Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-27T10:33:05.660Z Has data issue: false hasContentIssue false
  • English
  • Français

A genealogical approach to variable-population-size models in population genetics

Published online by Cambridge University Press:  14 July 2016

Peter Donnelly
Peter Donnelly*
Affiliation:
University College of Swansea
*
∗Present address: Department of Statistical Science, University College London, London, WC1E 6BT, U.K.
Get access Rights & Permissions [Opens in a new window]

Abstract

A general exchangeable model is introduced to study gene survival in populations whose size changes without density dependence. Necessary and sufficient conditions for the occurrence of fixation (that is the proportion of one of the types tending to 1 with probability 1) are obtained. These are then applied to the Wright–Fisher model, the Moran model, and conditioned branching-process models. For the Wright–Fisher model it is shown that certain fixation is equivalent to certain extinction of one of the types, but that this is not the case for the Moran model.

Keywords

GENE SURVIVALGENEALOGICAL PROCESSESWRIGHT–FISHER MODELMORAN MODELEXCHANGEABLE MODELSOCCUPANCY PROBLEMS
Type
Research Paper
Information
Journal of Applied Probability , Volume 23 , Issue 2 , June 1986 , pp. 283 - 296
Copyright
Copyright © Applied Probability Trust 1986 

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.)

References

Billingsley, P. (1979) Probability and Measure. Wiley, New York.Google Scholar
Buckley, M. J. and Seneta, E. (1983) The genetic balance between varying population size and selective neutrality. J. Math. Biol. 17, 217–222.CrossRefGoogle Scholar
Cannings, C. (1974) The latent roots of certain Markov chains arising in genetics: a new approach 1. Haploid models. Adv. Appl. Prob. 6, 260–290.CrossRefGoogle Scholar
Daley, D. J., Hall, P., and Heyde, C. C. (1982) Further results on the survival of a gene represented in a founder population. J. Math. Biol. 14, 355–363.CrossRefGoogle Scholar
Donnelly, P. (1985) Dual processes and an invariance result for exchangeable models in population genetics. J. Math. Biol. 23, 103–118.CrossRefGoogle Scholar
Donnelly, P. and Weber, N. (1985) The Wright–Fisher model with temporally varying selection and population size. J. Math. Biol. 22, 21–29.CrossRefGoogle Scholar
Heyde, C. C. (1977) The effect of selection on genetic balance when the population size is varying. Theoret. Popn. Biol. 11, 249–251.CrossRefGoogle ScholarPubMed
Heyde, C. C. (1981) On the survival of a gene represented in a founder population. J. Math. Biol. 12, 91–99.CrossRefGoogle Scholar
Heyde, C. C. (1983a) On limit theorems for gene survival. Colloq. Math. J. Bolyai. Google Scholar
Heyde, C. C. (1983b) An alternative approach to asymptotic results on genetic composition when the population size is varying. J. Math. Biol. 18, 163–168.CrossRefGoogle ScholarPubMed
Heyde, C. C. and Seneta, E. (1975) The genetic balance between random sampling and random population size. J. Math. Biol. 1, 317–320.CrossRefGoogle Scholar
Holgate, P. (1966) A mathematical study of the founder principle of evolutionary genetics. J. Appl. Prob. 3, 115–128.CrossRefGoogle Scholar
Karlin, S. (1968) Rates of approach to homozygosity for finite stochastic models with variable population size. Amer. Naturalist 102, 443–455.CrossRefGoogle Scholar
Kesten, H. (1971) Sums of random variables with infinite expectation. Amer. Math. Monthly 78, 305–308.CrossRefGoogle Scholar
Kingman, J. F. C. (1976) Coherent random walks arising in some genetical problems. Proc. R. Soc. London A 351, 19–31.Google Scholar
Kingman, J. F. C. (1980) Mathematics of Genetic Diversity. CBMS-NSF Regional Conference in Applied Mathematics 34. Society for Industrial and Applied Mathematics, Philadelphia.CrossRefGoogle Scholar
Mayr, E. (1942) Systematics and the Origin of Species. Columbia University Press, New York (Reprinted by Dover Publications, New York, 1964).Google Scholar
Moran, P. A. P. (1958) Random processes in genetics. Proc. Camb. Phil. Soc. 54, 60–71.CrossRefGoogle Scholar
Riordan, J. (1958) An Introduction to Combinatorial Analysis. Wiley, New York.Google Scholar
Seneta, E. (1974) A note on the balance between random sampling and population size. Genetics 77, 607–610.CrossRefGoogle ScholarPubMed
Tavare, S. (1984) Line-of-descent and genealogical processes, and their applications in population genetics models. Theoret. Popn Biol. 26, 119–164.CrossRefGoogle ScholarPubMed