A probabilistic and algebraic treatment of regular inbreeding systems
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.
scientific article; zbMATH DE number 3939158