具有Lévy跳跃驱动的Rosenzweig-MacArthur捕食者–食饵模型的渐近行为

Asymptotic Behavior of Rosenzweig-MacArthur Predator-Prey Model with Lévy Jump Drive

摘要: 本文主要利用随机分析等相关知识，根据进化稳定策略框架对Rosenzweig-MacArthur捕食者–食饵模型进行改进，证明了具有Lévy跳跃驱动的Rosenzweig-MacArthur捕食者–食饵模型解的存在唯一性和随机有界性，并探究了该模型的灭绝性、持久性和平稳分布。

Abstract: In this paper, we use stochastic analysis and other related knowledge to improve the Rosenzweig-MacArthur predator-prey model according to the evolutionary stable strategy framework, prove the existence, uniqueness and random boundedness of the solution of the Rosenzweig-MacArthur predator-prey model driven by Lévy jump, and explore the extinction, persistence and stationary distribution of the model.

1. 引言

众所周知，1956年Lotka和Volterra提出了经典的Lotka-Volterra模型，用于描述捕食者和食饵种群的动态发展 [1] [2] [3] [4] ，但是Lotka-Volterra模型并没有考虑到现实生态环境的复杂性和变化性，无法那么贴切的描述真实情况。在1963年M. Rosenzweig和R. MacArthur提出的Rosenzweig-MacArthur捕食者–食饵模型考虑了更多现实影响因素，对Lotka-Volterra模型进行了扩展和改进，使模型更贴近实际情况 [5] ，研究了生态稳定以及捕食者与食饵种群之间的相互影响和动态进化发展，吸引了许多研究者的注意 [6] - [11] 。

Rosenzweig-MacArthur的猎物–捕食者系统建立在Lotka-Volterra模型的基础上，考虑了更多的生态因素，使得模型更适用于描述复杂的生态系统动态。让x₁和y₁如分别表示猎物和捕食者的种群大小。假设系统的动态遵循逻辑–高斯型模型：

$\{\begin{array}{l}\frac{\text{d}{x}_1}{\text{d}t}=x_1\left(f_1\left(x_1\right)-y_1{\varphi }_{11}\left(x_1\right)\right),\\ \frac{\text{d}{x}_2}{\text{d}t}=y_1\left[-c_1+k{x}_1{\varphi }_{11}\left(x_1\right)\right].\end{array}$ (1.1)

在这里， $x_1{f}_1\left(x_1\right)$ 为猎物的生长率， $x_1{y}_1{\varphi }_{11}\left(x_1\right)$ 为捕食率， $c_1{y}_1$ 为捕食者的死亡率，且 $c_1>0$ ，其中：

${\varphi }_{11}\left(x\right)=\frac{a}{b_{11}+x}$

$f_1\left(x\right)=r_1\left(1-\frac{x}{K}\right)$

a和b₁₁为正常数， $r_1>0$ 为速率常数和 $K>0$ 为猎物种群的承载能力。

考虑到生物多样性，在自然环境中，可能会出现突变体种群，并且突变体种群很可能拥有与原始种群不同的生存率和死亡率，对其他种群造成或大或小的影响。为了研究更适合现实情况的模型，2021年，Grunert等人 [12] 根据进化稳定策略框架对Rosenzweig-MacArthur捕食者–食饵模型进行改进，得到新的模型：

$\{\begin{array}{l}\text{d}{x}_1=x_1\left(f_1\left(x_1+x_2\right)-y_1{\phi }_{11}\left(x_1+x_2\right)-y_2{\phi }_{12}\left(x_1+x_2\right)\right)\text{d}t,\hfill \\ \text{d}{y}_1=y_1\left[-c_1+k{x}_1{\phi }_{11}\left(x_1+x_2\right)+k{x}_2{\phi }_{21}\left(x_1+x_2\right)\right]\text{d}t,\hfill \\ \text{d}{x}_2=x_2\left(f_2\left(x_1+x_2\right)-y_1{\phi }_{21}\left(x_1+x_2\right)-y_2{\phi }_{22}\left(x_1+x_2\right)\right)\text{d}t,\hfill \\ \text{d}{y}_2=y_2\left[-c_2+k{x}_1{\phi }_{12}\left(x_1+x_2\right)+k{x}_2{\phi }_{22}\left(x_1+x_2\right)\right]\text{d}t.\hfill \end{array}$ (1.2)

其中x₁和y₁如分别表示原始的食饵种群和捕食者种群；x₂和y₂分别表示相应的突变种群。 $f_i\left(x\right)=r_i\left(1-\frac{x}{K_i}\right)$ 和 ${\phi }_{ij}\left(x\right)=\frac{a}{b_{ij}+x}\left(i,j=1,2\right)$ ，a，k和c， $c_i,r_i,K_i,c_{ij}\left(i=1,2\right)$ 都是正常数。

然而，生物种群不可避免会遭受严重的环境扰动，如海、火山、鸟类流感、SARS、洪水、飓风、地震、有毒污染物等，这些现象不能用随机性来描述连续模型。因此，将跳跃过程引入到种群系统是可行的，吸引了许多研究者的注意 [13] - [22] 。在本文中，我们将Lévy跳跃引入到Rosenzweig-MacArthur捕食者–食饵模型：

$\{\begin{array}{l}\text{d}{x}_1=\left\{x_1\left[f_1\left(x_1+x_2\right)-y_1{\phi }_{11}\left(x_1+x_2\right)-y_2{\phi }_{12}\left(x_1+x_2\right)\right]\right\}\text{d}t\\ +{\sigma }_1{x}_1\left(t\right)\text{d}w\left(t\right)+{{{\displaystyle \int }}^{\text{}}}_z{\psi }_1\left(z\right)x_1\left(t\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\\ \text{d}{y}_1=\left\{y_1\left[-c_1+k{x}_1{\phi }_{11}\left(x_1+x_2\right)+k{x}_2{\phi }_{21}\left(x_1+x_2\right)\right]\right\}\text{d}t\\ +{\sigma }_2{y}_1\left(t\right)\text{d}w\left(t\right)+{{{\displaystyle \int }}^{\text{}}}_z{\psi }_2\left(z\right)y_1\left(t\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\\ \text{d}{x}_2=\left\{x_2\left[f_2\left(x_1+x_2\right)-y_1{\phi }_{21}\left(x_1+x_2\right)-y_2{\phi }_{22}\left(x_1+x_2\right)\right]\right\}\text{d}t\\ +{\sigma }_3{x}_2\left(t\right)\text{d}w\left(t\right)+{{{\displaystyle \int }}^{\text{}}}_z{\psi }_3\left(z\right)x_2\left(t\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\\ \text{d}{y}_2=\left\{y_2\left[-c_2+k{x}_1{\phi }_{12}\left(x_1+x_2\right)+k{x}_2{\phi }_{22}\left(x_1+x_2\right)\right]\right\}\text{d}t\\ +{\sigma }_4{y}_2\left(t\right)\text{d}w\left(t\right)+{{{\displaystyle \int }}^{\text{}}}_z{\psi }_4\left(z\right)y_2\left(t\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\end{array}$ (1.3)

其中 ${\sigma }_i\left(i=1,2\right)$ 是噪声强度， $w_i\left(t\right)\left(i=1,2\right)$ 是定义在完全概率空间 $\left(\Omega ,\mathcal{F},P\right)$ 上的标准维纳过程， ${\psi }_i\left(i=1,2\right)$ 为有界函数。N是在可测子集 $Z\subseteq \left[0,+\infty \right)$ 上具有特征测度入的泊松计数测度，其中 $\lambda \left(Z\right)<+\infty$ ，且 ${\tilde{N}}_i\left(\text{d}t,\text{d}z\right)=N\left(\text{d}t,\text{d}z\right)-\lambda \left(\text{d}z\right)\text{d}t$ 。

本文的的研究目的是研究具有Lévy驱动的Rosenzweig-MacArthur捕食者–猎物模型的动力学行为。通过加入Lévy驱动，模拟真实生态环境，探究Lévy驱动对捕食者和猎物和群的动态影响。因为在现实中的环境扰动往往是比较复杂的，Lévy驱动的非线性和跳跃性质可以更好的贴近真实的捕食者–猎物种群动态过程，提高了模型对生态系统的拟合度。同时，还能为生态学研究提供新的思路，具有一定的参考意义。

2. 随机有界性

本小节，主要考虑具有Lévy跳跃驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)的正解和有界性。

定理2.1 对于任意给定的初始条件 $\left(x_1\left(0\right),x_2\left(0\right),y_1\left(0\right),y_2\left(0\right)\right)$ ，则具有Lévy跳跃驱动的 Rosenzweig-MacArthur捕食者–食饵模型(1.3)存在唯一的全局正解 $\left(x_1\left(t\right),x_2\left(t\right),y_1\left(t\right),y_2\left(t\right)\right)\in {ℝ}_{+}^4$ ， $t\in \left(0,{\tau }_e\right)$ ，其中 ${\tau }_e$ 是爆炸时间。

证明：令

$\overline{x_1}\left(t\right)=\ln x_1\left(t\right),\overline{x_2}\left(t\right)=\ln x_2\left(t\right),\overline{y_1}\left(t\right)=\ln y_1\left(t\right),\overline{y_2}\left(t\right)=\ln y_2\left(t\right),$

运用Itô公式，得到：

$\{\begin{array}{l}\text{d}\overline{x_1}\left(t\right)=\left\{f_1\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)-{\text{e}}^{y_1\left(t\right)}{\phi }_{11}\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)+{\phi }_{12}\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)-\frac{{\sigma }_1^2}{2}\right\}\text{d}t\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[\ln \left(1+{\psi }_1\left(z\right)\right)-{\psi }_1\left(z\right)\right]v\left(\text{d}z\right)\text{d}t+{\sigma }_1\text{d}w\left(t\right)+{{{\displaystyle \int }}^{\text{}}}_z\left(1+{\psi }_1\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\\ \text{d}\overline{y_1}\left(t\right)=\left\{-c_1+k{\text{e}}^{x_1\left(t\right)}{\phi }_{11}\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)+k{\text{e}}^{x_2\left(t\right)}{\phi }_{21}\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)-\frac{{\sigma }_2^2}{2}\right\}\text{d}t\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[\ln \left(1+{\psi }_2\left(z\right)\right)-{\psi }_2\left(z\right)\right]v\left(\text{d}z\right)\text{d}t+{\sigma }_2\text{d}w\left(t\right)+{{{\displaystyle \int }}^{\text{}}}_z\left(1+{\psi }_2\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\\ \text{d}\overline{x_2}\left(t\right)=\left\{f_2\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)-{\text{e}}^{y_1\left(t\right)}{\phi }_{21}\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)+{\phi }_{22}\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)-\frac{{\sigma }_3^2}{2}\right\}\text{d}t\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[\ln \left(1+{\psi }_3\left(z\right)\right)-{\psi }_3\left(z\right)\right]v\left(\text{d}z\right)\text{d}t+{\sigma }_3\text{d}w\left(t\right)+{{{\displaystyle \int }}^{\text{}}}_z\left(1+{\psi }_3\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\\ \text{d}\overline{y_2}\left(t\right)=\left\{-c_2+k{\text{e}}^{x_1\left(t\right)}{\phi }_{12}\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)+k{\text{e}}^{x_2\left(t\right)}{\phi }_{22}\left({\text{e}}^{x_1\left(t\right)}+{\text{e}}^{x_2\left(t\right)}\right)-\frac{{\sigma }_4^2}{2}\right\}\text{d}t\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[\ln \left(1+{\psi }_4\left(z\right)\right)-{\psi }_4\left(z\right)\right]v\left(\text{d}z\right)\text{d}t+{\sigma }_4\text{d}w\left(t\right)+{{{\displaystyle \int }}^{\text{}}}_z\left(1+{\psi }_4\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (2.1)

显然上面式子(2.1)的系数是满足局部Lipschitz条件的，故对于任意给定的初始条件 $\left(x_1\left(0\right),x_2\left(0\right),y_1\left(0\right),y_2\left(0\right)\right)$ ，对于 $t\in \left(0,{\tau }_e\right)$ ，存在唯一的局部解 $\left(x_1\left(t\right),x_2\left(t\right),y_1\left(t\right),y_2\left(t\right)\right)$ ，将局部解推广到全局解，只需证明 ${\tau }_e=\infty$ 几乎处处存在即可。

令 $n_0>0$ 充分大，使得解 $\left(x_1\left(t\right),x_2\left(t\right),y_1\left(t\right),y_2\left(t\right)\right)$ 在区间 $\left[\frac{1}{n_0},k{n}_0\right]$ 中。对每个 $n>n_0$ ，定义停止时间为

${\tau }_n=\inf \left\{t\in \left[0,{\tau }_e\right):x_1\left(t\right)\notin \left(\frac{1}{n_0},n_0\right)\text{or}{y}_1\left(t\right)\notin \left(\frac{1}{n_0},n_0\right)\text{or}{x}_2\left(t\right)\notin \left(\frac{1}{n_0},n_0\right)\text{or}{y}_2\left(t\right)\notin \left(\frac{1}{n_0},n_0\right)\right\}$

显然，当 $n\to \infty$ 时， ${\tau }_n$ 是单调增加的。令 ${\tau }_n={\lim }_{n\to \infty }{\tau }_n$ ，则有 ${\tau }_{\infty }\le {\tau }_e$ 几乎是处处成立的。我们需要证明 ${\tau }_{\infty }=\infty$ ，即 ${\tau }_e=\infty$ 且 $\left(x_1\left(t\right),x_2\left(t\right),y_1\left(t\right),y_2\left(t\right)\right)\in R_{+}^4$ 几乎处处成立。如若不然，则存在两个常数 $K_1>0$ 和 $e\in \left(0,1\right)$ 使得 $P\left({\tau }_{\infty }\le K_1\right)\ge e$ 。于是存在 $n_1\in N$ ，且 $n_1>n_0$ 使得 $P\left({\tau }_n\le K_1\right)\ge e$ ， $n\ge n_1$ 。

让

$\begin{array}{l}V\left(x_1\left(t\right),x_2\left(t\right),y_1\left(t\right),y_2\left(t\right)\right)\\ =\left(k{x}_1-1-k_1\ln x_1\right)+\left(y_1-1-\ln x_1\right)+\left(k{x}_2-1-k_2\ln x_2\right)+\left(y_2-1-\ln y_2\right).\end{array}$

运用Itô公式，有

$\begin{array}{l}\text{d}V\left(x_1\left(t\right),x_2\left(t\right),y_1\left(t\right),y_2\left(t\right)\right)\\ =LV\text{d}t+{\sigma }_1\left(k{x}_1\left(t\right)-k_1\right)\text{d}w\left(t\right)+{\sigma }_2\left(y_1\left(t\right)-1\right)\text{d}w\left(t\right)+{\sigma }_3\left(k{x}_2\left(t\right)-k_2\right)\text{d}w\left(t\right)+{\sigma }_4\left(y_2\left(t\right)-1\right)\text{d}w\left(t\right)\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[k{\psi }_1\left(z\right)x_1\left(t\right)-k_1\ln \left(1+\left(\text{d}t,\text{d}z\right)\right)\right]+{{{\displaystyle \int }}^{\text{}}}_z\left[{\psi }_2\left(z\right)y_1\left(t\right)-\ln \left(1+\left(\text{d}t,\text{d}z\right)\right)\right]\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[k{\psi }_3\left(z\right)x_2\left(t\right)-k_2\ln \left(1+\left(\text{d}t,\text{d}z\right)\right)\right]+{{{\displaystyle \int }}^{\text{}}}_z\left[{\psi }_4\left(z\right)y_2\left(t\right)-\ln \left(1+\left(\text{d}t,\text{d}z\right)\right)\right]\end{array}$ (2.2)

其中

$\begin{array}{c}LV\le -\frac{k{r}_1}{K_1}{x}_1^2+\left(k{r}_1+\frac{k_1{r}_1}{K_1}+\frac{k_2{r}_2}{K_2}\right)x_1+\left(\frac{a{k}_1}{b_{11}}+\frac{a{k}_2}{b_{21}}-c_1\right)y_1+c_1-k_1{r}_1\\ -\frac{k{r}_2}{K_2}{x}_2^2+\left(k{r}_2+\frac{k_1{r}_1}{K_1}+\frac{k_2{r}_2}{K_2}\right)x_2+\left(\frac{a{k}_1}{b_{12}}+\frac{a{k}_2}{b_{22}}-c_2\right)y_2+c_2-k_2{r}_2\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[k_1{\psi }_1-k_1\ln \left(1+{\psi }_1\right)\right]v\left(\text{d}z\right)+{{{\displaystyle \int }}^{\text{}}}_z\left[{\psi }_2-\ln \left(1+{\psi }_2\right)\right]v\left(\text{d}z\right)+\frac{k_1{\sigma }_1^2+{\sigma }_2^2}{2}\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[k_2{\psi }_3-k_2\ln \left(1+{\psi }_3\right)\right]v\left(\text{d}z\right)+{{{\displaystyle \int }}^{\text{}}}_z\left[{\psi }_4-\ln \left(1+{\psi }_4\right)\right]v\left(\text{d}z\right)+\frac{k_2{\sigma }_3^2+{\sigma }_4^2}{2}.\end{array}$

选取适当 $k,k_1,k_2$ 从而，获得

$\begin{array}{c}LV\le \frac{K_1^2}{4k^2{r}_1^2}{\left(k{r}_1+\frac{k_1{r}_1}{K_1}+\frac{k_2{r}_2}{K_2}\right)}^2+c_1-k_1{r}_1+\frac{K_2^2}{4k^2{r}_2^2}\left(k{r}_2+\frac{k_1{r}_1}{K_1}+\frac{k_2{r}_2}{K_2}\right)+c_2-k_2{r}_2\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[k_1{\psi }_1-k_1\ln \left(1+{\psi }_1\right)\right]v\left(\text{d}z\right)+{{{\displaystyle \int }}^{\text{}}}_z\left[{\psi }_2-\ln \left(1+{\psi }_2\right)\right]v\left(\text{d}z\right)+\frac{k_1{\sigma }_1^2+{\sigma }_2^2}{2}\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[k_2{\psi }_3-k_2\ln \left(1+{\psi }_3\right)\right]v\left(\text{d}z\right)+{{{\displaystyle \int }}^{\text{}}}_z\left[{\psi }_4-\ln \left(1+{\psi }_4\right)\right]v\left(\text{d}z\right)+\frac{k_2{\sigma }_3^2+{\sigma }_4^2}{2}\end{array}$ (2.3)

从而，可得 $LV\le C$ ，其中

$\begin{array}{c}C=\frac{K_1^2}{4k^2{r}_1^2}{\left(k{r}_1+\frac{k_1{r}_1}{K_1}+\frac{k_2{r}_2}{K_2}\right)}^2+c_1-k_1{r}_1+\frac{K_2^2}{4k^2{r}_2^2}\left(k{r}_2+\frac{k_1{r}_1}{K_1}+\frac{k_2{r}_2}{K_2}\right)+c_2-k_2{r}_2\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[k_1{\psi }_1-k_1\ln \left(1+{\psi }_1\right)\right]v\left(\text{d}z\right)+{{{\displaystyle \int }}^{\text{}}}_z\left[{\psi }_2-\ln \left(1+{\psi }_2\right)\right]v\left(\text{d}z\right)+\frac{k_1{\sigma }_1^2+{\sigma }_2^2}{2}\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[k_2{\psi }_3-k_2\ln \left(1+{\psi }_3\right)\right]v\left(\text{d}z\right)+{{{\displaystyle \int }}^{\text{}}}_z\left[{\psi }_4-\ln \left(1+{\psi }_4\right)\right]v\left(\text{d}z\right)+\frac{k_2{\sigma }_3^2+{\sigma }_4^2}{2}\end{array}$

对式子(2.2)两边从0到 ${\tau }_k\wedge T$ 之间积分，得到

$\begin{array}{c}0\le E\left(V\left(x_1\left({\tau }_k\wedge T\right),y_1\left({\tau }_k\wedge T\right),x_2\left({\tau }_k\wedge T\right),y_2\left({\tau }_k\wedge T\right)\right)\right)\\ \le V\left(x_1\left(0\right),y_1\left(0\right),x_2\left(0\right),y_2\left(0\right)\right)+CT.\end{array}$ (2.4)

对每一个g>0，定义

$G\left(g\right)=\inf \left\{V\left(x_1,y_1,x_2,y_2\right):x_i\ge g,y_i\ge g\vee x_i\le \frac{1}{g},y_i\le \frac{1}{g},i=1,2\right\}$

则有 ${\lim }_{g\to \infty }G\left(g\right)=\infty$ 。因此，令 $k\to \infty$ ，有 $\infty >V\left(x_1\left(0\right),y_1\left(0\right),x_2\left(0\right),y_2\left(0\right)\right)+CT=\infty$ ，这与之前的假设矛盾。因此， ${\tau }_{\infty }=\infty$ ，则具有Lévy驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)存在唯一的全局解 $\left(x_1\left(t\right),y_1\left(t\right),x_2\left(t\right),y_2\left(t\right)\right)$ 。定理证明完毕。

定理2.2 假设对于任何初始条件 $\left(x_1\left(0\right),y_1\left(0\right),x_2\left(0\right),y_2\left(0\right)\right)$ ，如果下列条件成立

$c_1>2ka+{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_2\left(z\right)\right)}^p-1-p{\psi }_2\left(z\right)\right]v\left(\text{d}z\right),$

$c_2>2ka+{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_4\left(z\right)\right)}^p-1-p{\psi }_4\left(z\right)\right]$

则具有Lévy跳跃驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)解是随机最终有界。

证明：定义

$V\left(x_1\left(t\right),x_2\left(t\right),y_1\left(t\right),y_2\left(t\right)\right)=x_1^p+y_1^p+x_2^p+y_2^p,\left(x_1\left(t\right),x_2\left(t\right),y_1\left(t\right),y_2\left(t\right)\right)\in R_{+}^4.$

运用Itô公式，得到

$\begin{array}{c}\text{d}V\left(x_1,y_1,x_2,y_2\right)=LV\text{d}t+{\sigma }_{1}p{x}_1^p\text{d}w+{\sigma }_{2}p{y}_1^p\text{d}w+{\sigma }_{3}p{x}_2^p\text{d}w+{\sigma }_{4}p{y}_2^p\text{d}w\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_1\left(z\right)\right)}^p-1\right]x_1^p\tilde{N}\left(\text{d}t,\text{d}z\right)+{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_2\left(z\right)\right)}^p-1\right]y_1^p\tilde{N}\left(\text{d}t,\text{d}z\right)\\ +{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_3\left(z\right)\right)}^p-1\right]x_2^p\tilde{N}\left(\text{d}t,\text{d}z\right)+{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_4\left(z\right)\right)}^p-1\right]y_2^p\tilde{N}\left(\text{d}t,\text{d}z\right),\end{array}$

其中

$\begin{array}{c}LV\le -x_1^p+\left[-c_1+2ka+{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_2\left(z\right)\right)}^p-1-p{\psi }_2\left(z\right)\right]v\left(\text{d}z\right)\right]y_1^p\\ -x_2^p+\left[-c_2+2ka+{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_4\left(z\right)\right)}^p-1-p{\psi }_4\left(z\right)\right]v\left(\text{d}z\right)\right]y_2^p+2N\\ \le -\alpha \left(x_1^p+y_1^p+x_2^p+y_2^p\right)+2N=-\alpha V\left(t\right)+2N,\end{array}$ (2.5)

且

$-\frac{r_1}{K_1}{x}_1^{p+1}+\left(r_1+1+{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_1\left(z\right)\right)}^p-1-p{\psi }_1\left(z\right)\right]v\left(\text{d}z\right)\right)x_1^p<N,$

$-\frac{r_2}{K_2}{x}_2^{p+1}+\left(r_2+1+{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_3\left(z\right)\right)}^p-1-p{\psi }_3\left(z\right)\right]v\left(\text{d}z\right)\right)x_2^p<N,$

$\begin{array}{l}\alpha =\min \{1,c_1-2ka-{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_2\left(z\right)\right)}^p-1-p{\psi }_2\left(z\right)\right]v\left(\text{d}z\right),\\ c_2-2ka-{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_4\left(z\right)\right)}^p-1-p{\psi }_4\left(z\right)\right]v\left(\text{d}z\right)\}.\end{array}$

运用Itô公式，得到

$\begin{array}{l}\text{d}\left[{\text{e}}^{\alpha t}V\right]=\alpha {\text{e}}^{\alpha t}V\text{d}t+{\text{e}}^t\text{d}V\\ \le N{\text{e}}^{\alpha t}\text{d}t+{\text{e}}^{\alpha t}\left({\sigma }_{1}p{x}_1^p\text{d}w+{\sigma }_{2}p{y}_1^p\text{d}w+{\sigma }_{3}p{x}_2^p\text{d}w+{\sigma }_{4}p{y}_2^p\text{d}w\right)\\ +{\text{e}}^{\alpha t}{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_1\left(z\right)\right)}^p-1\right]x_1^p\tilde{N}\left(\text{d}t,\text{d}z\right)+{\text{e}}^{\alpha t}{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_2\left(z\right)\right)}^p-1\right]y_1^p\tilde{N}\left(\text{d}t,\text{d}z\right)\\ +{\text{e}}^{\alpha t}{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_3\left(z\right)\right)}^p-1\right]x_2^p\tilde{N}\left(\text{d}t,\text{d}z\right)+{\text{e}}^{\alpha t}{{{\displaystyle \int }}^{\text{}}}_z\left[{\left(1+{\psi }_4\left(z\right)\right)}^p-1\right]y_2^p\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$

对 $\text{d}\left[{\text{e}}^{\alpha t}V\right]$ 两边从0到t积分，然后取期望，则有

${\text{e}}^{t}E\left[V\left(X\right)\right]\le V\left(x_1\left(0\right),x_2\left(0\right),y_1\left(0\right),y_2\left(0\right)\right)+N{\text{e}}^t-N.$ (2.6)

从而，则有

${\lim }_{t+\infty }\sup E\left[V\left(x\right)\right]\le N$

定理证明完毕。

3. 灭绝性与持久性

本小节，主要讨论具有Lévy驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)的灭绝性和持久性。

定义：种群物种具有均值强持久性，如果 ${\langle x\left(t\right)\rangle }_{*}>0$ ，其中

${\langle x\left(t\right)\rangle }_{*}=\frac{1}{t}{\displaystyle {\int }_0^{t}x\left(s\right)\text{d}s},{\langle x\left(t\right)\rangle }_{*}=\lim {\inf }_{t\to \infty }\frac{1}{t}{\displaystyle {\int }_0^{t}x\left(s\right)\text{d}s}$

定理3.1

(1) 如果满足下列条件之一

$\langle \mathrm{Ⅰ}\rangle$ $r_1+{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_1\left(z\right)\right)-{\psi }_1\left(z\right)\right]\left(\text{d}z\right)}>\frac{{\sigma }_1^2}{2}$

$\langle \mathrm{Ⅱ}\rangle$ $r_2+{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_3\left(z\right)\right)-{\psi }_3\left(z\right)\right]\left(\text{d}z\right)}>\frac{{\sigma }_3^2}{2}.$

则具有Lévy驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)有如下性质：

${\langle x_1\left(t\right)\rangle }_{*}>0,{\langle x_2\left(t\right)\rangle }_{*}>0,$

${\langle y_1\left(t\right)\rangle }_{*}>0,{\langle y_2\left(t\right)\rangle }_{*}>0.$

(2) 如果满足下列条件之一

$\langle \mathrm{Ⅰ}\rangle$ $c_1+\frac{{\sigma }_2^2}{2}>{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_2\left(z\right)\right)-{\psi }_2\left(z\right)\right](\; d\; z\; )}$

$\langle \mathrm{Ⅱ}\rangle$ $c_2+\frac{{\sigma }_4^2}{2}>{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_4\left(z\right)\right)-{\psi }_4\left(z\right)\right]}\left(\text{d}z\right).$

则具有Lévy驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)有如下性质：

${\langle x_1\left(t\right)\rangle }_{*}>0,{\langle x_2\left(t\right)\rangle }_{*}>0.$

(3) 此外，如果满足下列条件

$c_1+\frac{{\sigma }_2^2}{2}>2ka+{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_2\left(z\right)\right)-{\psi }_2\left(z\right)\right]}\left(\text{d}z\right).$

则具有Lévy驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)有如下性质：

$\begin{array}{c}{\lim }_{t\to \infty }{y}_1\left(t\right)=0.\end{array}$

和

如果满足下列条件

$\begin{array}{c}{c}_2+\frac{{\sigma }_4^2}{2}>2ka+{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_4\left(z\right)\right)-{\psi }_4\left(z\right)\right]}\left(\text{d}z\right).\end{array}$

则具有Lévy驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)有如下性质：

${\lim }_{t\to \infty }{y}_2\left(t\right)=0.$

证明：定义

$V\left(x_1\right)=\ln \left(x_1\left(t\right)\right).$ (3.1)

运用Itô公式，得到

$\begin{array}{c}\text{d}V\left(x_1\right)=\left[r_1-\frac{r_1}{K_1}\left(x_1+x_2\right)-y_1\frac{a}{b_{11}+x_1+x_2}-y_2\frac{a}{b_{12}+x_1+x_2}\right]\text{d}t\\ -\left(\frac{{\sigma }_1^2}{2}-{\displaystyle {\int }_z\ln }\left(1+{\psi }_1\left(z\right)\right)-{\psi }_1\left(z\right)v\left(\text{d}z\right)\right)\text{d}t\\ +{\sigma }_1\text{d}w\left(t\right)+{\displaystyle {\int }_z\ln }\left(1+{\psi }_1\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (3.2)

对式子(3.2)两边积分且除以t，可获得

$\begin{array}{c}\frac{\ln \frac{x_1\left(t\right)}{x_1\left(0\right)}}{t}\ge r_1-\frac{{\sigma }_1^2}{2}+{\displaystyle {\int }_z\ln }\left(1+{\psi }_1\left(z\right)\right)-{\psi }_1\left(z\right)v\left(\text{d}z\right)-\frac{r_1}{K_1}\langle x_1+x_2\rangle -\frac{a}{b_{11}}\langle y_1\rangle \\ -\frac{a}{b_{12}}\langle y_2\rangle +{\sigma }_1\frac{w\left(t\right)}{t}+\frac{1}{t}{\displaystyle {\int }_0^t{\displaystyle {\int }_z\ln }}\left(1+{\psi }_1\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (3.3)

从而，可获得

$\begin{array}{c}\frac{r_1}{K_1}{\langle x_1\rangle }_{*}+\frac{r_1}{K_1}{\langle x_2\rangle }_{*}+\frac{a}{b_{11}}{\langle y_1\rangle }_{*}+\frac{a}{b_{12}}{\langle y_2\rangle }_{*}\ge r_1-\frac{{\sigma }_1^2}{2}+{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_1\left(z\right)\right)-{\psi }_1\left(z\right)\right]}\left(\text{d}z\right).\end{array}$

定义

$V\left(y_1\right)=\ln \left(y_1(\; t\; )\right)$

运用Itô公式，可得到

$\begin{array}{c}\text{d}V\left(y_1\right)=\left[-c_1+k{x}_1\frac{a}{b_{11}+x_1+x_2}+k{x}_2\frac{a}{b_{21}+x_1+x_2}\right]\text{d}t\\ -\left(\frac{{\sigma }_2^2}{2}-{\displaystyle {\int }_z\ln }\left(1+{\psi }_2\left(z\right)\right)-{\psi }_2\left(z\right)v\left(\text{d}z\right)\right)\text{d}t\\ +{\sigma }_2\text{d}w\left(t\right)+{\displaystyle {\int }_z\ln }\left(1+{\psi }_2\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (3.5)

式子(3.4)两边积分且除以t，可获得

$\begin{array}{c}\frac{\ln \frac{y_1\left(t\right)}{y_1\left(0\right)}}{t}\le \frac{ka}{b_{11}}\langle x_1\rangle +\frac{ka}{b_{21}}\langle x_2\rangle -c_1-\frac{{\sigma }_2^2}{2}+{\displaystyle {\int }_z\ln }\left(1+{\psi }_2\left(z\right)\right)-{\psi }_2\left(z\right)v\left(\text{d}z\right)\\ +{\sigma }_2\frac{w\left(t\right)}{t}+\frac{1}{t}{\displaystyle {\int }_0^t{\displaystyle {\int }_z\ln }}\left(1+{\psi }_2\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (3.6)

从而，我们有

$c_1+\frac{{\sigma }_2^2}{2}-{\displaystyle {\int }_z\ln }\left(1+{\psi }_2\left(z\right)\right)-{\psi }_2\left(z\right)v\left(\text{d}z\right)\le \frac{ka}{b_{11}}{\langle x_1\rangle }_{*}+\frac{ka}{b_{21}}{\langle x_2\rangle }_{*}$

定义

$V\left(x_2\right)=\ln \left(x_2(\; t\; )\right)$

运用Itô公式，可得到：

$\begin{array}{c}\text{d}V\left(x_2\right)=\left[r_2-\frac{r_2}{K_2}\left(x_1+x_2\right)-y_1\frac{a}{b_{21}+x_1+x_2}-y_2\frac{a}{b_{22}+x_1+x_2}\right]\text{d}t\\ -\left(\frac{{\sigma }_3^2}{2}-{\displaystyle {\int }_z\ln }\left(1+{\psi }_3\left(z\right)\right)-{\psi }_3\left(z\right)v\left(\text{d}z\right)\right)\text{d}t\\ +{\sigma }_3\text{d}w\left(t\right)+{\displaystyle {\int }_z\ln }\left(1+{\psi }_3\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (3.8)

对式子(3.8)两边积分且除以t，可获得

$\begin{array}{c}\frac{\ln \frac{x_2\left(t\right)}{x_2\left(0\right)}}{t}\ge r_2-\frac{r_2}{K_2}\langle x_1+x_2\rangle -\frac{a}{b_{21}}\langle y_1\rangle -\frac{a}{b_{22}}\langle y_2\rangle \\ -\frac{{\sigma }_3^2}{2}+{\displaystyle {\int }_z\ln }\left(1+{\psi }_3\left(z\right)\right)-{\psi }_3\left(z\right)v\left(\text{d}z\right)\\ +{\sigma }_3\frac{w\left(t\right)}{t}+\frac{1}{t}{\displaystyle {\int }_0^t{\displaystyle {\int }_z\ln }}\left(1+{\psi }_3\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (3.9)

从而，可获得

$\begin{array}{c}\frac{r_2}{K_2}{\langle x_1\rangle }_{*}+\frac{r_2}{K_2}{\langle x_2\rangle }_{*}+\frac{a}{b_{21}}{\langle y_1\rangle }_{*}+\frac{a}{b_{22}}{\langle y_2\rangle }_{*}\ge r_2-\frac{{\sigma }_3^2}{2}+{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_3\left(z\right)\right)-{\psi }_3\left(z\right)\right]}\left(\text{d}z\right).\end{array}$

定义

$V\left(y_2\right)=\ln \left(y_2\left(t\right)\right)$ (3.10)

运用Itô公式，可得到：

$\begin{array}{c}\text{d}V\left(y_2\right)=\left[-c_2+k{x}_1\frac{a}{b_{12}+x_1+x_2}+k{x}_2\frac{a}{b_{22}+x_1+x_2}\right]\text{d}t\\ -\left(\frac{{\sigma }_4^2}{2}-{\displaystyle {\int }_z\ln }\left(1+{\psi }_4\left(z\right)\right)-{\psi }_4\left(z\right)v\left(\text{d}z\right)\right)\text{d}t\\ +{\sigma }_4\text{d}w\left(t\right)+{\displaystyle {\int }_z\ln }\left(1+{\psi }_4\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (3.11)

对式子(3.11)两边积分且除以t，可获得

$\begin{array}{c}\frac{\ln \frac{y_2\left(t\right)}{y_2\left(0\right)}}{t}\ge -c_2+\frac{ka}{b_{12}}\langle x_1\rangle +\frac{ka}{b_{22}}\langle x_2\rangle -\frac{{\sigma }_4^2}{2}+{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_4\left(z\right)\right)-{\psi }_4\left(z\right)\right]\left(\text{d}z\right)}\\ +{\sigma }_4\frac{w\left(t\right)}{t}+\frac{1}{t}{\displaystyle {\int }_0^t{\displaystyle {\int }_z\ln }}\left(1+{\psi }_4\left(z\right)\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (3.12)

从而，我们有

$c_2+\frac{{\sigma }_4^2}{2}-{\displaystyle {\int }_z\left[\ln \left(1+{\psi }_4\left(z\right)\right)-{\psi }_4\left(z\right)\right]v\left(\text{d}z\right)}\le \frac{ka}{b_{12}}{\langle x_1\rangle }_{*}+\frac{ka}{b_{22}}{\langle x_2\rangle }_{*}.$

定理证明完毕。

4. 平稳分布

本小节，主要讨论具有Lévy跳跃驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)存在唯一的平稳分布。

引理4.1 设 $X_1\left(t\right)=X_1\left(x_1,y_1,x_2,y_2,t\right)$ 和 $X_2\left(t\right)=X_2\left({\tilde{x}}_1,{\tilde{y}}_1,{\tilde{x}}_2,{\tilde{y}}_2,t\right)$ 是模型(1.3)的任意两个解，如果满足下列条件

$\frac{1}{K_1}>\frac{2ka}{b_{11}}+\frac{ka}{b_{21}}+\frac{2ka}{b_{12}}+\frac{ka}{b_{22}},\frac{1}{K_2}>\frac{ka}{b_{11}}+\frac{2ka}{b_{21}}+\frac{ka}{b_{12}}+\frac{2ka}{b_{22}}$

和

$k>\max \left\{\frac{1}{b_{12}}-\frac{1}{b_{22}},\frac{1}{b_{11}}-\frac{1}{b_{21}}\right\}$

则有

${\lim }_{t\to +\infty }\left(\Vert X_1\left(t\right)-X_2\left(t\right)\Vert \right)=0$

即

${\lim }_{t\to +\infty }\left(E\Vert x_1\left(t\right)-{\tilde{x}}_1\left(t\right)\Vert +E\Vert y_1\left(t\right)-{\tilde{y}}_1\left(t\right)\Vert +E\Vert x_2\left(t\right)-{\tilde{x}}_2\left(t\right)\Vert +E\Vert y_2\left(t\right)-{\tilde{y}}_2\left(t\right)\Vert \right)=0$

证明：定义

$V\left(x_1,y_1,x_2,y_2\right)=\alpha \left|\ln x_1-\ln {\tilde{x}}_1\right|+\left|\ln y_1-\ln {\tilde{y}}_1\right|+\beta \left|\ln x_2-\ln {\tilde{x}}_2\right|+\left|\ln y_2-\ln {\tilde{y}}_2\right|,$

则，我们有

$\begin{array}{l}\text{d}V\left(x_1,y_1,x_2,y_2,{\tilde{x}}_1,{\tilde{y}}_1,{\tilde{x}}_2,{\tilde{y}}_2\right)\\ =\text{d}\left[\alpha \left|\ln x_1-\ln {\tilde{x}}_1\right|+\left|\ln y_1-\ln {\tilde{y}}_1\right|+\beta \left|\ln x_2-\ln {\tilde{x}}_2\right|+\left|\ln y_2-\ln {\tilde{y}}_2\right|\right]\\ \le \{\left(-\frac{1}{K_1}+\frac{1}{K_2}+\frac{2ka}{b_{11}}+\frac{ka}{b_{21}}+\frac{2ka}{b_{12}}+\frac{ka}{b_{22}}\right)\left|x_1-{\tilde{x}}_1\right|\\ +\left(\frac{1}{K_1}-\frac{1}{K_2}+\frac{ka}{b_{11}}+\frac{2ka}{b_{21}}+\frac{ka}{b_{12}}+\frac{2ka}{b_{22}}\right)\left|x_2-{\tilde{x}}_2\right|\\ +\left(\frac{a}{b_{11}}+\frac{a}{b_{21}}-ka\right)\left|y_1-{\tilde{y}}_1\right|+\left(\frac{a}{b_{11}}+\frac{a}{b_{21}}+ka\right)\left(y_1+{\tilde{y}}_1\right)\\ +\left(\frac{a}{b_{12}}+\frac{a}{b_{22}}-ka\right)\left|y_2-{\tilde{y}}_2\right|+\left(\frac{a}{b_{12}}+\frac{a}{b_{22}}+ka\right)\left(y_2+{\tilde{y}}_2\right)\}\text{d}t.\end{array}$ (4.1)

因此，我们有

$\begin{array}{c}EV\left(x_1,y_1,x_2,y_2\right)\le \left(-\frac{1}{K_1}+\frac{1}{K_2}+\frac{2ka}{b_{11}}+\frac{ka}{b_{21}}+\frac{2ka}{b_{12}}+\frac{ka}{b_{22}}\right){\displaystyle {\int }_0^{t}E\left|x_1-{\tilde{x}}_1\right|\text{d}t}\\ +\left(\frac{1}{K_1}-\frac{1}{K_2}+\frac{ka}{b_{11}}+\frac{2ka}{b_{21}}+\frac{ka}{b_{12}}+\frac{2ka}{b_{22}}\right){\displaystyle {\int }_0^{t}E\left|x_2-{\tilde{x}}_2\right|\text{d}t}\\ +\left(\frac{a}{b_{11}}+\frac{a}{b_{21}}-ka\right){\displaystyle {\int }_0^{t}E\left|y_1-{\tilde{y}}_1\right|\text{d}t}+\left(\frac{a}{b_{11}}+\frac{a}{b_{21}}+ka\right)Nt\\ +\left(\frac{a}{b_{12}}+\frac{a}{b_{22}}-ka\right){\displaystyle {\int }_0^{t}E\left|y_2-{\tilde{y}}_2\right|\text{d}t}+\left(\frac{a}{b_{12}}+\frac{a}{b_{22}}+ka\right)Nt.\end{array}$ (4.2)

由于 $EV\left(x_1,y_1,x_2,y_2\right)>0$ ，因此，我们获得

$\begin{array}{l}\left(\frac{1}{K_1}-\frac{2ka}{b_{11}}-\frac{ka}{b_{21}}-\frac{2ka}{b_{12}}-\frac{ka}{b_{22}}\right){\displaystyle {\int }_0^{t}E\left|x_1-{\tilde{x}}_1\right|\text{d}t}\\ +\left(\frac{1}{K_2}-\frac{ka}{b_{11}}-\frac{2ka}{b_{21}}-\frac{ka}{b_{12}}-\frac{2ka}{b_{22}}\right){\displaystyle {\int }_0^{t}E\left|x_2-{\tilde{x}}_2\right|\text{d}t}\\ +\left(ka-\frac{a}{b_{11}}-\frac{a}{b_{21}}\right){\displaystyle {\int }_0^{t}E\left|y_1-{\tilde{y}}_1\right|\text{d}t}+\left(ka-\frac{a}{b_{12}}-\frac{a}{b_{22}}\right){\displaystyle {\int }_0^{t}E\left|y_2-{\tilde{y}}_2\right|\text{d}t}\\ \le V\left(0\right)+\left(\frac{a}{b_{11}}+\frac{a}{b_{21}}+ka\right)Nt+\left(\frac{a}{b_{12}}+\frac{a}{b_{22}}+ka+\frac{2}{K_1}+\frac{2}{K_2}\right)Nt<\infty .\end{array}$ (4.3)

因此

$E\left|x_1-{\tilde{x}}_1\right|,E\left|y_1-{\tilde{y}}_1\right|,E\left|x_2-{\tilde{x}}_2\right|,E\left|y_2-{\tilde{y}}_2\right|\in L\left[0,+\infty \right).$

类似文献 [23] 定理6.2的证明，我们可得

${\lim }_{t\to +\infty }\left(E\Vert x_1\left(t\right)-{\tilde{x}}_1\left(t\right)\Vert +E\Vert y_1\left(t\right)-{\tilde{y}}_1\left(t\right)\Vert +E\Vert x_2\left(t\right)-{\tilde{x}}_2\left(t\right)\Vert +E\Vert y_2\left(t\right)-{\tilde{y}}_2\left(t\right)\Vert \right)=0$

几乎必然成立。引理证明完毕。

引理4.2 对于任意p>0，任意的紧子集 $D\subseteq {\Re }_{+}^4$ ，则有

${\sup }_{D\subseteq {\Re }_{+}^4}E\left[{\sup }_{0\le s\le t}{\left|x_1\left(t\right)\right|}^p\right]<+\infty ,{\sup }_{D\subseteq {\Re }_{+}^4}E\left[{\sup }_{0\le s\le t}{\left|y_1\left(t\right)\right|}^p\right]<+\infty ,$

${\sup }_{D\subseteq {\Re }_{+}^4}E\left[{\sup }_{0\le s\le t}{\left|x_2\left(t\right)\right|}^p\right]<+\infty ,{\sup }_{D\subseteq {\Re }_{+}^4}E\left[{\sup }_{0\le s\le t}{\left|y_2\left(t\right)\right|}^p\right]<+\infty .$

证明：根据系统(1.3)，我们有

$\begin{array}{c}{x}_1\left(t\right)=x_1\left(0\right)+{\displaystyle {\int }_0^t{x}_1}\left(s\right)\left[r_1\left(1-\frac{x_1\left(s\right)+x_2}{K_1\left(s\right)}\right)-y_1\left(s\right)\frac{a}{b_{11}+x_1\left(s\right)+x_2\left(s\right)}-y_2\left(s\right)\frac{a}{b_{12}+x_1\left(s\right)+x_2\left(s\right)}\right]\text{d}s\\ +{\displaystyle {\int }_0^t{\sigma }_1{x}_1}\left(s\right)\text{d}w\left(s\right)+{\displaystyle {\int }_0^t{\displaystyle {\int }_z{\psi }_1}}\left(z\right)x_1\left(s\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\end{array}$

$\begin{array}{c}{y}_1\left(t\right)=y_1\left(0\right)+{\displaystyle {\int }_0^t{y}_1}\left(s\right)\left[-c_1+k{x}_1\left(s\right)\frac{a}{b_{11}+x_1\left(s\right)+x_2\left(s\right)}+k{x}_2\left(s\right)\frac{a}{b_{21}+x_1\left(s\right)+x_2\left(s\right)}\right]\text{d}s\\ +{\displaystyle {\int }_0^t{\sigma }_2{y}_1}\left(s\right)\text{d}w\left(t\right)+{\displaystyle {\int }_0^t{\displaystyle {\int }_z{\psi }_2}}\left(z\right)y_1\left(s\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\end{array}$

$\begin{array}{c}{x}_2\left(t\right)=x_2\left(0\right)+{\displaystyle {\int }_0^t{x}_2}\left(s\right)\left[r_2\left(1-\frac{x_1\left(s\right)+x_2\left(s\right)}{K_2}\right)y_1\left(s\right)\frac{a}{b_{21}+x_1\left(s\right)+x_2\left(s\right)}-y_2\left(s\right)\frac{a}{b_{22}+x_1\left(s\right)+x_2\left(s\right)}\right]\text{d}s\\ +{\displaystyle {\int }_0^t{\sigma }_3{x}_2}\left(s\right)\text{d}w\left(t\right)+{\displaystyle {\int }_0^t{\displaystyle {\int }_z{\psi }_3}}\left(z\right)x_2\left(s\right)\tilde{N}\left(\text{d}t,\text{d}z\right),\end{array}$

$\begin{array}{c}{y}_2\left(t\right)=y_2\left(0\right)+{\displaystyle {\int }_0^t{y}_2}\left(s\right)\left[-c_2+k{x}_1\left(s\right)\frac{a}{b_{12}+x_1\left(s\right)+x_2\left(s\right)}+k{x}_2\left(s\right)\frac{a}{b_{22}+x_1\left(s\right)+x_2\left(s\right)}\right]\text{d}s\\ +{\displaystyle {\int }_0^t{\sigma }_4{y}_2}\left(s\right)\text{d}w\left(t\right)+{\displaystyle {\int }_0^t{\displaystyle {\int }_z{\psi }_4}}\left(z\right)y_2\left(s\right)\tilde{N}\left(\text{d}t,\text{d}z\right).\end{array}$ (4.4)

结合Hölder不等式，对于 $k=1,2,\cdots$ ，我们获得

$\begin{array}{l}E\left[\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|x_1\left(s\right)\right|}^p\right]\\ \le 4^{p-1}{\left|x_1\left(\left(k-1\right)\xi \right)\right|}^p+4^{p-1}E[\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }|{\displaystyle {\int }_{\left(k-1\right)\xi }^{k\xi }{x}_1\left(s\right)}[r_1\left(1-\frac{x_1\left(s\right)+x_2\left(s\right)}{K_1}\right)\\ -y_1\left(s\right)\frac{a}{b_{11}+x_1\left(s\right)+x_2\left(s\right)}-{y_2\left(s\right)\frac{a}{b_{12}+x_1\left(s\right)+x_2\left(s\right)}]\text{d}s|}^p]\\ +4^{p-1}E\left[\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|{\displaystyle {\int }_{\left(k-1\right)\xi }^{k\xi }{\sigma }_1{x}_1\left(s\right)\text{d}w\left(s\right)}\right|}^p\right]\\ +4^{p-1}E\left[\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|{\displaystyle {\int }_{\left(k-1\right)\xi }^{k\xi }{\displaystyle {\int }_z{\psi }_1\left(z\right)x_1\left(s\right)\tilde{N}\left(\text{d}t,\text{d}z\right)}}\right|}^p\right].\end{array}$ (4.5)

根据定理2.2，存在一个正数K(p)，使得 $E{\left|x_1\left(t\right)\right|}^p,E{\left|y_1\left(t\right)\right|}^p,E{\left|x_2\left(t\right)\right|}^p,E{\left|y_2\left(t\right)\right|}^p\le K\left(p\right)$ ，从而，我们获得

$\begin{array}{l}E[\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }|{\displaystyle {\int }_{\left(k-1\right)\xi }^{k\xi }{x}_1\left(s\right)}[r_1\left(1-\frac{x_1\left(s\right)+x_2\left(s\right)}{K_1}\right)\\ -{y_1\left(s\right)\frac{a}{b_{11}+x_1\left(s\right)+x_2\left(s\right)}-y_2\left(s\right)\frac{a}{b_{12}+x_1\left(s\right)+x_2\left(s\right)}]\text{d}s|}^p]\\ \le \frac{1}{2}{\xi }^{p}E{\left|x_1\left(s\right)\right|}^{2p}+\frac{1}{2}{\xi }^p{5}^{2p-1}[r_1^{2p}+\frac{r_1^{2p}}{K_1^{2p}}E\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|x_1\left(s\right)\right|}^{2p}+\frac{r_1^{2p}}{K_1^{2p}}E\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|x_2\left(s\right)\right|}^{2p}\\ +\frac{a}{b_{11}}E\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|y_1\left(s\right)\right|}^{2p}+\frac{a}{b_{12}}\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|y_2\left(s\right)\right|}^{2p}]\\ :=M_1\left(p\right){\xi }^p.\end{array}$ (4.6)

根据Burkholder-Davis-Gundy不等式，我们有

$E\left[\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|{\displaystyle {\int }_{\left(k-1\right)\xi }^{k\xi }{\sigma }_1{x}_1\left(s\right)\text{d}w\left(s\right)}\right|}^p\right]\le C_p{\xi }^{\frac{p}{2}}{\sigma }_1^{p}E\left(\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|x_1\left(s\right)\right|}^p\right):=M_2\left(p\right){\xi }^{\frac{p}{2}}.$ (4.7)

根据Kunita’s第一不等式，我们有

$\begin{array}{l}E\left[\underset{\left(k-1\right)\xi \le s\le k\xi }{\sup }{\left|{\displaystyle {\int }_{\left(k-1\right)\xi }^{k\xi }{\displaystyle {\int }_z{\psi }_1\left(z\right)x_1\left(s\right)\tilde{N}\left(\text{d}s,\text{d}z\right)}}\right|}^p\right]\\ \le D_p\left\{E{\left({\displaystyle {\int }_{\left(k-1\right)\xi }^{k\xi }{\displaystyle {\int }_z{\left|{\psi }_1\left(z\right)x_1\left(s\right)\right|}^{2}v\left(\text{d}z\right)\text{d}s}}\right)}^{\frac{p}{2}}+E\left({\displaystyle {\int }_{\left(k-1\right)\xi }^{k\xi }{\displaystyle {\int }_z{\left|{\psi }_1\left(z\right)x_1\left(s\right)\right|}^{p}v\left(\text{d}z\right)\text{d}s}}\right)\right\}\\ \le D_p{\xi }^{\frac{p}{2}}{G}_1^{\frac{p}{2}}K\left(p\right)+D_p\xi G_2^{\frac{p}{2}}K\left(p\right).\end{array}$ (4.8)

根据(4.5)~(4.8)，我们有

$\underset{D\subseteq {\Re }_{+}^4}{\sup }E\left[\underset{0\le s\le t}{\sup }{\left|x_1\left(t\right)\right|}^p\right]<+\infty ,\forall s\in \left[0,t\right],\forall t\ge 0.$

类似的证明，我们可以获得

$\underset{D\subseteq {\Re }_{+}^4}{\sup }E\left[\underset{0\le s\le t}{\sup }{\left|y_1\left(t\right)\right|}^p\right]<+\infty ,\underset{D\subseteq {\Re }_{+}^4}{\sup }E\left[\underset{0\le s\le t}{\sup }{\left|x_2\left(t\right)\right|}^p\right]<+\infty ,\underset{D\subseteq {\Re }_{+}^4}{\sup }E\left[\underset{0\le s\le t}{\sup }{\left|y_2\left(t\right)\right|}^p\right]<+\infty .$ .

引理证明完毕。

让 $C_b\left({\Re }_{+}^4\right)$ 表示在 ${\Re }_{+}^4$ 上所有有界连续实值函数的集合， $P\left({\Re }_{+}^4\right)$ 是在 $\left({\Re }_{+}^4,\mathcal{B}\left({\Re }_{+}^4\right)\right)$ 上所有概率测度的空间，其中 $\mathcal{B}\left({\Re }_{+}^4\right)$ 是在 ${\Re }_{+}^4$ 上的Borel σ-代数。对于系统(1.3)的任意解 $X\left(x_1,y_1,x_2,y_2\right)$ ，众所周知， $X\left(x_1\left(t\right),y_1\left(t\right),x_2\left(t\right),y_2\left(t\right)\right)$ 在 ${\Re }_{+}^4$ 上是一个马尔可夫过程。

定义4.1 [24] ：系统(1.3)的解 $X=\left(t,\varphi \right)X\left(x_1,y_1,x_2,y_2\right)$ 是平稳分布，如果存在概率测度 $\mu \in P\left({\Re }_{+}^4\right)$ 满足

$\mu \left(f\right)=\mu \left({\mathbb{P}}_{t}f\right),t\ge 0$

其中

$\mu \left(f\right):={\displaystyle {\int }_{{\Re }_{+}^4}f\left(\varphi \right)\mu \left(\text{d}\varphi \right)}$ 和 ${\mathbb{P}}_{t}f\left(\varphi \right)=Ef\left(X\left(t,\varphi \right)\right),f\in C_b\left({\Re }_{+}^4\right)$

对于 ${\mu }_1,{\mu }_2\in P\left({\Re }_{+}^4\right)$ ，在 $P\left({\Re }_{+}^4\right)$ 上定义度量空间

$d\left({\mu }_1,{\mu }_2\right)=\underset{f\in \mathcal{M}}{\sup }\left|{\displaystyle {\int }_{{\Re }_{+}^4}f\left(\varphi \right){\mu }_1\left(\text{d}\varphi \right)}-{\displaystyle {\int }_{{\Re }_{+}^4}f\left(\psi \right){\mu }_2\left(\text{d}\psi \right)}\right|$

其中

$\mathcal{M}:=\left\{f:{\Re }_{+}^4\to \Re ,\left|f\left(\varphi \right)-f\left(\psi \right)\right|\le {\Vert \varphi -\psi \Vert }_{{\Re }_{+}^4},\forall \phi ,\psi \in {\Re }_{+}^4,\left|f(\cdot )\right|\le 1\right\}$

则众所周知，在度量 $d\left(\cdot ,\cdot \right)$ 上 $P\left({\Re }_{+}^4\right)$ 是完备的。

定理4.3 如果满足下列条件

$\frac{1}{K_1}>\frac{2ka}{b_{11}}+\frac{ka}{b_{21}}+\frac{2ka}{b_{12}}+\frac{ka}{b_{22}},\frac{1}{K_2}>\frac{ka}{b_{11}}+\frac{2ka}{b_{21}}+\frac{ka}{b_{12}}+\frac{2ka}{b_{22}}$

和

$k>\max \left\{\frac{1}{b_{12}}-\frac{1}{b_{22}},\frac{1}{b_{11}}-\frac{1}{b_{21}}\right\}.$

则具有Lévy跳跃驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)存在唯一的平稳分布。

证明：对任意给定 $\varphi \in {\Re }_{+}^4$ 和 $\epsilon >0$ ，下面证明存在一个时间 $T>0$ 使得

$d\left(\mathbb{P}\left(\varphi ,t+s,\cdot \right),\mathbb{P}\left(\varphi ,t,\cdot \right)\right)=\underset{f\in \mathcal{M}}{\sup }\left|Ef\left(X\left(t+s,\varphi \right)\right)-Ef\left(X\left(t,\varphi \right)\right)\right|\le \epsilon ,$ (4.9)

对任意 $t\ge T$ 和 $s>0$ 。

事实上，对于任意 $f\in \mathcal{M}$ 和 $t,s>0$ ，我们获得

$\begin{array}{l}\left|E\left[Ef\left(X\left(t+s,\varphi \right)\right)|{\mathcal{F}}_s\right]-Ef\left(X\left(t,\varphi \right)\right)\right|\\ =\left|Ef\left(X\left(t+s,\varphi \right)\right)-Ef\left(X\left(t,\varphi \right)\right)\right|\\ =\left|{\displaystyle {\int }_{{\Re }_{+}^4}E}\left(X\left(t,\psi \right)\right)\mathbb{P}\left(\varphi ,s,\text{d}\psi \right)-Ef\left(X\left(t,\varphi \right)\right)\right|\\ \le {\displaystyle {\int }_{{\Re }_{+}^4}\left|Ef\left(X\left(t,\psi \right)\right)-Ef\left(X\left(t,\varphi \right)\right)\right|\mathbb{P}\left(\varphi ,s,\text{d}\psi \right)}\\ \le 2\mathbb{P}\left(\varphi ,s,{\Re }_{+}^{4,c}\right)+{\displaystyle {\int }_{{\Re }_{+,R}^4}\left|Ef\left(X\left(t,\psi \right)\right)-Ef\left(X\left(t,\varphi \right)\right)\right|\mathbb{P}\left(\varphi ,s,\text{d}\psi \right)}\end{array}$ (4.10)

其中 ${\Re }_{+,R}^4=\left\{\varphi \in {\Re }_{+}^4:\Vert \varphi \Vert \le R\right\}$ 和 ${\Re }_{+,R}^{4,C}={\Re }_{+}^4-{\Re }_{+,R}^4$ ，根据引理3.2以及引理5.1 (参考文献 [24] )，存在一个充分大的R使得

$\mathbb{P}\left(\varphi ,s,{\Re }_{+}^{4,c}\right)\le \frac{\epsilon }{4},\forall s>0.$ (4.11)

另一方面，根据引理3.1以及引理5.1 (参考文献 [24] )，存在一个时间 $T_2$ 使得

$\underset{f\in \mathcal{M}}{\sup }\left|Ef\left(X\left(t,\psi \right)\right)-Ef\left(X\left(t,\varphi \right)\right)\right|\le \frac{\epsilon }{4},\varphi ,\psi \in {\Re }_{+}^4,\forall t>T_2$ (4.12)

因此，把(4.11)和(4.12)代入(4.10)，得到

$\left|E\left[Ef\left(X\left(t+s,\varphi \right)\right)|{\mathcal{F}}_s\right]-Ef\left(X\left(t,\varphi \right)\right)\right|\le \epsilon ,\forall t>T_2,s>0$ (4.13)

由于 $f\in \mathcal{M}$ 的任意性，我们获得式子(4.9)成立，即

$d\left(\mathbb{P}\left(\varphi ,t+s,\cdot \right),\mathbb{P}\left(\varphi ,t,\cdot \right)\right)\le \epsilon .$

因此，系统(1.3)的解 $X=\left(t,\varphi \right)X\left(x_1,y_1,x_2,y_2\right)$ 的转移概率 $\mathbb{P}\left(\varphi ,t,\cdot \right)$ 弱收敛到某些测度 $\mu \in P\left({\Re }_{+}^4\right)$ 。另一方面，对于任意 $f\in C_b\left({\Re }_{+}^4\right)$ ， $X\left(x_1\left(t\right),y_1\left(t\right),x_2\left(t\right),y_2\left(t\right)\right)$ 具有马尔可夫性质使得

${\mathbb{P}}_{t+s}f\left(\varphi \right)={\mathbb{P}}_t{\mathbb{P}}_{s}f\left(\varphi \right),t,s\ge 0,\varphi \in {\Re }_{+}^4.$

因此，对于给定 $t\ge 0$ ，当 $s\to \infty$ ，则有

$\mu \left(f\right)=\mu \left({\mathbb{P}}_{t}f\right),f\in C_b\left({\Re }_{+}^4\right)$

故 $\mu$ 是系统(1.3)解 $X=\left(t,\varphi \right)X\left(x_1,y_1,x_2,y_2\right)$ 的一个平稳分布。

下面我们将证明平稳分布的唯一性，如果 ${\mu }^{\prime }$ 是系统(1.3)解 $X=\left(t,\varphi \right)X\left(x_1,y_1,x_2,y_2\right)$ 的另一个平稳分布。让 $f\in C_{LB}\left({\Re }_{+}^4\right)$ ，其中 $C_{LB}\left({\Re }_{+}^4\right)$ 表示在 ${\Re }_{+}^4$ 上所有有界和Lipschitz连续函数族。根据引理3.1，引理3.2和hölder不等式以及 $\mu ,{\mu }^{\prime }\in P\left({\Re }_{+}^4\right)$ 的不变性，我们有

$\left|\mu \left(f\right)-{\mu }^{\prime }\left(f\right)\right|\le {\displaystyle {\int }_{{\Re }_{+}^4}\left|{\mathbb{P}}_{t}f\left(\varphi \right)-{\mathbb{P}}_{t}f\left(\psi \right)\right|\text{d}\mu \left(\varphi \right)\text{d}{\mu }^{\prime }\left(\psi \right)}<\epsilon$

这意味着存在唯一性的平稳分布，即具有Lévy跳跃驱动的Rosenzweig-MacArthur捕食者–食饵模型(1.3)存在唯一的平稳分布。定理证明完毕。