ABSTRACT

In a discussion of double stochastic population processes in continuous time, attention is concentrated on transition matrices, or equivalent operators, which are linear in the variable parameters. Difficulties with the extreme case of ‘white noise’ variability for the parameters are recalled by reference to ‘stochasticized’ deterministic models, but discussed here also in relation to a general ‘switching’ model.

The use of the ‘backward’ equations for determining extinction probabilities is illustrated by deriving various formulae for the (infinite) birth-and-death process with ‘white-noise’ variability for the birth-and-death coefficients.

INTRODUCTION

It seems very apt to discuss doubly stochastic (d.s.) population processes in this volume, as it recalls the mutual interest of David Kendall and myself in population processes many years ago (e.g. Bartlett, 1947, 1949; Kendall, 1948, 1949; Bartlett and Kendall, 1951). In more recent years d.s. population processes have been considered in discrete time in connection with extinction probabilities for random environments (see, for example, references in Bartlett, 1978, 2.31) and with genetic problems (e.g. Gillespie, 1974); but the concept of d.s processes in continuous time has also become of obvious interest to mathematicians as well as to biologists (e.g. Kaplan, 1973; Keiding, 1975). I might note that my own interest in d.s. processes in continuous time first arose during a visit to Australia in early 1980, when I began to notice in the literature the use of such processes as approximations to genetic or other biological problems for discrete generations.