Journal of Mathematical Biology

, Volume 29, Issue 1, pp 59–75 | Cite as

The coalescent and the genealogical process in geographically structured population

  • M. Notohara
Article

Abstract

We shall extend Kingman's coalescent to the geographically structured population model with migration among colonies. It is described by a continuous-time Markov chain, which is proved to be a dual process of the diffusion process of stepping-stone model. We shall derive a system of equations for the spatial distribution of a common ancestor of sampled genes from colonies and the mean time to getting to one common ancestor. These equations are solved in three particular models; a two-population model, the island model and the one-dimensional stepping-stone model with symmetric nearest-neighbour migration.

Key words

Coalescent Genealogical process Geographical structure Migration Markov chain 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cox, J. T., Griffeath, D.: Diffusive clustering in the two dimensional voter model. Ann. Probab. 14, 347–370 (1986)Google Scholar
  2. 2.
    Darden, T., Kaplan, N. L., Hudson, R. R.: A numerical method for calculating moments of coalescent times in finite populations with selection. J. Math. Biol. 27, 355–368 (1989)Google Scholar
  3. 3.
    Donnelly, P.: The transient behaviour of the Moran model in population genetics. Math. Proc. Camb. Phil. Soc. 95, 349–358 (1984)Google Scholar
  4. 4.
    Donnelly, P.: Dual processes and an invariance result for exchangeable models in population genetics. J. Math. Biol. 23, 103–118 (1985)Google Scholar
  5. 5.
    Donnelly, P., Tavare, S.: The population genealogy of the infinitely-many neutral alleles model. J. Math. Biol. 25, 381–391 (1987)Google Scholar
  6. 6.
    Ethier, S. N., Griffiths, R. C.: The infinitely-many-sites model as a measure-valued diffusion. Ann. Probab. 15, 515–545 (1987)Google Scholar
  7. 7.
    Gladstien, K.: The characteristic values and vectors for a class of stochastic matrices arising in genetics. SIAM. J. Appl. Math. 34, 630–642 (1978)Google Scholar
  8. 8.
    Griffeath, D.: Additive and cancellative interacting particle system. (Lect. Notes Math., vol. 724) Berlin Heidelberg New York: Springer 1979Google Scholar
  9. 9.
    Griffiths, R. C.: Lines of descent in the diffusion approximation of neutral Wright-Fisher models. Theor. Popul. Biol. 17, 37–55 (1980)Google Scholar
  10. 10.
    Griffiths, R. C.: The number of alleles and segregating sites in a sample from the infinite alleles model. Adv. Appl. Probab. 14, 225–239 (1982)Google Scholar
  11. 11.
    Griffiths, R. C.: Counting genealogical trees. J. Math. Biol. 25, 423–431 (1987)Google Scholar
  12. 12.
    Hudson, R. R., Kaplan, N. L.: The coalescent process in models with selection and recombination. Genetics 120, 831–840 (1988)Google Scholar
  13. 13.
    Kaplan, N. L., Darden, T., Hudson, R. R.: The coalescent process in models with selection. Genetics 120, 819–829 (1988)Google Scholar
  14. 14.
    Karlin, S., Taylor, H. M.: A second course in stochastic processes. New York: Academic Press 1980Google Scholar
  15. 15.
    Kingman, J. F. C.: On the genealogy of large populations. J. Appl. Probab. 19A, 27–43 (1982)Google Scholar
  16. 16.
    Kingman, J. F. C.: The coalescent. Stochastic Processes Appl. 13, 235–248 (1982)Google Scholar
  17. 17.
    Liggett, T. M.: Interacting particle system. Berlin Heidelberg New York: Springer 1985Google Scholar
  18. 18.
    Nagylaki, T.: The strong-migration limit in geographically structured populations. J. Math. Biol. 9, 101–114 (1980)Google Scholar
  19. 19.
    Nagylaki, T.: The robustness of neutral models of geographical variation. Theor. Popul. Biol. 24, 268–294 (1983)Google Scholar
  20. 20.
    Nagylaki, T.: Neutral models of geographical variation stochastic spatial process. (Lest. Notes Math., vol. 1212, pp. 216–237) Berlin Heidelberg New York: Springer 1986Google Scholar
  21. 21.
    Notohara, M., Shiga, T.: Convergence to genetically uniform state in stepping stone models of population genetics. J. Math. Biol. 10, 281–294 (1980)Google Scholar
  22. 22.
    Padmadisastra, S.: The genetic divergence of three populations. Theor. Popul. Biol. 32, 347–365 (1987)Google Scholar
  23. 23.
    Sawyer, S.: Results for the stepping stone model for migration in population genetics. Ann. Probab. 4, 699–728 (1976)Google Scholar
  24. 24.
    Shiga, T.: An interacting system in population genetics I, and II. J. Math. Kyoto Univ. 20, 212–242 and 723–733 (1980)Google Scholar
  25. 25.
    Shiga, T.: Continuous time multi-allelic stepping stone models in population genetics. J. Math. Kyoto Univ. 22, 1–40 (1982)Google Scholar
  26. 26.
    Takahata, N.: The coalescent in two partially isolated diffusion populations. Genet. Res., Camb. 52, 213–222 (1988)Google Scholar
  27. 27.
    Takahata, N.: Gene genealogy in three related populations: consistency probability between gene and population trees. Genetics 122, 957–966 (1989)Google Scholar
  28. 28.
    Tavaré, S.: Line of descent and genealogical process and their application in population genetics model. Theor. Popul. Biol. 26, 119–164 (1984)Google Scholar
  29. 29.
    Tavaré, S.: The birth process with immigration and the genealogical structure of large populations. J. Math. Biol. 25, 161–168 (1987)Google Scholar
  30. 30.
    Watterson, G. A.: Lines of descent and the coalescent. Theor. Popul. Biol. 26, 77–92 (1984)Google Scholar
  31. 31.
    Watterson, G. A.: The genetic divergence of two populations. Theor. Popul. Biol. 27, 298–317 (1985)Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • M. Notohara
    • 1
  1. 1.Department of Biology, Faculty of ScienceKyushu UniversityFukuokaJapan

Personalised recommendations