PREDATOR-PREY MODELS WITH DISTRIBUTED TIME DELAY

Abstract

Rich dynamics have been demonstrated when a discrete time delay is introduced in a simple predator-prey system. For example, Hopf bifurcations and a sequence of period doubling bifurcations that appear to lead to chaotic dynamics have been observed. In this thesis we consider two different predator-prey models: the classical Gause-type predator-prey model and the chemostat predator-prey model. In both cases, we explore how different ways of modeling the time between the first contact of the predator with the prey and its eventual conversion to predator biomass affects the possible range of dynamics predicted by the models. The models we explore are systems of integro-differential equations with delay kernels from various distributions including the gamma distribution of different orders, the uniform distribution, and the Dirac delta distribution. We study the models using bifurcation theory taking the mean delay as the main bifurcation parameter. We use both an analytical approach and a computational approach using the numerical continuation software XPPAUT and DDE-BIFTOOL. First, general results common to all the models are established. Then, the differences due to the selection of particular delay kernels are considered. In particular, the differences in regions of stability of the coexistence equilibrium are investigated. Finally, the effects on the predicted range of dynamics between the classical Gause-type and the chemostat predator-prey models are compared.