几类具有状态依赖时滞微分方程的定性分析

It is a challenge in studying the stability and Hopf-bifurcation theories for partial differential equations and neutral differential equations with state-dependent delays, and applications in stage structured population models. Firstly, the stability and Hopf-bifurcation theories for partial differential equations with state-dependent delays are established by considering the differentiability and compactness of semiflow, linearization at equilibria and local invariantmanifolds at stationary points. It is helpful to study the existence and stability of traveling waves for reaction diffusive equations with state-dependent delays, which can be transformed into the homoclinic solutions of ordinary differential equations with state-dependent delays by constructing a pair of upper-lower solutions and Schauder's fixed point theorem. Furthermore, we develop the local and global Hopf-bifurcation theory for neutral differential equations with state-dependent delays using the equivariant degree theory. According to the theories obtained, we finally analyze some stage structured population models, such as the boundedness of solutions, positivity, stability and bifurcations，reveale the effect of state-dependent delays on the population model.

研究状态依赖时滞偏微分方程和中立型状态依赖时滞微分方程的定性理论及其在年龄结构种群模型中的应用是一项挑战性的工作。本项目将通过对半流的可微性和紧性、稳态处的线性化和稳态处的局部不变流形的研究建立状态依赖时滞偏微分方程的稳定性和 Hopf 分支理论。进而考虑状态依赖时滞反应扩散方程行波解的存在性和稳定性这一问题，并将该问题转化为求一个状态依赖时滞常微分方程的异宿轨，然后借助构造适定的上下解和 Schauder不动点定理来解决。本课题还将借助等变度理论建立一类中立型状态依赖时滞微分方程的局部和全局 Hopf 分支。最后，基于所建立的理论分析一类年龄结构种群模型解的有界性、正性，稳态的稳定性和分支行为等，揭示状态依赖时滞对种群模型的影响。

该项目研究了状态依赖时滞抛物型微分方程解半流的光滑性；中立型状态依赖时滞微分方程的全局Hopf分支；状态依赖时滞阶段结构种群模型的波前解；状态依赖时滞竞争模型和捕食模型的全局稳定性；Gause型、年龄结构型和食饵释放毒素型捕食扩散模型的全局稳定性和Hopf分支；具有防御行为的时滞捕食模型的群聚效应；食草动物--浮游植物模型中反馈控制诱发的周期性；一个尺度结构种群模型的优化收获问题；时滞杂交系统的优化问题及其在农业生态系统中的应用。

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