Spatial subdivision of populations and estimates of genetic variation

Measures of variation in space are strongly affected by correlations between subdivisions used for sampling. Here we consider variation in gene frequencies across populations. Usually the variance of gene frequencies is standardized by dividing it by the mean gene frequency times one minus the mean (FST). Under the model of isolation by distance (usually called the "stepping stone" model), at the stationary state the correlation between the gene frequencies of two populations falls exponentially with the geographic distance between them. Using this model, we derive formulas for variances of blocks of populations of different sizes in one- and two-dimensional space and suggest that the theoretical results may be useful for understanding real observations, some examples of which are presented. We demonstrate how FST increases with the degree of subdivision among populations. We also show the effect of gaps between the sampled populations. Our results are valid, however, for traits other than gene frequencies, as long as their correlation with geographic distance falls exponentially. In the extension to 2-dimensional spaces, we present in closed form the distributions of distances between nodes of a lattice or of two lattices. These distributions might have applications in ecology.