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Limit diffusions of some stepping-stone models

Published online by Cambridge University Press:  14 July 2016

Ken-Iti Sato
Ken-Iti Sato*
Affiliation:
Kanazawa University
*
∗ Postal address: Department of Mathematics, College of Liberal Arts, Kanazawa University, Kanazawa, Japan.
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Abstract

A Markov chain model of a population consisting of a finite or countably infinite number of colonies with N particles at each colony is considered. There are d types of particle and transition from the nth generation to the (n + 1)th is made up of three stages: reproduction, migration, and sampling. Natural selection works in the reproduction stage. The limiting diffusion operator (as N→∞) for the proportion of types at colonies is found. Convergence to the diffusion is proved under certain conditions.

Keywords

MARKOV CHAINOFFSPRING DISTRIBUTIONSTEPPING-STONE STRUCTUREINFINITE-DIMENSIONAL DIFFUSIONMARTINGALE PROBLEM
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 3 , September 1983 , pp. 460 - 471
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

This work was done while the author was visiting Carleton University, supported by the Natural Science and Engineering Research Council Canada, the Japan Society for the Promotion of Science, and the NSERC operating grant of D. A. Dawson.

References

Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, New York.Google Scholar
Ethier, S. N. (1981) A class of infinite-dimensional diffusions occurring in population genetics. Indiana Univ. Math. J. 30, 925–935.CrossRefGoogle Scholar
Ethier, S. N. and Kurtz, T. G. (1981) The infinitely-many-neutral-alleles diffusion model. Adv. Appl. Prob. 13, 429–452.Google Scholar
Fleming, W. H. and Su, C. (1974) Some one-dimensional migration models in population genetics theory. Theoret. Popn Biol. 5, 431–449.Google Scholar
Gillespie, J. H. (1974), (1975) Natural selection for within-generation variance in offspring number. Genetics 76, 601–606; 81, 403–413.Google Scholar
Itatsu, S. (1981) Ergodic properties of the equilibrium measure of the stepping stone model in population genetics. Nagoya J. Math. 83, 37–51.CrossRefGoogle Scholar
Kimura, M. and Weiss, G. H. (1964) The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 49, 561–576.Google Scholar
Maruyama, T. (1969) Genetic correlation in the stepping stone model with non-symmetrical migration rates. J. Appl. Prob. 6, 463–477.Google Scholar
Nagylaki, T. (1978) Random genetic drift in a cline. Proc. Nat. Acad. Sci. U.S.A. 75, 423–426.CrossRefGoogle Scholar
Nagylaki, T. and Moody, M. (1980) Diffusion model for genotype-dependent migration. Proc. Nat. Acad. Sci. U.S.A. 77, 4842–4846.CrossRefGoogle ScholarPubMed
Okada, N. (1981) On the uniqueness problem of two dimensional diffusion processes occurring in population genetics. Z. Wahrscheinlichkeitsth. 74, 63–74.Google Scholar
Sato, K. (1976a) Diffusion processes and a class of Markov chains related to population genetics. Osaka J. Math. 13, 631–659.Google Scholar
Sato, K. (1976b) A class of Markov chains related to selection in population genetics. J. Math. Soc. Japan 28, 621–637.Google Scholar
Sato, K. (1978) Convergence to a diffusion of a multi-allelic model in population genetics. Adv. Appl. Prob. 10, 538–562.Google Scholar
Sawyer, S. (1976) Results for the stepping stone model for migration in population genetics. Ann. Prob. 4, 699–728.Google Scholar
Sawyer, S. (1977) Asymptotic properties of the equilibrium probability of identity in a geographically structured population. Adv. Appl. Prob. 9, 268–282.Google Scholar
Shiga, T. (1980) An interacting system in population genetics. J. Math. Kyoto Univ. 20, 213–242; 723–733.Google Scholar
Shiga, T. (1981) Diffusion processes in population genetics. J. Math. Kyoto Univ. 21, 133–151.Google Scholar
Shiga, T. (1982) Continuous time multi-allelic stepping stone models in population genetics. J. Math. Kyoto Univ. 22, 1–40.Google Scholar
Shiga, T. and Shimizu, A. (1980) Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20, 395–416.Google Scholar
Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin.Google Scholar