非线性泛函微分方程的分支问题及应用

In this project, we aim to investigate the bifurcations for delay differential equations. We will study the dynamics behavior of differential equations with delay and diffusion, and some size and spatially structured population dynamics models with (or without) delays. In particularly, we will study Turing-Hopf bifurcation, Bogdanov-Takens bifurcation and so on. We will study the dynamics behavior for a functional differential equation with infinite delay and a system of state dependent delay differential equations. The project is also devoted to study some dynamical properties of a class of abstract Cauchy problems with a non-densely defined linear operator on a Banach spaceＸ that is assumed to be an almost sectorial operator. The goal of this proposal is to study the center manifold theory, normal form theory and bifurcations of the non-densely defined abstract Cauchy problems by using the integrated semigroup approach. The motivation for using integrated semigroup theory here comes from the fact that we have successfully used the integrated semigroup theory to develop a bifurcation theory for the abstract non-densely defined Cauchy problems involving Hille-Yosida or weak Hille-Yosida operator. Several applications, coming from population dynamics, will then be studied. Based on our works on non-densely defined semi-linear equations recent years, one could expect to extend some new results for this challenging problem.The study can not only enrich its own theories, but also be applied to improve the understanding of nosocomial infections and cancer cell population dynamics.

本项目主要研究时滞微分方程的分支问题。研究时滞反应扩散方程，具有时滞的结构种群模型的动力学行为，特别是高余维的Turing-Hopf分支，Bogdanov-Takens分支等。研究时滞和空间结构对结构种群模型的分支等现象的影响。将对一类无穷时滞微分方程和一类源自大小结构种群模型的状态依赖时滞微分方程的动力学行为进行研究。基于近年来我们利用积分半群理论对具有Hille-Yosida和弱Hille-Yosida算子的非稠定柯西问题给出的正规型和分支结果，我们试图利用积分半群理论来考虑一类具有非稠定几乎扇形算子的半线性抽象Cauchy问题的动力学性质，特别是中心流形，正规型，分支问题及其在生物数学中的应用。在我们多年对非稠定半线性方程研究的基础上，相信可以对这个挑战性的问题给出一些令人满意的理论结果。而理论研究也必将推动我们在应用领域的新的发现。

研究了结构种群模型（如年龄结构，大小结构，空间结构），恒化器模型，各类时滞微分方程模型（如状态依赖时滞，无穷时滞），2维格上的非局部周期系统，空间异质环境的具有对流项的非局部的反映扩散方程，脉冲微分方程等的动力学行为。主要研究了这些系统的分支问题（如Hopf分支，Takens-Bogdanov分支），行波解问题，最优捕获问题，随机近最优控制和随机有界性问题，主特征值及全局渐近稳定性和周期解的存在性，整体解的存在性问题，概自守问题等。此外我们建立了一系列COVID-19传染病模型，并应用到中国，韩国，意大利，法国等, 通过报告的累计确诊病例数据确定模型的参数和初始条件, 预测了疫情的拐点，已报告和未报告的病例数量，疫情的趋势等，模型和研究结果得到了广泛关注，SIAM News 向我们团队约稿报告了我们的研究结果。