Covariation of gene frequencies in a stepping-stone lattice of populations

For a one- or two-dimensional lattice of finite length consisting of populations, each of which has the same population size, the classical stepping-stone model has been used to approximate the patterns of variation at neutral loci in geographic regions. In the pioneering papers by Maruyama (1970a, 1970b, 1971) the changes of gene frequency at a locus subject to neutral mutation between two alleles, migration, and random genetic drift were modeled by a vector autoregression model. Maruyama was able to use the spectrum of the migration matrix, but to do this he had to introduce approximations in which there was either extra mutation in the terminal populations, or extra migration from the subterminal population into the terminal population. In this paper a similar vector autoregression model is used, but it proves possible to obtain the eigenvalues and eigenvectors of the migration matrix without those approximations. Approximate formulas for the variances and covariances of gene frequencies in different populations are obtained, and checked by numerical iteration of the exact covariances of the vector autoregression model.