几类微分方程的标准型计算及分支分析

发布时间：2018-08-27 09:17

【摘要】：本文主要研究了三个微分方程模型的动力学性质,一个是带非线性收获的捕食被捕食模型,另一个是含有时滞的改进的Lesslie-Gower型捕食被捕食模型,最后一个是含两个时滞的有正阻尼的振荡模型.对三个模型从不同角度进行分析,得到各模型在相应平衡点处的开拆标准型及分支情况.第一章,介绍了数学生态学研究的背景和现状及一些相关概念等.第二章,主要研究具有非线性收获的比率依赖的捕食被捕食模型的多种分支问题.在这个模型中引入了与被捕食者有关的非线性收获p(x)=qEX/m1E+mwx,通过分析相应的特征方程,边界平衡点和内部平衡点的稳定性得以研究.通过应用常微分方程的定性理论和微分流形定理,我们可以得到系统在不同平衡点处的分支情况.第三章,考虑含有时滞的改进的Leslie-Gower型捕食被捕食模型的分支问题.我们首先给出平衡点是Bogdanov-Takenss(B-T)奇点或triple-zero奇点的存在条件.在这些奇点处选择合适的分支参数,通过中心流形理论和开拆标准型方法,分别给出系统在B-T奇点和triple-zero奇点处的开拆标准型及相应的分支结果.最后,通过把时滞作为分支参数,系统在另一个平衡点处的Hopf分支的性质得到分析.第四章,主要讨论含两个时滞的有正阻尼的振荡系统的B-T分支.通过讨论特征方程的特征根的分布,我们给出原点是B-T奇点的存在条件,进而由中心流形理论给出系统B-T分支的二次和三次标准型.
[Abstract]:In this paper, the dynamic properties of three differential equation models are studied, one is a predator-prey model with nonlinear harvesting, the other is an improved Lesslie-Gower predator-prey model with time delay. The last one is a positively damped oscillation model with two delays. The three models are analyzed from different angles, and the opening and breaking standard form and branch of each model at the corresponding equilibrium point are obtained. The first chapter introduces the background and present situation of mathematical ecology and some related concepts. In chapter 2, we mainly study the branching problem of predator-prey model with ratio dependence of nonlinear harvest. In this model, the nonlinear harvesting p (x) / qEX-m1E mwx, related to the prey is introduced. The stability of the boundary equilibrium point and the internal equilibrium point is studied by analyzing the corresponding characteristic equations. By applying the qualitative theory of ordinary differential equations and the differential manifold theorem, we can obtain the bifurcation of the system at different equilibrium points. In chapter 3, we consider the bifurcation problem of an improved Leslie-Gower predator-prey model with time delay. We first give the existence condition that the equilibrium point is Bogdanov-Takenss (B-T) singularities or triple-zero singularities. The proper bifurcation parameters are selected at these singularities. By means of the center manifold theory 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 are given, respectively, and the corresponding bifurcation results are obtained. Finally, the properties of the Hopf bifurcation of the system at another equilibrium point are analyzed by using time delay as a bifurcation parameter. In chapter 4, the B-T bifurcation of oscillatory systems with positive damping with two delays is discussed. By discussing the distribution of characteristic root of the characteristic equation, we give the existence condition that the origin is B-T singularity, and then give the quadratic and cubic normal forms of B-T bifurcation from the center manifold theory.
【学位授予单位】：河南师范大学
【学位级别】：硕士
【学位授予年份】：2015
【分类号】：O175

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