External noise in stochastic population dynamics: the case of binary environmental switching

Understanding the factors that affect species survival and coexistence is a problem of significant importance. Species survival and reproduction can be affected by intrinsic (i.e. the composition of the population) and environmental factors (e.g. light, moisture, heat), and analysing their combined effect is a difficult task. This thesis will investigate the combined effect of internal (demographic) noise, caused by random birth-death events in a finite population, and external (environmental) noise on two models of microbial competition. These models are inspired by the well known Prisoner's Dilemma and Rock-Paper-Scissors games of Evolutionary Game Theory, and their dynamics in a static environment is well known. In well mixed (i.e. non-spatial), finite populations without mutation between species, internal noise ultimately leads to the death of all but one species. The strength of these demographic fluctuations is dependent on the population size, hence the probability that a certain species takes over (fixates) the population depends on the structure of the game and the size of the population. The majority of this thesis will focus on the case where external noise is modelled as a randomly switching carrying capacity, following a dichotomous Markov process to mimic periods of abundant and scarce resources. This results in a fluctuating population size, coupling the demographic noise to the environmental noise, leading to interesting, complex effects on the population dynamics. The effects of this coupling within these models is analysed using numerical and analytical techniques, and in general it is found that external noise promotes the fixation of the species that is the least likely to fixate in a static environment, but does not prolong species coexistence.