Research Profile: Dynamical systems, bifurcation theory, population dynamics, mathematical ecology and epidemiology
My students and I have been formulating and analyzing models motivated by questions
in ecology and epidemiology. For example, one goal is to better understand basic population dynamics
so that measurable criteria can be developed, enabling scientists to
predict combinations of cultures of micro-organisms most effective and
safest for use in such processes as water purification and biological
waste decomposition. Other applications include pest control, the prevention
of species' extinction or the 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, ordinary, integro-
and functional differential equations, and bifurcation theory.