Mathematical Sciences: Stability, Oscillations, and Persistence in Differential Equation Models for Structured Populations.

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This project studies whether including population structures can change unstructured models of ecological or epidemiological systems that converge to equilibrium states into models that undergo damped oscillations. Mathematically this amounts to converting systems of ordinary differential equations into nonlinear partial differential equations with nonlinear boundary conditions and functional terms. These equations are studied as evolution equations in an appropriate Banach space. Specific examples are age-structured models of childhood diseases, infection-age-structured models of fatal infectious diseases, and size-structured models of waterflea populations (waterfleas are an important laboratory animal). The dynamics of structured populations and the spread of infectious diseases are more complicated than simple models predict. An understanding of the ways oscillations occur in mathematical models may well be important to public health, by helping to predict the course of epidemic or endemic infections.***