【作者】 王朝阳；

【导师】 刘宣亮；

【作者基本信息】 华南理工大学 ， 应用数学， 2010， 硕士

【摘要】 本文考虑了疾病仅在食饵中传播且具非线性发生率的一类捕食者-食饵系统,众所周知,关于单个种群中的传染病模型,已有大量文献进行研究.同时,对捕食者-食饵两种群模型的研究,人们也作了大量工作,但是对于在捕食者-食饵系统中有传染病流行的情况,研究工作相对较少.本文主要利用微分方程的定性与分支理论的知识,对于此类生态-流行病模型进行了研究,得到了模型在第一卦限的平衡点及局部稳定性,解的有界性,并对平衡点的分支情况进行了讨论.我们的结果表明,由于维数的升高,这类模型具有更加复杂的动力学现象.全文共分四章.第一章,介绍研究动力系统的分支与生态-流行病模型的意义和本文内容安排.第二章,介绍本文所涉及到的基本概念及方法.第三章,研究具非线性发生率的生态-流行病模型,主要讨论此三维系统的平衡点及其局部稳定性,解的有界性,然后讨论在一个边界平衡点的Bogdanov-Takens分支问题,得到了相应的鞍结点分支曲线,Hopf分支曲线,同宿分支曲线,并给出了分支图.最后,讨论了正平衡点附近的Hopf分支与广义Hopf分支,在某组参数下,得到了系统出现一个或两个极限环的条件,在本章中对一些分支情形进行了数值模拟.第四章,研究食饵种群有Logistic增长的生态-流行病模型,主要讨论了系统的平衡点及其局部稳定性,然后利用分支方法与技巧,分析了在一个边界平衡点附近的Bogdanov-Takens分支,得到了鞍结点分支曲线,Hopf分支曲线,同宿分支曲线,最后利用中心流形计算的投影方法研究了系统在正平衡点附近的Hopf分支,得到了某种情况下极限环存在的条件.更多还原

【Abstract】 This paper considers predator-prey systems with disease in the prey, Assume that the predator eats only the infected prey, and the incidence rate is nonlinear. As we know, there are many references on predator-prey models and epidemiological models, but little attention has been paid so far to merge these two important areas of research. In this paper, by means of qualitative theory and bifurcation theory of differential equations, we study the eco-epidemiological models of this type. We study the dynamics of the models in terms of local analysis of equilibria and bifurcation analysis of a boundary equilibrium and a positive equilibrium. Our results show a much wider range of dynamical behaviors than do those with predator-prey systems or epidemiological models due to higher dimension of the systems. This artical is divided into four chapters.The first chapter introduces the significance of studying bifurcation and eco-epidemiological models and arrangements for this article.The second chapter describes the basic concepts and methods of this paper.In Chapter III, we study an eco-epidemiological model with nonlinear incidence rate. The main purpose of this chapter is to present local analysis and bifurcation analysis of the model, we analyse the boundedness of solutions and local stability of equilibria, by using bifurcation methods and techniques, we study Bogdanov-Takens bifurcation near a boundary equilibrium, and obtain a saddle-node bifurcation curve, a Hopf bifurcation curve and a homoclinic bifurcation curve. The Hopf and generalized Hopf bifurcation near the positive equilibrium is analyzed, the existence of one or two limit cycles is also discussed. Numerical simulation results are given to support the theoretical predictions.In Chapter IV, we analysis an eco-epidemiological model with Logistic growth in prey. we analyse the existence and local stability of equilibria, by using bifurcation methods and techniques, we study Bogdanov-Takens bifurcation near a boundary equilibrium including a saddle-node bifurcation, a Hopf bifurcation and a homoclinic bifurcation. The Hopf bifucation near the positive equilibrium is analyzed by using the projection method for center manifold computation. In some case,the conditions for the existence of a limit cycle are obtained.更多还原