Hostname: page-component-7b7567c8c-fvjsw Total loading time: 0 Render date: 2026-11-12T23:19:50.878Z Has data issue: false hasContentIssue false
  • English
  • Français

Fixation in bisexual models with variable population sizes

Published online by Cambridge University Press:  14 July 2016

M. Möhle
M. Möhle*
Affiliation:
University of Chicago
*
∗ Postal address: The University of Chicago, Department of Statistics, 5734 University Avenue, Chicago, Illinois 60637, USA. Also at Johannes Gutenberg-Universität Mainz, Fachbereich 17 Mathematik, Saarstraβe 21, 55099 Mainz, Germany.
Get accessRights & Permissions [Opens in a new window]

Abstract

A general exchangeable bisexual model with variable population sizes is introduced. First the forward process, i.e. the number of certain descending pairs, is studied. For the bisexual Wright-Fisher model fixation of the descendants occurs, i.e. their proportion tends to 0 or 1 almost surely.

The main part of this article deals with necessary and sufficient conditions for ultimate homozygosity, i.e. the proportion of an arbitrarily chosen allelic type tends to 0 or 1 almost surely. The results are applied to a bisexual Wright-Fisher model and to a bisexual Moran model.

Keywords

EXCHANGEABLE PROCESSESFIXATIONHOMOZYGOSITYMORAN MODELWRIGHT-FISHER MODEL

MSC classification

Primary:92D25: Population dynamics (general)

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 34 , Issue 2 , June 1997, pp. 436 - 448
Copyright
Copyright © Applied Probability Trust 1997 

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] Donnelly, P. (1986) Genealogical approach to variable-population-size models in population genetics. J. Appl. Prob. 23, 283–296.CrossRefGoogle Scholar
[2] Hall, P. G. and Heyde, C. C. (1980) Martingale Limit Theory and its Application. Academic Press, New York.Google Scholar
[3] Heyde, C. C. (1983) On limit theorems for gene survival. Colloq. Math. J. Bolyai. Google Scholar
[4] Heyde, C. C. (1983) An alternative approach to asymptotic results on genetic composition when the population size is varying. J. Math. Biol. 18, 163–168.CrossRefGoogle ScholarPubMed
[5] 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
[6] Kämmerle, K. (1989) Vertauschbare bisexuelle Populationsmodelle. Dissertation. Johannes Gutenberg-Universität Mainz.Google Scholar
[7] Kämmerle, K. (1989) Looking forwards and backwards in a bisexual Moran model. J. Appl. Prob. 26, 880–885.CrossRefGoogle Scholar
[8] Kämmerle, K. (1991) The extinction probability of descendants in bisexual models of fixed population size. J. Appl. Prob. 28, 489–502.CrossRefGoogle Scholar
[9] Möhle, M. (1994) Forward and backward processes in bisexual models with fixed population sizes. J. Appl. Prob. 31, 309–332.CrossRefGoogle Scholar
[10] Seneta, E. (1974) A note on the balance between random sampling and population size. Genetics 77, 607–610.CrossRefGoogle ScholarPubMed