# The strong-migration limit in geographically structured populations

## Authors

- Revised:

DOI: 10.1007/BF00275916

- Cite this article as:
- Nagylaki, T. J. Math. Biology (1980) 9: 101. doi:10.1007/BF00275916

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## 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.