Random genetic drift in a cline

Abstract

A diffusion model is derived for random genetic drift in a cline. The monoecious organism occupies an unbounded linear habitat with constant, uniform population density. Migration is homogeneous, symmetric, and independent of genotype. The analysis applies to a single diallelic locus without dominance or mutation. Assuming that Hardy-Weinberg proportions obtain locally, partial differential equations are deduced for the expected gene frequency at each location and the covariance between gene frequencies at any two points of the habitat. For most selection gradients, the population density, rho, selection intensity, s, and migration variance, sigma(2), appear only combined in the single dimensionless parameter, beta, the ratio of the characteristic length for migration and random drift to the natural distance for migration and selection. Random drift is highly significant if beta << 1; it causes only relatively small variations of order 1/beta around the deterministic cline if beta >> 1. With beta >> 1, the correlation between the gene frequencies at any two points is very nearly independent of beta, and hence is parameter-free for particular forms of the selection gradient. For a very steep selection gradient, beta = 2 [unk] 2srhosigma, i.e., essentially the product of the square root of the selection intensity and the neighborhood size.