几类微分方程的定性分析及其在人口动力系统中的应用

项目名称： 几类微分方程的定性分析及其在人口动力系统中的应用

项目编号： No.11471044

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 刘志华

作者单位： 北京师范大学

项目金额： 65万元

中文摘要： 本项目主要研究时滞微分方程，时滞反应扩散方程, 结构种群模型的分支，吸引子和行波解等问题及其在传染病学和癌细胞人口动力系统中的应用。我们将对非稠定半线性方程建立对称性分支理论.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,特别是研究高余维的Turing-Hopf分支，Bogdanov-Takens分支，Hopf-Hopf分支现象等。此外我们将致力于研究周期激励或随机激励对这些方程或种群模型的吸引子,行波解和分支现象(如Hopf分支)等动力学行为的影响.我们试图通过将其转化成具有周期激励或随机激励的非稠定半线性方程来研究.在我们多年对非稠定半线性方程研究的基础上,相信从这样一个角度可以对这个挑战性的问题给出一些令人满意的新结果. 而后者的研究也将帮助我们更好的理解大家密切关注的传染病学和癌细胞人口动力系统中的许多问题.

中文关键词： 分支；时滞；结构种群模型

英文摘要： In this project, we aim to investigate the bifurcations, attractors and travelling waves for delay differential equations, differential equations with delay and diffusion, structured population dynamics models. Firstly, we will consider symmetric bifurcation theory for non-densely defined semi-linear equations and applications to structured population dynamics models. Then we will study the dynamics behavior of some age and spatially structured population dynamics models with (or without) delays. In particularly, we will study Turing-Hopf bifurcation, Bogdanov-Takens bifurcation, Hopf-Hopf bifurcation and so on.The project is also devoted to the effects of periodic forcing and random forcing driven by a Brownian motion on the above mentioned equations and models. The main idea of this proposal is to transform the periodically forced and randomly forced differential equations or models to semi-linear equations with non-dense domain under periodic forcing and random forcing. 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.

英文关键词： Bifurcations;Delay;Structured population dynamics