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

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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.