几类多时滞捕食系统的余维分支分析

发布时间：2018-06-06 07:28

本文选题：时滞 + Allee效应　；参考：《河南师范大学》2017年硕士论文

【摘要】：本文主要分析了两类捕食被捕食系统的分支问题:一类是带有双Allee效应的单时滞捕食被捕食系统;另一类是含有两个时滞的改进的Leslie-Gower型捕食被捕食系统.本文主要通过推广应用时滞微分方程的标准型理论和中心流形定理,得到对应系统的开拆标准型,进而得到一系列有趣的分支现象.第一章,介绍了本文的研究背景及发展现状,列出一些所需的理论知识.第二章,考虑被捕食者带有双Allee效应的单时滞捕食被捕食模型的分支问题.首先给出模型在其正平衡点处为Bogdanov-Takens(B-T)或triple-zero型奇点的充分条件.然后选择合适的分支参数,通过推广使用中心流形定理和开拆标准型方法,分别推导出系统在B-T型奇点和triple-zero型奇点处的开拆标准型及相应的分支结果.第三章,主要研究具有两个时滞改进的Leslie-Gower型的捕食被捕食模型在其正平衡点处的余维分支问题.通过分析相应奇点处的特征方程的根的分布,得到奇点为B-T型或triple-zero型奇点的存在条件.通过应用微分方程的定性理论和中心流形定理,可以得到系统在平衡点处的分支情况.
[Abstract]:In this paper, the bifurcation problems of two kinds of predator-prey systems are analyzed: one is a single-delay predator-prey system with double Allee effect, the other is an improved Leslie-Gower predator-prey system with two delays. In this paper, we generalize the canonical form theory of delay differential equation and the central manifold theorem, and obtain the open and disassembly canonical form of the corresponding system, and then obtain a series of interesting bifurcation phenomena. The first chapter introduces the research background and present situation of this paper, and lists some necessary theoretical knowledge. In chapter 2, the bifurcation problem of predator-prey model with double Allee effect is considered. The sufficient conditions for the model to be Bogdanov-Takenson B-T or triple-zero singularity at its positive equilibrium point are given. Then the proper bifurcation parameters are selected, and by using the center manifold theorem and the open and disassembly canonical form method, the open and disassembly canonical forms of the system at B-T singularities and triple-zero singularities and the corresponding bifurcation results are derived respectively. In chapter 3, we study the codimensional bifurcation of a predator-prey model with two delays at the positive equilibrium point of the predator-prey model. By analyzing the root distribution of the characteristic equations at the corresponding singularities, the existence conditions of the singularities of B-T type or triple-zero type are obtained. By applying the qualitative theory of differential equations and the central manifold theorem, the bifurcation of the system at the equilibrium point can be obtained.
【学位授予单位】：河南师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O175

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