Title: Linear and uniform in time estimates for the binary branching model with Moran interactions
Abstract: Branching processes are pertinent models for a range of different applications such as cell division, neutron transport and population dynamics. In many cases, they exhibit a so-called Perron Frobenius decomposition which gives the existence of a leading eigentriple that characterises the leading order behaviour of the first moment of the branching process. However, obtaining accurate estimates of this leading eigentriple can be challenging. In this talk we introduce an interacting particle system that approximates the branching process and can be used to estimate this eigentriple. We show that under some natural assumptions, the particle system admits linear and uniform in time estimates.
Title: Genealogical processes of non-neutral population models under rapid mutation
Abstract: The evolution of size-regulated populations is often modelled via discrete-time interacting particle systems in which offspring choose parents according to their fitnesses, and genetic types are inherited subject to mutation. Similar processes cycling through resampling and mutation steps arise in statistical computing for e.g. particle filtering and smoothing tasks, and for sampling from quasi-stationary distributions. The genealogical trees embedded into these models are objects of direct interest in genetics, and relevant for e.g. mixing and estimator variance of particle filtering algorithms. I will show that suitably scaled genealogical trees converge to the Kingman coalescent in the infinite-population limit under conditions which are strong, but standard in particle filtering. Similar convergence results are well-known for so-called neutral models, where parents are always equally fit. The non-neutral extension relies on rapid mutation to break down fitness correlations between generations, and covers a wide range of resampling schemes for allocating offspring to parents. Surprisingly, so-called minimum variance resampling schemes do not always result in a slower rate of coalescence than simple multinomial resampling.
Title: Scaling limits for genealogies of critical branching Markov processes
Title: On the survival of branching processes, generalised principal eigenvalues and stationary solutions of the FKPP equation
Abstract: H. Berestycki and Rossi introduced two notions of generalised principal eigenvalue, λ' and λ'' for non-divergence form uniformly elliptic operators, extending the more classical generalised principal eigenvalue. They studied the relationship between these different notions of generalised principal eigenvalue, and their relationship with the maximum principle. Here we relate the global survival or global extinction of branching processes to the positivity or negativity (respectively) of λ' , and moreover show that λ'' corresponds to the asymptotics of the expected number of particles.
Berestycki and Rossi established in this setting that λ'≤λ'', conjectured that one always has equality, and proved it for self-adjoint L in either one dimension or which is radially symmetric. Using our probabilistic interpretation, we prove the Berestycki-Rossi conjecture for general self-adjoint L, but provide a counterexample for non self-adjoint L. This provides a sharp characterisation of the validity of the maximum principle for self-adjoint L in unbounded domains.
Moreover we will connect the survival of branching processes with the existence and uniqueness of stationary solutions of the FKPP equation. Further, we will introduce a forthcoming work with Rossi on the uniqueness of stationary solutions of the FKPP equation in dimension at most 3, and explain its implications for branching Brownian motion.
This is joint work with Pascal Maillard and Luca Rossi.