Branching processes and neutral evolution.

*(English)*Zbl 0748.60081
Lecture Notes in Biomathematics. 93. Berlin: Springer-Verlag. viii, 112 p. (1992).

A general branching process models an asexually reproducing population. A point process gives the birth times of the initial ancestor’s children. Independent copies of this process give the birth times of these children’s children, relative to their parent’s birth time, and so on. Here the process is supercritical, a parent on average produces more than one child, and so with positive probability grows indefinitely. Stimulated by the neutral theory of molecular evolution the people are labelled; essentially children may either be the same type as their parent or of a new mutant type that has never occured before. Type has no effect on reproduction (so the effect of mutation is neutral). Many interesting questions about the resulting process, arising from the genetic context, are addressed in this monograph. As examples, there is discussion of: the asymptotics of the number of labels seen by time \(t\), and its characterization as a general branching process; the properties of the process formed by looking at the family tree centred on an individual randomly chosen from an old population, including the age of that person’s type; and, the probability two randomly sampled individuals share the same label (are “identical by descent”) and extensions of this.

The theory of general branching processes has been very well developed in recent years, mostly by P. Jagers and O. Nerman, and this monograph is a sustained application of that theory and as such is well worth reading. The author also gives a nice application of another important theoretical development in branching process, the definition of the process on the space of trees rather than on the usual Ulam-Harris sample space, due to J. Neveu [Ann. Inst. Henri PoincarĂ©, Probab. Stat. 22, 199-207 (1986; Zbl 0601.60082)]. It is this that allows the author to show that, treating the sub-tree formed by all individuals sharing a particular label as a single “particle”, the collection of these particles forms another general branching process.

The theory of general branching processes has been very well developed in recent years, mostly by P. Jagers and O. Nerman, and this monograph is a sustained application of that theory and as such is well worth reading. The author also gives a nice application of another important theoretical development in branching process, the definition of the process on the space of trees rather than on the usual Ulam-Harris sample space, due to J. Neveu [Ann. Inst. Henri PoincarĂ©, Probab. Stat. 22, 199-207 (1986; Zbl 0601.60082)]. It is this that allows the author to show that, treating the sub-tree formed by all individuals sharing a particular label as a single “particle”, the collection of these particles forms another general branching process.

Reviewer: J.D.Biggins (Sheffield)