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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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References

Bodmer, W. F. (1965) Differential fertility in population genetics models. Genetics 51, 411–424.CrossRefGoogle ScholarPubMed
Ethier, S. N. (1976) A class of degenerate diffusion processes occurring in population genetics. Comm. Pure Appl. Math. 29, 483–493.CrossRefGoogle Scholar
Ethier, S. N. (1979) A limit theorem for two-locus diffusion models in population genetics. J. Appl. Prob. 16, 402–408.CrossRefGoogle Scholar
Feldman, M. W. (1966) On the offspring number distribution in a genetic population. J. Appl. Prob. 3, 129–141.CrossRefGoogle Scholar
Guess, H. A. (1973) On the weak convergence of Wright–Fisher models. Stoch. Proc. Appl. 1, 287–306.CrossRefGoogle Scholar
Kertz, R. P. (1974) Perturbed semigroup limit theorems with applications to discontinuous random evolutions. Trans. Amer. Math. Soc. 199, 29–53.CrossRefGoogle Scholar
Kertz, R. P. (1979) A semigroup perturbation theorem with application to a singular perturbation problem in nonlinear ordinary differential equations. J. Differential Equations 31, 1–15.CrossRefGoogle Scholar
Kimura, M. (1964) Diffusion models in population genetics. J. Appl. Prob. 1, 177–232.CrossRefGoogle Scholar
Kurtz, T. G. (1969) Extensions of Trotter's operator semigroup approximation theorems. J. Functional Analysis 3, 354–375.CrossRefGoogle Scholar
Kurtz, T. G. (1973) A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Functional Analysis 12, 55–67.CrossRefGoogle Scholar
Kurtz, T. G. (1975) Semigroups of conditioned shifts and approximation of Markov processes. Ann. Prob. 3, 618–642.CrossRefGoogle Scholar
Kurtz, T. G. (1977) Applications of an abstract perturbation theorem to ordinary differential equations. Houston J. Math. 3, 67–82.Google Scholar
Latter, B. D. H. (1959) Genetic sampling in a random mating population of constant size and sex ratio. Austral. J. Biol. Sci. 12, 500–505.CrossRefGoogle Scholar
Lindvall, T. (1973) Weak convergence of probability measures and random functions in the function space D[0, ∞). J. Appl. Prob. 10, 109–121.CrossRefGoogle Scholar
Moran, P. A. P. (1958a) Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.CrossRefGoogle Scholar
Moran, P. A. P. (1958b) A general theory of the distribution of gene frequencies. I. Overlapping generations. Proc. R. Soc. London B 149, 102–111.Google ScholarPubMed
Moran, P. A. P. (1958c) A general theory of the distribution of gene frequencies. II. Non-overlapping generations. Proc. R. Soc. London B 149, 113–116.Google ScholarPubMed
Moran, P. A. P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
Moran, P. A. P. and Watterson, G. A. (1959) The genetic effects of family structure in natural populations. Austral. J. Biol. Sci. 12, 1–15.CrossRefGoogle Scholar
Nagylaki, T. (1977) Selection in One- and Two-Locus Systems. Lecture Notes in Biomathematics, 15, Springer-Verlag, Berlin.CrossRefGoogle ScholarPubMed
Nagylaki, T. (1980) The strong-migration limit in geographically structured populations. J. Math. Biol. To appear.CrossRefGoogle Scholar
Norman, M. F. (1975) Diffusion approximation of non-Markovian processes. Ann. Prob. 3, 358–364.CrossRefGoogle Scholar
Papanicolaou, G. C., Stroock, D., and Varadhan, S. R. S. (1977) Martingale approach to some limit theorems. In Statistical Mechanics and Dynamical Systems (by Ruelle, D.) and papers from the 1976 Duke Turbulence Conference. Duke University, Durham, N.C. Google Scholar
Searle, A. G. (1974) Mutation induction in mice. Adv. Radiation Biol. 4, 131–207.CrossRefGoogle Scholar
Vogel, F. (1977) A probable sex difference in some mutation rates. Amer. J. Hum. Genet. 29, 312–319.Google ScholarPubMed
Watterson, G. A. (1962) Some theoretical aspects of diffusion theory in population genetics. Ann. Math. Statist. 33, 939–957.CrossRefGoogle Scholar
Watterson, G. A. (1964) The application of diffusion theory to two population genetic models of Moran. J. Appl. Prob. 1, 233–246.CrossRefGoogle Scholar
Watterson, G. A. (1970) On the equivalence of random mating and random union of gametes models in finite, monoecious populations. Theoret. Popn. Biol. 1, 233–250.CrossRefGoogle ScholarPubMed
Wright, S. (1939) Statistical genetics in relation to evolution. In Actualités scientifiques et industrielles. 802, 5–64. Exposés de biométrie et de la statistique biologique XIII. Hermann et Cie, Paris.Google Scholar