The influence of partial panmixia on neutral models of spatial variation

Partial panmixia can be regarded as the limiting case of long-distance migration. The effect of incorporating partial panmixia into neutral models of geographical variation is investigated. The monoecious, diploid population is subdivided into randomly mating colonies that exchange gametes independently of genotype. The gametes fuse wholly at random, including self-fertilization. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus; every allele mutates to new alleles at the same rate. Introducing some panmixia intensifies sufficiently weak migration. A general formula is derived for the migration effective population number, N(e), and N(e) is evaluated explicitly in a number of models with nonconservative migration. Usually, N(e) increases as the panmictic rate, b, increases; in particular, this result holds for two demes, and generically if the underlying migration is either sufficiently weak or panmixia is sufficiently strong. However, in an analytic model, there exists an open set of parameters for which N(e) decreases as b increases. Migration is conservative in the island and circular-habitat models, which are studied in detail. In the former, including some panmixia simply alters the underlying migration rate, increasing (decreasing) it if it is less (greater) than the panmictic value. For the circular habitat, the probability of identity in allelic state at equilibrium is calculated in a nonlocal, continuous-space, continuous-time approximation. In both models, by an efficient, general method, the expected homozygosity, effective number of alleles, and differentiation of gene frequencies are evaluated and discussed; their monotonicity properties with respect to all the parameters are determined; and in the model of infinitely many sites, the mean coalescence times and nucleotide diversities are studied similarly. For the probability of identity at equilibrium in the unbounded stepping-stone model in arbitrarily many dimensions, introducing some panmixia merely replaces the mutation rate by a larger parameter. If the average probability of identity is initially zero, as for identity by descent, then the same conclusion holds for all time.