SMILE

Stochastic Models for the Inference of Life Evolution

Birth and Death Processes with Neutral Mutations

Champagnat, N., Lambert, A., Richard, M.

International Journal of Stochastic Analysis

2012

In this paper, we review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at birth of individuals or at a constant rate during their lives. In both models, we study the allelic partition of the population at time t. We give closed formulae for the expected frequency spectrum at t and prove pathwise convergence to an explicit limit, as t goes to infinity, of the relative numbers of types younger than some given age and carried by a given number of individuals (small families). We also provide convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models.

Bibtex

@article{champagnat_birth_2012,
Author = {Champagnat, Nicolas and Lambert, Amaury and Richard,
Mathieu},
Title = {Birth and {Death} {Processes} with {Neutral}
{Mutations}},
Journal = {International Journal of Stochastic Analysis},
Volume = {2012},
Pages = {1--20},
abstract = {In this paper, we review recent results of ours
concerning branching processes with general lifetimes
and neutral mutations, under the infinitely many
alleles model, where mutations can occur either at
birth of individuals or at a constant rate during their
lives. In both models, we study the allelic partition
of the population at time t. We give closed formulae
for the expected frequency spectrum at t and prove
pathwise convergence to an explicit limit, as t goes to
infinity, of the relative numbers of types younger than
some given age and carried by a given number of
individuals (small families). We also provide
convergences in distribution of the sizes or ages of
the largest families and of the oldest families. In the
case of exponential lifetimes, population dynamics are
given by linear birth and death processes, and we can
most of the time provide general formulations of our
results unifying both models.},
doi = {10.1155/2012/569081},
issn = {2090-3332, 2090-3340},
language = {en},
url = {http://www.hindawi.com/journals/ijsa/2012/569081/},
urldate = {2014-08-31},
year = 2012
}

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