Genetic structure of a population occupying a circular habitat

The geographical structure of a finite population distributed continuously and homogeneously along a circular habit is explored. Selection is supposed to be absent, and the analysis is restricted to a single locus with discrete, non-overlapping generations. Assuming every mutant is new to the population, the rate of decay of genetic variability is obtained, and the probability that two homologous genes separated by a given distance are different alleles is calculated. If moments of the migration function higher than second are neglected, the eigenvalue equation is shown to be a simple trigonometric one, and the Fourier series giving the transient and stationary probabilities of allelism are summed in terms of elementary functions. The proportion of homozygotes, the effective number of alleles maintained in the population, and the amount of local differentiation of gene frequencies are discussed.