【作者】 李莉；

【导师】 王玉文； 史峻平；

【作者基本信息】 哈尔滨师范大学 ， 基础数学， 2010， 硕士

【摘要】 生态数学是研究生物之间及其周围环境之间的一门学科,而捕食者和食饵之间的动力学行为是生态学和生物数学中的重要课题之一.随着现代数学的发展,我们更多的是用定性理论和稳定性理论来定性分析捕食-食饵模型,动力系统定性理论的研究就变得更为重要了,而平衡解,周期解的存在性以及稳定性,极限环渐近性态等一直都是动力系统研究中的几个重要分支.在现实世界里,一般的捕食-食饵模型不仅只是用状态随时间变化来刻画,同时也要考虑在空间中的扩散影响和随之产生的时空形态.而在半线性偏微分方程的研究领域中有一类非常重要的非线性现象,即分支现象.它反映的是由方程的解生成的流的拓扑结构随参数的变化而引起质的变异.分支问题主要包括局部分支,半局部分支和全局分支问题.对偏微分方程分支问题的研究不仅要用到经典的动力系统理论,而且又要用到代数,拓扑,泛函等相关知识,其研究具有比较强烈的实际背景和重大的理论意义.本文利用局部线性化分析Poincare-Bendixson定理,构造Lyapunov函数以及全局稳态分支定理等数学理论与数学方法,针对一类具有Holling-Ⅲ形式的捕食-食饵模型对应的常微分方程系统和反应扩散方程组,进行了系统的研究,给出了平衡解的局部Hopf分支,全局稳定性及全局稳态分支等结果,也为今后有关的研究提供了一定的理论依据.本文将就如下模型进行分析：其中Ω是一个有界空间区域,A, B, C, D, H均为正实数,p≥2.主要工作如下：首先,利用局部线性化分析和构造Lyapunov函数给出相应常微系统动力学行为分析：其次,利用一般的半线性偏微分方程的局部Hopf分支定理,对该模型进行局部Hopf分支的研究；最后,利用椭圆方程的正则性理论给出相应稳态方程解的有界性估计,并考虑该模型的全局稳态分支情况.更多还原

【Abstract】 Mathematical biology is a subject that discusses the interaction of species and surrounding environment, and the dynamical behavior of predator-prey system has been one of the important topics in ecology and mathematical biology. Along with the development of modern mathematics, we analyze the predator-prey system by using qualitative analysis and stability theory. Research on the qualitative theory of dynamical systems has become important in recent years, and the existence and stability of equilibrium and periodic solution are among important branches of the research of dynamical systems.In the real world predator-prey model, not only we should consider the time evolution of the population, but we also need to consider the spatial diffusive effect. In the field of nonlinear partial differential equations, there exists an important nonlinear phenomenon, which is bifurcation. Bifurcation phenomenon means that, when the parameters cross through certain critical values, there exist changes of some structural properties in the system. Bifurcation could be local bifurcation, semi-local bifurcation and global bifurcation. The study of bifurcation of partial dif-ferential equations is not only related to the theories of classical dynamical systems, but also related to the other knowledge such as topology, algebra and functional analysis. The study is of great theoretical significance and practical background.This paper mainly performs ODE equational system and reaction-diffusion equations of the predator-prey system with Holling typeⅢfunctional response by using local linear analysis, Poincare-Bendixson theorem, constructing the Lya-punov function, global steady state bifurcation theorems and a few other mathemat-ical theory and method. Obtained results including the local stability analysis of equilibrium solutions, analysis of Hopf bifurcations, global steady state bifurcations and so on. It also provides some theoretical basis for future research.The paper considers a general equation in the following form: whereΩis a bounded domain, A, B, C, D, H are positive constants, p≥2. The main content of the paper isas follows:First, we give the dynamical analysis to the according ODE system by using local linear analysis and constructing the Lyapunov function;Second, using local Hopf bifurcation theorem of the semilinear partial differential equations, this paper mainly performs local Hopf bifurcation analysis to this reaction diffusion system; Lastly, a priori estimates of solutions to the corresponding steady state system are obtained by using the theory of elliptic equations regularity and analyzing the global steady state bifurcation.更多还原