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\backslashgeq 0) is a homogeneous, binary Crump--Mode--Jagers process. We assume that individuals independently experience mutations at constant rate \backslashtheta 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.
@article{champagnat_splitting_2012,
Author = {Champagnat, Nicolas and Lambert, Amaury},
Title = {Splitting trees with neutral {Poissonian} mutations
{I}: {Small} families},
Journal = {Stochastic Processes and their Applications},
Volume = {122},
Number = {3},
Pages = {1003--1033},
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\backslash{}geq 0) is a homogeneous, binary
Crump--Mode--Jagers process. We assume that individuals
independently experience mutations at constant rate
\backslash{}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.},
doi = {10.1016/j.spa.2011.11.002},
issn = {03044149},
language = {en},
month = mar,
shorttitle = {Splitting trees with neutral {Poissonian} mutations
{I}},
url = {http://linkinghub.elsevier.com/retrieve/pii/S0304414911002778},
urldate = {2014-08-31},
year = 2012
}
