Journal of Mathematical Biology

, Volume 6, Issue 4, pp 375–382 | Cite as

A diffusion model for geographically structured populations

  • Thomas Nagylaki


A diffusion model is derived for the evolution of a diploid monoecious population under the influence of migration, mutation, selection, and random genetic drift. The population occupies an unbounded linear habitat; migration is independent of genotype, symmetric, and homogeneous. The treatment is restricted to a single diallelic locus without dominance. With the customary diffusion hypotheses for migration and the assumption that the mutation rates, selection coefficient, variance of the migrational displacement, and reciprocal of the population density are all small and of the same order of magnitude, a boundary value problem is deduced for the mean gene frequency and the covariance between the gene frequencies at any two points in the habitat.


Diffusion Model Gene Frequency Diffusion Approximation Migration Model Selection Coefficient 
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  1. Feller, W.: An Introduction to Probability Theory and Its Applications. Vol. II, 2nd edition. New York: Wiley 1971zbMATHGoogle Scholar
  2. Felsenstein, J.: A pain in the torus: some difficulties with models of isolation by distance. Am. Nat.109, 359–368 (1975).CrossRefGoogle Scholar
  3. Fife, P. C., McLeod, J. B.: The approach of solutions of nonlinear diffusion equations to travelling front solutions. Arch. Rat. Mech. Anal.65, 335–361 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  4. Fleming, W. H.: A selection-migration model in population genetics. J. Math. Biol.2, 191–207 (1975)MathSciNetCrossRefGoogle Scholar
  5. Fleming, W. H., Su, C.-H.: Some one-dimensional migration models in population genetics theory. Theor. Pop. Biol.5, 431–449 (1974)zbMATHCrossRefGoogle Scholar
  6. Luskin, M., Nagylaki, T.: Numerical analysis of weak random drift in a cline. In press (1978)Google Scholar
  7. Malécot, G.: Les mathématiques de l'hérédité. Paris: Masson 1948. Extended translation: The mathematics of Heredity. San Francisco: Freeman 1969zbMATHGoogle Scholar
  8. Maruyama, T.: The rate of decrease of heterozygosity in a population occupying a circular or a linear habitat. Genetics67, 437–454 (1971)MathSciNetGoogle Scholar
  9. Nagylaki, T.: Genetic structure of a population occupying a circular habitat. Genetics78, 777–790 (1974)MathSciNetGoogle Scholar
  10. Nagylaki, T.: The decay of genetic variability in geographically structured populations Proc. Natl. Acad. Sci. USA71, 2932–2936 (1974a)zbMATHMathSciNetCrossRefGoogle Scholar
  11. Nagylaki, T.: Conditions for the existence of clines. Genetics80, 595–615 (1975)Google Scholar
  12. Nagylaki, T.: The evolution of one- and two-locus systems. Genetics83, 583–600 (1976)MathSciNetGoogle Scholar
  13. Nagylaki, T.: Selection in One- and Two-Locus Systems. Berlin: Springer-Verlag 1977zbMATHGoogle Scholar
  14. Nagylaki, T.: Decay of genetic variability in geographically structured populations. Proc. Natl. Acad. Sci. USA74, 2523–2525 (1977a)zbMATHCrossRefGoogle Scholar
  15. Nagylaki, T.: The geographical structure of populations. In: Studies in Mathematical Biology (S. Levin, Ed.). Washington, D.C.: Mathematical Association of America. In press (1977b)Google Scholar
  16. Nagylaki, T.: The evolution of one- and two-locus systems. II. Genetics85, 347–354 (1977c)MathSciNetGoogle Scholar
  17. Nagylaki, T.: Clines with asymmetric migration. Genetics88, 813–827 (1978)Google Scholar
  18. Nagylaki, T.: Random genetic drift in a cline. Proc. Natl. Acad. Sci. USA75, 423–426 (1978a)zbMATHCrossRefGoogle Scholar
  19. Protter, M. H., Weinberger, H. F.: Maximum Principles in Differential Equations. Englewood Cliffs, N.J.: Prentice-Hall 1967Google Scholar
  20. Sawyer, S.: Asymptotic properties of the equilibrium probability of identity in a geographically structured population. Adv. Appl. Prob.9, 268–282 (1977)zbMATHMathSciNetCrossRefGoogle Scholar
  21. Sawyer, S.: Rates of consolidation in a selectively neutral migration model. Ann. Prob.5, 486–493 (1977a)zbMATHMathSciNetGoogle Scholar
  22. Sawyer, S.: A limit theorem for patch sizes in a selectively neutral migration model. J. Appl. Prob. In press (1978)Google Scholar
  23. Sawyer, S.: A. continuous migration model with stable demography. In press (1978a)Google Scholar
  24. Sawyer, S.: Results for inhomogeneous selection-migration models. In press (1978b)Google Scholar
  25. Sawyer, S.: Felsenstein, J.: A continuous migration model with stable demography. In press (1978)Google Scholar
  26. Wright, S.: Evolution in Mendelian populations. Genetics16, 97–159 (1931)Google Scholar
  27. Wright, S.: Evolution and the Genetics of Populations. Vol. III. Chicago: The University of Chicago Press 1977Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • Thomas Nagylaki
    • 1
  1. 1.Department of Biophysics and Theoretical BiologyThe University of ChicagoChicagoUSA

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