The decay of genetic variability in geographically structured populations

The geographical structure of a population distributed continuously and homogeneously along an infinite linear habitat is explored. The analysis is restricted to a single locus in the absence of selection, and every mutant is assumed to be new to the population. An explicit formula is derived for the probability that two homologous genes separated by a given distance at any time t are the same allele. The ultimate rate of approach to equilibrium is shown to be t(-3/2)e(-2ut), where u is the mutation rate. An approximation is given for the stationary probability of allelism in an infinite two-dimensional population, which, unlike previous expressions, is finite everywhere. For a finite habitat of arbitrary shape and any number of dimensions, it is proved that if the population density is very high, then asymptotically the transient part of the probability of allelism is spatially uniform and decays at the rate e(-[2u+1/(2N)]t), where N is the total population size. Thus, in this respect the population behaves as if it were panmictic. The dependence of the amount of local gene frequency differentiation on population density and habitat size and dimensionality is discussed.