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Separable models for age-structured population genetics


This paper is concerned with the applications of nonlinear age-dependent dynamics to population genetics. Age-structured models are formulated for a single autosomal locus with an arbitrary number of alleles. The following cases are considered: a) haploid populations with selection and mutation; b) monoecious diploid populations with or without mutation reproducing by self-fertilization or by two types of random mating. The diploid models do not deal with selection. For these cases the genic and genotypic frequencies evolve towards time-persistent forms, whether the total population size tends towards exponential growth or not.

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  1. 1.

    Bellman, R., Cooke K. L.: Differential-difference equations, pp. 216–256. New York: Academic Press 1963

  2. 2.

    Busenberg, S., Iannelli, M: A class of nonlinear diffusion problems in age-dependent population dynamics. Nonlinear Analysis T.M.A. 7, 501–529 (1983a)

  3. 3.

    Busenberg, S., Iannelli, M.: A degenerate nonlinear diffusion problem in age-structured population dynamics. Nonlinear Analysis T.M.A. 7, 1411–1429 (1983b)

  4. 4.

    Busenberg, S., Iannelli, M.: Separable models in age-dependent population dynamics. J. Math. Biol. 22, 145–173 (1985)

  5. 5.

    Charlesworth, B.: Selection in populations with overlapping generations. I. The use of Malthusian parameters in population genetics. Theor. Popul. Biol. 1, 352–370 (1970)

  6. 6.

    Charlesworth, B.: Evolution in age-structured populations. Cambridge: Cambridge University Press 1980

  7. 7.

    Demetrius, L.: Statistical mechanics and population biology. J. Stat. Phys. 30, 709–753 (1983)

  8. 8.

    Demetrius, L., Ziehe, M.: The measurement of Darwinian fitness in human populations. Proc. R. Soc. London B222, 33–50 (1984)

  9. 9.

    Fisher, R. A.: The genetical theory of natural selection. Oxford: Clarendon Press 1930

  10. 10.

    Gurtin, M. E., MacCamy, R. C.: Product solutions and asymptotic behavior for age-dependent, dispersing populations. Math. Biosci. 62, 157–167 (1982)

  11. 11.

    Hadeler, K. P., Dietz, K.: Population dynamics of killing parasites which reproduce in the host. J. Math. Biol. 21, 45–65 (1984)

  12. 12.

    Keyfitz, N.: Introduction to the mathematics of population. Reading: Addison-Wesley 1968

  13. 13.

    Leslie, P. H.: On the use of matrices in certain population mathematics, Biometrika 33, 183–212 (1945)

  14. 14.

    Li, C. C.: Population genetics. Chicago: University of Chicago Press 1955

  15. 15.

    Lotka, A. J.: Théorie analytique des associations biologiques, vol. II, Paris: Hermann 1939

  16. 16.

    Pollak, E. Kempthorne, O.: Malthusian parameters in genetic populations. I. Haploid and selfing models. Theor. Popul. Biol. 1, 315–345 (1970).

  17. 17.

    Pollak, E., Kempthorne, O.: Malthusian parameters in genetic populations. Part II. Random mating populations in infinite habitats. Theor. Popul. Biol. 2, 357–390 (1971)

  18. 18.

    Pollak, E., Callanan, T.: Convergence of two-locus gamete frequencies in random-mating age-structured populations, Math. Biosci. 62, 179–199 (1982)

  19. 19.

    Pollak, E.: Gamete frequencies at two sex-linked loci in random mating, Math. Biosci. 70, 217–235 (1984)

  20. 20.

    Vlad, M. O., Popa, V. T.: A new nonlinear model of age-dependent population growth, Math. Biosci. 76, 161–184 (1985)

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Vlad, M.O. Separable models for age-structured population genetics. J. Math. Biology 26, 73–92 (1988).

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Key words

  • Age-dependence
  • Separable models
  • Population genetics