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Ergodic properties of the equilibrium measure of the stepping stone model in population genetics

Published online by Cambridge University Press:  22 January 2016

Seiichi Itatsu
Seiichi Itatsu*
Affiliation:
Department of Mathematics, Faculty of Science, Shizuoka University
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We shall present in this paper some ergodic properties of the stepping stone model. The model has been proposed by M. Kimura [2], to describe the evolution of a genetical population with mating and geographical structures. It has been investigated and developed by M. Kimura and G. H. Weiss [3], G. H. Weiss and M. Kimura [6], W. Fleming and C.-H. Su [1], S. Sawyer [5], and others.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 83 , October 1981 , pp. 37 - 51
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[1] Fleming, W. and Su, C.-H., Some one-dimensional migration models in population genetics theory, Theoret. Population Biology 5 (1974), 431–449.CrossRefGoogle ScholarPubMed
[2] Kimura, M., “Stepping stone” model of population, Annual Report of the National Institute of Genetics, Japan 3 (1953), 63–65.Google Scholar
[3] Kimura, M. and Weiss, G. H., The stepping stone model of population structure and the decrease of genetic correlation with distance, Genetics, 49 (1964), 561–576.CrossRefGoogle ScholarPubMed
[4] Revuz, D., “Markov Chains”, North Holland Publishing Company, Amsterdam, (1975).Google Scholar
[5] Sawyer, S., Results for the stepping stone model for migration in population genetics, Ann. Probability, 4 (1976), 699–728.CrossRefGoogle Scholar
[6] Weiss, G. H. and Kimura, M., A mathematical analysis of the stepping stone model of genetics correlation, J. Appl. Probability, 2 (1965), 129–149.CrossRefGoogle Scholar