## The Influence of Fear Effect on Stability Interval of Reaction-Diffusion Predator-Prey System with Time Delay

Sun Yue,, Zhang Daoxiang,, Zhou Wen

School of Mathematics and Statistics, Anhui Normal University, Anhui Wuhu 241002

 基金资助: 国家自然科学基金.  11671013国家自然科学基金.  11302002安徽省自然科学基金.  2008085MA13

 Fund supported: the NSFC.  11671013the NSFC.  11302002the NSF of Anhui Province.  2008085MA13

Abstract

This paper combines theoretical derivation and numerical simulation to study the dynamics of a delayed reaction-diffusion predator-prey model with fear effect. First, the existence and stability of the positive equilibrium point of the system are studied. Secondly, the Hopf bifurcation problem of the system is studied through linear stability analysis. The results show that the fear effect affects the Hopf bifurcation point, and then affects the stability interval of the system. Finally, the theoretical results are verified by numerical simulations, and the nonlinear relationship between the fear effect and the stability interval is found, that is, as the fear effect continues to increase, the system will change from a stable state to an unstable state, and then to a stable state.

Keywords： Fear effect ; Hopf bifurcation ; Delay ; Diffusion ; Predator-prey system

Sun Yue, Zhang Daoxiang, Zhou Wen. The Influence of Fear Effect on Stability Interval of Reaction-Diffusion Predator-Prey System with Time Delay. Acta Mathematica Scientia[J], 2021, 41(6): 1980-1992 doi:

## 1 引言

Wang和Zanette等[14]为了考虑恐惧效应对种群的影响, 首次将恐惧效应引入到捕食-食饵模型中. 基于实验数据, 在食饵出生率项上增加恐惧效应比例系数, 建立了如下具有恐惧效应的生态动力系统:

$$$\left\{ \begin{array}{l} { } \frac{{\rm d}u}{{\rm d}t} = ur_{0}f(k_{0}, v)-du-au^{2}-g(u)v, \\ { } \frac{{\rm d}v}{{\rm d}t} = v(-m+cg(u)), \end{array}\right.$$$

$$$\left\{ \begin{array}{ll} { } \frac{\partial u}{\partial t} = d_{1}\Delta u+\frac{ru}{1+kv}-du-au^{2}-\frac{buv}{1+qu}, {\quad} x\in\Omega, t>0, \\ { } \frac{\partial v}{\partial t} = d_{2}\Delta v-m_{1}v-m_{2}v^{2}+\frac{cuv}{1+qu}, {\quad} x\in\Omega, t>0, \\ { } \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0, {\quad} x\in\partial\Omega, t>0, \\ { } u(x, 0) = u_{0}(x)\geq(\not\equiv)0, v(x, 0) = v_{0}(x)\geq(\not\equiv)0, \end{array}\right.$$$

$u\rightarrow +\infty$时, $F(u)$与最高次项系数$a_{7}$同号. 而

$F(+\infty)<0.$

$u = u_{\star \min}$时, 有

$$$\left. \begin{array}{l} \left( \begin{array}{cc} { } \frac{\partial U(x, t)}{{\rm d}t}\\ { } \frac{\partial V(x, t)}{{\rm d}t} \end{array} \right) = {D}\left( \begin{array}{cc} \Delta U(x, t)\\ \Delta V(x, t) \end{array} \right) +{J_{1}}\left( \begin{array}{cc} U(x, t)\\ V(x, t) \end{array} \right) +{J_{2}}\left( \begin{array}{cc} U(x, t-\tau)\\ V(x, t-\tau) \end{array} \right), \\ \end{array} \right.$$$

$$$\lambda^{2}+\delta_{1k}\lambda+\delta_{2k}+\delta_{3}e^{-\lambda\tau} = 0,$$$

$H_{2}:a_{11}+a_{22}<0;$

$H_{3}:a_{11}d_{2}+a_{22}d_{1}<0;$

$H_{4}:a_{11}a_{22}-a_{12}a_{21}>0.$

当$\tau = 0$时, 特征方程(2.3)变为

$$$\lambda^{2}+\delta_{1k}\lambda+\delta_{2k}+\delta_{3} = 0,$$$

## 3 模型的Hopf分支

$$$-\omega^2+\delta_{2k} = -\delta_{3}\cos(\omega\tau),$$$

$$$\delta_{1k}\omega = \delta_{3}\sin(\omega\tau),$$$

$$$w^4+(\delta_{1k}^2-2\delta_{2k})w^2+\delta_{2k}^{2}-\delta_{3}^{2} = 0,$$$

$\omega^{2} = z,$则方程(3.3)变为

$$$z^{2}+(\delta_{1k}^2-2\delta_{2k})z+\delta_{2k}^{2}-\delta_{3}^{2} = 0,$$$

$H_{5}:a_{11}d_{1}+a_{22}d_{2}<0;$

$H_{6}:a_{11}a_{22}+a_{12}a_{21}>0;$

$H_{7}:a_{11}a_{22}+a_{12}a_{21}<0.$

我们设方程(3.4)有两个根$z_{1k}, z_{2k}$, 由韦达定理, 得到

当$k = 0$时, 方程(3.4)为

$$$z^{2}+(\delta_{10}^2-2\delta_{20})z+\delta_{20}^{2}-\delta_{3}^{2} = 0,$$$

$H_{4}, H_{7}$成立, $z_{1}z_{2}<0$, 方程(3.5)有唯一的正实根$z_{0}$.

当$k = 0$时, 由方程(3.1), 我们可以得到

$$$(\frac{{\rm d}\lambda(\tau)}{{\rm d}\tau})^{-1} = \frac{(2\lambda+\delta_{10})e^{\lambda\tau}}{\lambda\delta_{3}}-\frac{\tau}{\lambda},$$$

$\tau_{0}^{j}$代入方程(3.6)中, 可以得到

$\rm (1) $$\tau\in[0, \tau_{0}), 系统(1.3)的正平衡点 E_{\star} 是局部渐近稳定的; \rm (2)$$ \tau\in(\tau_{0}, +\infty)$, 系统(1.3)的正平衡点$E_{\star}$是不稳定的.

### 4 数值模拟

(a) $\tau = 1.4$, 食饵的密度演化图; (b) $\tau = 1.4$, 捕食者的密度演化图; (c) $\tau = 2$, 食饵的密度演化图; (d) $\tau = 2$, 捕食者的密度演化图

### 图 3

(a) $t = 400$, (b) $t = 2000$, (c) $t = 4000$, (d) $t = 20000.$ 捕食者的空间斑图

## 5 结论

(1) 不论有无时滞, 恐惧效应均会影响系统正平衡点$E_{\star}$的稳定性;

(2) 以时滞为参数时, 恐惧效应将影响系统的Hopf分支临界值$\tau_{0}$, 在一定参数条件下, 当恐惧效应持续增加时, 稳定区间$[0, \tau_{0})$先缩小后增大, 恐惧效应与稳定区间的大小呈现非线性关系;

(3) 数值模拟结果显示, 系统具有复杂的动力学行为, 如出现了极限环, 螺旋波等现象.

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