Diffusion Processes in Population Biology

Abstract

Mathematical models in population biology deal with the dynamics of natural populations, i.e., with temporal changes in the numbers of individuals and with the composition of populations containing several types of individuals. In population genetics theory, one is concerned with individuals of different genetic types. In ecological models several species interact in some way, for instance by competing for the same resource. A third example is a population in which individuals are categorized by age. We are concerned here with stochastic population models, in particular models involving Markov diffusion processes. The random effects may arise in several distinct ways. One is from random fluctuations in an environmental parameter, for instance the population growth rate (Section 2). A second is from chance fluctuations in population numbers, or in the frequencies of different types, as individuals die and new ones are born (Section 3). In models of such phenomena, diffusion processes appear as the limits of certain Markov chains (e.g., multitype branching processes or birth-death processes) after a suitable rescaling. In Section 4 geographically structured populations are considered. In some models of geographically structured populations, the movement of individuals within the habitat where the population lives is also treated as random. In such models this introduces yet another stochastic effect Before proceeding further, two comments should be made. First, while this paper is part of a conference on 'Directions of Mathematical Statistics', its topic belongs to applied probability. Population biology has been a source of inspiration in statistics since the early days of Pearson and Fisher, but we do not treat statistical aspects here. Among recent work one should mention Ewens's sampling theory of selectively neutral alleles [6]. The second comment is that stochastic population models are regarded by some as a frill, those holding this view believing that progress toward understanding natural evolution comes instead by studying deterministic