Birth and death processes with neutral mutations

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 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 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 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.