【作者】 黄玉梅；

【导师】 李树勇；

【作者基本信息】 四川师范大学 ， 基础数学， 2004， 硕士

【摘要】 本文所研究的问题涉及到生态系统中有关捕食-被捕食系统、流行病模型等几个常见的生态模型的定性分析，所采用的研究方法是通过构造合适的Lyapunov泛函以及对模型的线性近似系统在其平衡态处的特征根分析，获得模型解或正周期解的一致持久和稳定的渐近性质。 在第二章中，我们分别对一类含分布时滞的扩散周期Lokta-Volterra竞争模型和具Holling Ⅱ类功能反应的时滞扩散Lokta-Volterra模型进行分析，通过构造Lyapunov泛函，得到系统解或正周期解的渐近性质。 在第二章中，我们研究的扩散行为是离散的，而在第三章中，我们研究两类含连续扩散行为的非线性时滞Gilpin-Ayala模型(反应扩散方程)，通过构造Lyapunov泛函获得其平衡态全局渐近稳定、渐近稳定的一些充分条件。 第四章的研究对象是一类具有阶段结构和时滞的SIRS传染病模型，通过对模型的线性近似系统在其平衡态处的特征根分析，得到模型平衡点的局部渐近稳定性等渐近性质。更多还原

【Abstract】 In this paper, we consider several classes of common ecological models such as predator prey model and epidemic model, and obtain uniform persistence and stability of the solution or the periodic solution by constructing appropriate Lyapunov functional and using eigenvalue analysis to model’s linear similar system at the equilibrium.In chapter 2, we deal with a non-autonomous periodic competitive Lotka-Volterra model with distributive delay and a Lotka-Volterra model with distributive time delay and Rolling type II functional response, and get the asymptotic quality of models by means of using Lyapunov functional method. Both of them have discrete diffusion.In chapter 3, two nonlinear modified Gilpin-Ayala models with distributive delay and continuous diffusion are studied. Some simple sufficient conditions for globally asymptotically stable and asymptotically stable of the unique positive equilibrium point are established by constructing Lyapunov functional.In chapter 4, a SIRS epidemic model with stage-structure consisting immature and mature stage and time delay is studied. By using eigenvalue analysis we obtain that the asymptotic behavior of the model and the local asymptotic stability of its equilibrium.更多还原