Splitting trees with neutral Poissonian mutations I: Small families

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:122

Article

Champagnat, Nicolas¹  Lambert, Amaury^2,3 

[1] Nancy Univ, TOSCA Project Team, INRIA Nancy Grand Est, IECN UMR 7502, F-54506 Vandoeuvre Les Nancy, France

[2] Univ Paris 06, F-75252 Paris 05, France

[3] UMR 7599 CNRS, Lab Probabilites & Modeles Aleatoires, F-75252 Paris 05, France

关键词: Branching process;    Coalescent point process;    Splitting tree;    Crump-Mode-Jagers process;    Linear birth-death process;    Allelic partition;    Infinite alleles model;    Poisson point process;    Levy process;    Scale function;    Regenerative set;    Random characteristic;   

DOI  :  10.1016/j.spa.2011.11.002

来源: Elsevier

【 摘 要 】

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 [9], and the population counting process (N-t: t >= 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 an 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, as defined in [15], and branching processes counted by random characteristics, as defined in [11,13]. We provide explicit formulae for the expectation of A(k, t) conditional on population size in a coalescent point process, 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, in the same vein as in [19]. Last, we separately compute the expected homozygosity by applying a method introduced in [14], 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. C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

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