Abstract/Summary

This thesis concerns the modelling of partial differential equations in population dynamics and examines two approaches to the choice of meshes for the numerical approximation of the equations. The first part describes a fixed-mesh approach to the predator-prey Lotka-Volterra equations and applies it to a two-dimensional temperature-dependent parasitism system. A climate function is used to incorporate the effect of climate change with the aim to explore the impact of ongoing warming on the spatial distribution and community formation of the interacting species. The second part concerns an adaptive approach to meshes for populations with moving boundaries, internal and external. In particular, a velocity-based moving mesh method based on conservation is applied to systems of competitive species with moving boundaries by which species interact, arise or disappear. The application of the method to epidemic models is also examined where the spreading of the disease through the domain is modelled by a moving front. This is achieved using a moving mesh technique governed by conservation in which the masses of coexisting species are combined. We conclude that fixed mesh methods are adequate for many problems in population dynamics, but where there are significant moving boundaries, as in evolving population domains, the moving mesh method based on conservation is an advantageous choice, as it allows the accurate handling of the moving boundaries and the efficient treatment of topological changes.