Bulletin of Mathematical Biology

, Volume 48, Issue 1, pp 87–95 | Cite as

Existence of steady-state probability distributions in multilocus models for genotype evolution

  • Gerald Rosen


It is shown that a representative Fisher-Wright model withn(≥3) diallelic loci admits a necessary condition for existence of a time-independent steady-state probability distribution. This necessary condition states that a global integral depending on the phenotype fitness functions of natural selection must be larger than a certain quantity depending on the parameters associated with genetic drift.


Genetic Drift Kolmogorov Equation Diallelic Locus Relative Allele Frequency Multilocus Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Campbell, R. B. 1984. “Manifestation of Phenotypic Selection at Constituent Loci. I. Stabilizing Selection.”Evolution 38, 1033–1038.CrossRefGoogle Scholar
  2. Cohan, F. M. 1984. “Can Uniform Selection Retard Random Genetic Divergence Between Isolated Conspecific Populations?”Evolution 38, 495–504.CrossRefGoogle Scholar
  3. Curtsinger, J. W. 1984. “Evolutionary Landscapes for Complex Selection.”Evolution 38, 359–367.CrossRefGoogle Scholar
  4. Ethier, S. N. and T. Nagylaki. 1980. “Diffusion Approximations of Markov Chains with Two Time Scales and Applications to Population Genetics.”Adv. appl. Prob. 12, 14–19.MATHMathSciNetCrossRefGoogle Scholar
  5. Fisher, R. A. 1922. “On the Dominance Ratio.”Proc. R. Soc. Edinb. 42, 321–341.Google Scholar
  6. Lieb, E. H. 1983. “Sharp Constants in the Hardy-Littlewood-Sobolev and Related Inequalities.”Ann. Math. 118, 349–374.MATHMathSciNetCrossRefGoogle Scholar
  7. Lions, P. L. 1983. “Applications de la Méthode de Concentration-compacité à l'Existence de Fonctions Extrémales.”C. r. Acad. Sci. Paris 296, 645–648.MATHGoogle Scholar
  8. Maruyama, T. 1983. “Stochastic Theory of Population Genetics.”Bull. math. Biol. 45, 521–554.MATHMathSciNetCrossRefGoogle Scholar
  9. Protter, M. H. and H. F. Weinberger. 1967.Maximum Principles in Differential Equations, p. 187. Englewood Cliffs, NJ: Prentice-Hall.Google Scholar
  10. Rosen, G. 1984. “Necessary Condition for Fokker-Planck Statistical Equilibrium.”Phys. Rev. A30, 3359–3363.MathSciNetCrossRefGoogle Scholar
  11. Roughgarden, J. 1979.Theory of Population Genetics and Evolutionary Ecology: An Introduction, p. 111–113. New York: Macmillan.Google Scholar
  12. Stenseth, N. C. and J. Maynard Smith. 1984. “Coevolution in Ecosystems: Red Queen Evolution or Stasis?”Evolution 38, 870–880.CrossRefGoogle Scholar
  13. Wang, M. C. and G. E. Uhlenbeck. 1945. “On the Theory of the Brownian Motion II.”Rev. mod. Phys. 17, 323–342.MATHMathSciNetCrossRefGoogle Scholar
  14. Wright, S. 1931. “Evolution in Mendelian Populations.”Genetics 16, 97–159.Google Scholar

Copyright information

© Society for Mathematical Biology 1986

Authors and Affiliations

  • Gerald Rosen
    • 1
  1. 1.Department of PhysicsDrexel UniversityPhiladelphiaU.S.A.

Personalised recommendations