Stochasticity and randomness in eco-evolutionary modelling

Evolution is an inherently stochastic process comprised of many randomly occurring events. The evolutionary fate of a population depends largely on its underlying ecological interactions, while ecological interactions can also be influenced by evolutionary change in turn. This phenomenon is known as an eco-evolutionary feedback loop. Ecological models tend to have a strong focus on how complex ecological features such as interaction structures influence the behaviour of ecosystems, rather than their consequences on long-term evolutionary fates. Evolutionary models on the other hand tend to overlook the complexities associated with their underlying ecological features. This is of particular importance since many evolutionary problems in nature, particularly those associated to the evolution of sexual reproduction, are underpinned by myriad ecological factors. The aim of this thesis is to develop mathematical models for ecological and evolutionary problems in biology, with particular focus on problems surrounding the evolution of sexual reproduction. We begin in chapter 2 by developing an analytical prediction for the stability of generalised Lotka-Volterra systems with biologically motivated interaction structures. In chapter 3, we develop an eco-evolutionary model for the evolution of gamete size and motility to study the evolution of male and female sexes. Chapter 4 repurposes the model of chapter 3 to look at how binary cell fusion can evolve in response to environmental stress. Chapter 5 investigates how genetic recombination evolves in response to environmental stress using an integrative mathematical model that incorporates aspects of population dynamics, population genetics and eco-evolutionary feedback. This allows us to explain analytically how recombination and hibernation evolved to occur together, as well as why they both occur shortly before the onset of environmental stress.