Maximum geographic range of a mutant allele considered as a subtype of a Brownian branching random field

Abstract

A rare allelic type is modeled as a field of [animal] individuals diffusing independently in d-dimensional space (d = 1,2,...), in which individuals are replaced by random numbers of offspring at a constant rate. In an infinite-allele model with selectively identical alleles, the offspring distribution would have mean 1 - u, in which u is the mutation rate; otherwise mean (1 - u)w, in which w is the relative fitness of the allele. Let p(x) be the probability that some descendent of an individual initially at zero diffuses unilaterally as far as x(d = 1), or else the probability that some descendent of an individual at x diffuses within a > 0 of the origin (d .gtoreq. 2). A nonlinear differential equation is found for p(x) that is solvable for d = 1 and related to Emden''s equation for d .gtoreq. 2. For p(x) .gtoreq. 10-4 and u .ltoreq. 10-5, genetic drift is more important than mutation in the behavior of p(x)(d = 1). If u = 0 and w = 1, p(x) .apprx. C/x2 as x .fwdarw. .infin. for d .ltoreq. 3. As a mathematical application, if the initial distribution is uniform Poisson, a bounded open set K is visited by individuals in the field at arbitrarily large times if d .gtoreq. 2 but not if d = 1.