Neutral models of geographical variation

  • Thomas Nagylaki
Workshop Contributions
Part of the Lecture Notes in Mathematics book series (LNM, volume 1212)


The amount and pattern of genetic variability in a geographically structured population under the joint action of migration, mutation, and random genetic drift is studied. The monoecious, diploid population is subdivided into panmictic colonies that exchange gametes. In each deme, the rate of self-fertilization is equal to the reciprocal of the number of individuals in that deme. Generations are discrete and nonoverlapping; the analysis is restricted to a single locus in the absence of selection; every allele mutates to new alleles at the same rate. It is shown that if the population is at equilibrium, the number of demes is finite, and migration does not alter the deme sizes, then population subdivision produces interdeme differentiation and the mean homozygosity and the effective number of alleles exceed their panmictic values. The equilibrium and transient states of the island, circular stepping-stone, and infinite linear stepping-stone models are investigated in detail.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Apostol, T. M. 1974. Mathematical Analysis, 2nd edition. Addison-Wesley, Reading, Mass.MATHGoogle Scholar
  2. Crow, J. F., and Maruyama, T. 1971. The number of neutral alleles maintained in a finite, geographically structured population. Theor. Pop. Biol. 2, 437–453.CrossRefGoogle Scholar
  3. Erdélyi, A. 1954. Tables of Integral Transforms, Vol. I. McGraw-Hill, New York.MATHGoogle Scholar
  4. Feller, W. 1968. An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edition. Wiley, New York.MATHGoogle Scholar
  5. Feller, W. 1971. An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edition. Wiley, New York.MATHGoogle Scholar
  6. Felsenstein, J. 1975. A pain in the torus: some difficulties with models of isolation by distance. Am. Nat. 109, 359–368.CrossRefGoogle Scholar
  7. Fleming, W. H., and Su, C.-H. 1974. Some one-dimensional migration models in population genetics theory. Theor. Pop. Biol. 5, 431–449.CrossRefMATHGoogle Scholar
  8. Kimura, M. 1963. A probability method for treating inbreeding systems, especially with linked genes. Biometrics 19, 1–17.CrossRefMATHGoogle Scholar
  9. Kimura, M., and Crow, J. F. 1964. The number of alleles that can be maintained in a finite population. Genetics 49, 725–738.Google Scholar
  10. Kimura, M., and Maruyama, T. 1971. Pattern of neutral polymorphism in a geographically structured population. Genet. Res. 18, 125–131.CrossRefGoogle Scholar
  11. Kingman, J. F. C. 1977. Remarks on the spatial distribution of a reproducing population. J. Appl. Prob. 14, 577–583.MathSciNetCrossRefMATHGoogle Scholar
  12. Latter, B. D. H. 1973. The island model of population differentiation: A general solution. Genetics 73, 147–157.MathSciNetGoogle Scholar
  13. Li, W.-H. 1976. Effect of migration on genetic distance. Am. Nat. 110, 841–847.CrossRefGoogle Scholar
  14. Malécot, G. 1946. La consanguinité dans une population limitée. Comp. Rend. Acad. Sci. 222, 841–843.Google Scholar
  15. Malécot, G. 1948. Les mathématiques de l'hérédité. Masson, Paris. (Extended translation: The Mathematics of Heredity. Freeman, San Francisco, 1969.)MATHGoogle Scholar
  16. Malécot, G. 1950. Quelques schémas probabilistes sur la variabilité des populations naturelles. Ann. Univ. Lyon, Sci., Sect. A, 13, 37–60.MATHGoogle Scholar
  17. Malécot, G. 1951. Un traitement stochastiques des problèmes linéaires (mutation, linkage, migration) en Génétique de Population. Ann. Univ. Lyon, Sci., Sect. A, 14, 79–117.MATHGoogle Scholar
  18. Malécot, G. 1965. Évolution continue des fréquences d'un gène mendélien (dans le cas de migration homogène entre groupes d'effectif fini constant). Ann. Inst. H. Poincaré, Sect. B, 2, 137–150.MATHGoogle Scholar
  19. Malécot, G. 1967. Identical loci and relationship. Proc. Fifth Berk. Symp. Math. Stat. Prob. 4, 317–332.MATHGoogle Scholar
  20. Malécot, G. 1975. Heterozygosity and relationship in regularly subdivided populations. Theor. Pop. Biol. 8, 212–241.MathSciNetCrossRefGoogle Scholar
  21. Maruyama, T. 1970. Effective number of alleles in a subdivided population. Theor. Pop. Biol. 1, 273–306.MathSciNetCrossRefMATHGoogle Scholar
  22. Maruyama, T. 1971. The rate of decrease of heterozygosity in a population occupying a circular or linear habitat. Genetics 67, 437–454.MathSciNetGoogle Scholar
  23. Maynard Smith, J. 1970. Population size, polymorphism, and the rate of non-Darwinian evolution. Am. Nat. 104, 231–237.CrossRefGoogle Scholar
  24. Moran, P. A. P. 1959. The theory of some genetical effects of population subdivision. Aust. J. Biol. Sci. 12, 109–116.CrossRefMATHGoogle Scholar
  25. Nagylaki, T. 1974a. Genetic structure of a population occupying a circular habitat. Genetics 78, 777–790.MathSciNetGoogle Scholar
  26. Nagylaki, T. 1974b. The decay of genetic variability in geographically structured populations. Proc. Natl. Acad. Sci. USA 71, 2932–2936.MathSciNetCrossRefMATHGoogle Scholar
  27. Nagylaki, T. 1976. The decay of genetic variability in geographically structured populations. II. Theor. Pop. Biol. 10, 70–82.MathSciNetCrossRefMATHGoogle Scholar
  28. Nagylaki, T. 1977. Selection in One-and Two-Locus Systems. Springer, Berlin.CrossRefMATHGoogle Scholar
  29. Nagylaki, T. 1978a. The geographical structure of populations. In Studies in Mathematics. Vol. 16: Studies in Mathematical Biology. Part II (S. A. Levin, ed.). Pp. 588–624. The Mathematical Association of America, Washington.Google Scholar
  30. Nagylaki, T. 1978b. A diffusion model for geographically structured populations. J. Math. Biol. 6, 375–382.MathSciNetCrossRefMATHGoogle Scholar
  31. Nagylaki, T. 1980. The strong-migration limit in geographically structured populations. J. Math. Biol. 9, 101–114.MathSciNetCrossRefMATHGoogle Scholar
  32. Nagylaki, T. 1982. Geographical invariance in population genetics. J. Theor. Biol. 99, 159–172.MathSciNetCrossRefGoogle Scholar
  33. Nagylaki, T. 1983. The robustness of neutral models of geographical variation. Theor. Pop. Biol. 24, 268–294.CrossRefMATHGoogle Scholar
  34. Nagylaki, T. 1984. Some mathematical problems in population genetics. In Proc. Symp. Appl. Math. Vol. 30: Population Biology (S. A. Levin, ed.). Pp. 19–36. American Mathematical Society, Providence, R. I.Google Scholar
  35. Nei, M. 1975. Molecular Population Genetics and Evolution. North-Holland, Amsterdam.Google Scholar
  36. Protter, M. H., and Weinberger, H. F. 1967. Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs, N. J.MATHGoogle Scholar
  37. Sawyer, S. 1976. Results for the stepping-stone model for migration in population genetics. Ann. Prob. 4, 699–728.MathSciNetCrossRefMATHGoogle Scholar
  38. Sawyer, S. 1977. Asymptotic properties of the equilibrium probability of identity in a geographically structured population. Adv. Appl. Prob. 9, 268–282.MathSciNetCrossRefMATHGoogle Scholar
  39. Sawyer, S., and Felsenstein, J. 1981. A continuous migration model with stable demography. J. Math. Biol. 11, 193–205.MathSciNetCrossRefMATHGoogle Scholar
  40. Sudbury, A. 1977. Clumping effects in models of isolation by distance. J. Appl. Prob. 14, 391–395.MathSciNetCrossRefMATHGoogle Scholar
  41. Weiss, G.H., and Kimura, M. 1965. A mathematical analysis of the stepping-stone model of genetic correlation. J. Appl. Prob. 2, 129–149.MathSciNetCrossRefMATHGoogle Scholar
  42. Wright, S. 1931. Evolution in Mendelian populations. Genetics 16, 97–159.Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • Thomas Nagylaki
    • 1
  1. 1.Department of Molecular Genetics and Cell BiologyThe University of ChicagoChicagoUSA

Personalised recommendations