Hostname: page-component-cb9f654ff-65tv2 Total loading time: 0 Render date: 2025-08-10T09:57:25.379Z Has data issue: false hasContentIssue false
  • English
  • Français

Coalescent theory for a Monoecious Random Mating Population with a Varying Size

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Edward Pollak
Edward Pollak*
Affiliation:
Iowa State University
*
∗ Postal address: Department of Statistics, Iowa State University, Ames, IA 50011-1210, USA. Email address: pllk@iastate.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Consider a monoecious diploid population with nonoverlapping generations, whose size varies with time according to an irreducible, aperiodic Markov chain with states x 1 N,…,x K N, where K ≪ N. It is assumed that all matings except for selfing are possible and equally probable. At time 0 a random sample of n ≪ N genes is taken. Given two successive population sizes x j N and x i N, the numbers of gametes that individual parents contribute to offspring can be shown to be exchangeable random variables distributed as G ij . Under minimal conditions on the first three moments of G ij for all i and j, a suitable effective population size N e is derived. Then if time is recorded in a backward direction in units of 2N e generations, it can be shown that coalescent theory holds.

Keywords

Coalescentexchangeabilitypopulation genetics

MSC classification

Secondary: 92D10: Genetics 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 92D15: Problems related to evolution 60G09: Exchangeability

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 1 , March 2010 , pp. 41 - 57
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Ethier, S. N. and Nagylaki, T. (1980). Diffusion approximations of Markov chains with two time scales and applications to population genetics. Adv. Appl. Prob. 12, 14–49.CrossRefGoogle Scholar
[2] Franklin, J. N. (1968). Matrix Theory. Prentice Hall, Englewood Cliffs, NJ.Google Scholar
[3] Jagers, P. and Sagitov, S. (2004). Convergence to the coalescent in populations of substantially varying size. J. Appl. Prob. 41, 368–378.CrossRefGoogle Scholar
[4] Kingman, J. F. C. (1982). Exchangeability and the evolution of large populations. In Exchangeability in Probability and Statistics, eds Koch, G. and Spizzichino, F., North-Holland, Amsterdam, pp. 97–112.Google Scholar
[5] Kingman, J. F. C. (1982). On the genealogy of large populations. In Essays in Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 27–43.Google Scholar
[6] Kingman, J. F. C. (1982). The coalescent. Stoch. Process. Appl. 13, 235–248.CrossRefGoogle Scholar
[7] Möhle, M. (1998). A convergence theorem for Markov chains arising in population genetics and the coalescent with selfing. Adv. Appl. Prob. 30, 493–512.CrossRefGoogle Scholar
[8] Möhle, M. (1998). Coalescent results for two-sex population models. Adv. Appl. Prob. 30, 513–520.CrossRefGoogle Scholar
[9] Möhle, M. (2000). Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models. Adv. Appl. Prob. 32, 983–993.CrossRefGoogle Scholar
[10] Möhle, M. and Sagitov, S. (2003). Coalescent patterns in diploid exchangeable population models. J. Math. Biol. 47, 337–352.CrossRefGoogle ScholarPubMed
[11] Nagylaki, T. (1995). The inbreeding effective population number in dioecious populations. Genetics 139, 473–485.CrossRefGoogle ScholarPubMed
[12] Nordborg, M. (2001). Coalescent theory. In Handbook of Statistical Genetics, eds Balding, D. J., Bishop, M. J. and Cannings, C., John Wiley, Chichester, pp. 179–212.Google Scholar
[13] Nordborg, M. and Donnelly, P. (1997). The coalescent process with selfing. Genetics 146, 1185–1195.CrossRefGoogle ScholarPubMed
[14] Pollak, E. (2006). Genealogical theory for random mating populations with two sexes. Math. Biosci. 202, 133–155.CrossRefGoogle ScholarPubMed
[15] Pollak, E. (2007). Coalescent theory for a completely random mating monoecious population. Math. Biosci. 205, 315–324.CrossRefGoogle ScholarPubMed