【作者】 童欢；

【导师】 傅朝金；

【作者基本信息】 湖北师范学院 ， 应用数学， 2012， 硕士

【摘要】 本文运用常微分方程定性理论和常微分方程稳定性理论，研究了两种带有竞争项的时滞系统，首先得到了系统存在正平衡点的条件，然后对该点处的特征方程进行讨论分析，从而得到该系统产生Hopf分支现象时的条件，最后运用中心流形定理和规范型理论进行讨论分析，得到了分支的相关性质.全文分为五章：第一章是引言，先介绍了捕食者-食饵模型的研究背景和研究现状，随后介绍了本文的主要工作.第二章介绍了一些理论工具，包括常微分方程定性和稳定性理论和Hopf支理论的相关引理和定理.第三章讨论了具有竞争项的双时滞系统的Hopf分支模型，利用常微分方程定性和稳定性理论及方法，选取时滞τ为分支参数，讨论正平衡点的稳定性和Hopf分支的存在性.最后给出了一个数值举例.第四章讨论了具有竞争项的分布时滞系统的Hopf分支模型，并运用常微分方程定性和稳定性理论及方法，得到了平衡点稳定的条件.第五章对本文研究的内容进行了总结，并对以后的研究方向和生物数学的未来进行了展望.更多还原

【Abstract】 In this paper, by using qualitative method and stability theory of ordinarydifferential equations, Hopf bifurcation for two classes with time delay in competitionmodels are studied. First of all, we can get the existence of the positive equilibriumsystem conditions. By analyzing the characteristic equation, we can get condition ofthe system occurring Hopf bifurcation. Finally by using the center manifold theoremand normal form method, we can get the related properties of the bifurcation. Thepaper is divided into five parts.The first part is the introduction, the predator prey model background and theresearch present situation, then introduces the main work of this paper.The second part is some theoretical tools, including qualitative and stabilitytheory of ordinary differential equations and Hopf bifurcation theory related lemmasand theorems.The third part is a discussion with two delays in competition Hopf bifurcationmodel, by using qualitative method and stability theory of ordinary differentialequations, selecting delay for bifurcation parameter, and discuss the existence of Hopfand the positive equilibrium of the stability. Finally, we give a numerical example.The fourth part discusses a distributed delay in competition Hopf bifurcationmodel, and get the balance of the stability conditions.The last part has an summarize of this paper, and we have a future prospects ofthe future research direction of mathematical biology.更多还原