Evolution of competing populations in time-varying environments

Populations evolve subject to the conditions of their environment, which vary in time. Changes in the environment directly impact the evolution of the population, making an understanding of population evolution under environmental variability crucial to uncovering the evolutionary dynamics of natural populations. This thesis investigates the effect of the coupling between ecological dynamics, driven by varying environmental conditions, and population evolution. In particular, it considers the combined effect of demographic fluctuations (randomness caused by stochastic births and deaths in a finite population) and environmental variability on four models of competition in two-species microbial populations. These models are inspired by real-world issues such as antimicrobial resistance evolution and the establishment of unwanted mutants in healthy populations. In a constant environment, the behaviour in each model is understood and, in the motivating contexts considered, often leads to undesirable evolutionary outcomes. However, these dynamics change dramatically upon the introduction of environmental variability. The majority of this thesis focuses on environmental variability modelled by a dichotomous Markov noise controlling the carrying capacity of the population (the number of individuals it can typically support). This drives the population size of the community and thus directly impacts the strength of demographic fluctuations, providing a coupling between ecological and evolutionary dynamics. The case where environmental variability impacts both the reproductive capabilities of species and the carrying capacity is also considered. Due to the eco-evolutionary coupling, driven by environmental changes, interesting novel phenomena arise at the evolutionary level. In particular, in each model a preferred evolutionary outcome is motivated, and it is shown how appropriate conditions on environmental variability can promote those outcomes. These behaviours are investigated through extensive stochastic simulations and the development of analytical techniques.