Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-27T02:20:06.182Z Has data issue: false hasContentIssue false
  • English
  • Français

On convergence to diffusion processes of Markov chains related to population genetics

Published online by Cambridge University Press:  01 July 2016

Norio Okada
Norio Okada*
Affiliation:
Tokyo Institute of Technology
*
∗Postal address: Department of Applied Physics, Tokyo Institute of Technology, Oh-Okayama, Tokyo, Japan.
Get access Rights & Permissions [Opens in a new window]

Abstract

A Markov chain model related to population genetics and its convergence to a diffusion process on the multi-dimensional bounded domain are treated. We discuss the case where natural selection is random and the different selection effects over successive generations are independent. Our model is a multi-allelic version of the haploid model of Karlin and Levikson. The asymptotic properties of the limiting diffusion are stated.

Keywords

MARKOV CHAINPOPULATION GENETICSRANDOM NATURAL SELECTIONDIFFUSION APPROXIMATIONMARTINGALE PROBLEMFIXATION PROBABILITY
Type
Research Article
Information
Advances in Applied Probability , Volume 11 , Issue 4 , December 1979 , pp. 673 - 700
Copyright
Copyright © Applied Probability Trust 1979 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Crow, J. F. and Kimura, M. (1970) An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
2. Ethier, S. N. (1976) A class of degenerate diffusion processes occurring in population genetics. Comm. Pure Appl. Math. 29, 483–493.CrossRefGoogle Scholar
3. Feller, W. (1951) Diffusion processes in genetics. Proc. 2nd Berkeley Symp. Math. Statist. Prob., 227–246.Google Scholar
4. Felsenstein, J. (1976) The theoretical population genetics of variable selection and migration. Ann. Rev. Genet. 10, 253–280.Google Scholar
5. Friedman, A. (1975) Stochastic Differential Equations and Applications, Vol. 1. Academic Press, New York.Google Scholar
6. Karlin, S. and Levikson, B. (1974) Temporal fluctuations in selection intensities: Case of small population size. Theoret. Popn Biol. 6, 383–412.Google Scholar
7. Karlin, S. and McGregor, J. (1964) Direct product branching processes and related Markov chains. Proc. Nat. Acad. Sci. USA 51, 598–602.Google Scholar
8. Kimura, M. (1960) Introduction to Population Genetics (in Japanese). Baifūkan, Tokyo.Google Scholar
9. Sato, K. (1976) Asymptotic properties of eigenvalues of a class of Markov chains induced by direct product branching processes. J. Math. Soc. Japan 28, 192–211.Google Scholar
10. Sato, K. (1976) Diffusion processes and a class of Markov chains related to population genetics. Osaka J. Math. 13, 631–659.Google Scholar
11. Sato, K. (1976) A class of Markov chains related to selection in population genetics. J. Math. Soc. Japan 28, 621–637.Google Scholar
12. Sato, K. (1978) Convergence to a diffusion of a multi-allelic model in population genetics. Adv. Appl. Prob. 10, 538–562.Google Scholar
13. Strook, D. W. and Varadhan, S. R. S. (1969) Diffusion processes with continuous coefficients. Comm. Pure Appl. Math. 22, 345–400.Google Scholar
14. Strook, D. W. and Varadhan, S. R. S. (1972) On the support of diffusion processes with applications to the strong maximum principle. Proc. 6th Berkeley Symp. Math. Statist. Prob. 4, 333–359.Google Scholar
15. Takahata, N., Ishii, K. and Matsuda, H. (1975) Effect of temporal fluctuation of selection coefficient on gene frequency in a population. Proc. Nat. Acad. Sci. USA 72, 4541–4545.Google Scholar
16. Yanagihara, J. (1962) Series (in Japanese). Asakura-shoten.Google Scholar