Splitting trees with neutral Poissonian mutations I: Small families

Nicolas Champagnat 1, 2 Amaury Lambert 3
INRIA Lorraine, CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, INPL - Institut National Polytechnique de Lorraine, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : We consider a neutral dynamical model of biological diversity, where individuals live and reproduce independently. They have i.i.d. lifetime durations (which are not necessarily exponentially distributed) and give birth (singly) at constant rate b. Such a genealogical tree is usually called a splitting tree, and the population counting process (N_t;t\geq 0) is a homogeneous, binary Crump--Mode--Jagers process. We assume that individuals independently experience mutations at constant rate \theta during their lifetimes, under the infinite-alleles assumption: each mutation instantaneously confers a brand new type, called allele, to its carrier. We are interested in the allele frequency spectrum at time t, i.e., the number A(t) of distinct alleles represented in the population at time t, and more specifically, the numbers A(k,t) of alleles represented by k individuals at time t, k=1,2,...,N_t. We mainly use two classes of tools: coalescent point processes and branching processes counted by random characteristics. We provide explicit formulae for the expectation of A(k,t) in a coalescent point process conditional on population size, which apply to the special case of splitting trees. We separately derive the a.s. limits of A(k,t)/N_t and of A(t)/N_t thanks to random characteristics. Last, we separately compute the expected homozygosity by applying a method characterizing the dynamics of the tree distribution as the origination time of the tree moves back in time, in the spirit of backward Kolmogorov equations.
Type de document :
Article dans une revue
Stochastic Processes and their Applications, Elsevier, 2012, 122 (3), pp.1003-1033. 〈10.1016/j.spa.2011.11.002〉
Liste complète des métadonnées

Littérature citée [20 références]  Voir  Masquer  Télécharger

Contributeur : Nicolas Champagnat <>
Soumis le : jeudi 10 mars 2016 - 16:30:13
Dernière modification le : vendredi 4 janvier 2019 - 17:32:34
Document(s) archivé(s) le : lundi 13 juin 2016 - 09:23:10


Fichiers produits par l'(les) auteur(s)


Distributed under a Creative Commons Paternité 4.0 International License



Nicolas Champagnat, Amaury Lambert. Splitting trees with neutral Poissonian mutations I: Small families. Stochastic Processes and their Applications, Elsevier, 2012, 122 (3), pp.1003-1033. 〈10.1016/j.spa.2011.11.002〉. 〈inria-00515481〉



Consultations de la notice


Téléchargements de fichiers