Hostname: page-component-7b7567c8c-2n9c9 Total loading time: 0 Render date: 2026-11-15T10:19:26.597Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence to a diffusion of a multi-allelic model in population genetics

Published online by Cambridge University Press:  01 July 2016

Ken-Iti Sato
Ken-Iti Sato*
Affiliation:
Kanazawa University
Get accessRights & Permissions [Opens in a new window]

Abstract

We consider a Markov chain on the d-dimensional (d-allelî) non-negative lattice points with the sum of components being N, for which one-step transition consists of two stages—independent reproduction and random sampling. Convergence to a degenerate diffusion process when N → ∞ is proved. We show how difference among alleles in means and variances of offspring numbers affects the limit diffusion, giving a rigorous multi-allelic version of a result of Gillespie.

Keywords

MARKOV CHAINPOPULATION GENETICSOFFSPRING DISTRIBUTIONDEGENERATE DIFFUSIONSTOCHASTIC DIFFERENTIAL EQUATION

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 3 , September 1978, pp. 538 - 562
Copyright
Copyright © Applied Probability Trust 1978 

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.)

Article purchase

Temporarily unavailable

References

Chia, A. B. and Watterson, G.A. (1969) Demographic effects on the rate of genetic evolution I. Constant size populations with two genotypes. J. Appl. Prob. 6, 231–248.Google Scholar
Ethier, S. N. (1976) A class of degenerate diffusion processes occurring in population genetics. Comm. Pure Appl. Math. 29, 483–493.CrossRefGoogle Scholar
Feller, W. (1968) An Introduction to Probability Theory and its Applications, Vol. I, 3rd edn. Wiley, New York.Google Scholar
Fisher, R. A. (1939) Stage of development as a factor influencing the variance in the number of offspring, frequency of mutants and related quantities. Ann. Eugen. 9, 406–408.Google Scholar
Freidlin, M. I. (1968) On the factorization of non-negative definite matrices. Theory Prob. Appl. 13, 354–356.Google Scholar
Gillespie, J. H. (1974) Natural selection for within-generation variance in offspring number. Genetics 76, 601–606.Google Scholar
Gillespie, J. H. (1975) Natural selection for within-generation variance in offspring number. II. Discrete haploid models. Genetics 81, 403–413.Google Scholar
Johnson, N. L. and Kotz, S. (1969) Distributions in Statistics. Discrete Distributions. Houghton Mifflin, Boston.Google Scholar
Karlin, S. and McGregor, J. (1964a) On some stochastic models in genetics. In Stochastic Models in Medicine and Biology, ed. Gurland, J.. University of Wisconsin Press. Madison, 245–279.Google Scholar
Karlin, S. and McGregor, J. (1964b) Direct product branching processes and related Markov chains. Proc. Natn. Acad. Sci. USA 51, 598–602.CrossRefGoogle ScholarPubMed
Kimura, M. (1955) Stochastic processes and distribution of gene frequencies under natural selection. Cold Spring Harbor Symposia on Quantitative Biology 20, 33–53.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.Google Scholar
Okada, N. (to appear) On convergence to diffusion processes of Markov chains related to population genetics.Google Scholar
Phillips, R. S. and Sarason, L. (1968) Elliptic-parabolic equations of the second order. J. Math. Mech. 17, 891–917.Google Scholar
Priouret, P. (1974) Prosessus de diffusion et équations différentielles stochastiques. In Ecole d'Eté de Probabilités de Saint-Flour III, 1973. Lecture Notes in Mathematics 390, Springer–Verlag, Berlin, 37–113.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.CrossRefGoogle Scholar
Sato, K. (1977) A note on convergence of probability measures on C and D. Ann. Sci. Kanazawa Univ. 14, 1–5.Google Scholar
Skorokhod, A. V. (1965) Studies in the Theory of Random Processes. Addison–Wesley, Reading.Google Scholar
Stroock, D. W. and Varadhan, S. R. S. (1969) Diffusion processes with continuous coefficients. Comm. Pure Appl. Math. 22, 345–400 and 479–530.Google Scholar
Wright, S. (1949) Adaptation and selection. In Genetics, Paleontology and Evolution, ed. Jepsen, G. L., Simpson, G. G. and Mayr, E.. Princeton University Press, Princeton, 365–389.Google Scholar
Yamada, T. and Watanabe, S. (1971) On the uniqueness of solutions of stochastic differential equations J. Math. Kyoto Univ. 11, 155–167.Google Scholar