Applied Maths Seminar: Harris -- Genealogies of samples from stochastic population models

Simon Harris (University of Auckland) is visiting the SMRI at the moment and will give a
seminar next Wednesday 21 January in Carslaw 451 at 12pm.  

Title: Genealogies of samples from stochastic population models Abstract: Consider some
population evolving stochastically in time.  Conditional on the population surviving
until some large time T, take a sample of individuals from those alive.  What does the
ancestral tree drawn out by this sample look like? Some special cases were known, e.g.
Durrett (1978), O'Connell (1995), but we will discuss some more recent advances for
Bienyamé-Galton-Watson (BGW) branching processes conditioned to survive.  

In near-critical or in critical varying environment BGW settings, the same universal
limiting sample genealogy always appears up to some deterministic time change (which
only depends on the mean and variance of the offspring distributions).  This
genealogical tree has the same binary tree topology as the classical Kingman coalescent,
but where the coalescent (or split) times are quite different due to stochastic
population size effects, with a representation as a mixture of independent identically
distributed times.  In contrast, in critical infinite variance offspring settings, we
find that more complex universal limiting sample genealogies emerge that exhibit
multiple-mergers, these being driven by rare but massive birth events within the
underlying population (eg.  "superspreaders" in an epidemic).  Some ongoing work, open
problems, and potential downstream applications will be discussed.  

This talk is based on collaborative works in Annals of Applied Probability (2020, 2024)
and Annals of Probability (2024) with collaborators S.Palau (UNAM), J.C.Pardo (CIMAT),
S.Johnston (Kings College London), and M.Roberts (Bath).  

I would also like to acknowledge the support of the New Zealand Royal Society Te
Apārangi Marsden fund.