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Conditions for fixation of an allele in the density-dependent wright–Fisher models

Published online by Cambridge University Press:  14 July 2016

Fima C. Klebaner
Fima C. Klebaner*
Affiliation:
Monash University
*
∗Postal address: Department of Mathematics, Monash University, Clayton, VIC 3168, Australia.
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Abstract

A density-dependent Wright–Fisher model is a model where the population size changes randomly depending on the genetic composition process. If population sizes Mn vary without density dependence then the condition ΣMn–1 = ∞ is necessary and sufficient for fixation. It is shown that the above condition is no longer necessary for fixation in the density dependent models. Another necessary condition for fixation is given. Some known results on series of functions of sums of i.i.d. random variables are generalized to weighted sums.

Keywords

DENSITY DEPENDENCEWEIGHTED SUMS OF RANDOM VARIABLES
Type
Research Papers
Information
Journal of Applied Probability , Volume 25 , Issue 2 , June 1988 , pp. 247 - 256
Copyright
Copyright © Applied Probability Trust 1988 

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References

Doney, R. A. (1984) A note on some results of Schuh. J. Appl. Prob. 21, 192–196.Google Scholar
Donnelly, P. and Weber, N. (1985) The Wright-Fisher model with temporally varying selection and population size. J. Math. Biol. 22, 21–29.Google Scholar
Hall, P. and Heyde, C. C. (1980) Martingale Limit Theory and its Applications. Academic Press, New York.Google Scholar
Heyde, C. C. (1977) The effect of selection on genetic balance when the population size is varying. Theoret. Popn Biol. 11, 249–251.Google Scholar
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
Heyde, C. C. and Seneta, E. (1975) The genetic balance between random sampling and random population size. J. Math. Biol. 1, 317–320.Google Scholar
Schuh, H.-J. (1982) Sums of i.i.d. random variables and an application to the explosion criterion for Markov branching processes. J. Appl. Prob. 19, 29–38.Google Scholar
Seneta, E. (1974) A note on the balance between random sampling and population size. Genetics 77, 607–610.Google Scholar