Dynamics of Functional Differential Equations with Applications to Biology and Ecology

许多生物和生态过程涉及时间滞后。例如，捕食者需要时间将猎物的消耗量转化为生物量；对于许多传染病来说，由于种群的分散和疾病状态的变化，存在时间滞后。为了模拟这些涉及时滞的过程，自然的选择是使用泛函微分方程（即，受延迟影响的微分方程）。与常微分方程所描述的系统不同，泛函微分方程的初始状态是在一个区间上指定的函数，其相空间是无限维的。由于所得到的动力系统是无限维的，因此分析泛函微分方程是极具挑战性的，特别是对于具有时滞相关系数、非线性和多状态变量的方程，而这些方程恰好是生物学和生态学中产生的方程的典型特征。

Many biological and ecological processes involve time lags. For example, a predator needs time to convert its consumption of prey to its biomass; for many infectious diseases, time lags exist due to dispersal of populations and disease state changes. To model these time-lag involving processes, a natural choice is to use functional differential equations (i.e., differential equations subject to delays). In contrast to systems described by ordinary differential equations, the initial state of a functional differential equation is a function specified on an interval and the associated phase space is infinite dimensional. Since the resulting dynamical systems are infinite dimensional, the analysis of functional differential equations is extremely challenging, especially for equations with delay-dependent coefficients, nonlinearity and multiple state variables, which happen to be typical features of equations risen from biology and ecology.