双捕食者与双食饵的随机捕食系统动力学 Dynamic of Stochastic Predator-Prey Sys-tem with Two Predators and Two Prey

doi:10.12677/ORF.2023.132069

Operations Research and Fuzziology
Vol. 13 No. 02 ( 2023 ), Article ID: 63857 , 15 pages
10.12677/ORF.2023.132069

双捕食者与双食饵的随机捕食系统动力学

王欣琦，张天四^*

●How to Cite this Article

上海理工大学理学院，上海

收稿日期：2023年2月18日；录用日期：2023年4月6日；发布日期：2023年4月13日

摘要

本文研究了一类具有两个捕食者和两个食饵的随机捕食食饵模型，证明了系统具有唯一的全局正解，并利用随机李雅普诺夫函数给出了系统具有平稳分布的充分条件。最后分别在食饵种群存活和捕食者种群灭绝，及所有食饵和捕食者种群均灭绝这两种情况下，给出了捕食者种群均灭绝的充分条件。

关键词

捕食食饵模型，随机，平稳分布，遍历性，灭绝

Dynamic of Stochastic Predator-Prey System with Two Predators and Two Prey

Xinqi Wang, Tiansi Zhang^*

College of Science, University of Shanghai for Science and Technology, Shanghai

Received: Feb. 18^th, 2023; accepted: Apr. 6^th, 2023; published: Apr. 13^th, 2023

ABSTRACT

The paper studies a stochastic predator-prey model with two predators and two prey. Firstly, we prove that the system has a unique global positive solution. Then, by using the stochastic Lyapunov function method, we obtain sufficient criteria for the existence of stationary distribution and ergodicity. Finally, sufficient conditions for extinction of the predator population in two cases are shown, those are, the prey population survival and the predator population extinction, and all the prey and predator populations extinction.

Keywords:Predator-Prey Model, Stochastic, Stationary Distribution, Ergodicity, Extinction

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 模型的建立

自Lotake和Volterra提出的第一个捕食食饵模型以来，捕食食饵模型便得到了大家广泛的关注，及模型的不断完善，例如添加食饵和捕食者的自我竞争项 [1] [2] [3] 、加入不同的功能反应函数 [4] [5] [6] [7] 、考虑捕食者与食饵的年龄结构 [8] - [12] 等。

由于生态系统的复杂性，二维的捕食食饵模型已经无法准确的描述现实世界，许多关键行为只能由具有三个或更多物种的模型表现出来，因此捕食食饵模型逐渐的由二维模型转向高维模型的研究，多物种的食物网比双物种的食物网也更加的丰富与值得被关注。阮世贵等人 [13] 讨论了具有Holling II型功能反应函数的两个捕食者和一个食饵的捕食食饵模型，其中两个捕食者互为竞争关系。Jaume Llibre和肖冬梅 [14] 描述了两个捕食者竞争同一食饵的模型。在2021年发表的一篇文章中 [15] ，Abhijit Jana和Sankar Kumar Roy考虑了以下具有两个捕食者和两个食饵的四维的捕食食饵模型：

${\begin{cases} \frac{d x}{d t} = r x (1 - \frac{x}{K}) - a_{1} x y - ω_{1} x u - ω_{2} x v \\ \frac{d y}{d t} = s y (1 - \frac{y}{L}) - a_{2} x y - \frac{ω_{3} u y}{m + y} - ω_{4} v y \\ \frac{d u}{d t} = n_{1} ω_{1} x u + \frac{n_{2} ω_{3} u y}{m + y} - b_{1} u v - c_{1} u \\ \frac{d v}{d t} = n_{3} ω_{2} x v + n_{4} ω_{4} y v - b_{2} u v - c_{2} v \end{cases}$ (1.1)

其中 $x, y, u, v$ 分别表示两个不同的食饵种群和两个不同的捕食者种群在时刻t的种群密度， $r, s, K, L, a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2}, ω_{1}, ω_{2}, ω_{3}, ω_{4}, n_{1}, n_{2}, n_{3}, n_{4}$ 均为正的常数， $r, s$ 分别表示两个食饵种群的内在增长率， $K, L$ 分别表示两个食饵种群的环境容纳量，m表示半饱和常数， $ω_{1}, ω_{2}, ω_{3}, ω_{4}$ 表示捕食过程对食饵的影响， $a_{1}, a_{2}, b_{1}, b_{2}$ 分别表示两个食饵种群间的影响和两个捕食者种群间的影响， $n_{1}, n_{2}, n_{3}, n_{4}$ 表示捕食者食用食饵后转化为自身能量的转化率， $c_{1}, c_{2}$ 分别表示两个捕食者种群的自然死亡率。

在复杂的生态系统中，种群的生存会受到外界环境因素的干扰，例如白噪声、彩色噪声和马尔可夫转换等 [2] ，因此确定性模型已经不足以去研究捕食食饵系统，从而很多学者开始研究随机模型 [1] [2] [3] [16] [17] [18] [19] [20] ，即在确定性模型中加入白噪声参数，使模型更加贴合生态系统的自然现象。为了更好的讨论具有多捕食者种群和多食饵种群的捕食食饵模型，本文在 [15] 的确定性模型基础之上，引入了白噪声的影响，构造了以下的随机捕食食饵模型：

${\begin{cases} d x_{1} = [r x_{1} (1 - \frac{x_{1}}{K}) - a_{1} x_{1} x_{2} - ω_{1} x_{1} y_{1} - ω_{2} x_{1} y_{2}] d t + σ_{1} x_{1} d B_{1} (t), \\ d x_{2} = [s x_{2} (1 - \frac{x_{2}}{L}) - a_{2} x_{1} x_{2} - \frac{ω_{3} x_{2} y_{1}}{m + x_{2}} - ω_{4} x_{2} y_{2}] d t + σ_{2} x_{2} d B_{2} (t), \\ d y_{1} = y_{1} (n_{1} ω_{1} x_{1} + \frac{n_{2} ω_{3} x_{2}}{m + x_{2}} - b_{1} y_{2} - c_{1} - q_{1} y_{1}) d t + σ_{3} y_{1} d B_{3} (t), \\ d y_{2} = y_{2} (n_{3} ω_{2} x_{1} + n_{4} ω_{4} x_{2} - b_{2} y_{1} - c_{2} - q_{2} y_{2}) d t + σ_{4} y_{2} d B_{4} (t), \end{cases}$ (1.2)

其中 $x_{1}, x_{2}$ 表示两个不同的食饵种群在时刻t的种群密度， $y_{1}, y_{2}$ 表示两个不同的捕食者种群在时刻t的种群密度， $q_{1}, q_{2}$ 为正的常数分别表示两个捕食者种群的种内竞争率， $B_{1} (t), B_{2} (t), B_{3} (t), B_{4} (t)$ 为标准布朗运动， $σ_{1}, σ_{2}, σ_{3}, σ_{4}$ 表示噪声强度，其余参数的定义均与系统(1.1)相同。

在本文中， $(Ω, F, {F_{t}}_{t \geq 0}, ℙ)$ 表示一个带有过滤的完全概率空间，其中 ${F_{t}}_{t \geq 0}$ 满足通常的条件(即 $F_{0}$ 包含所有 $ℙ$ -空集，它是递增和右连续的)。设 $B_{i} (t), i = 1, 2, 3, 4$ 定义在这个完全概率空间上。我们定义 $ℝ_{+}^{d} = {x = (x_{1}, \dots, x_{d}) \in ℝ^{d} : x_{i} > 0, 1 \leq i \leq d}$ 。

本文其他部分的研究主要分为以下几个部分：第二部分，证明了系统(1.2)对任意的初始值都存在唯一的全局正解；第三部分，建立了系统(1.2)存在遍历平稳分布的充分条件；第四部分，证明了两种情况下捕食者种群灭绝的充分条件；最后对全文进行了总结。

2. 正解的存在性和唯一性

在本节中，我们先使用变量变换 [17] 证明系统(1.2)存在唯一的局部正解，然后使用李雅普诺夫分析法 [16] 证明该解是全局的。

定理2.1对于任意的初始值 $(x_{1} (0), x_{2} (0), y_{1} (0), y_{2} (0)) \in ℝ_{+}^{4}$ ，系统(1.2)在 $t \in [0, τ_{e})$ 时存在唯一的全局正解 $(x_{1} (t), x_{2} (t), y_{1} (t), y_{2} (t))$ 几乎处处成立，其中 $τ_{e}$ 是爆炸时间。

证明：对系统(1.2)使用变量变换 $u_{1} (t) = \ln x_{1} (t)$ ， $u_{2} (t) = \ln x_{2} (t)$ ， $v_{1} (t) = \ln y_{1} (t)$ ， $v_{2} (t) = \ln y_{2} (t)$ 和Itô’s公式，则可以得到

${\begin{cases} d u_{1} = (r - \frac{σ_{1}^{2}}{2} - \frac{r}{K} e^{u_{1}} - a_{1} e^{u_{2}} - ω_{1} e^{v_{1}} - ω_{2} e^{v_{2}}) d t + σ_{1} d B_{1} (t), \\ d u_{2} = (s - \frac{σ_{2}^{2}}{2} - \frac{s}{L} e^{u_{2}} - a_{2} e^{u_{1}} - \frac{ω_{3} e^{v_{1}}}{m + e^{u_{2}}} - ω_{4} e^{v_{2}}) d t + σ_{2} d B_{1} (t), \\ d v_{1} = (n_{1} ω_{1} e^{u_{1}} + \frac{n_{2} ω_{3} e^{u_{2}}}{m + e^{u_{2}}} - b_{1} e^{v_{2}} - c_{1} - \frac{σ_{3}^{2}}{2} - q_{1} e^{v_{1}}) d t + σ_{3} d B_{3} (t), \\ d v_{2} = (n_{3} ω_{2} e^{u_{1}} + n_{4} ω_{4} e^{u_{2}} - b_{2} e^{v_{1}} - c_{2} - \frac{σ_{4}^{2}}{2} - q_{2} e^{v_{2}}) d t + σ_{4} d B_{4} (t), \end{cases}$ (2.1)

其中，其满足初始条件 $u_{1} (0) = \ln x_{1} (0)$ ， $u_{2} (0) = \ln x_{2} (0)$ ， $v_{1} (0) = \ln y_{1} (0)$ ， $v_{2} (0) = \ln y_{2} (0)$ 。由于系统(2.1)满足线性增长条件和局部Lipschitz条件，则对于任意的初始值 $(u_{1} (0), u_{2} (0), v_{1} (0), v_{2} (0)) \in ℝ_{+}^{4}$ ，系统(2.1)在 $t \in [0, τ_{e})$ 时存在唯一的局部正解 $(u_{1} (t), u_{2} (t), v_{1} (t), v_{2} (t))$ 。因此 $(u_{1} (t), u_{2} (t), v_{1} (t), v_{2} (t)) = (e^{u_{1}}, e^{u_{2}}, e^{v_{1}}, e^{v_{2}})$ 是随机系统(1.2)从第一象限内开始的唯一的正局部解。此即证得系统(1.2)存在唯一的局部正解。下面，我们来证明上面证出的解是全局的，为了证明这一点，我们只需要证明 $τ_{e} = \infty$ 几乎处处成立即可，我们引入以下定理。

定理2.2对于任意的初值 $(x_{1} (0), x_{2} (0), y_{1} (0), y_{2} (0)) \in ℝ_{+}^{4}$ ，对所有的 $t \geq 0$ ，随机系统(1.2)存在唯一解 $(x_{1} (t), x_{2} (t), y_{1} (t), y_{2} (t))$ ，并且该解将以概率1留在 $ℝ_{+}^{4}$ 中，即系统(1.2)的解 $(x_{1} (t), x_{2} (t), y_{1} (t), y_{2} (t)) \in ℝ_{+}^{4}$ 对于所有的 $t \geq 0$ 几乎处处成立。

证明：这个定理的证明方法 [17] 是常用的，在这里我们只给出证明的不同点即对李雅普诺夫函数使用Itô’s公式的计算过程。

定义一个 $C^{2}$ 函数 $V : ℝ_{+}^{4} \to ℝ_{+} \cup {0}$

$V (x_{1}, x_{2}, y_{1}, y_{2}) = (x_{1} - 1 - \ln x_{1}) + (x_{2} - 1 - \ln x_{2}) + f_{1} (y_{1} - 1 - \ln y_{1}) + f_{2} (y_{2} - 1 - \ln y_{2}),$

其中 $f_{1} = \min {\frac{1}{n_{1}}, \frac{1}{n_{2}}}$ ， $f_{2} = \min {\frac{1}{n_{3}}, \frac{1}{n_{4}}}$ 。这个函数的非负性可以由不等式 $u - 1 - \ln u \geq 0$ ， $\forall u > 0$ 得到。

我们对上面的V函数使用Itô’s公式，可以得到

$\begin{matrix} d V (x_{1}, x_{2}, y_{1}, y_{2}) = L V (x_{1}, x_{2}, y_{1}, y_{2}) d t + σ_{1} (x_{1} - 1) d B_{1} (t) + σ_{2} (x_{2} - 1) d B_{2} (t) \\ + f_{1} σ_{3} (y_{1} - 1) d B_{3} (t) + f_{2} σ_{4} (y_{2} - 1) d B_{4} (t), \end{matrix}$

这里 $L V : ℝ_{+}^{4} \to ℝ$ 为

$\begin{matrix} L V = (x_{1} - 1) (r - \frac{r}{K} x_{1} - a_{1} x_{2} - ω_{1} y_{1} - ω_{2} y_{2}) + \frac{1}{2} \frac{1}{x_{1}^{2}} σ_{1}^{2} x_{1}^{2} \\ + (x_{2} - 1) (s - \frac{s}{L} x_{2} - a_{2} x_{1} - \frac{ω_{3} y_{1}}{m + x_{2}} - ω_{4} y_{2}) + \frac{1}{2} \frac{1}{x_{2}^{2}} σ_{2}^{2} x_{2}^{2} \\ + f_{1} (y_{1} - 1) (n_{1} ω_{1} x_{1} + \frac{n_{2} ω_{3} x_{2}}{m + x_{2}} - b_{1} y_{2} - c_{1} - q_{1} y_{1}) + \frac{1}{2} \frac{1}{y_{1}^{2}} σ_{1}^{2} y_{1}^{2} \\ + f_{2} (y_{2} - 1) (n_{3} ω_{2} x_{1} + n_{4} ω_{4} x_{2} - b_{2} y_{1} - c_{2} - q_{2} y_{2}) + \frac{1}{2} \frac{1}{y_{2}^{2}} σ_{2}^{2} y_{2}^{2} \end{matrix}$

$\begin{matrix} \leq \sup_{x_{1} \in R_{+}} {- \frac{r}{K} x_{1}^{2} + (r + \frac{r}{K} + a_{2}) x_{1}} + \sup_{x_{2} \in R_{+}} {- \frac{s}{L} x_{2}^{2} + (s + \frac{s}{L} + a_{1}) x_{2}} \\ + \sup_{y_{1} \in R_{+}} {- f_{1} q_{1} y_{1}^{2} + (ω_{1} + f_{2} b_{2} + f_{1} q_{1} + \frac{ω_{3}}{m}) y_{1}} \\ + \sup_{y_{2} \in R_{+}} {- f_{2} q_{2} y_{2}^{2} + (ω_{2} + ω_{4} + f_{1} b_{1} + f_{2} q_{2}) y_{2}} \\ + f_{1} c_{1} + f_{2} c_{2} + \frac{σ_{1}^{2}}{2} + \frac{σ_{2}^{2}}{2} + \frac{f_{1} σ_{3}^{2}}{2} + \frac{f_{2} σ_{4}^{2}}{2} \\ : = C_{0}, \end{matrix}$

其中 $C_{0}$ 是一个正常数。

3. 遍历平稳分布的存在

在本节中，我们将建立系统(1.2)正解的遍历平稳分布存在唯一性的充分条件。

设 $X (t)$ 是在 $ℝ^{d}$ 中由随机微分方程描述的正则时间齐次马尔可夫过程

$d X (t) = f (X (t)) d t + \sum_{r = 1}^{k} g_{r} (X (t)) d B_{r} (t),$

$X (t)$ 的扩散矩阵定义为 $A (x) = (a_{i j} (x)), a_{i j} (x) = \sum_{r = 1}^{k} g_{r}^{i} (x) g_{r}^{j} (x)$ 。

引理3.1 [17] 如果一个马尔可夫过程存在一个带有常规边界 $Γ$ 的有界开域 $U \subset ℝ^{d}$ ，且满足性质：

$A_{1}$ ：存在一个正数M使得 $\sum_{i, j = 1}^{d} a_{i j} (x) ξ_{i} ξ_{j} \geq M {| ξ |}^{2}$ ， $x \in U$ ， $ξ \in ℝ^{d}$ 。

$A_{2}$ ：存在一个非负 $C^{2} - V$ 函数，使得 $L V$ 在任何 $ℝ^{d} \ U$ 上均为负的。

则该马尔可夫过程 $X (t)$ 有唯一的遍历平稳分布 $π (\cdot)$ 。

定理3.1如果满足条件

$\frac{s}{L} > \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}$ (3.1)

和

$B + 2 \leq \min {\frac{r}{K} {(x_{1}^{*})}^{2}, (\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) {(x_{2}^{*})}^{2}, e_{1} q_{1} {(y_{1}^{*})}^{2}, e_{2} q_{2} {(y_{2}^{*})}^{2}},$ (3.2)

其中

$B : = \frac{1}{2} σ_{1}^{2} x_{1}^{*} + \frac{1}{2} σ_{2}^{2} x_{2}^{*} + \frac{e_{1}}{2} σ_{3}^{2} y_{1}^{*} + \frac{e_{2}}{2} σ_{4}^{2} y_{2}^{*},$

$(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})$ 满足

${\begin{cases} r = \frac{r}{K} x_{1}^{*} + a_{1} x_{2}^{*} + ω_{1} y_{1}^{*} + ω_{2} y_{2}^{*}, \\ s = \frac{s}{L} x_{2}^{*} + a_{2} x_{1}^{*} + \frac{ω_{3} y_{1}^{*}}{m + x_{2}^{*}} + ω_{4} y_{2}^{*}, \\ - c_{1} = - n_{1} ω_{1} x_{1}^{*} - \frac{n_{2} ω_{3} x_{2}^{*}}{m + x_{2}^{*}} + b_{1} y_{2}^{*} + q_{1} y_{1}^{*}, \\ - c_{2} = - n_{3} ω_{2} x_{1}^{*} - n_{4} ω_{4} x_{2}^{*} + b_{2} y_{1}^{*} + q_{2} y_{2}^{*}, \end{cases}$

则对任意的初始值 $(x_{1} (0), x_{2} (0), y_{1} (0), y_{2} (0)) \in ℝ_{+}^{4}$ ，系统(1.2)存在唯一的遍历平稳分布 $π (\cdot)$ 。

证明：为了证明系统(1.2)存在遍历平稳分布，只需证明引理3.1的两个条件 $A_{1}$ 、 $A_{2}$ 均成立即可。下面我们证明条件 $A_{1}$ 。

易知，对任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in {\bar{U}}_{σ} \subset ℝ_{+}^{4}$ ， $ξ = (ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}) \in ℝ_{+}^{4}$ ，系统(1.2)的扩散矩阵为

$\sum_{i, j = 1}^{4} a_{i j} (x_{1}, x_{2}, y_{1}, y_{2}) ξ_{i} ξ_{j} = σ_{1}^{2} x_{1}^{2} ξ_{1}^{2} + σ_{2}^{2} x_{2}^{2} ξ_{2}^{2} + σ_{3}^{2} y_{1}^{2} ξ_{3}^{2} + σ_{4}^{2} y_{2}^{2} ξ_{4}^{2} \geq M_{0} {‖ ξ ‖}^{2},$

其中 $M_{0} = \min_{(x_{1}, x_{2}, y_{1}, y_{2}) \in {\bar{U}}_{σ}} {σ_{1}^{2} x_{1}^{2}, σ_{2}^{2} x_{2}^{2}, σ_{3}^{2} y_{1}^{2}, σ_{4}^{2} y_{2}^{2}}$ 。则引理3.1中的条件 $A_{1}$ 得证。

对于条件 $A_{2}$ ，定义 $\overset{⌢}{V}$ 函数

$\begin{matrix} \overset{⌢}{V} (x_{1}, x_{2}, y_{1}, y_{2}) = (x_{1} - x_{1}^{*} - x_{1}^{*} \ln \frac{x_{1}}{x_{1}^{*}}) + (x_{2} - x_{2}^{*} - x_{2}^{*} \ln \frac{x_{2}}{x_{2}^{*}}) \\ + e_{1} (y_{1} - y_{1}^{*} - y_{1}^{*} \ln \frac{y_{1}}{y_{1}^{*}}) + e_{2} (y_{2} - y_{2}^{*} - y_{2}^{*} \ln \frac{y_{2}}{y_{2}^{*}}), \end{matrix}$

其中 $e_{1} = \min {\frac{1}{n_{1}}, \frac{m + x_{2}^{*}}{n_{2} m}}$ ， $e_{2} = \min {\frac{1}{n_{3}}, \frac{1}{n_{4}}}$ 。由Itô’s公式，可得

$\begin{matrix} L \overset{⌢}{V} = - \frac{r}{K} {(x_{1} - x_{1}^{*})}^{2} - \frac{s}{L} {(x_{2} - x_{2}^{*})}^{2} + \frac{ω_{3} y_{1}^{*} {(x_{2} - x_{2}^{*})}^{2}}{(m + x_{2}^{*}) (m + x_{2})} - e_{1} q_{1} {(y_{1} - y_{1}^{*})}^{2} \\ - e_{2} q_{2} {(y_{2} - y_{2}^{*})}^{2} - (a_{1} + a_{2}) (x_{1} - x_{1}^{*}) (x_{2} - x_{2}^{*}) - ω_{1} (1 - e_{1} n_{1}) (x_{1} - x_{1}^{*}) (y_{1} - y_{1}^{*}) \\ - ω_{2} (1 - e_{2} n_{3}) (x_{1} - x_{1}^{*}) (y_{2} - y_{2}^{*}) - ω_{3} (m + x_{2}^{*} - e_{1} n_{2} m) \frac{(x_{2} - x_{2}^{*}) (y_{1} - y_{1}^{*})}{(m + x_{2}^{*}) (m + x_{2})} \\ - ω_{4} (1 - e_{2} n_{4}) (x_{2} - x_{2}^{*}) (y_{2} - y_{2}^{*}) - (e_{1} b_{1} + e_{2} b_{2}) (y_{1} - y_{1}^{*}) (y_{2} - y_{2}^{*}) + B \end{matrix}$

$\begin{matrix} \leq - \frac{r}{K} {(x_{1} - x_{1}^{*})}^{2} - \frac{s}{L} {(x_{2} - x_{2}^{*})}^{2} + \frac{ω_{3} y_{1}^{*} {(x_{2} - x_{2}^{*})}^{2}}{(m + x_{2}^{*}) (m + x_{2})} - e_{1} q_{1} {(y_{1} - y_{1}^{*})}^{2} - e_{2} q_{2} {(y_{2} - y_{2}^{*})}^{2} + B \\ \leq - \frac{r}{K} {(x_{1} - x_{1}^{*})}^{2} - (\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) {(x_{2} - x_{2}^{*})}^{2} - e_{1} q_{1} {(y_{1} - y_{1}^{*})}^{2} - e_{2} q_{2} {(y_{2} - y_{2}^{*})}^{2} + B . \end{matrix}$ (3.3)

为了证明引理3.1中的条件 $A_{2}$ ，定义有界开集 $U_{ε}$

$U_{ε} = {(x_{1}, x_{2}, y_{1}, y_{2}) \in R_{+}^{4} : ε < x_{1} < \frac{1}{ε}, ε < x_{2} < \frac{1}{ε}, ε < y_{1} < \frac{1}{ε}, ε < y_{2} < \frac{1}{ε}},$

其中 $0 < ε < 1$ 是足够小的常数。在集合 $U_{ε}^{c}$ 中，可选择足够小的常数 $ε$ 使其满足如下条件：

$B < 1$ (3.4)

$ε < \frac{1}{4 x_{1}^{*}} [\frac{r}{K} {(x_{1}^{*})}^{2} - B] \cdot \frac{K}{r}$ (3.5)

$ε < \frac{1}{4 x_{2}^{*}} [(\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) {(x_{2}^{*})}^{2} - B] \cdot {(\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})})}^{- 1}$ (3.6)

$ε < \frac{1}{4 y_{1}^{*}} [e_{1} q_{1} {(y_{1}^{*})}^{2} - B] \cdot \frac{1}{e_{1} q_{1}}$ (3.7)

$ε < \frac{1}{4 y_{2}^{*}} [e_{2} q_{2} {(y_{2}^{*})}^{2} - B] \cdot \frac{1}{e_{2} q_{2}}$ (3.8)

$\frac{1}{ε} > \sqrt{\frac{K}{r} (1 - B)} + x_{1}^{*}$ (3.9)

$\frac{1}{ε} > \sqrt{(1 - B) {(\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})})}^{- 1}} + x_{2}^{*}$ (3.10)

$\frac{1}{ε} > \sqrt{\frac{1}{e_{1} q_{1}} (1 - B)} + y_{1}^{*}$ (3.11)

$\frac{1}{ε} > \sqrt{\frac{1}{e_{2} q_{2}} (1 - B)} + y_{2}^{*}$ (3.12)

为了方便，我们把 $U_{ε}^{c}$ 分为以下8个区域，

$U_{1} = {(x_{1}, x_{2}, y_{1}, y_{2}) \in R_{+}^{4} : x_{1} \leq ε}$ ， $U_{2} = {(x_{1}, x_{2}, y_{1}, y_{2}) \in R_{+}^{4} : x_{2} \leq ε}$ ，

$U_{3} = {(x_{1}, x_{2}, y_{1}, y_{2}) \in R_{+}^{4} : y_{1} \leq ε}$ ， $U_{4} = {(x_{1}, x_{2}, y_{1}, y_{2}) \in R_{+}^{4} : y_{2} \leq ε}$ ，

$U_{5} = {(x_{1}, x_{2}, y_{1}, y_{2}) \in R_{+}^{4} : x_{1} \geq \frac{1}{ε}}$ ， $U_{6} = {(x_{1}, x_{2}, y_{1}, y_{2}) \in R_{+}^{4} : x_{2} \geq \frac{1}{ε}}$ ，

$U_{7} = {(x_{1}, x_{2}, y_{1}, y_{2}) \in R_{+}^{4} : y_{1} \geq \frac{1}{ε}}$ ， $U_{8} = {(x_{1}, x_{2}, y_{1}, y_{2}) \in R_{+}^{4} : y_{2} \geq \frac{1}{ε}}$ 。

显然， $U_{ε}^{c} = U_{1} \cup U_{2} \cup U_{3} \cup U_{4} \cup U_{5} \cup U_{6} \cup U_{7} \cup U_{8}$ 。接下来证明对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{ε}^{c}$ 有 $L \overset{⌢}{V} < - 1$ ，这等价于分别在上述八个区域上证明它。

情况1：对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{1}$ ，由式子(3.2)、(3.3)和(3.5)可得

$\begin{matrix} L \overset{⌢}{V} \leq - \frac{r}{K} {(x_{1} - x_{1}^{*})}^{2} + B \leq \frac{r}{K} \cdot 2 x_{1}^{*} x_{1} - \frac{r}{K} {(x_{1}^{*})}^{2} + B \\ \leq \frac{2 r x_{1}^{*}}{K} ε - \frac{r}{K} {(x_{1}^{*})}^{2} + B < - \frac{1}{2} (\frac{r}{K} {(x_{1}^{*})}^{2} - B) < - 1, \end{matrix}$ (3.13)

因此可以得到对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{1}$ 有 $L \overset{⌢}{V} < - 1$ 。

情况2：对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{2}$ ，由式子(3.1)、(3.2)、(3.3)和(3.6)可得

$\begin{matrix} L \overset{⌢}{V} \leq - (\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) {(x_{2} - x_{2}^{*})}^{2} + B \\ \leq (\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) 2 x_{2}^{*} x_{2} - (\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) {(x_{2}^{*})}^{2} + B \\ \leq (\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) 2 x_{2}^{*} ε - (\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) {(x_{2}^{*})}^{2} + B \\ < - \frac{1}{2} [(\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) {(x_{2}^{*})}^{2} - B] < - 1, \end{matrix}$ (3.14)

因此可以得到对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{2}$ 有 $L \overset{⌢}{V} < - 1$ 。

情况3：对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{3}$ ，由式子(3.2)、(3.3)和(3.7)可得

$\begin{matrix} L \overset{⌢}{V} \leq - e_{1} q_{1} {(y_{1} - y_{1}^{*})}^{2} + B \leq 2 e_{1} q_{1} y_{1}^{*} y_{1} - e_{1} q_{1} {(y_{1}^{*})}^{2} + B \\ \leq 2 e_{1} q_{1} y_{1}^{*} ε - e_{1} q_{1} {(y_{1}^{*})}^{2} + B < - \frac{1}{2} (e_{1} q_{1} {(y_{1}^{*})}^{2} - B) < - 1, \end{matrix}$ (3.15)

因此可以得到对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{3}$ 有 $L \overset{⌢}{V} < - 1$ 。

情况4：对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{4}$ ，由式子(3.2)、(3.3)和(3.8)可得

$\begin{matrix} L \overset{⌢}{V} \leq - e_{2} q_{2} {(y_{2} - y_{2}^{*})}^{2} + B \leq 2 e_{2} q_{2} y_{2}^{*} y_{2} - e_{2} q_{2} {(y_{2}^{*})}^{2} + B \\ \leq 2 e_{2} q_{2} y_{2}^{*} ε - e_{2} q_{2} {(y_{2}^{*})}^{2} + B < - \frac{1}{2} (e_{2} q_{2} {(y_{2}^{*})}^{2} - B) < - 1, \end{matrix}$ (3.16)

因此可以得到对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{4}$ 有 $L \overset{⌢}{V} < - 1$ 。

情况5：对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{5}$ ，由式子(3.2)、(3.3)、(3.4)和(3.9)可得

$L \overset{⌢}{V} \leq - \frac{r}{K} {(x_{1} - x_{1}^{*})}^{2} + B \leq - \frac{r}{K} {(\frac{1}{ε} - x_{1}^{*})}^{2} + B < - 1,$ (3.17)

因此可以得到对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{5}$ 有 $L \overset{⌢}{V} < - 1$ 。

情况6：对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{6}$ ，由式子(3.1) (3.2)、(3.3)、(3.4)和(3.10)可得

$L \overset{⌢}{V} \leq - (\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) {(x_{2} - x_{2}^{*})}^{2} + B \leq - (\frac{s}{L} - \frac{ω_{3} y_{1}^{*}}{m (m + x_{2}^{*})}) {(\frac{1}{ε} - x_{2}^{*})}^{2} + B < - 1,$ (3.18)

因此可以得到对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{6}$ 有 $L \overset{⌢}{V} < - 1$ 。

情况7：对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{7}$ ，由式子(3.2)、(3.3)、(3.4)和(3.11)可得

$L \overset{⌢}{V} \leq - e_{1} q_{1} {(y_{1} - y_{1}^{*})}^{2} + B \leq - e_{1} q_{1} {(\frac{1}{ε} - y_{1}^{*})}^{2} + B < - 1,$ (3.19)

因此可以得到对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{7}$ 有 $L \overset{⌢}{V} < - 1$ 。

情况8：对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{8}$ ，由式子(3.2)、(3.3)、(3.4)和(3.12)可得

$L \overset{⌢}{V} \leq - e_{2} q_{2} {(y_{2} - y_{2}^{*})}^{2} + B \leq - e_{2} q_{2} {(\frac{1}{ε} - y_{2}^{*})}^{2} + B < - 1,$ (3.20)

因此可以得到对于任意的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{8}$ 有 $L \overset{⌢}{V} < - 1$ 。

显然，通过式子(3.13)~(3.20)可以得到，对于足够小的 $ε$ ，对所有的 $(x_{1}, x_{2}, y_{1}, y_{2}) \in U_{ε}^{c}$ 均满足 $L \overset{⌢}{V} < - 1$ ，因此满足引理3.1中的条件 $A_{2}$ ，故通过引理3.1可知系统(1.2)具有唯一的遍历平稳分布。这就完成了证明。

4. 灭绝

在本节中，我们将在两种情况下建立捕食者种群灭绝的充分条件。首先，我们给出了以下引理，它将用于随后的分析。

引理4.1 [21] 设 $f \in C [[0, \infty) \times Ω, (0, \infty)]$ ，若对任意的 $t \geq 0$ 存在正常数 $λ_{0}, λ$ 使得

$\ln f (t) \geq λ t - λ_{0} \int_{0}^{t} f (ξ) d ξ + F ( t )$

几乎处处成立，其中 $F \in C [[0, \infty) \times Ω, (0, \infty)]$ ， $\lim_{t \to \infty} \frac{F (t)}{t} = 0$ 几乎处处成立，则有

$\underset{t \to \infty}{\lim \inf} \frac{1}{t} \int_{0}^{t} f (ξ) d ξ \geq \frac{λ}{λ_{0}}$

几乎处处成立。这个引理的证明类似于季春艳和蒋达清 [21] 的证明，这里省略。

定理4.1设 $(x_{1} (t), x_{2} (t), y_{1} (t), y_{2} (t))$ 是系统(1.2)的满足任意初值 $(x_{1} (0), x_{2} (0), y_{1} (0), y_{2} (0)) \in R_{+}^{4}$ 的解，如果满足条件

$- c_{1} - \frac{1}{2} σ_{3}^{2} + \frac{n_{2} ω_{3} r}{a_{1} m} - \frac{n_{2} ω_{3}}{2 a_{1} m} σ_{1}^{2} + \frac{n_{1} ω_{1} s}{a_{2}} - \frac{n_{1} ω_{1}}{2 a_{2}} σ_{2}^{2} < 0,$

$r + \frac{L a_{1} σ_{2}^{2}}{2 s} - \frac{σ_{1}^{2}}{2} - L a_{1} > 0,$

$- c_{2} - \frac{1}{2} σ_{4}^{2} + \frac{n_{4} ω_{4} r}{a_{1}} - \frac{n_{4} ω_{4}}{2 a_{1}} σ_{1}^{2} + \frac{n_{3} ω_{2} s}{a_{2}} - \frac{n_{3} ω_{2}}{2 a_{2}} σ_{2}^{2} < 0,$

$s + \frac{K a_{2} σ_{1}^{2}}{2 r} - \frac{σ_{2}^{2}}{2} - K a_{2} > 0,$

则捕食者种群 $y_{1}$ 和捕食者种群 $y_{2}$ 均以概率1的指数形式灭绝且食饵种群均存活，有

$\underset{t \to \infty}{\lim \inf} \frac{1}{t} \int_{0}^{t} x_{1} (ξ) d ξ \geq \frac{r + \frac{L a_{1} σ_{2}^{2}}{2 s} - \frac{σ_{1}^{2}}{2} - L a_{1}}{\frac{r}{K}}, \underset{t \to \infty}{\lim \inf} \frac{1}{t} \int_{0}^{t} x_{2} (ξ) d ξ \geq \frac{s + \frac{K a_{2} σ_{1}^{2}}{2 r} - \frac{σ_{2}^{2}}{2} - K a_{2}}{\frac{s}{L}},$

$\lim_{t \to \infty} y_{1} (t) = 0, \lim_{t \to \infty} y_{2} (t) = 0$ 几乎处处成立。

证明：通过系统(1.2)可得

$L (\ln x_{1}) = r - \frac{r}{K} x_{1} - a_{1} x_{2} - ω_{1} y_{1} - ω_{2} y_{2} - \frac{1}{2} σ_{1}^{2},$ (4.1)

$L (\ln x_{2}) = s - \frac{s}{L} x_{2} - a_{2} x_{1} - \frac{ω_{3} y_{1}}{m + x_{2}} - ω_{4} y_{2} - \frac{1}{2} σ_{2}^{2},$ (4.2)

$L (\ln y_{1}) = - c_{1} - \frac{1}{2} σ_{3}^{2} + n_{1} ω_{1} x_{1} + \frac{n_{2} ω_{3} x_{2}}{m + x_{2}} - b_{1} y_{2} - q_{1} y_{1} \leq - c_{1} - \frac{1}{2} σ_{3}^{2} + n_{1} ω_{1} x_{1} + \frac{n_{2} ω_{3} x_{2}}{m + x_{2}},$ (4.3)

$L (\ln y_{2}) = - c_{2} - \frac{1}{2} σ_{4}^{2} + n_{3} ω_{2} x_{1} + n_{4} ω_{4} x_{2} - b_{2} y_{1} - q_{2} y_{2} \leq - c_{2} - \frac{1}{2} σ_{4}^{2} + n_{3} ω_{2} x_{1} + n_{4} ω_{4} x_{2} .$ (4.4)

定义

$V_{1} (x_{1} (t), x_{2} (t), y_{1} (t), y_{2} (t)) = \ln y_{1} + \frac{n_{2} ω_{3}}{a_{1} m} \ln x_{1} + \frac{n_{1} ω_{1}}{a_{2}} \ln x_{2} ，$

则由式子(4.1)，(4.2)和(4.3)可得

$L V_{1} \leq - c_{1} - \frac{1}{2} σ_{3}^{2} + \frac{n_{2} ω_{3} r}{a_{1} m} - \frac{n_{2} ω_{3}}{2 a_{1} m} σ_{1}^{2} + \frac{n_{1} ω_{1} s}{a_{2}} - \frac{n_{1} ω_{1}}{2 a_{2}} σ_{2}^{2},$

几乎处处成立。因此有

$\begin{matrix} d V_{1} \leq - (c_{1} - \frac{n_{2} ω_{3} r}{a_{1} m} - \frac{n_{1} ω_{1} s}{a_{2}} + \frac{1}{2} σ_{3}^{2} + \frac{n_{2} ω_{3}}{2 a_{1} m} σ_{1}^{2} + \frac{n_{1} ω_{1}}{2 a_{2}} σ_{2}^{2}) d t \\ + σ_{3} d B_{3} (t) + \frac{n_{2} ω_{3}}{a_{1} m} σ_{1} d B_{1} (t) + \frac{n_{1} ω_{1}}{a_{2}} σ_{2} d B_{2} (t) . \end{matrix}$ (4.5)

定义

$V_{2} (x_{1} (t), x_{2} (t), y_{1} (t), y_{2} (t)) = \ln y_{2} + \frac{n_{4} ω_{4}}{a_{1}} \ln x_{1} + \frac{n_{3} ω_{2}}{a_{2}} \ln x_{2}$

则由式子(4.1)，(4.2)和(4.4)可得

$L V_{2} \leq - c_{2} - \frac{1}{2} σ_{4}^{2} + \frac{n_{4} ω_{4} r}{a_{1}} - \frac{n_{4} ω_{4}}{2 a_{1}} σ_{1}^{2} + \frac{n_{3} ω_{2} s}{a_{2}} - \frac{n_{3} ω_{2}}{2 a_{2}} σ_{2}^{2},$

几乎处处成立。因此有

$\begin{matrix} d V_{2} \leq - (c_{2} - \frac{n_{4} ω_{4} r}{a_{1}} - \frac{n_{3} ω_{2} s}{a_{2}} + \frac{1}{2} σ_{4}^{2} + \frac{n_{4} ω_{4}}{2 a_{1}} σ_{1}^{2} + \frac{n_{3} ω_{2}}{2 a_{2}} σ_{2}^{2}) d t \\ + σ_{4} d B_{4} (t) + \frac{n_{4} ω_{4}}{a_{1}} σ_{1} d B_{1} (t) + \frac{n_{3} ω_{2}}{a_{2}} σ_{2} d B_{2} (t) . \end{matrix}$ (4.6)

对式子(4.5)和(4.6)等式两边从0到t积分，并且两边同时除以t可得

$\begin{matrix} \frac{V_{1} (t) - V_{1} (0)}{t} \leq - (c_{1} - \frac{n_{2} ω_{3} r}{a_{1} m} - \frac{n_{1} ω_{1} s}{a_{2}} + \frac{1}{2} σ_{3}^{2} + \frac{n_{2} ω_{3}}{2 a_{1} m} σ_{1}^{2} + \frac{n_{1} ω_{1}}{2 a_{2}} σ_{2}^{2}) \\ + \frac{σ_{3} B_{3} (t)}{t} + \frac{n_{2} ω_{3} σ_{1} B_{1} (t)}{a_{1} m t} + \frac{n_{1} ω_{1} σ_{2} B_{2} (t)}{a_{2} t}, \end{matrix}$ (4.7)

$\begin{matrix} \frac{V_{2} (t) - V_{2} (0)}{t} \leq - (c_{2} - \frac{n_{4} ω_{4} r}{a_{1}} - \frac{n_{3} ω_{2} s}{a_{2}} + \frac{1}{2} σ_{4}^{2} + \frac{n_{4} ω_{4}}{2 a_{1}} σ_{1}^{2} + \frac{n_{3} ω_{2}}{2 a_{2}} σ_{2}^{2}) \\ + \frac{σ_{4} B_{4} (t)}{t} + \frac{n_{4} ω_{4} σ_{1} B_{1} (t)}{t} + \frac{n_{3} ω_{2} σ_{2} B_{2} (t)}{a_{2} t} . \end{matrix}$ (4.8)

对式子(4.7)和(4.8)两边取上确界，利用局部鞅的强大数定理 [22] 可以得到 $\lim_{t \to \infty} \frac{B_{1} (t)}{t} = 0$ ， $\lim_{t \to \infty} \frac{B_{2} (t)}{t} = 0$ ， $\lim_{t \to \infty} \frac{B_{3} (t)}{t} = 0$ 和 $\lim_{t \to \infty} \frac{B_{4} (t)}{t} = 0$ 几乎是处处成立的，因此有

$\underset{t \to \infty}{\lim \sup} \frac{V_{1} (t)}{t} \leq - (c_{1} - \frac{n_{2} ω_{3} r}{a_{1} m} - \frac{n_{1} ω_{1} s}{a_{2}} + \frac{1}{2} σ_{3}^{2} + \frac{n_{2} ω_{3}}{2 a_{1} m} σ_{1}^{2} + \frac{n_{1} ω_{1}}{2 a_{2}} σ_{2}^{2}) < 0,$

$\underset{t \to \infty}{\lim \sup} \frac{V_{2} (t)}{t} \leq - (c_{2} - \frac{n_{4} ω_{4} r}{a_{1}} - \frac{n_{3} ω_{2} s}{a_{2}} + \frac{1}{2} σ_{4}^{2} + \frac{n_{4} ω_{4}}{2 a_{1}} σ_{1}^{2} + \frac{n_{3} ω_{2}}{2 a_{2}} σ_{2}^{2}) < 0.$

几乎处处成立。因为此时两个食饵种群 $x_{1}$ 和 $x_{2}$ 均为存活的，故有 $\ln x_{1} > 0$ ， $\ln x_{2} > 0$ 。所以有

$\underset{t \to \infty}{\lim \sup} \frac{\ln y_{1} (t)}{t} \leq - (c_{1} - \frac{n_{2} ω_{3} r}{a_{1} m} - \frac{n_{1} ω_{1} s}{a_{2}} + \frac{1}{2} σ_{3}^{2} + \frac{n_{2} ω_{3}}{2 a_{1} m} σ_{1}^{2} + \frac{n_{1} ω_{1}}{2 a_{2}} σ_{2}^{2}) < 0,$

$\underset{t \to \infty}{\lim \sup} \frac{\ln y_{2} (t)}{t} \leq - (c_{2} - \frac{n_{4} ω_{4} r}{a_{1}} - \frac{n_{3} ω_{2} s}{a_{2}} + \frac{1}{2} σ_{4}^{2} + \frac{n_{4} ω_{4}}{2 a_{1}} σ_{1}^{2} + \frac{n_{3} ω_{2}}{2 a_{2}} σ_{2}^{2}) < 0,$

几乎处处成立。则有 $\lim_{t \to \infty} y_{1} (t) = 0$ ， $\lim_{t \to \infty} y_{2} (t) = 0$ 几乎处处成立，即捕食者种群 $y_{1}$ 和捕食者种群 $y_{2}$ 均灭绝。因此对所有满足条件 $0 < ε_{1} < r + \frac{L a_{1} σ_{2}^{2}}{2 s} - \frac{σ_{1}^{2}}{2} - L a_{1}$ 的 $ε_{1}$ ，存在 $t_{1}$ 和一个集合 $Ω_{ε_{1}} \subset Ω$ ，对任意的 $t \geq t_{1}$ 和 $ε_{1} \subset Ω_{ε_{1}}$ 满足 $ℙ (Ω_{ε_{1}}) > 1 - ε_{1}, y_{1} \leq ε_{1}$ 和 $y_{2} \leq ε_{1}$ ，所以有

$\begin{array}{l} d (\ln x_{1} - \frac{L a_{1}}{s} \ln x_{2}) \\ = [r - \frac{σ_{1}^{2}}{2} - \frac{r}{K} x_{1} - ω_{1} y_{1} - ω_{2} y_{2} - L a_{1} + \frac{L a_{1} a_{2}}{s} x_{1} + \frac{L a_{1} ω_{3} y_{1}}{s (m + x_{2})} \\ + \frac{L a_{1} ω_{4}}{s} y_{2} + \frac{L a_{1} σ_{2}^{2}}{2 s}] d t + σ_{1} d B_{1} (t) - \frac{L a_{1} σ_{2}}{s} d B_{2} (t) \\ \geq (r + \frac{L a_{1} σ_{2}^{2}}{2 s} - \frac{σ_{1}^{2}}{2} - L a_{1} - \frac{r}{K} x_{1} - ω_{1} y_{1} - ω_{2} y_{2}) d t + σ_{1} d B_{1} (t) - \frac{L a_{1} σ_{2}}{s} d B_{2} (t) \\ \geq (r + \frac{L a_{1} σ_{2}^{2}}{2 s} - \frac{σ_{1}^{2}}{2} - L a_{1} - \frac{r}{K} x_{1} - ω_{1} ε_{1} - ω_{2} ε_{1}) d t + σ_{1} d B_{1} (t) - \frac{L a_{1} σ_{2}}{s} d B_{2} (t), \end{array}$

$\begin{array}{l} d (\ln x_{2} - \frac{K a_{2}}{r} \ln x_{1}) \\ = (s - \frac{σ_{2}^{2}}{2} - \frac{s}{L} x_{2} - \frac{ω_{3} y_{1}}{m + x_{2}} - ω_{4} y_{2} - K a_{2} + \frac{K a_{2} σ_{1}^{2}}{2 r} + \frac{K a_{1} a_{2}}{r} x_{2} \\ + \frac{K a_{2} ω_{1}}{r} y_{1} + \frac{K a_{2} ω_{2}}{r} y_{2}) d t - \frac{K a_{2} σ_{1}}{r} d B_{1} (t) + σ_{2} d B_{2} (t) \\ \geq (s + \frac{K a_{2} σ_{1}^{2}}{2 r} - \frac{σ_{2}^{2}}{2} - K a_{2} - \frac{s}{L} x_{2} - \frac{ω_{3}}{m} y_{1} - ω_{4} y_{2}) d t - \frac{K a_{2} σ_{1}}{r} d B_{1} (t) + σ_{2} d B_{2} (t) \\ \geq (s + \frac{K a_{2} σ_{1}^{2}}{2 r} - \frac{σ_{2}^{2}}{2} - K a_{2} - \frac{s}{L} x_{2} - \frac{ω_{3}}{m} ε_{1} - ω_{4} ε_{1}) d t - \frac{K a_{2} σ_{1}}{r} d B_{1} (t) + σ_{2} d B_{2} (t), \end{array}$

令 $ε_{1} \to 0$ ，则有

$d (\ln x_{1} - \frac{L a_{1}}{s} \ln x_{2}) \geq (r + \frac{L a_{1} σ_{2}^{2}}{2 s} - \frac{σ_{1}^{2}}{2} - L a_{1} - \frac{r}{K} x_{1}) d t + σ_{1} d B_{1} (t) - \frac{L a_{1} σ_{2}}{s} d B_{2} (t),$ (4.9)

$d (\ln x_{2} - \frac{K a_{2}}{r} \ln x_{1}) \geq (s + \frac{K a_{2} σ_{1}^{2}}{2 r} - \frac{σ_{2}^{2}}{2} - K a_{2} - \frac{s}{L} x_{2}) d t - \frac{K a_{2} σ_{1}}{r} d B_{1} (t) + σ_{2} d B_{2} (t),$ (4.10)

对式子(4.9)和(4.10)关于0到t积分可得

$\begin{array}{l} \ln x_{1} (t) - \frac{L a_{1}}{s} \ln x_{2} (t) - \ln x_{1} (0) + \frac{L a_{1}}{s} \ln x_{2} (0) \\ \geq (r + \frac{L a_{1} σ_{2}^{2}}{2 s} - \frac{σ_{1}^{2}}{2} - L a_{1}) t - \frac{r}{K} \int_{0}^{t} x_{1} (ξ) d ξ + σ_{1} B_{1} (t) - \frac{L a_{1} σ_{2} B_{2} (t)}{s}, \end{array}$

$\begin{array}{l} \ln x_{2} (t) - \frac{K a_{2}}{r} \ln x_{1} (t) - \ln x_{2} (0) + \frac{K a_{2}}{r} \ln x_{1} (0) \\ \geq (s + \frac{K a_{2} σ_{1}^{2}}{2 r} - \frac{σ_{2}^{2}}{2} - K a_{2}) t - \frac{s}{L} \int_{0}^{t} x_{2} (ξ) d ξ - \frac{K a_{2} σ_{1} B_{1} (t)}{r} + σ_{2} B_{2} (t), \end{array}$

因此可以得到

$\ln x_{1} (t) + \frac{L a_{1}}{s} \ln x_{2} (0) \geq (r + \frac{L a_{1} σ_{2}^{2}}{2 s} - \frac{σ_{1}^{2}}{2} - L a_{1}) t - \frac{r}{K} \int_{0}^{t} x_{1} (ξ) d ξ + σ_{1} B_{1} (t) - \frac{L a_{1} σ_{2} B_{2} (t)}{s},$ (4.11)

$\ln x_{2} (t) + \frac{K a_{2}}{r} \ln x_{1} (0) \geq (s + \frac{K a_{2} σ_{1}^{2}}{2 r} - \frac{σ_{2}^{2}}{2} - K a_{2}) t - \frac{s}{L} \int_{0}^{t} x_{2} (ξ) d ξ - \frac{K a_{2} σ_{1} B_{1} (t)}{r} + σ_{2} B_{2} (t),$ (4.12)

即式子(4.11)和(4.12)均满足引理4.1的条件，因此有

此即证得此时食饵均存活。这便完成了证明。

定理4.2设 $(x_{1} (t), x_{2} (t), y_{1} (t), y_{2} (t))$ 是系统(1.2)的满足任意初值 $(x_{1} (0), x_{2} (0), y_{1} (0), y_{2} (0)) \in R_{+}^{4}$ 的解，如果满足 $\max {r, s} - \frac{1}{2 (\frac{1}{σ_{1}^{2}} + \frac{1}{σ_{2}^{2}})} < 0$ ，则食饵种群与捕食者种群均灭绝，即有 $\lim_{t \to \infty} x_{1} (t) = 0$ ， $\lim_{t \to \infty} x_{2} (t) = 0$ ， $\lim_{t \to \infty} y_{1} (t) = 0$ ， $\lim_{t \to \infty} y_{2} (t) = 0$ 。

证明：定义一个 $C^{2}$ -函数 $\tilde{V} (x_{1}, x_{2}) = x_{1} + x_{2}$ ，对 $\ln \tilde{V}$ 使用Itô’s公式可得

$\begin{matrix} d (\ln \tilde{V}) = [\frac{1}{\tilde{V}} (r x_{1} - \frac{r}{K} x_{1}^{2} - a_{1} x_{1} x_{2} - ω_{1} x_{1} y_{1} - ω_{2} x_{1} y_{2}) - \frac{1}{2} \frac{1}{{\tilde{V}}^{2}} σ_{1}^{2} x_{1}^{2} \\ + \frac{1}{\tilde{V}} (s x_{2} - \frac{s}{L} x_{2}^{2} - a_{2} x_{1} x_{2} - \frac{ω_{3} x_{2} y_{1}}{m + x_{2}} - ω_{4} x_{2} y_{2}) - \frac{1}{2} \frac{1}{{\tilde{V}}^{2}} σ_{2}^{2} x_{2}^{2}] d t \\ + \frac{1}{\tilde{V}} σ_{1} x_{1} d B_{1} (t) + \frac{1}{\tilde{V}} σ_{2} x_{2} d B_{2} (t) \\ = L (\ln \tilde{V}) d t + \frac{1}{\tilde{V}} (σ_{1} x_{1} d B_{1} (t) + σ_{2} x_{2} d B_{2} (t)), \end{matrix}$ (4.13)

其中

$\begin{matrix} L (\ln \tilde{V}) = \frac{1}{\tilde{V}} (r x_{1} - \frac{r}{K} x_{1}^{2} - a_{1} x_{1} x_{2} - ω_{1} x_{1} y_{1} - ω_{2} x_{1} y_{2}) \\ + \frac{1}{\tilde{V}} (s x_{2} - \frac{s}{L} x_{2}^{2} - a_{2} x_{1} x_{2} - \frac{ω_{3} x_{2} y_{1}}{m + x_{2}} - ω_{4} x_{2} y_{2}) \\ - \frac{1}{2 {\tilde{V}}^{2}} (σ_{1}^{2} x_{1}^{2} + σ_{2}^{2} x_{2}^{2}) . \end{matrix}$ (4.14)

此外，我们可以得到

${\tilde{V}}^{2} = {(σ_{1} x_{1} \cdot \frac{1}{σ_{1}} + σ_{2} x_{2} \cdot \frac{1}{σ_{2}})}^{2} \leq (σ_{1}^{2} x_{1}^{2} + σ_{2}^{2} x_{2}^{2}) (\frac{1}{σ_{1}^{2}} + \frac{1}{σ_{2}^{2}}),$ (4.15)

和

$\begin{array}{l} \frac{1}{\tilde{V}} (r x_{1} - \frac{r}{K} x_{1}^{2} - a_{1} x_{1} x_{2} - ω_{1} x_{1} y_{1} - ω_{2} x_{1} y_{2}) + \frac{1}{\tilde{V}} (s x_{2} - \frac{s}{L} x_{2}^{2} - a_{2} x_{1} x_{2} - \frac{ω_{3} x_{2} y_{1}}{m + x_{2}} - ω_{4} x_{2} y_{2}) \\ \leq \frac{1}{\tilde{V}} (r x_{1} + s x_{2}) \leq \max {r, s}, \end{array}$ (4.16)

把式子(4.15)、(4.16)带入(4.14)可得

$L (\ln \tilde{V}) \leq \max {r, s} - \frac{1}{2 (\frac{1}{σ_{1}^{2}} + \frac{1}{σ_{2}^{2}})} .$ (4.17)

由式子(4.13)和(4.17)可以得到

$d (\ln \tilde{V}) \leq [\max {r, s} - \frac{1}{2 (\frac{1}{σ_{1}^{2}} + \frac{1}{σ_{2}^{2}})}] d t + \frac{1}{\tilde{V}} (σ_{1} x_{1} d B_{1} (t) + σ_{2} x_{2} d B_{2} (t)) .$ (4.18)

对式子(4.18)两边从0到t积分，并且两边同时除以t可以得到

$\begin{matrix} \frac{\ln \tilde{V} (t) - \ln \tilde{V} (0)}{t} \leq \max {r, s} - \frac{1}{2 (\frac{1}{σ_{1}^{2}} + \frac{1}{σ_{2}^{2}})} + \frac{1}{t} \int_{0}^{t} \frac{σ_{1} x_{1} (z)}{\tilde{V} (z)} d B_{1} (z) + \frac{1}{t} \int_{0}^{t} \frac{σ_{2} x_{2} (z)}{\tilde{V} (z)} d B_{2} (z) \\ = \max {r, s} - \frac{1}{2 (\frac{1}{σ_{1}^{2}} + \frac{1}{σ_{2}^{2}})} + \frac{\tilde{M} (t)}{t} + \frac{\tilde{N} (t)}{t}, \end{matrix}$ (4.19)

其中 $\tilde{M} (t) : = \int_{0}^{t} \frac{σ_{1} x_{1} (z)}{\tilde{V} (z)} d B_{1} (z)$ 、 $\tilde{N} (t) : = \int_{0}^{t} \frac{σ_{2} x_{2} (z)}{\tilde{V} (z)} d B_{2} (z)$ 是局部鞅，其二次变差分别为：

${〈 \tilde{M} (t), \tilde{M} (t) 〉}_{t} = σ_{1}^{2} {\int_{0}^{t} (\frac{x_{1} (z)}{\tilde{V} (z)})}^{2} d z \leq σ_{1}^{2} t, {〈 \tilde{N} (t), \tilde{N} (t) 〉}_{t} = σ_{2}^{2} \int_{0}^{t} {(\frac{x_{2} (z)}{\tilde{V} (z)})}^{2} d z \leq σ_{2}^{2} t .$

对式子(4.19)应用局部鞅的强大数定理，可以得到 $\lim_{t \to \infty} \frac{\tilde{M} (t)}{t} = 0$ ， $\lim_{t \to \infty} \frac{\tilde{N} (t)}{t} = 0$ 几乎是处处成立的。

对式子(4.19)两边取上确界可得

$\underset{t \to \infty}{\lim \sup} \frac{\ln \tilde{V} (t)}{t} = \max {r, s} - \frac{1}{2 (\frac{1}{σ_{1}^{2}} + \frac{1}{σ_{2}^{2}})} < 0,$

几乎处处成立。根据 $x_{1}, x_{2}$ 的正性可以得到 $\underset{t \to \infty}{\lim \sup} \frac{\ln x_{1} (t)}{t} < 0$ ， $\underset{t \to \infty}{\lim \sup} \frac{\ln x_{2} (t)}{t} < 0$ ，几乎处处成立。则有 $\lim_{t \to \infty} x_{1} (t) = 0$ ， $\lim_{t \to \infty} x_{2} (t) = 0$ 几乎处处成立，即此时两个食饵种群 $x_{1}$ 和 $x_{2}$ 均灭绝。因此，存在 $t_{0}$ 和一个集合 $Ω_{ε} \subset Ω$ 使得对任意的 $t \geq t_{0}$ ， $ω \in Ω_{ε}$ 满足 $ℙ (Ω_{ε}) > 1 - ε$ 、 $n_{1} ω_{1} x_{1} \leq n_{1} ω_{1} ε$ ， $\frac{n_{2} ω_{3}}{m} x_{2} \leq \frac{n_{2} ω_{3}}{m} ε$ ， $n_{3} ω_{2} x_{1} \leq n_{3} ω_{2} ε$ 和 $n_{4} ω_{4} x_{2} \leq n_{4} ω_{4} ε$ 。

对 $\ln y_{1}$ 和 $\ln y_{2}$ 使用Itô’s公式可以得到

$\begin{matrix} d (\ln y_{1} (t)) = (- c_{1} - \frac{1}{2} σ_{3}^{2} + n_{1} ω_{1} x_{1} + \frac{n_{2} ω_{3} x_{2}}{m + x_{2}} - b_{1} y_{2} - q_{1} y_{1}) d t + σ_{3} d B_{3} (t) \\ \leq (- c_{1} - \frac{1}{2} σ_{3}^{2} + n_{1} ω_{1} x_{1} + \frac{n_{2} ω_{3}}{m} x_{2} - b_{1} y_{2} - q_{1} y_{1}) d t + σ_{3} d B_{3} (t) \\ \leq [- c_{1} - \frac{1}{2} σ_{3}^{2} + (n_{1} ω_{1} + \frac{n_{2} ω_{3}}{m}) ε] d t + σ_{3} d B_{3} (t), \end{matrix}$ (4.20)

$\begin{matrix} d (\ln y_{2} (t)) = (- c_{2} - \frac{1}{2} σ_{4}^{2} + n_{3} ω_{2} x_{1} + n_{4} ω_{4} x_{2} - b_{2} y_{1} - q_{2} y_{2}) d t + σ_{4} d B_{4} (t) \\ \leq [- c_{2} - \frac{1}{2} σ_{4}^{2} + (n_{3} ω_{2} + n_{4} ω_{4}) ε] d t + σ_{4} d B_{4} (t) . \end{matrix}$ (4.21)

对式子(4.20)和(4.21)两边从0到t积分，并且两边同时除以t可以得到

$\frac{\ln y_{1} (t) - \ln y_{1} (0)}{t} \leq - c_{1} - \frac{1}{2} σ_{3}^{2} + (n_{1} ω_{1} + \frac{n_{2} ω_{3}}{m}) ε + \frac{σ_{3} B_{3} (t)}{t},$ (4.22)

$\frac{\ln y_{2} (t) - \ln y_{2} (0)}{t} \leq - c_{2} - \frac{1}{2} σ_{4}^{2} + (n_{3} ω_{2} + n_{4} ω_{4}) ε + \frac{σ_{4} B_{4} (t)}{t} .$ (4.23)

对式子(4.22)和(4.23)两边取上确界，注意 $\lim_{t \to \infty} \frac{B_{3} (t)}{t} = 0, \lim_{t \to \infty} \frac{B_{4} (t)}{t} = 0$ 几乎处处成立，则有

$\underset{t \to \infty}{\lim \sup} \frac{\ln y_{1} (t)}{t} \leq - c_{1} - \frac{1}{2} σ_{3}^{2} + (n_{1} ω_{1} + \frac{n_{2} ω_{3}}{m}) ε, \underset{t \to \infty}{\lim \sup} \frac{\ln y_{2} (t)}{t} \leq - c_{2} - \frac{1}{2} σ_{4}^{2} + (n_{3} ω_{2} + n_{4} ω_{4}) ε .$

令 $ε \to 0$ ，则有

$\underset{t \to \infty}{\lim \sup} \frac{\ln y_{1} (t)}{t} \leq - c_{1} - \frac{1}{2} σ_{3}^{2} < 0, \underset{t \to \infty}{\lim \sup} \frac{\ln y_{2} (t)}{t} \leq - c_{2} - \frac{1}{2} σ_{4}^{2} < 0.$

根据捕食者种群 $y_{1}, y_{2}$ 的正性可以得到 $\underset{t \to \infty}{\lim \sup} \frac{\ln y_{1} (t)}{t} < 0$ ， $\underset{t \to \infty}{\lim \sup} \frac{\ln y_{2} (t)}{t} < 0$ 几乎处处成立，则有

$\lim_{t \to \infty} y_{1} (t) = 0$ ， $\lim_{t \to \infty} y_{2} (t) = 0$ ，即两个捕食者种群 $y_{1}$ 和 $y_{2}$ 均灭绝。此便完成食饵种群与捕食者种群均灭绝的证明。

5. 结论

本文主要研究了具有两个食饵种群和两个捕食者种群的随机捕食食饵模型的基本特征，了解其存在白噪声情形下的行为动力学。即研究了系统(1.2)正解的存在唯一性和其平稳分布。同时还在两个捕食者种群均灭绝，及食饵种群与捕食者种群均灭绝这两种情况下讨论了捕食者种群均灭绝的条件。

众所周知，在整个生态系统中，影响生物生存的因素有很多，捕食食饵模型还有很多值得研究的问题。例如，可以考虑在系统(1.2)的基础之上再加入新的捕食者种群或食饵种群，扩大种群数量的研究，增加方程的维数，更加地贴近复杂是生物圈；另一方面，可以在系统(1.2)中引入彩色噪声，以考虑到生活中可能会遭受到的突然的环境变化的影响，如温度、湿度、收获等。这些问题都可以在后续继续进行研究。

文章引用

王欣琦,张天四. 双捕食者与双食饵的随机捕食系统动力学
Dynamic of Stochastic Predator-Prey Sys-tem with Two Predators and Two Prey[J]. 运筹与模糊学, 2023, 13(02): 666-680. https://doi.org/10.12677/ORF.2023.132069

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22. Has’Minskii, R.Z. (1980) Stochastic Stability of Differential Equations. Sijthoff and Noordhoff, Alphen aan den Rijn.

NOTES

^*通讯作者。

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