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Diffusion approximations of Markov chains with two time scales and applications to population genetics

Published online by Cambridge University Press:  01 July 2016

S. N. Ethier  and
Thomas Nagylaki
S. N. Ethier*
Affiliation:
Michigan State University
Thomas Nagylaki*
Affiliation:
The University of Chicago
*
∗Postal address: Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, U.S.A. Supported in part by NSF grant MCS78-02878.
∗∗Postal address: Department of Biophysics and Theoretical Biology. The University of Chicago, 920 East 58th St, Chicago, IL 60637, U.S.A. Supported in part by NSF grant DEB77-21494.
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Abstract

For N = 1, 2, ···, let {(XN(k), YN(k)), k = 0, 1, ···} be a homogeneous Markov chain in ℝm x ℝn. Suppose that, asymptotically as N → ∞, the ‘infinitesimal’ covariances and means of XN([·/∊N]) are aij(x, y) and bi(x, y), and those of YN([·/δN]) are 0 and cl(x, y). Assume limN→∞ δN = limN→∞ ∊N/δN = 0 and the zero solution of ý = c(x, y) is globally asymptotically stable. Then, under some technical conditions, it is shown that (i) XN([·/∊N]) converges weakly to a diffusion process with coefficients aij(x, 0) and bi(x, 0), and (ii) YN([t/∊N]) → 0 in probability for every t > 0. (The case limN→∞ δN = δ∞ > 0 = limN→∞ ∊N is also treated.) The proof is based on the discrete-parameter analogue of a generalization of Kurtz's limit theorems for perturbed operator semigroups.

The results are applied to three classes of stochastic models for random genetic drift at a multiallelic locus in a finite diploid population. The three classes involve multinomial sampling, overlapping generations, and general progeny distributions. Within each class, the monoecious, dioecious autosomal, and X-linked cases are analyzed. It is found that results for a monoecious population obtained from a diffusion approximation can be applied at once to the dioecious cases by using the appropriate effective population size and averaging allelic frequencies, selection intensities, and mutation rates, weighting each sex by the number of genes carried by an individual at the locus under consideration.

Keywords

MARKOV CHAINSDIFFUSION APPROXIMATIONSOPERATOR SEMIGROUPSPOPULATION GENETICSRANDOM GENETIC DRIFTHARDY-WEINBERG PROPORTIONSEFFECTIVE POPULATION SIZE
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 1 , March 1980 , pp. 14 - 49
Copyright
Copyright © Applied Probability Trust 1980 

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