Abstract
The aim of this article is to study lattice models of neutral multi-alleles including Ohta-Kimura's step-wise mutation model. We shall show an outline of the construction of a unique strongly continuous non-negative semi-group associated with the infinite dimensional generator and show a general and straightforward method of obtaining the time dependent and equilibrium solutions of all polynomial moments of the gene frequencies. We shall discuss the spectrum of the diffusion processes and as an application we obtain all higher moments of the homozygosity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ethier, S. N.: A class of degenerate diffusion processes occurring in population genetics. Comm. Pure Appl. Math. 29, 483–493 (1976)
Ethier, S. N.: A class of infinite-dimensional diffusions occurring in population genetics. Indiana Univ. Math. J. (in press) (1982)
Ethier, S. N., Kurtz, T. G.: The infinitely many neutral alleles diffusion model (in press) (1982)
Fleming, W. H., Viot, M.: Some measure-valued Markov processes in population genetics theory. Indiana Univ. Math. J. 28, 817–843 (1979)
Griffiths, R. C.: A transition density expansion for a multi-alleles diffusion model. Adv. Appl. Prob. 11, 310–325 (1979)
Kesten, H.: The number of distinguishable alleles according to the Ohta-Kimura model of neutral mutation. J. Math. Biol. 10, 167–187 (1980)
Kingman, J. F. C.: Coherent random walks arising in some genetical models. Proc. R. Soc. London A: 351, 19–31 (1976)
Maruyama, T.: Stochastic problems in population genetics. Lecture notes in biomathematics, Vol. 17. Berlin-Heidelberg-New York: Springer 1977
Moran, P. A. P.: Wandering distribution and the electrophoretic profile. Theoret. Population Biology 8, 318–330 (1975)
Notohara, M.: Eigenanalysis for the Kolmogorov backward equation for the neutral multi-allelic model. J. Math. Biol. 11, 235–244 (1981)
Notohara, M., Shiga, T.: Convergence to genetically uniform state in stepping stone models. J. Math. Biol. 10, 281–294 (1980)
Ohta, T., Kimura, M.: A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population. Genet. Res. 22, 201–204 (1973)
Perlow, J.: The transition density of multiple neutral alleles. Theoret. Population Biology 16, 223–232 (1979)
Sato, K.: Convergence to a diffusion of a multi-allelic model in population genetics. Adv. Appl. Prob. 10, 538–562 (1978)
Shiga, T.: An interacting system in population genetics. J. Math. Kyoto Univ. 20, 213–242 (1980)
Shiga, T.: An interacting system in population genetics II. J. Math. Kyoto Univ. (in press) (1982)
Shiga, T.: Diffusion processes in population genetics. J. Math. Kyoto Univ. (in press) (1982)
Shiga, T.: Continuous time multi-allelic stepping stone models in population genetics. J. Math. Kyoto Univ. (in press) (1982)
Shiga, T., Shimizu, A.: Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20, 395–416 (1980)
Shimakura, N.: Equations differentielles provenant de la genetique des population. Tohoku Math. J. 29, 287–318 (1977)
Stewart, F. M.: Variability in the amount of heterozygosity maintained by neutral mutations. Theoret. Population Biology 9, 188–201 (1976)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Notohara, M. The lattice models of neutral multi-alleles in population genetics theory. J. Math. Biology 15, 79–92 (1982). https://doi.org/10.1007/BF00275790
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00275790