Hostname: page-component-7dd5485656-glrdx Total loading time: 0 Render date: 2025-10-29T14:40:01.089Z Has data issue: false hasContentIssue false
  • English
  • Français

Diffusion models in population genetics

Published online by Cambridge University Press:  14 July 2016

Motoo Kimura
Motoo Kimura*
Affiliation:
National Institute of Genetics, Mishima, Japan
Get access Rights & Permissions [Opens in a new window]

Extract

Population genetics is that branch of genetics, whose object is the study of the genetical make-up of natural populations. By investigating the laws which govern the genetic structure of natural populations, we intend to clarify the mechanism of evolution.

Information

Type
Review Paper
Information
Journal of Applied Probability , Volume 1 , Issue 2 , December 1964 , pp. 177 - 232
Copyright
Copyright © Applied Probability Trust 

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

[1] Bodmer, W. F. (1960) Discrete stochastic processes in population genetics. J.R. Statist. Soc. B 22, 218–244.Google Scholar
[2] Cavalli-Sforza, L.L. and Conterio, F. (1960) Analisi della fluttuazione di frequenze geniche nella popolazione della. Val Parma. Atti A.G.I. 5, 333–344.Google Scholar
[3] Crow, J.F. (1954) Breeding structure of populations II. Effective population number. Statistics and Mathematics in Biology. Kempthorne, etal. (ed). Iowa State College Press, Ames, Iowa.Google Scholar
[4] Crow, J.F. and Morton, N. (1955) Measurement of gene frequency drift in small populations. Evolution 9, 202–214.CrossRefGoogle Scholar
[5] Crow, J. F. and Kimura, ?. (1956) Some genetic problems in natural populations. Proc. Third Berkeley Symp. on Math. Statist. and Prob. 4, 1–22.Google Scholar
[6] Ewens, W. J. (1963) Numerical results and diffusion approximations in a genetic process. Biometrika 50, 241–249.Google Scholar
[7] Ewens, W. J. (1964) The pseudo-transient distribution and its uses in genetics. J. Appl. Prob. 1, 141–156.Google Scholar
[8] Feller, W. (1951) Diffusion processes in genetics. Proc. Second Berkeley Symp. on Math. Statist. and Prob. 227–246.Google Scholar
[9] Feller, W. (1952) The parabolic differential equations and the associated semigroup of transformations. Ann. Math. 55, 468–519.Google Scholar
[10] Fisher, R.A. (1922) On the dominance ratio. Proc. Roy. Soc. Edin. 42, 321–341.Google Scholar
[11] Fisher, R.A. (1930) The distribution of gene ratios for rare mutations. Proc. Roy. Soc. Edin. 50, 205–220.Google Scholar
[12] Fisher, R.A. (1953) Population genetics. Proc. Roy. Soc. London B 141, 510–523.Google Scholar
[13] Fisher, R.A. (1958) The Genetical Theory of Natural Selection (2nd revised ed.). Dover, New York.Google Scholar
[14] Fokker, A.D. (1914) Die mittlere Energie rotierender elektrischer Dipole im Strahlungsfeld. Ann. d. Phys. 43, 810–820.CrossRefGoogle Scholar
[15] Goldberg, S. (1950) On a singular diffusion equation. Ph. D. thesis (unpublished). Cornell University.Google Scholar
[16] Haldane, J.B.S. (1924) A mathematical theory of natural and artificial selection. Part I. Trans. Camb. Phil. Soc. 23, 19–41.Google Scholar
[17] Haldane, J.B.S. (1927) A mathematical theory of natural and artificial selection. Part V: Selection and mutation. Proc. Camb. Phil. Soc. 23, 838–844.Google Scholar
[18] Haldane, J.B.S. (1932) The Causes of Evolution. Harper and Brothers, New York.Google Scholar
[19] Haldane, J.B.S. (1949) Suggestions as to quantitative measurement of rates of evolution. Evolution 3, 51–56.CrossRefGoogle ScholarPubMed
[20] Karlin, S. and Mcgregor, J. (1962) On a genetics model of Moran. Proc. Camb. Phil. Soc. 58, 299–311.Google Scholar
[21] Kimura, M. (1954) Process leading to quasi-fixation of genes in natural populations due to random fluctuation of selection intensities. Genetics 39, 280–295.Google Scholar
[22] Kimura, M. (1955a) Solution of a process of random genetic drift with a continuous model. Proc. Nat. Acad. Sci. 41, 144–150.CrossRefGoogle ScholarPubMed
[23] Kimura, M. (1955b) Random genetic drift in multi-allelic locus. Evolution 9, 419–435.CrossRefGoogle Scholar
[24] Kimura, M. (1955C) Stochastic processes and distribution of gene frequencies under natural selection. Cold Spring Harbor Symp. 20, 33–53.CrossRefGoogle ScholarPubMed
[25] Kimura, M. (1956a) Random genetic drift in a tri-allelic locus: Exact solution with a continuous model. Biometrics 12, 57–66.Google Scholar
[26] Kimura, M. (1956b) Stochastic processes in population genetics. Ph. D. thesis (unpublished). Univ. of Wisconsin.Google Scholar
[27] Kimura, M. (1957) Some problems of stochastic processes in genetics. Ann. Math. Statist. 28, 882–901.CrossRefGoogle Scholar
[28] Kimura, M. (1958) On the change of population fitness by natural selection. Heredity 12, 145–167.CrossRefGoogle Scholar
[29] Kimura, M. (1962) On the probability of fixation of mutant genes in a population. Genetics 47, 713–719.CrossRefGoogle ScholarPubMed
[30] Kimura, M. and Crow, J.F. (1963) The measurement of effective population number. Evolution 17, 279–288.CrossRefGoogle Scholar
[31] Kolmogorov, A. (1931) über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Math. Ann. 104, 415–458.CrossRefGoogle Scholar
[32] Malecot, G. (1948) Les Mathématiques de l'Hérédité. Masson et Cie, Paris.Google Scholar
[33] 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.CrossRefGoogle Scholar
[34] Moran, P.A.P. (1958a) A general theory of the distribution of gene frequencies. I. Overlapping generations. Proc. Roy. Soc. London. B 149, 102–112.Google ScholarPubMed
[35] Moran, P.A.P. (1958b) A general theory of the distribution of gene frequencies. II. Nonoverlapping generations. Proc. Roy. Soc. London B 149, 113–116.Google Scholar
[36] Moran, P.A.P. (1958c) Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.Google Scholar
[37] Moran, P.A.P. (1961) The survival of a mutant under general conditions. Proc. Camb. Phil. Soc. 57, 304–314.CrossRefGoogle Scholar
[38] Moran, P.A.P. (1962) The Statistical Processes of Evolutionary Theory. Clarendon Press, Oxford.Google Scholar
[39] Morse, P.M. and Feshbach, H. (1953) Methods of Theoretical Physics. McGraw-Hill, New York.Google Scholar
[40] Planck, M. (1917) über einen Satz der statistischen Dynamik und seine Erweiterung in der Quantentheorie. Sitz. der.preuss. Akad. 324–341.Google Scholar
[41] Robertson, A. (1960) A theory of limits in artificial selection. Proc. Roy. Soc. London B 153, 234–249.Google Scholar
[42] Robertson, A. (1962) Selection for heterozygotes in small populations. Genetics 47, 1291–1300.CrossRefGoogle ScholarPubMed
[43] Stratton, J.A., Morse, P.M., Chu, L.J., and Hutner, R.A. (1941) Elliptic Cylinder and Spheroidal Wave Functions. John Wiley, New York.Google Scholar
[44] Stratton, J.A., Morse, P.M., Chu, L.J., Little, J.D.C., and Corbató, F.J. (1956) Spheroidal Wave Functions. Technology Press of M.I.T. & John Wiley, New York.Google Scholar
[45] Watterson, G.A. (1962) Some theoretical aspects of diffusion theory in population genetics. Ann. Math. Stat. 33, 939–957.Google Scholar
[46] Watterson, G.A. (1964) The application of diffusion theory to two population genetic models of Moran. J. Appl. Prob. 1, 233–246.CrossRefGoogle Scholar
[47] Wright, S. (1931) Evolution in Mendelian populations. Genetics 16, 97–159.CrossRefGoogle ScholarPubMed
[48] Wright, S. (1937) The distribution of gene frequencies in populations. Proc. Nat, Acad. Sci. 23, 307–320.CrossRefGoogle ScholarPubMed
[49] Wright, S. (1938a) The distribution of gene frequencies under irreversible mutation. Proc. Nat. Acad. Sci. 24, 253–259.CrossRefGoogle ScholarPubMed
[50] Wright, S. (1938b) The distribution of gene frequencies in populations of polyploids. Proc. Nat. Acad. Sci. 24, 372–377.CrossRefGoogle ScholarPubMed
[51] Wright, S. (1939) The distribution of self-sterility alleles in populations. Genetics 24, 538–552.CrossRefGoogle ScholarPubMed
[52] Wright, S. (1942) Statistical genetics and evolution. Bull. Amer. Math. Soc. 48, 223–246.CrossRefGoogle Scholar
[53] Wright, S. (1945) The differential equation of the distribution of gene frequencies. Proc. Nat. Acad. Sci. 31, 382–389.CrossRefGoogle ScholarPubMed
[54] Wright, S. (1948) On the roles of directed and random changes in gene frequency in the genetics of populations. Evolution 2, 279–294.CrossRefGoogle ScholarPubMed
[55] Wright, S. (1949) Adaptation and selection. In Genetics, Paleontology, and Evolution. Jepsen, et al. (ed.), Princeton Univ. Press.Google Scholar
[56] Wright, S. (1950) Population structure as a factor in evolution. In Moderne Biologie, Festschrift für Hans Nachtsheim, F. W. Peters, Berlin.Google Scholar
[57] Wright, S. (1952) The theoretical variance within and among subdivisions of a population that is in a steady state. Genetics 37, 312–321.Google Scholar
[58] Wright, S. and Kerr, W.E. (1954) Experimental studies of the distribution of gene frequencies in very small populations of Drosophila melanogaster. II. Bar. Evolution 8, 225–240.Google Scholar