【作者】 王娓；

【导师】 熊佐亮；

【作者基本信息】 南昌大学 ， 应用数学， 2010， 硕士

【摘要】 种群动力学是生物数学的一个重要分支.本文在传统常微分模型的基础上,讨论加入了脉冲和时滞作用的种群动力学模型.本文中所讨论的内容简单安排如下：第一章简单介绍种群动力系统的发展概况,并给出本文所要做的主要工作.第二章研究一类具阶段结构及捕食者有脉冲扰动的食饵-捕食者模型(其中食饵幼体有父母照顾),利用脉冲比较定理讨论食饵灭绝周期解的稳定性和系统的持续生存,利用数值模拟揭示系统的复杂动力学行为,并分析阶段结构对系统复杂动力行为的影响.第三章讨论一类具时滞的食饵-捕食者系统的稳定性：时滞广泛存在于生物种群的活动中,时滞在生物动力系统中有着重要的作用,它可能会影响到正平衡点的稳定性.由Roth-Hurwitz判据讨论系统正平衡点的存在性和稳定性.第四章讨论一类具时滞及修改的Holling-Tanner食饵-捕食者系统的Hopf分支：利用特征方程讨论正平衡点的渐近稳定性.运用Hopf分支理论,以时滞τ为参数给出系统发生Hopf分支的条件；利用规范型定理和中心流形定理,计算出Hopf分支在分支临界值τi处的性质.运用数值模拟揭示系统正平衡点的局部稳定性以及由Hopf分支产生的复杂动力学行为.更多还原

【Abstract】 Population dynamics is a very important branch of biomathematics. In this paper, the population dynamics models with impulsive and time delay on the base of traditional differential model have been considered. This paper arranged as follows:In the first chapter, the development of population dynamics system are given, and the major work of this paper is introduced.In the second chapter, a delayed stage-structured predator-prey model with the immature prey are taken care of by their parents and biological control strategies (that is releasing natural enemies at different fixed time) is considered. By using comparison theorem, the existence of the globally asymptotically stable prey-eradication periodic solution when the number of releasing natural enemies more than some critical value is proved. The conditions for the permanence of the system are given. Numerical simulation proved the academic results.In the third chapter, the stability of a delayed stage-structured predator-prey model is considered. Delay is existed popularly on the action of populations, and it operated important role with populations, it may be affect the stability of positive equilibrium. By using Roth-Hurwitz criterion discussing the existence and stability of positive equilibrium.In the last chapter, a modified Holling-Tanner predator-prey model is studied. By analyzing the associated characteristic equation, it is found that Hopf bifurcation occurs whenτcrosses some critical value. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and center manifold theorem. From calculate, the characters of Hopf bifurcation on the critical valueτj are obtained. By using Numerical simulation explained the stability of a positive equilibrium, and the rich dynamic behaviors of Hopf bifurcation.更多还原