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.