Introduction

This book has a dual purpose. The first is to provide an introductory text that collects various discrete stochastic population models. This will furnish the reader with a systematic methodology for the formulation of the models, and it will explore their dynamical properties and the ways of characterising their behaviours in terms of customary measurements that can be made upon them. The second purpose is to then use these models as a tool for generating a series of events, or point processes, that have distinct properties according to which population model is used as a motor. The two qualifiers, discrete and stochastic, simultaneously provide the subject with a richness of phenomenology and technical challenges for formulating, describing, and extracting those behaviours. This is because the population can only change by an integer amount, and those changes are triggered to occur at times not governed by the invariable ticks of a clock. These strictures are absent when adopting a more straightforward continuous and deterministic approach, whereby the population is described as a continuous density and time marches uniformly onward at a regular pace. This distinction between the two approaches also delineates between the source, character and strength of fluctuations. In the stochastic formulation the fluctuations are an intrinsic property of the population itself. The mechanisms causing the changes in population size are essentially non-perturbative in nature because they change the state of the population by values of finite size, and this is true even if the mathematical formulation of a particular mechanism is linear. In the deterministic approach, intrinsic fluctuations can only arise through non-linearity. Both approaches can be affected by the presence of extrinsic noise, but again this is essentially non-perturbative in the stochastic formulation.