2010/09/04 by Yu-Ting Chen, Chen, Yu-Ting, Jean‐François Delmas +2 · 2 citations
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Bayesian Methods and Mixture Models #FOS: Biological sciences #FOS: Mathematics #Populations and Evolution (q-bio.PE) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #math.PR #q-bio.PE
paper · pdf · doi:10.48550/arxiv.1009.0814
arxiv created 2010/09/04 · openalex publication_date 2010/09/04 · arxiv updated 2010/09/07 · openalex created_date 2019/07/30 · openalex updated_date 2026/07/28
We present an elementary model of random size varying population given by a stationary continuous state branching process. For this model we compute the joint distribution of: the time to the most recent common ancestor, the size of the current population and the size of the population just before the most recent common ancestor (MRCA). In particular we show a natural mild bottleneck effect as the size of the population just before the MRCA is stochastically smaller than the size of the current population. We also compute the number of old families which corresponds to the number of individuals involved in the last coalescent event of the genealogical tree. By studying more precisely the genealogical structure of the population, we get asymptotics for the number of ancestors just before the current time. We give explicit computations in the case of the quadratic branching mechanism. In this case, the size of the population at the MRCA is, in mean, less by 1/3 than size of the current population size. We also provide in this case the fluctuations for the renormalized number of ancestors.