Population Dynamics in a stochastic environment

When modelling population dynamics within an ecosystem, there are many factors that need to be considered, these include; birth and death rates, intra-species competition, migration, resources available, environmental conditions and predation. Withso many aspects that need to be considered, many of which are constantly changing, population models quickly become very complicated. Often, in an attempt to reduce complexity, many of these factors are either ignored or overly simplified. One of the most fundamental approaches to population modelling, based on the logistic differential equation, bundles many of these factors as demographic (the intrinsic growth rate) and environmental (through the carrying capacity). Current models treat both the demographic and environmental factors as constants. Whilst this simplifies the modelling process, it is well known that both demographic and environmental factors change with time, which in turn affects the population dynamics. Of particular importance are environmental fluctuations. Recent attempts to model environmental fluctuations as time-dependent variables have demonstrated an increased complexity in population dynamics. However, environmental fluctuations cannot completely be characterised by time-dependent functions alone, the environment is also subjected to random and unpredictable perturbations and should be modelled accordingly. This thesis is an investigation into population dynamics that are the result of random environmental fluctuations. Here, the carrying capacity (the maximum population an environment can sustain) is treated as a proxy for the state of the environment. The environment is allowed to vary according to a Wiener and as an Ornstein-Uhlenbeck process with an appropriate absorbing boundary condition ensuring that the carrying capacity must remain positive for the population to remain viable. For both processes, the exact probability density function for the carrying capacity is found. The statistical properties of the carrying capacity and the population are analysed using the Monte Carlo method giving: the expected time evolution of the population and its variance, the probability distribution of the population and the mean-time to extinction. Finally, future developments are discussed that include, among others, the effectsof environmental stochasticity on the strength of the interaction between competing populations