Abstract
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 the birth of particles 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-form formulae for the expected frequency spectrum at t and prove a pathwise convergence to an explicit limit, as t→+∞, of the relative numbers of types younger than some given age and carried by a given number of particles (small families). We also provide the 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.
Highlights
We consider a general branching model, where particles have i.i.d. not necessarily exponential life lengths and give birth at a constant rate b during their lives to independent copies of themselves
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 the birth of particles or at a constant rate during their lives
The process that counts the number of the alive particles through time is a Crump-Mode-Jagers process or general branching process 4 which is binary births occur singly and homogeneous constant birth rate
Summary
We consider a general branching model, where particles have i.i.d. not necessarily exponential life lengths and give birth at a constant rate b during their lives to independent copies of themselves. We are first interested in the allelic partition of the population and more precisely in properties about the frequency spectrum Mti,a, i ≥ 1 , where Mti,a is the number of distinct types younger than a i.e., whose original mutation appeared after t − a carried by exactly i particles at time t This kind of question was first studied by Ewens 18 who discovered the well-known “sampling formula” named after him and which describes the law of the allelic partition for a Wright-Fisher model with neutral mutations. In the monography , Taıb is interested in general branching processes known as Crump-Mode-Jagers processes see 4, and references therein where mutations still occur at birth but with a probability that may depend, for example, on the age of the mother He obtained limited theorems about the frequency spectrum by using random characteristics techniques but in most cases, limits cannot be explicitly computed. Specific effort has been put on finding a unifying formulation for our results as soon as it seemed possible
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