Ph.D. Public Lecture (Math) - Ao Li

"Mathematical Modelling of Ecological Systems in Patchy Environment"

In this thesis, we incorporate spatial structure into different ecological/epidemiological systems by applying the patch model. Firstly, we consider two specific costs of dispersal: (i) the period of time spent for migration; (ii) deaths during the dispersal process. Together with the delayed logistic growth, we propose a two-patch model in terms of delay differential equation with two constant time delays. The costs of dispersal, by themselves, only affect the population sizes at equilibrium and may even drive the populations to extinction. With oscillations induced by the delay in logistic growth, numerical examples are provided to illustrate the impact of loss by dispersal.

Secondly, we study a predator-prey system in a two-patch environment with indirect effect (fear) considered. When perceiving a risk from predators, a prey may respond by reducing its reproduction and decreasing or increasing (depending on the species) its mobility. The benefit of an anti-predation response is also included. We investigate the effect of anti-predation response on population dynamics by analyzing the model with a fixed response level and study the anti-predation strategies from an evolutionary perspective by applying adaptive dynamics.

Thirdly, we explore the short-term or transient dynamics of some SIR infectious disease models over a patchy environment. Employing the measurements of reactivity of equilibrium and amplification rates previously used in ecology to study the response of an ecological system to perturbations to an equilibrium, we analyze the impact of the dispersals/travels between patches and other disease-related parameters on short term dynamics of these spatially structured disease models. This contrasts with most existing works on modelling the dynamics of infectious disease which are only interested in long-term disease dynamics in terms of the basic reproduction number.