## Complex Dynamics and Stochastic Sensitivity Analysis of a Predator-Prey Model with Crowley-Martin Type Functional Response

Wang Tengfei,, Feng Tao,, Meng Xinzhu,

 基金资助: 山东省自然科学基金项目.  ZR2019MA003山东省泰山学者项目研究基金

 Fund supported: the NSF of Shandong Province.  ZR2019MA003the Research Fund for the Taishan Scholar Project of Shandong Province

Abstract

Predator-prey interactions serve a pivotal role in protecting species diversity. In this study, the parameter $λ$ is presented to show the stochastic dynamics of a predator-prey model. The results show that the predator population tends to become extinct if $λ$ < 0. Furthermore, the solution of the prey population converges weakly to an invariant probability distribution. If $λ$>0, we find that the stochastic system admits a unique ergodic stationary distribution. In addition, stochastic sensitivity analysis is proposed to study the effects of noise on the dynamics of predator-prey populations. Our findings suggest that small-intensity noise can suppress population size enlargement, while large-intensity noise can lead to the extinction of populations.

Keywords： Predator-prey interaction ; Stochastic model ; Stochastic sensitivity analysis ; Ergodic property ; Stationary distribution

Wang Tengfei, Feng Tao, Meng Xinzhu. Complex Dynamics and Stochastic Sensitivity Analysis of a Predator-Prey Model with Crowley-Martin Type Functional Response. Acta Mathematica Scientia[J], 2021, 41(4): 1192-1203 doi:

## 1 引言

$\begin{eqnarray} \left\{\begin{array}{ll} {\rm d}x(t) = x(t)\left(a-x(t)-\frac{by(t)}{(1+\alpha x(t))(1+\beta y(t))}\right){\rm d}t+\sigma_1x(t){\rm d}B_1(t), \\ {\rm d}y(t) = y(t)\left(c-y(t)+\frac{fx(t)}{(1+\alpha x(t))(1+\beta y(t))}\right){\rm d}t+\sigma_2y(t){\rm d}B_2(t), \\ \end{array}\right. \end{eqnarray}$

 符号 生物学意义 $a$ 在没有捕食的情况下, 食饵种群的内在增长率 $b$ 攻击系数 $\alpha$ 处理时间 $\beta$ 捕食者个体之间的干扰程度 $c$ 在没有捕食的情况下, 捕食者种群的内在增长率 $f$ 捕食的转化率 $\sigma_i^2$ 布朗运动$B_i$的强度$(i = 1, 2)$

## 2 长期的随机动力学

$\begin{eqnarray} {\rm d}\tilde{x}(t) = \tilde{x}(t)\left(a-\tilde{x}(t)\right){\rm d}t+\sigma_1\tilde{x}(t){\rm d}B_1(t), \; \tilde{x}(0) = x(0). \end{eqnarray}$

(ⅰ) 当$\alpha\leq\frac{1}{2}\sigma_1^2$时, 有$\lim\limits_{t\rightarrow \infty}\tilde{x}(t) = 0\; \; {\rm a.s.};$

(ⅱ) 当$\alpha>\frac{1}{2}\sigma_1^2$时, 随机过程$\tilde{x}$存在一个唯一的不变概率测度(IPM) $\pi(\cdot)$. 此外有$\lim\limits_{t\rightarrow \infty}\frac{1}{t}\int_0^t\frac{\tilde{x}(s)}{1+\alpha \tilde{x}(s)}{\rm d}s = \int_0^\infty\frac{\nu}{1+\alpha\nu}\pi({\rm d}\nu)\; \; {\rm a.s.}, $$\lim\limits_{t\rightarrow \infty}\frac{\ln \tilde{x}(t)}{t} = 0\; \; {\rm a.s.} . 为了研究随机系统(1.1)的动力学, 定义 {\cal B}({\Bbb R}_+^{n})$$ {\Bbb R}_+^{n}$的Borel可测子集, 空间$({\Bbb R}_+^{n}, {\cal B}({\Bbb R}_+^{n}))$上的总变异范数$\|\cdot, \cdot\|_{TV}$被定义为$\|\varphi, \psi\|_{TV} = \sup\limits_{A\in{\cal B}({\Bbb R}_+^{n})}|\varphi(A)-\psi(A)|.$

(ⅰ) 当$\lambda<0$时, 捕食者$y(t)$的数量以概率1指数收敛到0, 而食饵$x(t)$的水平会弱收敛到IPM $\pi(\cdot)$.

(ⅱ) 当$\lambda>0$, 解$(x(t), y(t))$存在一个唯一的IPM$\varphi\in{\Bbb R}_+^{2, o}$, 这表明系统(1.1)是遍历的. 而且

(Ⅰ) 情形$\lambda<0$. 根据伊藤公式可得

$\begin{eqnarray} {\rm d}\ln y = \left(c-\frac{1}{2}\sigma_1^2-y+\frac{fx}{(1+\alpha x)(1+\beta y)}\right){\rm d}t+\sigma_2{\rm d}B_2(t). \end{eqnarray}$

$\begin{eqnarray} \frac{\ln y(t)}{t} = c-\frac{1}{2}\sigma_2^2+\frac{1}{t}\int_0^t\left(\frac{fx(s)}{(1+\alpha x(s))(1+\beta y(s))}-y(s)\right){\rm d}s+\frac{\sigma_2B_2(t)}{t}+\frac{\ln y(0)}{t}. \end{eqnarray}$

$\begin{eqnarray} \limsup\limits_{t\rightarrow \infty}\frac{\ln y(t)}{t}& = &c-\frac{1}{2}\sigma_2^2+\limsup\limits_{t\rightarrow \infty}\frac{1}{t}\int_0^t\left(\frac{fx(s)}{(1+\alpha x(s))(1+\beta y(s))}-y(s)\right){\rm d}s\\ & \leq&c-\frac{1}{2}\sigma_2^2+\limsup\limits_{t\rightarrow \infty}\frac{1}{t}\int_0^t\frac{fx(s)}{1+\alpha x(s)}{\rm d}s. \end{eqnarray}$

$\begin{eqnarray} \limsup\limits_{t\rightarrow \infty}\frac{\ln y(t)}{t} \leq c-\frac{1}{2}\sigma_2^2+\int_0^\infty\frac{f\nu}{1+\alpha\nu}\pi({\rm d}\nu) = \lambda<0\; \; {\rm a.s.}. \end{eqnarray}$

$\Omega_\epsilon = \left\{\omega:\ln y(t)\leq\frac{\lambda t}{2}\right\}$, $\epsilon>0$是任意的. 根据(2.5)式, 选择$T>0$使得

$\ln \tilde{x}(t)-\ln x(t)$应用伊藤公式得

$\begin{eqnarray} \ln\tilde{x}(t)-\ln x(t)& = &\int_{T}^t(x(s)-\tilde{x}(s)){\rm d}s+\int_{T}^t\frac{by(s)}{(1+\alpha x(s)(1+\beta y(s))}{\rm d}s\\ & \leq&b\int_{T}^ty(s){\rm d}s\leq-\frac{2b}{\lambda}{\rm e}^{\frac{\lambda T}{2}}. \end{eqnarray}$

$\begin{eqnarray} {\Bbb P}\{|\ln\tilde{x}(t)-\ln x(t)|>\epsilon\}\leq1-{\Bbb P}(\Omega_\epsilon)<\epsilon, \; \; \forall\; t\geq T. \end{eqnarray}$

$\begin{eqnarray} {\Bbb E}g(\ln x(t))\rightarrow \bar{g}: = \int_{\Bbb R}g(\omega)\pi^*({\rm d}\omega) = \int_0^\infty g(\ln \omega)\pi({\rm d}\omega), \; \; t\rightarrow \infty. \end{eqnarray}$

### 图 2

1) 小的噪声抑制种群的大小, 使种群大小浮动在一定范围之内. 从图 1中可以看出, 当随机噪声的强度较小时, 随机模型的解在相应的确定性模型解的周围浮动, 而且随机解的样本均值小于确定性模型解的值. 此外, 图 1(e)1(f)展示了随机模型(1.1)有一个平稳分布.

2) 大的噪声导致种群灭绝. 由图 2可以看出, 确定性模型的解是持久的, 而当噪声强度较大时, 随机模型的解$y$变为0, 这说明较大的随机噪声会导致种群的灭绝.

## 3 随机敏感性分析

$\begin{eqnarray} \left\{\begin{array}{ll} \frac{{\rm d}x(t)}{{\rm d}t} = x(t)\left(a-x(t)-\frac{by(t)}{(1+\alpha x(t))(1+\beta y(t))}\right), \\ \frac{{\rm d}y(t)}{{\rm d}t} = y(t)\left(c-y(t)+\frac{fx(t)}{(1+\alpha x(t))(1+\beta y(t))}\right). \end{array}\right. \end{eqnarray}$

(3) 确定性模型(3.1)在点$E_{10}$的稳定和不稳定流形分别是$W^s(E_{10}) = 1 $$W^u(E_{10}) = 1 , 即边界平衡点 E_{10} 是一个鞍点, 因此不稳定; (4) 确定性模型(3.1)在点 E_{11} 的稳定流形是 W^s(E_{11}) = 2 , 即内部平衡点 E_{11} 是一个结点, 因此是局部渐近稳定的. 确定性模型(3.1)的平衡点应满足以下条件 $$\begin{array}{ll} 0 = x(t)\left(a-x(t)-\frac{by(t)}{(1+\alpha x(t))(1+\beta y(t))}\right), \\ 0 = y(t)\left(c-y(t)+\frac{fx(t)}{(1+\alpha x(t))(1+\beta y(t))}\right). \end{array}$$ 通过解方程(3.2), 可知确定性模型(3.1)有三个边界平衡点 E_{00} , E_{01} , 和 E_{10} , 而且确定性模型(3.1)的内部平衡点应该是方程(3.2)的正根. 通过Matlab数值模拟发现确定性模型(3.1)在第一象限只有一个内部平衡点 E_{11}(0.5761, 0.4157) , 见图 3. ### 图 3 图 3 确定性模型(3.1)平衡点的存在性, 参数: a = 0.6, b = 0.1, c = 0.3, f = 0.35, \alpha = 0.4, \beta = 1 . 下面分析确定性模型(3.1)平衡点的稳定性. 为了方便研究, 设为模型(3.1)对应的平衡点为 E^\#(x^\#, y^\#) . 在点 E^\# 相应的雅可比矩阵的取值为 因此, 确定性模型(3.1)在 E_{00} 处的特征值取值为 \lambda_1(E_{00}) = a>0$$ \lambda_2(E_{00}) = c>0$. 也就是说, 确定性模型(3.1)在点$E_{00}$的不稳定流形是$W^u(E_{00}) = 2$, 即边界平衡点$E_{00}$是不稳定. 情形(2)、(3)、(4)的证明与情形(1)的证明非常相似, 所以在此省略.

$\begin{eqnarray} \dot{x} = f(x)+\varepsilon \sigma(x) \dot{w}. \end{eqnarray}$

$\begin{eqnarray} \rho(x, \varepsilon) \approx K \cdot \exp \left(-\frac{v(x)}{\varepsilon^{2}}\right). \end{eqnarray}$

$\begin{eqnarray} \rho(x, \varepsilon) \approx K \cdot \exp \left(-\frac{\left(x-\overline{x}, W^{-1}(x-\overline{x})\right)}{2 \varepsilon^{2}}\right), \end{eqnarray}$

$\begin{eqnarray} {F W+W F^{\top} = -S, \quad F = \frac{\partial f}{\partial x}(\overline{x})}, {S = G G^{\top}, \quad G = \sigma(\overline{x})}. \end{eqnarray}$

$\begin{eqnarray} \left(x-\overline{x}, W^{-1}(x-\overline{x})\right) = 2 k^{2} \varepsilon^{2}, \end{eqnarray}$

$\begin{eqnarray} \frac{z_{1}^{2}}{\mu_{1}}+\frac{z_{2}^{2}}{\mu_{2}} = 2 k^{2} \varepsilon^{2}. \end{eqnarray}$

定义$W = \left( \begin{array}{cc} w_{11}\ &w_{12}\\ w_{21}\ &w_{22}\\ \end{array}\right)$为随机敏感性矩阵, 根据文献[35]得

$\begin{eqnarray} FW+WF^{\top} = -S, \end{eqnarray}$

### 图 4

(1) 在没有噪声的情况下, 置信椭圆变为一个点, 即确定性模型(3.1)的稳定内部平衡点$E_{11}$. 在噪声存在的情况下, 随着噪声强度的增大, 置信椭圆的半径逐渐增大, 确定性模型(3.1)的内部平衡点$E_{11}$始终处于置信椭圆的中心位置. 这意味着在一定情况下, 较大的噪声会导致随机模型(1.1)的解在确定性模型(3.1)的解附近波动, 且波动幅度与噪声强度呈正相关.

(2) 随着置信概率的增加(从95%增加到99%), 置信椭圆中包含的样本也越来越多. 这说明在实际应用中, 要使统计结果更加准确, 应选择合理的置信概率.

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