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

Abstract

Let I(x, u) be the probability that two genes found a vector distance x apart are the same type in an infinite-allele selectively-neutral migration model with mutation rate u. The creatures involved inhabit an infinite of colonies, are diploid and are held at N per colony. Set in one dimension and in higher dimensions, where σ² is the covariance matrix of the migration law (which is assumed to have finite fifth moments). Then in one dimension, in two dimensions, and in three dimensions uniformly for Here C ₀ is a constant depending on the migration law, K₀ (y) is the Bessel function of the second kind of order zero, and are the eigenvalues of σ². For symmetric nearest-neighbor migrations, in one dimension and log m _i in two. For is known in one dimension and C ₀ does not appear. In two dimensions, These results extend and make more precise earlier work of Malécot, Weiss and Kimura and Nagylaki.