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The strong-migration limit in geographically structured populations

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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.

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Supported by the National Science Foundation (Grant No. DEB77-21494)

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Nagylaki, T. The strong-migration limit in geographically structured populations. J. Math. Biology 9, 101–114 (1980). https://doi.org/10.1007/BF00275916

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Key words

  • Migration
  • Random drift
  • Geographical structure
  • Markov chains
  • Limit theorems