Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

The strong-migration limit in geographically structured populations

  • 234 Accesses

  • 150 Citations

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.

This is a preview of subscription content, log in to check access.

References

  1. Dempster, E. R.: Maintenance of genetic heterogeneity. Cold Spring Harbor Symp. Quant.Biol.20, 25–32 (1955)

  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)

  3. Feller, W.: An Introduction to Probability Theory and Its Applications, 3rd edition, Vol. I. New York: Wiley, 1968

  4. Franklin, J. N.: Matrix Theory. Englewood Cliffs, N.J.: Prentice-Hall, 1968

  5. Gantmacher, F. R.: The Theory of Matrices, 2 vol. New York: Chelsea, 1959

  6. Karlin, S., Taylor, H. M.: A First Course in Stochastic Processes, 2nd edition. New York:Academic Press, 1975

  7. Kimura, M.: Diffusion models in population genetics. J. Appl. Prob, 1, 177–232 (1964)

  8. Kimura, M., Crow, J. F.: The number of alleles that can be maintained in a finite populationGenetics 49, 725–738 (1964)

  9. Malécot, G.: Les mathématiques de l'hérédité. Paris: Masson 1948. Extended translation Malécot, G.: The Mathematics of Heredity. San Francisco: Freeman, 1969

  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)

  11. Nagylaki, T.: Selection in One and Two-Locus Systems. Berlin: Springer-Verlag, 1977

  12. Nagylaki, T.: Decay of genetic variability in geographically structured populations. Proc. Natl.Acad. Sci. USA 74, 2523–2525 (1977a)

  13. Nagylaki, T.: A diffusion model for geographically structured populations. J. Math. Biol. 6, 375–382 (1978)

  14. Sawyer, S.: Results for the stepping-stone model for migration in population genetics. Ann.Prob. 4, 699–728 (1976)

  15. Wallace, B.: Topics in Population Genetics. New York: Norton 1968

  16. Wright, S.: Evolution in Mendelian populations. Genetics 16, 97–159 (1931)

Download references

Author information

Additional information

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

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Key words

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