Splitting trees with neutral mutations at birth

Mathieu Richard (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We consider a population model where individuals behave independently from each other and whose genealogy is described by a chronological tree called splitting tree. The individuals have i.i.d. (non-exponential) lifetime durations and give birth at constant rate to clonal or mutant children in an infinitely many alleles model with neutral mutations. First, to study the allelic partition of the population, we are interested in its frequency spectrum, which, at a fixed time, describes the number of alleles carried by a given number of individuals and with a given age. We compute the expected value of this spectrum and obtain some almost sure convergence results thanks to classical properties of Crump-Mode-Jagers (CMJ) processes counted by random characteristics. Then, by using multitype CMJ-processes, we get asymptotic properties about the number of alleles that have undergone a fixed number of mutations with respect to the ancestral allele of the population.

Mots clés: Splitting tree; Crump-Mode-Jagers branching processes; neutral mutation; infinitely many alleles model; frequency spectrum; multitype branching processes; almost-sure limit theorem; epidemiology.; epidemiology

Domaines: Probabilités [math.PR]

Mathieu Richard. Splitting trees with neutral mutations at birth. Stochastic Processes and their Applications, 2014, pp.Volume 124, Issue 10, October 2014, Pages 3206-3230. ⟨10.1016/j.spa.2014.05.008⟩. ⟨hal-00834732v2⟩