Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T12:40:42.719Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Markov processes Stochastic 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 x1N,…,xKN, 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 xjN and xiN, the numbers of gametes that individual parents contribute to offspring can be shown to be exchangeable random variables distributed as Gij. Under minimal conditions on the first three moments of Gij for all i and j, a suitable effective population size Ne is derived. Then if time is recorded in a backward direction in units of 2Ne 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
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