Applications of stability theory to ecological problems

Abstract

The goal of ecology is to investigate the interactions among organisms and their environment. However, ecological systems often exhibit complex dynamics. The application of mathematics to ecological problems has made tremendous progress over the years and many mathematical methods and tools have been developed for the exploration, whether analytical or numerical, of these dynamics. Mathematicians often study ecological systems by modelling them with partial differential equations (PDEs). Calculating the stability of solutions to these PDE systems is a classical question. This thesis first explores the concept of stability in the context of predator-prey invasions. Many ecological systems exhibit multi-year cycles. In such systems, invasions have a complicated spatiotemporal structure. In particular, it is common for unstable steady states to exist as long-term transients behind the invasion front, a phenomenon known as dynamical stabilisation. We combine absolute stability theory and computation to predict how the width of the stabilised region depends on parameter values. We develop our calculations in the context of a model for a cyclic predator-prey system, in which the invasion front and spatiotemporal oscillations of predators and prey are separated by a region in which the coexistence steady state is dynamically stabilised. Vegetation pattern formation in water-limited environments is another topic where stability theory plays a key role; indeed in mathematical models, these patterns are often the results of the dynamics that arise from perturbations to an unstable homogeneous steady state. Vegetation patterns are widespread in semi-deserts and aerial photographs of arid and semi-arid ecosystems have shown several kilometers square of these patterns. On hillsides in particular, vegetation is organised into banded spatial patterns. We first choose a domain in parameter space and calculate the boundary of the region in parameter space where pattern solutions exist. Finally we conclude with investigating how changes in the mean annual rainfall affect the properties of pattern solutions. Our work also highlights the importance of research on the calculation of the absolute spectrum for non-constant solutions.