Mathematical studies of conservation and extinction in inhomogeneous environments

Abstract

A fragmented ecosystem contains communities of organisms that live in fragmented habitats. Understanding the way biological processes such as reproduction and dispersal over the fragmented habitats take place constitutes a major challenge in spatial ecology. In this thesis we discuss a number of mathematical models of density-dependent populations in inhomogeneous environments presenting growth, decay and diffusion amongst woodland patches of variable potential for reproductive success. These models include one- and two-dimensional analyses of single population systems in fragmented environments. We investigate and compute effective properties for single patch systems in one dimension, linking ecological features with landscape structure and size. A mathematical analysis of potential impacts on spread rates due to the behaviour of individuals in the population is then developed. For the analysis of the population dispersal between areas of plentiful resources and areas of scarce resources, we introduce a novel development that models individuals hazard sensitivity when outside plentiful regions. This sensitivity is modelled by introducing a term called endrotaxis that generates a dispersal gradient, resulting in realistically low migration between regions of plentiful resources. Numerical methods and semi-analytic results yield maximum patch separations for one and two dimensional systems and show that the velocity of spread depends on inter-patch distances and patch geometries. By introducing Allee effects (i.e., inverse density-dependent responses to the difficulty of finding mates at low density) over the population growth function, we find that dispersal is slowed down when combined with hazard sensitivity. In the final Chapter we sumarise the results of the previous chapters, concluding that the work performed in this thesis complements and enriches the current mathematical models of movement behaviour