Journal of Mathematical Biology

, Volume 9, Issue 2, pp 101–114 | Cite as

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Dempster, E. R.: Maintenance of genetic heterogeneity. Cold Spring Harbor Symp. Quant.Biol.20, 25–32 (1955)Google Scholar
  2. Ethier, S. N., Nagylaki, T.: Diffusion approximations of Markov chains with two time scales and applications to population genetics. Adv. Appl. Prob., in press (1980)Google Scholar
  3. Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edition, Vol. I. New York: Wiley, 1968Google Scholar
  4. Franklin, J. N.: Matrix Theory. Englewood Cliffs, N.J.: Prentice-Hall, 1968Google Scholar
  5. Gantmacher, F. R.: The Theory of Matrices, 2 vol. New York: Chelsea, 1959Google Scholar
  6. Karlin, S., Taylor, H. M.: A First Course in Stochastic Processes, 2nd edition. New York:Academic Press, 1975Google Scholar
  7. Kimura, M.: Diffusion models in population genetics. J. Appl. Prob, 1, 177–232 (1964)Google Scholar
  8. Kimura, M., Crow, J. F.: The number of alleles that can be maintained in a finite populationGenetics 49, 725–738 (1964)Google Scholar
  9. Malécot, G.: Les mathématiques de l'hérédité. Paris: Masson 1948. Extended translation Malécot, G.: The Mathematics of Heredity. San Francisco: Freeman, 1969Google Scholar
  10. Malécot, G.: Un traitement stochastique des problèmes linéaires (mutation, linkage, migration)en Génétique de Population. Ann. Univ. Lyon, Sciences, Sec. A 14, 79–117 (1951)Google Scholar
  11. Nagylaki, T.: Selection in One and Two-Locus Systems. Berlin: Springer-Verlag, 1977Google Scholar
  12. Nagylaki, T.: Decay of genetic variability in geographically structured populations. Proc. Natl.Acad. Sci. USA 74, 2523–2525 (1977a)Google Scholar
  13. Nagylaki, T.: A diffusion model for geographically structured populations. J. Math. Biol. 6, 375–382 (1978)Google Scholar
  14. Sawyer, S.: Results for the stepping-stone model for migration in population genetics. Ann.Prob. 4, 699–728 (1976)Google Scholar
  15. Wallace, B.: Topics in Population Genetics. New York: Norton 1968Google Scholar
  16. Wright, S.: Evolution in Mendelian populations. Genetics 16, 97–159 (1931)Google Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Thomas Nagylaki
    • 1
  1. 1.Department of Biophysics and Theoretical BiologyUniversity of ChicagoChicagoUSA

Personalised recommendations