反应扩散方程中时滞引起的Turing斑图及分支分析

The Project is to investigate the effect of delay and diffusion on the dynamics of the reaction-diffusion equations. For reaction-diffusion equations with application background, we firstly discuss the conditions of the existence and nonexistence of the non-constant steady state solutions by applying the degree theory, and then we investigate the Turing pattern induced by delay by using the bifurcation theory of delayed functional differential equations, and explain the effect of delay on Turing pattern. For general reaction-diffusion equations with delay, we investigate the existence of the codimension-two bifurcations, such as Turing-Hopf bifurcation, Hopf-Hopf bifurcation and so on, derive an algorithm for computing the normal form, and illustrate the influence of delay and diffusion coefficients on the topological structure of solutions of the delayed reaction-diffusion equations. Compared to the reaction-diffusion equations, the delay brings more difficulties in analyzing the eigenvalue problem, and moreover there are few theoretical results on high codimensional bifurcations of delayed reaction-diffusion equations. Hence, we not only need to enrich the existing theory, but also need to develop new methods.

本项目旨在研究时滞以及扩散对反应扩散方程的动力学性质的影响。主要研究内容包括：对具应用背景的反应扩散方程，应用度理论讨论系统中非常值稳态解的存在以及不存在的条件；应用偏泛函微分方程的分支理论，推导出时滞引起的Turing斑图的存在性条件，揭示时滞对Turing斑图的影响。对一般的时滞反应扩散方程推导出系统余维二分支（Turing-Hopf分支、Hopf-Hopf分支等）的存在性条件以及规范型的计算公式，揭示时滞和扩散系数对时滞反应扩散方程解的拓扑结构的影响。与一般的反应扩散方程相比，时滞的加入使得特征值的分布情况的分析变得非常复杂，而且目前关于时滞反应扩散方程高余维分支的理论结果还比较少。因此，本课题的研究不仅要丰富现有的理论，还需要寻求新的方法。