Asymptotic properties of the equilibrium probability of identity in a geographically structured population

Abstract

LetI(x, u) be the probability that two genes found a vector distancexapart are the same type in an infinite-allele selectively-neutral migration model with mutation rateu.The creatures involved inhabit an infinite of colonies, are diploid and are held atNper colony. Setin one dimension andin higher dimensions, where σ²is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Thenin one dimension,in two dimensions, andin three dimensions uniformly forHereC₀is a constant depending on the migration law,K₀(y) is the Bessel function of the second kind of order zero, andare the eigenvalues of σ². For symmetric nearest-neighbor migrations,in one dimension andlogm_iin two. Foris known in one dimension andC₀does not appear. In two dimensions,These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.