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Solving a singular diffusion equation occurring in population genetics

Published online by Cambridge University Press:  14 July 2016

Louis Jensen
Louis Jensen*
Affiliation:
Iowa State University, Ames
*
*Now at The University of Michigan, Ann Arbor.
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Abstract

A technique for solving a partial differential equation is presented. The technique is based upon the known solution of a similar equation. The method is used to attempt to solve the equation describing the change in the frequency of an allele in the presence of selection and random drift in a finite population. Two cases can be solved within a reasonable degree of approximation: (a) the viabilities are additive and (b) heterozygotes are symmetrically overdominant to the homozygotes. The solutions in both cases are compared with the exact discrete solutions found by powering the re evant transition matrix.

Keywords

DIFFUSION EQUATIONSCHANGE IN GENE FREQUENCY AND SELECTION
Type
Research Papers
Information
Journal of Applied Probability , Volume 11 , Issue 1 , March 1974 , pp. 1 - 15
Copyright
Copyright © Applied Probability Trust 1974 

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Footnotes

Journal Paper No. J-7168 of the Agriculture and Home Economics Experiment Station, Ames, Iowa. Project No. 1669. Research supported in part by NIH Grant GM-13827.

References

Courant, R. and Hilbert, D. (1937) Methods of Mathematical Physics. Vol. I. Interscience, New York.Google Scholar
Crow, J. and Kimura, M. (1970) An Introduction to Population Genetics Theory. Harper and Row, New York.Google Scholar
Ewens, W. J. (1963) Numerical results and diffusion approximations in a genetic process Biometrika 50, 241–249.Google Scholar
Feller, W. (1952) The parabolic differential equations and the associated semigroup of transformations Ann. of Math. 55, 468–519.Google Scholar
Gautschi, W. (1967) Computational aspects of three-term recurrence relations Soc. Industrial and Appl. Math. Review 9, 24–82.Google Scholar
Hill, W. G. and Robertson, A. (1968) The effects of inbreeding at loci with heterozygote advantage Genetics 60, 615–628.Google Scholar
Kantorovich, L. V. and Krylov, V. I. (1958) Approximate Methods of Higher Analysis. Interscience, New York.Google Scholar
Kimura, M. (1955a) Solution of a process of random genetic drift with a continuous model Proc. Nat. Acad. Sci. U.S.A. 41, 144–150.Google Scholar
Kimura, M. (1955b) Stochastic processes and distribution of gene frequencies under natural selection Cold Spring Harbor Symposium on Quantatative Biology 20, 33–53.Google Scholar
Kimura, M. (1964) Diffusion models in population genetics J. Appl. Prob. 1, 177–232.Google Scholar
Miller, G. F. (1962) The evaluation of eigenvalues of a differential equation arising in a problem in genetics Proc. Camb. Phil. Soc. 58, 588–593.Google Scholar
Morse, P. M. and Feshbach, H. (1953) Methods of Theoretical Physics. Part I. McGraw Hill. New York.Google Scholar
Stratton, J. A., Morse, P. M., Chu, L. J. and Hunter, R. A. (1941) Elliptic Cylinder and Spheroidal Wave Functions. John Wiley, New York.Google Scholar
Watterson, G. A. (1962) Some theoretical aspects of diffusion theory in population genetics Ann. Math. Statist. 33, 939–957.CrossRefGoogle Scholar