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Branching processes and neutral mutations. (English) Zbl 0676.92003

Mathematical statistics, theory and applications, Proc. World Congr. Bernoulli Soc., Tashkent/USSR 1986, Vol. 2, 683-692 (1987).
[For the entire collection see Zbl 0671.00013.]
A single type Crump-Mode-Jagers asexually branching population (\(\Omega\),\({\mathcal F},P)\) is considered [see P. Jagers and the author, Adv. Appl. Probab. 16, 221-259 (1984; Zbl 0535.60075)], where the “life” \(\omega\in \Omega\) tells us the life span \(\lambda\) and for all \(i=1,2,...\), the reproduction ages \(\tau_ i\) and the genetic status \(\rho_ i\) of the off-spring. Here \(\rho_ i=1\) if the i-th child carries the same allele and \(\rho_ i=0\) if it is a mutant with an entirely new allele.
Branching process theory is applied to study the asymptotics of the genetical composition of this exponentially growing population. A measure of evolutionary speed is given since the proportions of different family constellations stabilize. Also the conditional probability that two individuals sampled at random carry the same allele as well as the asymptotics of the times when the oldest allele in the population first appeared are calculated. The proofs will appear in a forthcoming dissertation of Taib.
Reviewer: G.Jetschke

MSC:

92D10 Genetics and epigenetics
60J85 Applications of branching processes