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

Stochastic selection in both haplophase and diplophase

  • 39 Accesses

  • 1 Citations

Summary

A population genetic model incorporating the evolutionary forces of zygotic selection, gametic selection and non-Mendelian segregation has been analyzed for the case in which all selection coefficients and the segregation parameter are assumed to be random variables that are uncorrelated from generation to generation. The diffusion approximation of the model is developed, and the subsequent analysis shows that one of four limiting outcomes of the stochastic process may obtain — an allele may be fixed or lost almost surely and irrespective of the initial gene frequency, the gene frequency may converge to a unique stationary distribution, or an allele may be fixed or lost with probabilities depending on the initial gene frequency. These outcomes correspond rather closely with the possible outcomes of the deterministic model — fixation or loss of an allele, convergence to a stable equilibrium, or the existence of an unstable equilibrium.

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

References

  1. Cook, R. D., Hartl, D. L.: Uncorrelated random environments and their effects on gene frequency. Evolution 28, 265–274 (1974).

  2. Cook, R. D., Hartl, D. L.: Stochastic selection in large and small populations. Theoret. Population Biology 7, 55–63 (1975).

  3. Feller, W.: Diffusion processes in one dimension. Trans. Amer. Math. Soc. 77, 1–31 (1954).

  4. Gillespie, J.: Polymorphism in random environments. Theoret. Population Biology 4, 193–195 (1973).

  5. Hartl, D. L.: Stochastic selection in gametes and zygotes. In: Gamete Competition in Plants and Animals (Mulcahy, D. L., ed.), pp. 233–242. Amsterdam: North Holland Publishing Company 1975.

  6. Hartl, D. L., Cook, R. D.: Balanced polymorphisms of quasineutral alleles. Theoret. Population Biology 4, 163–172 (1973).

  7. Hartl, D. L., Cook, R. D.: Stochastic selection and the maintenance of genetic variation. In: Internal Conference on Population Genetics and Ecology (Karlin, S., Nevo, E., eds.), pp. 593–615. New York: Academic Press 1976.

  8. Hiraizumi, Y., Sandler, L., Crow, J. F.: Meiotic drive in natural populations of Drosophila melanogaster. III. Population implications of the segregation-distorter locus. Evolution 44, 433–444 (1960).

  9. Karlin, S., Lieberman, U.: Random temporal variation in selection intensities: Case of large population size. Theoret. Population Biology 6, 355–382 (1974).

  10. Karlin, S., Lieberman, U.: Random temporal variation in selection intensities: one-locus two-allele model. J. Math. Biol. 2, 1–17 (1975).

  11. Karlin, S., Levikson, B.: Temporal variation in selection intensities: Case of small population size. Theoret. Population Biology 6, 383–412 (1974).

  12. Levikson, B.: The effects of random environments on the evolutionary process of gene frequencies: A mathematical analysis. Ph. D. Thesis, Tel-Aviv University, Israel. 1974.

  13. Levikson, B., Karlin, S.: Random temporal variation in selection intensities acting on infinite diploid populations: Diffusion method analysis. Theoret. Population Biology 8, 292–300 (1975).

  14. Norman, M. F.: An ergodic theorem for evolution in a random environment. J. Applied Probability 12, 661–672 (1975).

  15. Scudo, F. M.: Selection on both haplo and diplophase. Genetics 56, 693–704 (1967).

Download references

Author information

Additional information

Work supported by N. I. H. grants GM21732 and GM21623. The author is supported by Research Career Award GM2301.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hartl, D.L. Stochastic selection in both haplophase and diplophase. J. Math. Biol. 3, 263–269 (1976). https://doi.org/10.1007/BF00275059

Download citation

Keywords

  • Stationary Distribution
  • Mathematical Biology
  • Genetic Model
  • Matrix Theory
  • Gene Frequency