具Allee效应的时滞反应扩散方程的分支理论

时滞反应扩散方程描述了带有时滞和空间异质性的微分方程形式，具有重要的实际应用价值。Allee效应刻画了在种群数量稀少或密度较低时个体适合度与种群数量或密度正相关的现象，在基础和应用生态学研究中具有重要意义。本项目拟研究具Allee效应的时滞反应扩散方程的分支理论，针对单种群和两种群模型，建立系统常值稳态解的存在性和稳定性，利用稳态分支定理，给出系统非常值稳态解的存在性与稳定性，建立Turing分支、Hopf分支及其分支性质，探讨Turing-Hopf分支和Hopf-Hopf分支等余维二分支的存在性。特别将着重讨论Allee效应、时滞和空间异质性对系统动力学性质的影响及其内在规律，同时阐明分支的发生在解决具体实际生态问题中的重要作用。本项目的研究有助于推动偏泛函微分方程分支理论的发展，并为生态系统可持续稳定发展提供依据。

Delayed reaction-diffusion equations describe differential equations with time delay and spatial heterogeneity, which have crucial practical application value. Allee effects are a density-dependent phenomenon in which individual fitness is positively correlated with population size or density when the size is scarce or the density is low. It has important significance in the study of basic and applied ecology. In this project, we systematically study the bifurcation theory of delayed reaction-diffusion equations with Allee effects. According to single-specie and two-species models, we establish the existence, stability of the constant steady state solutions and the nonconstant steady state solutions by steady state bifurcation theory, investigate the existence theorem of Turing bifurcation and Hopf bifurcation, and derive the bifurcating properties. And then we discuss the existence of the codimension-two bifurcation such as Turing-Hopf bifurcation, Hopf-Hopf bifurcation. In particular, we explore the effects and interior rules of Allee effects, time delay and spatial heterogeneity for the dynamical properties on delayed reaction-diffusion equations with Allee effects, and emphasize the important role of the occurrence of bifurcation in solving practical ecological problems. Therefore, this research project may promote the development of bifurcation theory for partial functional differential equations and provide some theoretical bases for the development of ecosystem.