The dynamics of ecological invasions and epidemics
The systems of interest in this study are the spread of epidemics and invasions from a small propagule introduced into an arena that was initially devoid of the given species or stage of illness. In reaction-diffusion models, populations are continuous. Populations at low densities have the same growth functions as populations at high densities. In nature, such low densities would signify extinction of a population or of a disease. This property can be removed from reaction-diffusion models by small changes in the formulation so that small populations become extinct. This can be achieved by the use of a threshold density or an Allee effect, so there is negative growth at low densities. Both these alterations were made to the Fisher model, a predator-prey model and a two stage and a three stage epidemic model. A semi-numerical method, termed the Shooting method, was developed to predict the shapes and velocities of these wave fronts. This was found to correctly predict the velocity, the peak density of the invading stage or species and the width of the wave front. It was found that in oscillatory cases of the multi species models, a high threshold can remove the wave train or wake which would normally follow the wave front, so the wave becomes a soliton. The next step is to investigate probable causes of persistence behind the initial wavefront. To do this, discrete time and space versions of the models were formulated so that experiments investigating persistence can be carried out in a two dimensional arena with less computational effort. The formulations were chosen so that at reasonable time and space steps the discrete models show no behaviour different to that of the reaction diffusion model, and so that the Shooting method could also be used to make predictions about these wavefronts. Three mechanisms of persistence are investigated; environmental heterogeneity, long range dispersal and self organised patterns.