Mathematical modelling of population and disease control in patchy environments

Abstract

Natural populations may be managed by humans for a number of reasons, with mathematical modelling playing an increasing role in the planning of such management and control strategies. In an increasingly heterogeneous, or `patchy' landscape, the interactions between distinct groups of individuals must be taken into account to predict meaningful management strategies. Invasive control strategies, involving reduction of populations, such as harvesting or culling have been shown to cause a level of disturbance, or spatial perturbation, to these groups, a factor which is largely ignored in the modelling literature. \\ In this thesis, we present a series of deterministic, differential equation models which are used to investigate the impact of this disturbance in response to control. We address this impact in two scenarios. Firstly, in terms of a harvested population, where extinction must be prevented whilst maximising the yield obtained. Secondly, we address the impact of disturbance in an epidemic model, where the aim of the control strategy is to eradicate an endemic pathogen, or to prevent the invasion of a pathogen into a susceptible population. The movement of individuals between patches is modelled as both a constant rate, and a function which is increasing with population density. Finally, we discuss the `optimal' control strategy in this context. \\ We find that, whilst a population harvested from a coupled system is able to produce an inflated yield, this coupling can also cause the population to be more resistant to higher harvesting efforts, increasing the effort required to drive the population to extinction. Spatial perturbation raises this extinction threshold further still, providing a survival mechanism not only for the individuals that avoid being killed, but for the population as a whole. \\ With regards to the eradication of disease, we show that disturbance may either raise or lower the pathogen exclusion threshold depending on the particular characteristics of the pathogen. In certain cases, we have shown that spatial perturbation may force a population to be susceptible to an infectious invasion where its natural carrying capacity would prevent this.