Stochastic Models for Structured Populations

© 2018 Elsevier B.V. This chapter deals with stochastic models for structured populations whose dynamics depend crucially on individual characteristics such as age, size, or location. We deal with linear stochastic models, and their analysis is also essential to the analysis of nonlinear stochastic models, particularly their boundary behavior which determines invasion, extinction, and persistence. The models discussed here operate in discrete time, and describe individuals who can be in one of the many finite stages. The analyses described here also apply to other discrete-time models, known as integral population models, under suitable conditions. These models are naturally framed in terms of products of random matrices, and so are connected to models used in a number of other areas of science. We explain the assumptions and structure of the models, how they are constructed, and how they may be analyzed. We focus on two kinds of random environments, those that are independently identically distributed (IID), and finite-state Markov chains, but many of the results also apply to other environmental processes. We give examples to illustrate the theory. We also discuss the computation of derivatives of the stochastic growth rate for our models. Code (in R) to compute these derivatives will soon appear as a package; until then, the code is available from the authors.