Mathematical Sciences: Analysis of Models with Time Delays and Spatial Dependence

8902712布森伯格这个项目将研究生物和物理系统的模型，这些模型是用带有非局部项的泛函和偏微分方程式表示的。这项工作的主要领域是：a)。具有空间扩散和时间延迟的隔室模型及其在理解大小、时间延迟和其他因素对细胞中某些生化过程稳定性的影响方面的应用；b)流行病模型，以实际疾病的动态为基础，特别强调垂直传播，以及人口的年龄结构和其他内部变量对这些模型的动态行为的影响；关于连续和离散动力系统振荡解周期的锐界的推导，以及排除周期和更复杂的动力学的可能性的条件的推导。该项目将采用分析调查和数值方法，旨在使数学结果与观测数据相吻合。所有这些问题都需要开发新的分析技术，这些技术可以应用于对某些类型的生物和物理现象进行建模时出现的广泛类型的方程。

8902712 Busenberg This project will study models of biological and physical systems which are formulated in terms of functional and partial differential equations with nonlocal terms. The main areas of this work are: a). Compartmental models with spatial diffusion and time delays and their application to the understanding of the effects of size, time delays, and other factors on the stability of certain biochemical processes in cells; b). models of epidemics which are based on the dynamics of actual diseases with particular emphasis on vertical transmission and on the effects of age-structure and other internal variables of a population on the dynamics behavior of these models; and c). the derivation of sharp bounds on the periods of oscillatory solutions of continuous and discrete dynamical systems and the derivation of conditions which rule out the possibility of periodic and more complicated dynamics. The project will employ both analytical investigations and numerical methods that will be aimed at fitting the mathematical results to observed data. All of these problems require the development of new techniques of analysis that can be applied to broad classes of equations that occur in modelling certain types of biological and physical phenomena.