Electronic Thesis and Dissertation Repository

Thesis Format

Integrated Article

Degree

Doctor of Philosophy

Program

Applied Mathematics

Supervisor

Zou, Xingfu

Abstract

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.

Summary for Lay Audience

Population dynamics is an important subject that has wide applications in areas such as ecology, microbiology, epidemiology, virology, and immunology. There are millions of species in the real world, some of them interacting with each other. Among all types of interactions between species, the predator-prey type is most interesting and complicated. Moreover and importantly, the transmission mechanism of infectious diseases is also of this type, adding more weight to its significance. On the other hand, many species including ourselves are mobile. It has been widely agreed that spatial dispersion is one of the main factors responsible for biodiversity. As for the spread of disease, dispersals/travels of infected individuals play a key role. This thesis aims to address some issues on population dynamics with the above-mentioned two main features: predator-prey type interaction and dispersal in spatially heterogeneous environments. For the latter, we only deal with discrete spatial variation, meaning that we use patch models. We start from the model of one species to study the impact of costs associated with dispersal. In predator-prey systems, we consider some indirect effects in contrast to predation, including the impact on the dispersal of prey. Finally, we investigate the transmission dynamics of infectious diseases over a patchy environment.

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