A probabilistic and algebraic treatment of regular inbreeding systems
type: Item
instance of: scholarly article
title: A probabilistic and algebraic treatment of regular inbreeding systems
zbMATH Open document ID: 0585.92016
DOI: 10.1007/BF00275714
published in: Journal of Mathematical Biology
publication date: 1985-01-01T00:00:00Z
review text: This paper studies regular inbreeding systems in population genetics, with a view to determining those for which the population tends to genetic uniformity after an infinite time. Let the measure of genetic uniformity be \(f(x)=f\), the inbreeding coefficient for one individual x, namely the probability that the 2 distinct genes in x are identical by descent. Then in theorem 5, it is proved that in a regular mating system, \(f=1\) if and only if \(\sum A_ n^{-1}=\infty\) where \(A_ n\) is the number of ancestors in the n-th generation into the past. The inbreeding models used are represented by graphs on whose nodes random walks are assumed to be performed. The author shows that the converse of theorem 5 is false in general; two counter-examples are given to prove this. However, the converse is shown to be true for graphs of certain finitely presented semigroups.
reviewed by: J. Gani
Mathematics Subject Classification ID: 92D10
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 3939158
zbMATH Keywords: renewal event
zbMATH Keywords: homogeneity property
zbMATH Keywords: graphs of finitely presented
zbMATH Keywords: semigroups
zbMATH Keywords: martingales
zbMATH Keywords: random walks on graphs
zbMATH Keywords: regular inbreeding systems
zbMATH Keywords: population genetics
zbMATH Keywords: genetic uniformity
zbMATH Keywords: infinite time
zbMATH Keywords: identical by descent
zbMATH Keywords: regular mating system
zbMATH Keywords: counter-examples
Wikidata QID: Q112879201
author: Peter W. Arzberger
MaRDI profile type: Publication
cites work: Q3848243
cites work: Q5659379
cites work: Q5624436
cites work: Ratio theorems for random walks. II
description: scientific article; zbMATH DE number 3939158