Journal of Mathematical Biology

, Volume 17, Issue 2, pp 241–251 | Cite as

Dynamics of games and genes: Discrete versus continuous time

  • V. Losert
  • E. Akin
Article

Abstract

It is shown that in the classical model of population genetics (Fisher-Wright-Haldane, discrete or continuous version) every solution p(t) converges to equilibrium for t → ∞. For related models of evolutionary games (with non-symmetric matrices) it is shown that the transformation that describes the dynamics is a diffeomorphism (in particular one-to-one).

Key words

Population genetics Evolutionary games Convergence to equilibrium 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Akin, E.: The metric theory of banach manifolds. Lecture notes in mathematics, vol. 622. Berlin- Heidelberg-New York: Springer 1978Google Scholar
  2. 2.
    Akin, E.: The geometry of population genetics. Lecture notes in biomathematics, vol. 31. Berlin- Heidelberg-New York: Springer 1979Google Scholar
  3. 3.
    Akin, E., Hofbauer, J.: Recurrence of the unfit. Math. Biosci. (to appear) (1982)Google Scholar
  4. 4.
    Ginzburg, L. R.: Diversity of fitness and generalized fitness. J. Gen. Biol. 33, 77–81 (1972)Google Scholar
  5. 5.
    an der Heiden, U.: On manifolds of equilibria in the selection model for multiple alleles. J. Math. Biol. 1, 321–330 (1975)Google Scholar
  6. 6.
    Hines, W. G. S.: Three characterizations of population strategy stability. J. Appl. Prob. 17, 333–340 (1980)Google Scholar
  7. 7.
    Karlin, S.: A first course in stochastic processes. New York: Academic Press 1966Google Scholar
  8. 8.
    Karlin, S., Feldman, M. W.: Linkage and selection: Two locus symmetric viability model. J. Theoret. Population Biology 1, 39–71 (1970)Google Scholar
  9. 9.
    Kingman, J. F. C.: A mathematical problem in population genetics. Proc. Camb. Phil. Soc. 57, 574–582 (1961)Google Scholar
  10. 10.
    Lang, S.: Differential manifolds. Reading, Mass.: Addison-Wesley 1972Google Scholar
  11. 11.
    May, R. M.: Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos. Science 186, 645–647 (1974)Google Scholar
  12. 12.
    Mulholland, H. P., Smith, C. A. B.: An inequality arising in genetical theory. Am. Math. Mon. 66, 673–683 (1959)Google Scholar
  13. 13.
    Scheuer, P. A. G., Mandel, S. P. H.: An inequality in population genetics. Heredity 13, 519–524 (1959)Google Scholar
  14. 14.
    Taylor, P., Jonker, L.: Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156 (1978)Google Scholar
  15. 15.
    Feller, W.: A geometrical analysis of fitness in triply allelic systems. Math. Biosci. 5, 19–38 (1969).Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • V. Losert
    • 1
  • E. Akin
    • 2
  1. 1.Institut für MathematikUniversität WienWienAustria
  2. 2.Mathematics DepartmentThe City CollegeNew YorkUSA

Personalised recommendations