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The geometry of random genetic drift VI. A random selection diffusion model

Published online by Cambridge University Press:  01 July 2016

Peter L. Antonelli ,
Jared Chapin  and
B. H. Voorhees
Peter L. Antonelli*
Affiliation:
University of Alberta
Jared Chapin*
Affiliation:
University of Alberta
B. H. Voorhees*
Affiliation:
University of Alberta
*
∗Postal address: Department of Mathematics, The University of Alberta, Edmonton, Canada T6G 2G1.
∗Postal address: Department of Mathematics, The University of Alberta, Edmonton, Canada T6G 2G1.
∗∗Postal address: The Center for Advanced Study in Theoretical Psychology, The University of Alberta, Edmonton, Canada, T6G 2E9.
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Abstract

The ray solution of Felsenstein's n-allele random selection diffusion process is given for small values of the selection parameters. This solution holds away from, but not near, the boundary of frequency space. The solution is possible only because the coefficients of the associated Jacobi field equations agree uniformly with those for the case of zero selection up to fourth powers in the selection parameters, whilst the covariance of the diffusion has only quadratic dependence.

Keywords

RANDOM SELECTION DIFFUSIONRAY SOLUTIONJACOBI FIELDSRIEMANNIAN GEOMETRYCURVATURE TENSORWFK PROCESSCHRISTOFFEL VELOCITY FIELDSWEAK CONVERGENCE OF FINITE MARKOV CHAINS
Type
Research Article
Information
Advances in Applied Probability , Volume 12 , Issue 1 , March 1980 , pp. 50 - 58
Copyright
Copyright © Applied Probability Trust 1980 

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Footnotes

Research partially supported by NRC-A-7667.

References

Antonelli, P. L. (1978) An asymptotic formula for the transition density of random genetic drift. J. Appl. Prob. 15, 184–186.Google Scholar
Antonelli, P. L. and Strobeck, C. (1977) The geometry of random drift I. Stochastic distance and diffusion. Adv. Appl. Prob. 9, 238–249.Google Scholar
Antonelli, P. L., Morgan, K. and Lathrop, G. M. (1977) The geometry of random drift III. Recombination and diffusion. Adv. Appl. Prob. 9, 260–267.Google Scholar
Felsenstein, J. Probability of fixation of a mutant when selection coefficients vary randomly with time. Unpublished.Google Scholar
Gillespie, J. (1973) Natural selection with varying selection coefficients—a haploid model. Genet. Res., Camb. 21, 115–120.Google Scholar
Jensen, L. and Pollak, E. (1969) Random selective advantages of a gene in a finite population. J. Appl. Prob. 6, 19–37.Google Scholar
Laugwitz, D. (1965) Differential and Riemannian Geometry. Academic Press, New York.Google Scholar
Molchanov, S. (1975) Diffusion processes and Riemannian geometry (in Russian). Usp. Math. Nauk. 30, 3–59. English translation: Russian Math. Surveys 30 (1975), 1–63.Google Scholar
Milnor, J. (1963) Morse Theory. Princeton University Study Series 51, Princeton University Press.Google Scholar
Pinsky, M. (1978) Stochastic Riemannian geometry. In Probabilistic Analysis and Related Topics, Vol. I. Academic Press, New York.Google Scholar