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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.
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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 

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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