Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-22T05:29:21.851Z Has data issue: false hasContentIssue false
  • English
  • Français

A limit theorem for two-locus diffusion models in population genetics

Published online by Cambridge University Press:  14 July 2016

S. N. Ethier
S. N. Ethier*
Affiliation:
Michigan State University
*
∗Postal address: Department of Statistics and Probability, Wells Hall, Michigan State University, East Lansing, Michigan 48824, U.S.A.
Get access Rights & Permissions [Opens in a new window]

Abstract

A limit theorem of Kurtz for perturbed operator semigroups is applied to show that the two-locus diffusion model in population genetics (allowing for selection, mutation, and migration) converges to a linkage-equilibrium diffusion model as Nc →∞, where N is the population size and c is the recombination fraction; in fact, with an appropriate change of variables, the limiting diffusion is what has been called the independent-loci diffusion model. This generalizes a result of Littler.

Keywords

POPULATION GENETICSRANDOM GENETIC DRIFTLINKAGEDIFFUSION PROCESSESOPERATOR SEMIGROUPS
Type
Short Communications
Information
Journal of Applied Probability , Volume 16 , Issue 2 , June 1979 , pp. 402 - 408
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ethier, S. N. (1976) A class of degenerate diffusion processes occurring in population genetics. Comm. Pure Appl. Math. 29, 483–493.Google Scholar
[2] Ethier, S. N. (1979) Limit theorems for absorption times of genetic models. Ann. Prob. To appear.Google Scholar
[3] Kurtz, T. G. (1973) A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Funct. Anal. 12, 55–67.Google Scholar
[4] Kurtz, T. G. (1975) Semigroups of conditioned shifts and approximation of Markov processes. Ann. Prob. 3, 618–642.Google Scholar
[5] Littler, R. A. (1976) The independent loci model as a limit of a two locus model. University of Waikato, Dept. of Math., Research Report No. 41.Google Scholar
[6] Littler, R. A. and Good, A. J. (1978) Fixation times and probabilities for an independent loci model in genetics. Theoret. Popn Biol. 14, 204–214.CrossRefGoogle ScholarPubMed
[7] Nagylaki, T. (1977) Selection in One- and Two-Locus Systems. Lecture Notes in Biomathematics, Vol. 15. Springer-Verlag, Berlin.Google Scholar
[8] Ohta, T. (1973) Effect of linkage on behavior of mutant genes in finite populations. Theoret. Popn. Biol. 4, 145–162.Google Scholar
[9] Ohta, T. and Kimura, M. (1969) Linkage disequilibrium due to random genetic drift. Genet. Res. 13, 47–55.Google Scholar
[10] Tier, C. and Keller, J. B. (1978) Asymptotic analysis of diffusion equations in population genetics. SIAM J. Appl. Math. 34, 549–576.CrossRefGoogle Scholar