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An infinite-alleles version of the simple branching process. (English) Zbl 0653.92009

This long paper is a branching process approach to the problem of neutral mutation. The population size is no longer constant as for example in the Wright-Fisher model but develops according to the laws of a (single type) Galton-Watson process given by the probability generating function (p.g.f.) f(s). At birth each individual mutates with probability u and the new mutant is of an entirely novel allelic type not currently or previously existing in the population. Because of this mutation mechanism there is no fixed labelling of allelic types and the model does not fit into the theory of branching processes with a finite or an infinite a number of types.

Nevertheless, branching process techniques can be used to investigate quantities of interest in genetics. The qualitative behavior of these quantities is mainly determined by $m={f}^{\text{'}}\left(1\right)$, the expected total number of offspring of one individual, and by $M=m\left(1-u\right)$, the expected number of offspring having the parental type.

In Section 2 (following the introduction), the number of alleles, especially ${K}_{n}$, the number of alleles in the n-th generation, is investigated. The authors give an expression for E ${K}_{n}$ and two derivations for the p.g.f. of ${K}_{n}$ leading to a strong limit result when $m>1$, to an exponential limit law when $m=1$, and to a conditional limit law when $m<1$. In the last three subsections a Poisson limit distribution for ${K}_{n}$ is found (when suitably scaling the initial population size and the mutation probability) and results on the total number of alleles seen up to generation n are given.

In Section 3, L, the index of the last generation containing new alleles is examined for $m\le 1$ when the initial population size becomes large. When $m=1$, the limit distribution of L is independent of the mutation probability and coincides with that of the extinction time T. But when $m<1$, the limit distribution of L depends on u and the difference T-L is difficult to investigate.

In the last section, the authors investigate the frequency spectrum ${\phi }^{\left(n\right)}\left(j\right)=E{\alpha }_{n}\left(j\right)$ and its limiting behavior $\left({\alpha }_{n}\left(j\right)$ denotes the number of alleles having j representatives in the -th generation). These investigations turn out to be very difficult and lead to two conjectures which are valid at least in the fractional linear case (for the p.g.f. f(s)). Despite the sign of m-1, one also has to take i A computerized fuzzy graphic rating scale which is an extension of a semantic differential is described. The scale allows respondents to provide an imprecise rating and lends itself to analysis using fuzzy set theory.

##### MSC:
 92D10 Genetics 60J85 Applications of branching processes 60J80 Branching processes