Outbreak and Extinction Dynamics in Stochastic Populations

Each piece of work herein examines nonlinear population dynamics using methods from deterministic dynamical systems, stochastic processes and statistical mechanics. This dissertation is the compilation of three independent – but related – pieces of work: the first investigates an isolated population that is capable of maintaining multiple carrying capacities; the second project examines a stochastic Ebola model with a zoonotic disease reservoir; and the third looks at a basic disease-invasion model to characterize outbreak vulnerability and the connectedness of supposedly separate populations. Each of these three chapters explore the interplay between interconnected systems, without explicitly modeling the elements that are external to the system of interest. The goal is to take a very large and complex lattice of interconnected biological systems and isolate the necessary components, so that modeling is both practical and utilitarian. These works are done in either an ecological or an epidemiological context, but the results in each chapter can be broadly applied to outbreak, invasion, extinction, and connectedness in stochastic population modeling. i