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.