Birth, death and migration processes

Abstract

SUMMARY The effect of migration between a finite number of colonies each of which undergoes a simple birth and death process is studied. The first two moments are obtained for the general process and deterministic solutions are developed for several special models including the finite linear model proposed by Bailey (1968). In a recent paper Bailey (1968) constructs a model for spatially distributed populations by considering a population to be composed of an infinite number of colonies situated at the integer points of a single co-ordinate axis represented by -xo < i < x. Each colony is assumed to be subject to a simple birth and death process with birth and death rates A and Iu respectively, and with migration rates to each of the two neighbouring colonies. He derives the mean number of individuals in each colony at time t together with the variances and covariances, and briefly considers the corresponding models where the population is extended to two and three dimensions. Clearly by assuming the existence of an infinite number of colonies he avoids the problem of 'edge effects' at the boundaries when the number of colonies is finite. We shall later examine this finite analogue of Bailey's model and show how his results for the infinite model follow as a special case. His paper has recently been extended by several authors. Adke (1969) generalizes Bailey's process to include time-dependent birth and death rates whilst Usher & Williamson (1970) use discrete time intervals to analyze the model when the number of colonies is finite. They consider the population to be split into migrants and nonmigrants, each group having different birth and death rates. Davis (1970) presents some results for a general Markov branching-diffusion process and then applies them to Bailey's model. Crump (1970) studies a general age-dependent branching process in which the population is distributed in N colonies with migration between them and he obtains asymptotic expressions for the first two moments in several special cases. The stochastic equations for the general parameter case of the simple birth-death-migration process are considered by Puri (1968).