Nagylaki, T. J. Math. Biology (1980) 9: 101. doi:10.1007/BF00275916

Summary

Some strong-migration limits are established for geographically structured populations. A diploid monoecious population is subdivided into a finite number of colonies, which exchange migrants. The migration pattern is fixed and ergodic, but otherwise arbitrary. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus. In all the limiting results, an effective population number N_{e} (⩽ N_{T}) appears instead of the actual total population number N_{T}. 1. If there is no selection, every allele mutates at rate u to types not preexisting in the population, and the (finite) subpopulation numbers N_{i} are very large, then the ultimate rate and pattern of convergence of the probabilities of allelic identity are approximately the same as for panmixia. If, in addition, the N_{i} are proportional to 1/u, as N_{T}→∼8, the equilibrium probabilities of identity converge to the panmictic value. 2. With a finite number of alleles, any mutation pattern, an arbitrary selection scheme for each colony, and the mutation rates and selection coefficients proportional to 1/N_{T}, let P_{j} be the frequency of the allele A_{j} in the entire population, averaged with respect to the stationary distribution of the backward migration matrix M. As N_{T} → ∼8, the deviations of the allelic frequencies in each of the subpopulations from P_{j} converge to zero; the usual panmictic mutation-selection diffusion is obtained for P_{j}, with the selection intensities averaged with respect to the stationary distribution of M. In both models, N_{e}= N_{T} and all effects of population subdivision disappear in the limit if, and only if, migration does not alter the subpopulation numbers.