两类生物动力系统的分岔问题

项目名称： 两类生物动力系统的分岔问题

项目编号： No.11201321

项目类型： 青年科学基金项目

立项/批准年度： 2013

项目学科： 数理科学和化学

项目作者： 邹兰

作者单位： 四川大学

项目金额： 22万元

中文摘要： 本项目主要研究两类生物动力系统的分岔问题，包括一类含有驻点的非线性功能反应函数的捕食者-食饵模型和一类传染病动力学模型。这两类从实际中建立的生物动力系统模型具有复杂的数学表达形式，例如系统独立参数多、含有超越函数、系统中的每个自由变量都具有较高次数、相空间维数较高等，有的还是具有周期函数或者拟周期函数的系数的非自治系统。在这种情况下的自治系统，决定平衡点的方程每一项的系数都是系统参数组成的复杂函数，平衡点坐标无法解析地求出，平衡点的特征值或者判定量无法具体表示出来，有时候甚至连平衡点的个数都很难得知。而非自治系统就连周期轨的存在性、位置都很难确定。本项目试图寻求若干方法避开对复杂形式动力系统求解平衡点的困难，研究两类生物动力系统的分岔问题。事实上，对定性性质和分岔现象判定的不等式条件，如横截性条件、非退化条件等，为我们寻求这样的方法提供了可能。

中文关键词： 周期轨；分岔；平衡点；基本再生数；

英文摘要： In this programmme, we focus on the bifurcation problems of two types of biological dynamical systems, including a predator-prey system whose functional response functions have critical points, and a dynamical model of infectious diseases. These dynamical systms are built from practical modelling, and described by complicated mathematical expressions with plenty of dependent parameters, transcendental functions, free variables with high degrees, high dimensional phase spaces, and some of them are even nonautonomous systems with periodic coefficients or almost periodic coefficients. For thoses systems, every coefficient of the equations, which deternmine the equilibria, is a complicated function of parameters. The coordinates of equilibria can not be solved analytically, and their eigenvalues as well as descriminants can not be described in details. Sometimes, it is even difficult to know the number of equilibria. Moreover, for nonautonomous systems, the existence and positions of periodic orbits are difficult to know. In this programme, we will try to find the methods to avoid the difficulties for finding equilibria of complicated dynamical systems, and study bifurcations of two types of biological dynamical systems. In fact, the inequilities for qualitative properties and discriminants for bifurcations, such a

英文关键词： periodic orbit；bifurcation；equilibrium；basic reproduction number；