Overview
My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. One goal is to better understand basic population dynamics so that measurable criteria can be developed, enabling scientists to predict combinations of cultures of microorganisms, most effective and safest for use in such processes as water purification and biological waste decomposition. Other applications include pest control, prevention of species' extinction, and control or eradication of certain diseases. In order to elicit all the potential dynamics, a bifurcation theory approach is used so that the full spectrum of behaviour can be predicted for all appropriate parameter ranges and initial states. Computer simulations are used to elucidate complicated dynamics, to test conjectures and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used to carry out complicated calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, including difference equations, ordinary, integro- and functional differential equations, and bifurcation theory.
