反应扩散方程中时滞引起的Turing不稳定性及Hopf分支分析

This Project is intended to study the effect of time delay on Turing instability and Hopf bifurcation in reaction-diffusion equations by using biological models as background. The main contents of the study include the following three aspects. First, the effect of time delay on the stability and Turing instability of steady states of reaction-diffusion system is investigated. Second, the local and global Hopf bifurcation are considered, including existence and property of of Hopf bifurcation. Finally, application of the research results in some specific models (such as population model, epidemic model, tumor cell growth model) is carried out. Analysis and forecast of the development trend of specific biological problems are given. The dynamic system generated by delayed reaction-diffusion equation is infinite dimensional and difficult to study. To our knowledge, there are surprisingly few studies considering the effect of delay on Turing instability and the global Hopf Bifurcation. To conduct this project, one not only needs to apply and develop the existing theory of bifurcation analysis, but also requires new ideas and methods.

本项目拟以生物模型为背景，来研究时滞反应扩散系统中时滞对 Turing 不稳定性及Hopf分支的影响。研究的内容包括：(1)时滞是如何影响反应扩散方程的稳定性与 Turing 不稳定性。(2)时滞反应扩散方程的局部以及全局 Hopf 分支产生的充分条件以及分支周期解的性质。(3)在某些具体模型（如种群模型、传染病模型、肿瘤细胞增长模型等）中应用研究结果，分析并预测具体问题的发展趋势，给出相应的数学解释。时滞反应扩散方程产生的动力系统是无穷维的，研究难度较大。目前，关于时滞如何影响 Turing 不稳定性以及时滞反应扩散方程中的全局 Hopf 分支的研究较少。本课题在研究过程中不仅需要应用和发展已有的分支研究工具，更需要新的研究思想和方法。

本项目以生物模型为背景，研究了时滞反应扩散系统中时滞对 Turing 不稳定性及Hopf分支的影响。研究的内容包括：(1)稳态解的全局稳定性分析。(2)Turing分支的存在性分析。(3)Hopf分支的存在性分析，以及分支周期解的性质，如周期解的稳定性、方向、周期等。(4)分析了Turing分支曲线和Hopf分支曲线相交时，系统的动力学性质。(5)在某些具体模型（如种群模型、传染病模型、肿瘤细胞增长模型等）中应用研究结果，分析并预测具体问题的发展趋势，给出相应的数学解释。