具趋化的浮游动植物反应扩散模型的动力学分析

Analysis of Dynamics in Toxin-phytoplankton-zooplankton Reaction-diffusion Model with Prey-taxis

本文研究趋化和时滞对一类浮游动植物反应扩散模型动力学的影响.首先,通过分析相关的特征方程,得到正稳态解的稳定性,借助Crandall-Rabinowitz局部分支理论,将趋化敏感系数和时滞分别作为分支参数,探究Turing分支和Hopf分支的存在性;接着利用中心流形定理和规范型方法研究Hopf分支的方向和稳定性;最后利用数值模拟展示趋化和时滞对系统的分支与模式形成的影响.我们的结果说明:在无时滞的系统中,当趋化敏感系数超过某临界值时,会使正常数稳态解由稳定变为不稳定(Turing不稳定性);在具时滞的系统中,当趋化敏感系数小于临界值时,如果时滞低于某个值,那么正常数稳态解是局部渐近稳定的;如果时滞越过某个值,则系统会在正常数稳态解经历Hopf分支,并且从正常数稳态解处分支出一个稳定的空间齐次周期解.

This paper investigates a time-delayed reaction-diffusion model with chemotaxis for phytoplankton.The stability of the positive steady states is obtained by analyzing the related characteristic equations.With the help of Crandall-Rabinowitz’s local partial bifurcation theory,the chemotaxis sensitivity coefficient and delay are taken as bifurcation parameters respectively to investigate the existence of Turing bifurcation and Hopf bifurcation.Then,the direction and stability of Hopf bifurcation are studied by using the central manifold theorem and the normal form method.Finally,numerical simulations are presented to show the influence of chemotaxis and time delay on the bifurcation and pattern formation of the system.Our results show that:in the system without time delay,when the chemotaxis sensitivity coefficient exceeds a critical value,the positive steady-state solution of the system will change from stable to unstable(Turing instability);in the system with delay,when the chemotaxis sensitivity coefficient is less than the critical value,if the delay is below a certain value,the positive steady-state solution is locally asymptotically stable;if the time delay exceeds a certain value,then the system will undergo Hopf bifurcation at the positive steady-state solution,and a stable spatially homogeneous periodic solution will be bifurcated from the positive steady-state solution.