits_{1\leq i\leq 2}x_i^p\leq 2^{\frac{p}{2}}\max\limits_{1\leq i\leq 2}V(x(t))$. 因此 $$E|X(t)|^p\leq 2^{\frac{p}{2}}({\rm e}^-t(V(X(0))+V_1(X(0)))+K). $$

所以 $\lim\limits_{t\rightarrow +\infty}\sup E|X(t)|^p\leq 2^{\frac{p}{2}}K\doteq C$成立.

根据引理3.1和引理4.1,下面结论成立.

引理4.4 如果系统(2.2)满足假设($H_1-H_6$),则系统(2.2)的解$U(t)=(u_1(t),u_2(t))$满足 $$\lim\limits_{t\rightarrow \infty}\sup E|U(t)|^p\leq M^pC,~~M=\max\{M_1,M_2\}. $$

定理4.5 如果系统(3.2)满足假设($H_1-H_6$),则系统(3.2)的解是随机最终有界的.

证 根据Chebyshev不等式和引理4.3,对于$H>0$有 $$P\{|X(t)|>H\}\leq \frac{E|X(t)|^2}{H^2}\leq \frac{C}{H^2}. $$

所以对充分大的$H$可得$\lim\limits_{t\rightarrow \infty}P\{|X(t)|>H\}<\varepsilon.$

从引理4.4和定理4.5的证明,容易获得系统(2.2)的解也是随机最终有界的.

本节研究系统(2.2)解的渐近性态.

令 $b_1(\gamma(t))=r_1(\gamma(t))-\frac{1}{2}\sigma_1^2(\gamma(t)),b_2(\gamma(t))=r_2(\gamma(t))+\frac{1}{2}\sigma_2^2(\gamma(t))$.

定理5.1 如果定理4.1的条件和$\frac{\hat{a}_{21}}{1-\tau_2^{'u}}<a_{11}(\gamma(t))$ 成立,则对给定初始值$x_i(\theta)=\varphi_i(\theta)\geq 0,-\tau\leq\theta\leq 0,i=1,2,$ 系统(3.2)的解满足 $$\lim\limits_{t\rightarrow +\infty}\sup\frac{\ln x_1+\ln x_2}{t}\leq \sum\limits_{i=1}^L(b_1(i)+b_2(i))\pi_i. \ \ {\rm a.s.} . $$

证 对$\ln x_1+\ln x_2$ 使用Itô's公式有\begin{eqnarray*}{\rm d}(\ln x_1+\ln x_2)&\leq&(r_1(\gamma(t))-r_2(\gamma(t))-\frac{\sigma_1^2(\gamma(t))}{2}-\frac{\sigma_2^2(\gamma(t))}{2}\\&&-a_{11}(\gamma(t))x_1-a_{22}(\gamma(t))x_2+a_{21}(\gamma(t))x_1(t-\tau_2(t))){\rm d}t\\&&+\sigma_1(\gamma(t)){\rm d}B_1(t)-\sigma_2(\gamma(t)){\rm d}B_2(t).\end{eqnarray*}

为了估计时滞项引进非负函数$W(X(t))=\frac{\hat{a}_{21}}{1-\tau_2^{'u}} \int_{t-\tau_2(t)}^tx_1(s){\rm d}s.$ 所以,

$\begin{gathered} \sum\limits_{i = 1}^2 {\ln } {x_i}(t) + W(X(t)) \leqslant \hfill \\ \sum\limits_{i = 1}^2 {\ln {x_i}} (0) + W(X(0)) + \int_0^t [{r_1}(\gamma (s)) - {r_2}(\gamma (s)) \hfill \\ - \frac{{\sigma _1^2(\gamma (s))}}{2} - \frac{{\sigma _2^2(\gamma (s))}}{2} + (\frac{{{a_{21}}(\gamma (s))}}{{1 - \tau _2^u}} - \hfill \\ {a_{11}}(\gamma (s))){x_1}(s) - {a_{22}}(\gamma (s)){x_2}(s)]{\text{d}}s + M(t),\hfill \\ \end{gathered} $

其中$M(t)=\int_0^t\sigma_1(\gamma(s)){\rm d}B_1(s)-\int_0^t\sigma_2(\gamma(s)){\rm d}B_2(s)$ 且$M(t)$的二次变分为 $$\langle M(t),M(t)\rangle=\int_0^t\sum\limits_{i=1}^2\sigma_i^2(\gamma(s)){\rm d}s. $$

根据指数鞅不等式^[30],对任意的正数$T,\alpha$ 和 $\beta$得到 $$ P(\sup\limits_{0\leq t\leq T}[M(t)-\frac{\alpha}{2}\langle M(t),M(t)\rangle]>\beta)\leq {\rm e}^{-\alpha\beta}. $$

取 $T=n,\alpha=1,\beta=2\ln 2$,从而有$ P(\sup\limits_{0\leq t\leq n}[M(t)-\frac{1}{2}\langle M(t),M(t)\rangle]>2\ln n)\leq \frac{1}{n^2}.$\\

由 $\sum\limits_{i=1}^{\infty}\frac{1}{n^2}<\infty,$ 利用Brorel-Cantelli 引理^[30],对几乎所有的$\omega\in \Omega,$存在一个随机整数$n_0=n_0(\omega)$,当$n\geq n_0$时有$ \sup\limits_{0\leq t\leq n}[M(t)-\frac{1}{2}\langle M(t),M(t)\rangle]\leq 2\ln n.$

当$0\leq t\leq n,n\geq n_0$时有$ M(t)\leq 2\ln n +\frac{1}{2}\langle M(t),M(t)\rangle,$所以得到

$\begin{gathered} \sum\limits_{i = 1}^2 {\ln } {x_i}(t) \leqslant \sum\limits_{i = 1}^2 {\ln {x_i}} (0) + W(X(0)) + \hfill \\ \int_0^t [{r_1}(\gamma (s)) - {r_2}(\gamma (s)) - \frac{{\sigma _1^2(\gamma (s))}}{2} - \frac{{\sigma _2^2(\gamma (s))}}{2}]{\text{d}}s + 2\ln n. \hfill \\ \end{gathered} $

(5.1)

对$n-1\leq t\leq n$,(5.1)式两端除以 $t$有

$\begin{gathered} {t^{ - 1}}\sum\limits_{i = 1}^2 {\ln } {x_i}(t) \leqslant {t^{ - 1}}(\sum\limits_{i = 1}^2 {\ln {x_i}} (0) + W(X(0))) \hfill \\ + \frac{1}{t}\int_0^t [{r_1}(\gamma (s)) - {r_2}(\gamma (s)) - \hfill \\ \frac{{\sigma _1^2(\gamma (s))}}{2} - \frac{{\sigma _2^2(\gamma (s))}}{2}]{\text{d}}s + \frac{{2\ln n}}{{n - 1}}. \hfill \\ \end{gathered} $

(5.2)

所以定理的结论成立.

定理5.2 如果定理4.1的条件和$\frac{{{{\text{e}}^{\alpha \tau }}{{\hat a}_{21}}}}{{1 - {\tau _2}'u}}{a_{11}}(\gamma (t))$ 成立,则对任意给定初始值$x_i(\theta)=\varphi_i(\theta)\geq 0,-\tau\leq\theta\leq 0,i=1,2$与充分小的$\alpha$,系统(3.2)的解满足 $$\lim\limits_{t\rightarrow \infty}\sup\frac{\ln x_1+\ln x_2}{\ln t}\leq 2 \ \ {\rm a.s.}. $$

证 对${{\text{e}}^{\alpha t}}(\sum\limits_{i = 1}^2 {\ln } {x_i}(t)) $使用Itô's 公式易得

$\begin{gathered} {\text{d}}({{\text{e}}^{\alpha t}}\sum\limits_{i = 1}^2 {\ln } {x_i}(t)) = \alpha {{\text{e}}^{\alpha t}}(\sum\limits_{i = 1}^2 {\ln {x_i}} (t)){\text{d}}t + {{\text{e}}^{\alpha t}}\sum\limits_{i = 1}^2 {\text{d}} \ln {x_i}(t) = \hfill \\ \alpha {{\text{e}}^{\alpha t}}(\sum\limits_{i = 1}^2 {\ln {x_i}} (t)){\text{d}}t + \hfill \\ {{\text{e}}^{\alpha t}}\{ ({r_1}(\gamma (t)) - {r_2}(\gamma (t)) - \frac{{\sigma _1^2(\gamma (t))}}{2} - \frac{{\sigma _2^2(\gamma (t))}}{2} \hfill \\ - {a_{11}}(\gamma (t)){x_1} - {a_{22}}(\gamma (t)){x_2} - {a_{12}}(\gamma (t)){x_2}(t - {\tau _1}(t)) \hfill \\ + {a_{21}}(\gamma (t)){x_1}(t - {\tau _2}(t))){\text{d}}t + {\sigma _1}(\gamma (t)){\text{d}}{B_1}(t) - {\sigma _2}(\gamma (t)){\text{d}}{B_2}(t)\} . \hfill \\ \end{gathered} $

为了估计时滞项引进非负函数$\widehat{W}(X(t))=\frac{{\rm e}^{\alpha\tau} \hat{a}_{21}}{1-\tau_2^{'u}} \int_{t-\tau_2(t)}^t{\rm e}^{\alpha s}x_1(s){\rm d}s$,因此,

$\begin{gathered} {\text{d}}({{\text{e}}^{\alpha t}}\sum\limits_{i = 1}^2 {\ln {x_i}} (t) + \hat W(X(t))) \leqslant {{\text{e}}^{\alpha t}}(\alpha \sum\limits_{i = 1}^2 {\ln {x_i}} (t)){\text{d}}t \hfill \\ + {{\text{e}}^{\alpha t}}\{ ({r_1}(\gamma (t)) - {r_2}(\gamma (t)) - \frac{{\sigma _1^2(\gamma (t))}}{2} - \frac{{\sigma _2^2(\gamma (t))}}{2} \hfill \\ - {a_{22}}(\gamma (t)){x_2} - ({a_{11}}(\gamma (t)) - \frac{{{{\text{e}}^{\alpha \tau }}{{\hat a}_{21}}}}{{1 - {\tau _2}'u}}){x_1}){\text{d}}t \hfill \\ + {\sigma _1}(\gamma (t)){\text{d}}{B_1}(t) - {\sigma _2}(\gamma (t)){\text{d}}{B_2}(t)\} . \hfill \\ \end{gathered} $

上式两端从$0$到$t$积分得

$\begin{gathered} {{\text{e}}^{\alpha t}}\sum\limits_{i = 1}^2 {\ln } {x_i}(t)) \leqslant \alpha \sum\limits_{i = 1}^2 {\ln {x_i}} (0) + \hat W(X(0)) + \hfill \\ \int_0^t {{{\text{e}}^{\alpha s}}} \{ \sum\limits_{i = 1}^2 {\ln {x_i}} (s)) + {r_1}(\gamma (s)) - {r_2}(\gamma (s)) - \frac{{\sigma _1^2(\gamma (s))}}{2} \hfill \\ - \frac{{\sigma _2^2(\gamma (s))}}{2} - ({a_{11}}(\gamma (t)) - \frac{{{{\text{e}}^{\alpha \tau }}{{\hat a}_{21}}}}{{1 - {\tau _2}'u}}){x_1} - {a_{22}}(\gamma (s)){x_2}\} {\text{d}}s \hfill \\ + {M_1}(t) - {M_2}(t),\hfill \\ \end{gathered}$

其中$M_i(t)=\int_0^t{\rm e}^s\sigma_1(\gamma(s)){\rm d}B_i(s),i=1,2.$ 令$H(t)=M_1(t)-M_2(t)$得 $$\langle H(t),H(t)\rangle=\int_0^t{\rm e}^{2\alpha s}(\sigma_1^2(\gamma(s))+\sigma_2^2(\gamma(s))){\rm d}s. $$

根据指数鞅不等式^[30],对每个整数$n\geq 1$可得 $$ P\bigg(\sup\limits_{0\leq t\leq n}\bigg[H(t)-\frac{\alpha}{2{\rm e}^{\alpha n}}\langle H(t),H(t)\rangle\bigg]\leq 2{\rm e}^{\alpha n}\ln n\bigg)\leq \frac{1}{n^2}. $$

由 $\sum\limits_{i=1}^{\infty}\frac{1}{n^2}<\infty,$ 利用Brorel-Cantelli 引理^[30],对几乎所有的$\omega\in \Omega,$存在一个随机整数$n_0=n_0(\omega)$,当$n\geq n_0$有 $$ \sup\limits_{0\leq t\leq n}\bigg[H(t)-\frac{1}{2{\rm e}^{\alpha n}}\langle H(t),H(t)\rangle\bigg]\leq 2{\rm e}^ {\alpha n}\ln n. $$

即当$0\leq t\leq n,n\geq n_0$时有$ H(t)\leq 2{\rm e}^{\alpha n}\ln n +\frac{1}{2{\rm e}^{\alpha n}}\langle H(t),H(t)\rangle,$所以得到

$\begin{gathered} {{\text{e}}^{\alpha t}}\sum\limits_{i = 1}^2 {\ln } {x_i}(t) \leqslant \sum\limits_{i = 1}^2 {\ln } {x_i}(0) + \hat W(X(0)) + \hfill \\ \int_0^t {{{\text{e}}^{\alpha s}}} \{ \sum\limits_{i = 1}^2 {\ln {x_i}} (s) + {r_1}(\gamma (s)) - {r_2}(\gamma (s)) - \hfill \\ \frac{{\sigma _1^2(\gamma (s))}}{2} - \frac{{\sigma _2^2(\gamma (s))}}{2} - ({a_{11}}(\gamma (t)) - \frac{{{{\text{e}}^{\alpha \tau }}{{\hat a}_{21}}}}{{1 - {\tau _2}'u}}){x_1} - \hfill \\ {a_{22}}(\gamma (s)){x_2}\} {\text{d}}s + 2{{\text{e}}^{\alpha n}}\ln n. \hfill \\ \end{gathered} $

(5.3)

根据定理的条件对系统(3.2)的解$X(t)$,存在一个正常数$\widetilde{C}$使得下式成立

$\begin{gathered} \sum\limits_{i = 1}^2 {\ln } {x_i}(s) + {r_1}(\gamma (s)) - {r_2}(\gamma (s)) - \hfill \\ \frac{{\sigma _1^2(\gamma (s))}}{2} - \frac{{\sigma _2^2(\gamma (s))}}{2} - ({a_{11}}(\gamma (t)) - \hfill \\ \frac{{{{\text{e}}^{\alpha \tau }}{{\hat a}_{21}}}}{{1 - {\tau _2}'u}}){x_1} - {a_{22}}(\gamma (s)){x_2} \leqslant C. \hfill \\ \end{gathered} $

(5.4)

由(5.3)和(5.4)式,对$0\leq t\leq n$和$n\geq n(\omega) $$${{\text{e}}^{\alpha t}}\sum\limits_{i = 1}^2 {\ln } {x_i}(t)) \leqslant \sum\limits_{i = 1}^2 {\ln } {x_i}(0) + \hat W(X(0)) + 2{{\text{e}}^{\alpha n}} + \tilde C({{\text{e}}^{\alpha t}} - 1). $$

若 $n-1 \leq t\leq n$ 和 $n>n_0(\omega)$,则 $$\frac{{\sum\limits_{i = 1}^2 {\ln } {x_i}(t))}}{{\ln n}} \leqslant \frac{{{{\text{e}}^{ - \alpha (n - 1)}}(\sum\limits_{i = 1}^2 {\ln {x_i}} (0) + \hat W(X(0))) + 2{{\text{e}}^\alpha }\ln n + \tilde C(1 - {{\text{e}}^{ - \alpha (n - 1)}})}}{{\ln (n - 1)}}, $$进而 $$\mathop {\lim }\limits_{t \to \infty } \frac{{\sum\limits_{i = 1}^2 {\ln } {x_i}(t)}}{{\ln t}} \leqslant 2{{\text{e}}^\alpha }.\;\{ {\text{a}}.{\text{s}}.. $$

令 $\alpha\rightarrow 0$,结论成立.

根据引理3.1和定理5.1和5.2,有下列定理成立.

定理 5.3 假设条件($H_1-H_6$)和$\frac{\hat{a}_{21}}{1-\tau_2^{'u}}<a_{11}(\gamma(t))$成立,则对于初值$u_i(\theta)=\varphi_i(\theta)\geq 0,-\tau\leq\theta\leq 0,i=1,2,$ 系统(2.2)的解$U(t)=(u_1(t),u_2(t))$ 满足 $$\lim\limits_{t\rightarrow +\infty}\sup\frac{\ln u_1+\ln u_2}{t}\leq \sum\limits_{i=1}^L(b_1(i)+b(i))\pi_i,\ {\rm a.s.,}~~\lim\limits_{t\rightarrow +\infty}\sup\frac{\ln u_1+\ln u_2}{\ln t}\leq 2,\ {\rm a.s.}. $$

这节讨论食饵种群与捕食者种群的灭绝与持续生存.

定理6.1 系统(2.2)满足($H_1-H_3,H_4-H_5$)和 $$b_j^*=\lim\limits_{t\rightarrow+\infty}\sup \frac{1}{t}\sum\limits_{0<\tau_k<t}\ln(1+a_k^j)+\sum\limits_{i=1}^{N}\pi_ib_j(i)<0,~~j=1,2. $$ 则种群$u_1(t)$ 和$u_2(t)$灭绝 a.s..

证 对系统(3.2)使用广义的Itô's 公式得

$\begin{gathered} {\text{d}}\ln {x_1} = [{b_1}(\gamma (t)) - {a_{11}}(\gamma (t)){x_1} - {a_{12}}(\gamma (t)){x_2}(t - {\tau _1}(t))]{\text{d}}t \hfill \\ + {\sigma _1}(\gamma (t)){\text{d}}{B_1}(t),{\text{d}}\ln {x_2} = [- {b_2}(\gamma (t)) - {a_{22}}(\gamma (t)){x_2} \hfill \\ + {a_{21}}(\gamma (t)){x_1}(t - {\tau _2}(t))]{\text{d}}t - {\sigma _2}(\gamma (t)){\text{d}}{B_1}(t). \hfill \\ \end{gathered} $

(6.1)

进而有\begin{eqnarray*}\ln x_1(t)-\ln x_1(0)&=&\int_0^tb_1(\gamma(s)){\rm d}s-\int_0^ta_{11}(\gamma(s))x_1{\rm d}s\\&&-\int_0^ta_{12}(\gamma(s))x_2(s-\tau_1(s)){\rm d}s+\int_0^t\sigma_1(\gamma(s)){\rm d}B_1(s),\\[2mm]\ln x_2(t)-\ln x_2(0)&=&-\int_0^tb_2(\gamma(s)){\rm d}s-\int_0^ta_{22}(\gamma(s))x_2{\rm d}s\\&&+\int_0^ta_{21}(\gamma(s))x_1(s-\tau_2(s)){\rm d}s-\int_0^t\sigma_2(\gamma(s)){\rm d}B_2(s).\end{eqnarray*}

从(3.2)-(3.4)式和引理3.1可推出

$\begin{gathered} \ln {u_1}(t) - \ln {u_1}(0) = \sum\limits_{0{\text{ < }}{t_k}{\text{ < }}t} {\ln (1 + a_k^1)} + \hfill \\ \int_0^t {{b_1}} (\gamma (s)){\text{d}}s - \int_0^t {{b_{11}}} (\gamma (s)){u_1}{\text{d}}s - \int_0^t {{b_{12}}} (\gamma (s)){u_2} \hfill \\ (s - {\tau _1}(s)){\text{d}}s + {M_1}(t), \hfill \\ \end{gathered} $

(6.3)

$\begin{gathered} \ln {u_2}(t) - \ln {u_2}(0) = \sum\limits_{0{\text{ < }}{t_k}{\text{ < }}t} {\ln (1 + a_k^2)} - \hfill \\ \int_0^t {{b_2}} (\gamma (s)){\text{d}}s - \int_0^t {{b_{22}}} (\gamma (s)){u_2}{\text{d}}s + \hfill \\ \int_0^t {{b_{21}}} (\gamma (s)){u_1}(s - {\tau _2}(s)){\text{d}}s - {M_2}(t), \hfill \\ \end{gathered} $

(6.4)

其中$M_i(t)=\int_0^t\sigma_i(\gamma(s)){\rm d}B_i(s),i=1,2.$ $M_i(t)$的二次变分为 $$\langle M_i(t),M_i(t)\rangle=\int_0^t\sigma_i^2(\gamma(t)){\rm d}s\leq \hat{\sigma_i}^2t. $$

利用鞅的强大数定律^[27]易得

$\lim\limits_{t\rightarrow +\infty}\frac{M_i(t)}{t}=0,\ \ {\rm a.s. }.$

(6.5)

从(6.3)式有 $$t^{-1}(\ln u_1(t)-\ln u_1(0))\leq t^{-1}\bigg(\sum\limits_{0<t_k<<t}\ln(1+a_k^1)+\int_0^tb_1(\gamma(s)){\rm d}s\bigg)+\frac{M_1(t)}{t}. $$

由$\gamma(t)$的遍历性和(6.5)式有 $$\lim\limits_{t\rightarrow +\infty}t^{-1}\ln u_1(t)\leq\lim\limits_{t\rightarrow +\infty}t^{-1}\bigg(\sum\limits_{0<t_k<t}\ln(1+a_k^1)+\int_0^tb_1(\gamma(s)){\rm d}s\bigg)=b_1^*<0. $$

所以有 $\lim\limits_{t\rightarrow +\infty}u_1(t)=0. $ a.s..

对于任意$\varepsilon>0$,存在一个充分大的$T$使得当$t>T$时有$u_1(t)<\frac{\varepsilon}{2\hat{b}_{21}},\frac{M_2(t)}{t}<\frac{\varepsilon}{2}.$

所以(6.4)式改写为 $$t^{-1}(\ln u_2(t)-\ln u_2(0))\leq t^{-1}\bigg(\sum\limits_{0<t_k<t}\ln(1+a_k^2)+\int_0^tb_2(\gamma(s)){\rm d}s\bigg)+\varepsilon, $$由$\gamma(t)$的遍历性和$\varepsilon$的任意小得到 $$\lim\limits_{t\rightarrow +\infty}t^{-1}\ln u_2(t)\leq\lim\limits_{t\rightarrow +\infty}t^{-1}\bigg(\sum\limits_{0<t_k<t}\ln(1+a_k^2)+\int_0^tb_2(\gamma(s)){\rm d}s\bigg)=b_2^*<0, $$所以定理结论成立,定理得证.

定理6.2 如果系统(2.2)满足$(H_1-H_3,H_5-H_6)$,则下列结论成立.

(1) 如果 $b_1^*(t)\geq 0$,那么 $$\lim\limits_{t\rightarrow+\infty}\sup\frac{1}{t}\int_0^tu_1(s){\rm d}s\leq b_1^*/\check{b}_{11}~~{\rm a.s..} $$ 特别的,若 $b_1^*=0,$ 则食饵 $u_1(t)$ 是平均非持续生存 a.s..

(2) 如果$b_2^*+\frac{\hat{b}_{21}b_1^*}{(1-\tau_2^{'u})\check{b}_{11}}\geq0,$那么 $$\lim\limits_{t\rightarrow+\infty}\sup\frac{1}{t}\int_0^tu_2(s){\rm d}s\leq\bigg(b_2^*+\frac{\hat{b}_{21}b_1^*}{(1-\tau_2^{'u})\check{b}_{11}}\bigg)/\check{b}_{22}~~{\rm a.s..} $$ 特别的,若$b_2^*+\frac{\hat{b}_{21}b_1^*}{(1-\tau_2^{'u})\check{b}_{11}}=0,$则捕食者 $u_2(t)$ 是平均非持续生存 a.s..

证 对任意$\varepsilon>0,$存在$T>0$使得 $$\ln u_1(0)/t\leq \varepsilon/3,\sum\limits_{0<t_k<t}\ln (1+a_k^1)+\int_0^tb_1(\gamma(s)){\rm d}s\leq b_1^*+\varepsilon/3,M_1(t)/t\leq \varepsilon/3, $$ 将上述的不等式代入(6.3)式可得\begin{eqnarray*}t^{-1}\ln u_1(t)&\leq& t^{-1}(\ln u_1(0)+\sum\limits_{0<t_k<t}\ln (1+a_k^1)\\&&+\int_0^tb_1(\gamma(s)){\rm d}s-\int_0^tb_{11}(\gamma(s))u_1 {\rm d}s+M_1(t))\\&\leq& b_1^*+\varepsilon-t^{-1}\check{b}_{11}\int_0^tu_1(s){\rm d}s.\end{eqnarray*}

利用文献[23,引理2]有

$\lim\limits_{t\rightarrow+\infty}\sup\frac{1}{t}\int_0^tu_1(s){\rm d}s\leq\frac{b_1^*}{\check{b}_{11}}.$

(6.6)

若$b_1^*=0$,则$\lim\limits_{t\rightarrow +\infty}\frac{1}{t}\int_0^tu_1(s){\rm d}s=\frac{b_1^*}{\check{b}_{11}}=0,$ 即 $u_1(t)$ 是平均非持续生存的a.s..

(2) (6.4)式的两边除以$t$获得

$\begin{align} & {{t}^{-1}}\ln {{u}_{2}}(t)\le {{t}^{-1}}\ln {{u}_{2}}(0)+{{t}^{-1}}\sum\limits_{0{{t}_{k}}t}{\ln (1+a_{k}^{2})}- \\ & {{t}^{-1}}\int_{0}^{t}{{{b}_{2}}}(\gamma (s))\text{d}s-{{t}^{-1}}{{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{b}}}_{22}}\int_{0}^{t}{{{u}_{2}}}\text{d}s+{{t}^{-1}}{{{\hat{b}}}_{21}} \\ & \int_{0}^{t}{{{u}_{1}}}(s-{{\tau }_{2}}(s))\text{d}s-{{t}^{-1}}{{M}_{2}}(t). \\ \end{align}$

(6.7)

进而可得

$\begin{gathered} \mathop {\lim }\limits_{t \to + \infty } \sup \frac{1}{t}\int_0^t {{u_1}} (s - {\tau _2}(s)){\text{d}}s = \hfill \\ \mathop {\lim }\limits_{t \to + \infty } \sup \frac{1}{t}\int_{ - {\tau _2}(0)}^{t - {\tau _2}(t)} {\frac{1}{{1 - {\tau _2}'(\theta )}}} {u_1}(\theta ){\text{d}}\theta \leqslant \hfill \\ \frac{1}{{1 - {\tau _2}'(\theta )}}(\mathop {\lim }\limits_{t \to + \infty } \sup \frac{1}{t}\int_0^t {{u_1}} (\theta ){\text{d}}\theta + \hfill \\ \mathop {\lim }\limits_{t \to + \infty } \sup (\frac{1}{t}\int_{ - {\tau _2}(0)}^0 {{u_1}} (\theta ){\text{d}}\theta + \frac{1}{t}\int_t^{t - {\tau _2}(t)} {{u_1}} (\theta ){\text{d}}\theta )). \hfill \\ \end{gathered} $

(6.8)

应用(6.6)式可得

$\begin{gathered} \mathop {\lim }\limits_{t \to + \infty } \sup \frac{1}{t}\int_t^{t - {\tau _2}(t)} {{u_1}} (\theta ){\text{d}}\theta = \hfill \\ \mathop {\lim }\limits_{t \to + \infty } \sup \frac{1}{t}\int_0^{t - {\tau _2}(t)} {{u_1}} (\theta ){\text{d}}\theta - \mathop {\lim }\limits_{t \to + \infty } \sup \frac{1}{t}\int_0^t {{u_1}} (\theta ){\text{d}}\theta = 0,\hfill \\ \end{gathered} $

(6.9)

$\lim\limits_{t\rightarrow+\infty}\sup\frac{1}{t}\int_{-\tau_2(0)}^{0}u_1(\theta){\rm d}\theta=0.$

(6.10)

根据文献[23,引理2]和表达式(6.7)-(6.10),有 $$\lim\limits_{t\rightarrow+\infty}\sup\frac{1}{t}\int_0^tu_2(s){\rm d}s\leq\bigg(b_2^*+\frac{\hat{b}_{21}b_1^*}{(1-\tau_2^{'u})\check{b}_{11}}\bigg)/\check{b}_{22}. $$

若$b_2^*+\frac{\hat{b}_{21}b_1^*}{(1-\tau_2^{'u})\check{b}_{11}}=0$,则 $\lim\limits_{t\rightarrow+\infty}\frac{1}{t}\int_0^tu_2(s){\rm d}s=0,$ 即 $u_2(t)$是平均非持续生存的 a.s..

本文研究了具有Markov转换和脉冲扰动的随机时滞的捕食-食饵系统.文章给出了具有Markov转换的脉冲随机微分方程解的定义,确定了具有脉冲的Markov转换 的随机微分系统(2.2)与无脉冲的具有Markov转换的随机微分系统(3.2)解的等价关系. 文章首先证明了系统存在全局正解,给出系统解的均值上极限的估计,讨论了系统的随机最终有界性和长时间渐近性态,最后得到系统解灭绝与平均非持续生存的充分条件.

文章的结论有明显的生物意义.我们从种群灭绝与平均非持续生存的充分条件看到了冲效应$a_k^{i}$和随机因素$\sigma_{i}$对捕食-食饵系统两种群命运的影响. 当脉冲扰动有界时,由表达式$b_j^*=\sum\limits_{i=1}^{N}\pi_ib_j(i),j=1,2$发现 脉冲对种群灭绝与非持续生存性没有影响. 然而脉冲扰动无界时种群灭绝与平均非持续生存受到了影响.

捕食-食饵系统由于其广泛存在性,使之成为数学生态学中重要的系统. 同时考虑随机噪声与脉冲扰动对种群的影响,所以 研究具有Markov转换和脉冲扰动的捕食-食饵系统的种群持续生存与灭绝更加有意义. 文章的研究方法对研究随机脉冲系统是基本的、有效的. 此方法可以用来研究更复杂数学模型.