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Isolation by distance in a hierarchically clustered population

Published online by Cambridge University Press:  14 July 2016

Stanley Sawyer  and
Joseph Felsenstein
Stanley Sawyer*
Affiliation:
Purdue University
Joseph Felsenstein*
Affiliation:
University of Washington
*
∗ Postal address: Department of Mathematics, Purdue University, W. Lafayette, IN 47907, U.S.A.
∗∗ Postal address: Department of Genetics SK-50, University of Washington, Seattle, WA 98105, U.S.A.
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Abstract

A biological population with local random mating, migration, and mutation is studied which exhibits clustering at several different levels. The migration is determined by the clustering rather than actual geographic or physical distance. Darwinian selection is assumed to be absent, and population densities are such that nearby individuals have a probability of being related. An expression is found for the equilibrium probability of genetic relatedness between any two individuals as a function of their clustering distance. Asymptotics for a small mutation rate u are discussed for both a finite number of clustering levels (and of total population size), and for an infinite number of levels. A natural example is discussed in which the probability of heterozygosity varies as u to a power times a periodic function of log(l/u).

Keywords

POPULATION GENETICSCLUSTERINGIDENTITY BY DESCENTFOURIER TRANSFORMHIERARCHICAL MODELKARAMATA'S THEOREMMIGRATION
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 1 , March 1983 , pp. 1 - 10
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Partially supported by NSF Grant MCS79-03472.

Partially supported by Task Agreement No. DE-AT06-76EV71005 under Contract No. DE-AM06-76RL02225 between the U.S. Department of Energy and the University of Washington.

References

Bunimovich, L. (1975) A model of human populations with hierarchical structure (in Russian). Soviet Genetics 11, 134–143.Google Scholar
Carmelli, D. and Cavalli-Sforza, L. (1976) Some models of population structure and evolution. Theoret. Popn Biol. 9, 329–359.CrossRefGoogle ScholarPubMed
Cavalli-Sforza, L. and Bodmer, W. (1971) The Genetics of Human Populations. Freeman, San Francisco.Google Scholar
Darling, D. and Erdös, P. (1968) On the recurrence of a certain chain. Proc. Amer. Math. Soc. 19, 336–338.Google Scholar
Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
Flatto, L. and Pitt, J. (1974) Recurrence criteria for random walks on countable Abelian groups. Illinois J. Math. 18, 1–19.Google Scholar
Kimura, M. and Weiss, G. (1964) The stepping stone model of population structure and the decrease of genetic correlation with distance. Genetics 49, 561–576.Google Scholar
Malecot, G. (1969) The Mathematics of Heredity. Freeman, San Francisco.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. (1978) Isotropic random walks in a tree. Z. Wahrscheinlichkeitsth. 42, 279–292.Google Scholar
Sawyer, S. (1979) A limit theorem for patch sizes in a selectively-neutral migration model. J. Appl. Prob. 16, 482–495.Google Scholar
Sawyer, S. and Felsenstein, J. (1981) A continuous migration model with stable demography. J. Math. Biol. 11, 193–205.Google Scholar