Journal of Mathematical Biology

, Volume 9, Issue 2, pp 101–114

# The strong-migration limit in geographically structured populations

• Thomas Nagylaki
Article

## 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 Ne (⩽ NT) appears instead of the actual total population number NT. 1. If there is no selection, every allele mutates at rate u to types not preexisting in the population, and the (finite) subpopulation numbers Ni 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 Ni are proportional to 1/u, as NT→∼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/NT, let Pj be the frequency of the allele Aj in the entire population, averaged with respect to the stationary distribution of the backward migration matrix M. As NT → ∼8, the deviations of the allelic frequencies in each of the subpopulations from Pj converge to zero; the usual panmictic mutation-selection diffusion is obtained for Pj, with the selection intensities averaged with respect to the stationary distribution of M. In both models, Ne= NT and all effects of population subdivision disappear in the limit if, and only if, migration does not alter the subpopulation numbers.

### Key words

Migration Random drift Geographical structure Markov chains Limit theorems

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