Dispersal and periodic travelling waves in ecology
Abstract
The research in this thesis is the culmination of four separate studies on periodic
travelling wave generation in three ecological systems: cyclic predator-prey interactions, intertidal mussel beds and semi-arid vegetation. Patterns in space and time
generated in mathematical models are analysed with the aim of identifying the underlying mechanisms in real-world ecosystems, and determining the impact of ecological change and variation. In particular, pattern formation theory is extended to
include more realistic and justifiable model assumptions about population dispersal.
In Chapter 1, we discuss the general ideas behind pattern formation theory; providing ecological examples and an overview of important mathematical techniques. In
Chapter 2, we derive an equation for the amplitude of periodic travelling waves generated by an ecological invasion in cyclic predator-prey systems when populations
disperse at different rates. In Chapter 3, we demonstrate how both stripe and spotted patterns can arise in intertidal mussel beds as a result of algal dispersal via tidal
flow by performing an analysis in two space dimensions. In Chapter 4, we discover
that the non-local seed dispersal of banded vegetation in semi-arid regions can increase ecosystem resilience to climate change via oscillating peaks of vegetation and
stationary patterns. In Chapter 5, we return to invasive predator-prey systems and
locate the transition from periodic travelling waves to spatio-temporal irregularity
when populations disperse at different rates. In Chapter 6, we provide a summary of
our conclusions.