Modèles stochastiques en dynamique des populations et génétique

mercredi 7 mai 2025, 14:00 → 17:30 Europe/Paris

Salle Pierre Grisvard (IHP - Bâtiment Borel)

 

1 Introduction to stochastic models of phylogenies and genealogies¶

We will review various stochastic models describing the phylogenies and genealogies for evolving populations: coalescent processes, branching processes, historical processes and processes valued in marked measured metric spaces.

Orateur: Prof. Viet Chi Tran (INRIA Lille)

15:00 Pause café (longue)

2 Genealogies in structured frequency-dependent branching processes¶

Genealogies in structured frequency-dependent branching processes
Our work delves into the universality class of a very celebrated entity in population genetics : Kingman’s coalescent. This object serves as baseline models for panmictic, neutral populations. More generally, Λ−coalescents catalog the genealogies of constant–sized exchangeable populations models known as Cannings models. We establish a broad class of regulated multitype individual-based models extending beyond exchangeable and fix-sized populations, yet for which the scaling limit of the genealogies sampled at a given time is still Kingman’s coalescent. Thus, our result strides towards refining the intuition that neutral, Cannings–like genealogies can arise from complex interactions. We prove convergence in distribution of the genealogies for the Gromov–Weak topology, using a change of measure and building on the multiple spine decomposition formalism developed by Foutel-Rodier and Schertzer (2023). The underlying purpose is to formulate a versatile methodology to derive scaling limits of genealogies within structured populations subject to type–frequency interactions. This method streamlines computations on forest-valued processes to a fine–grained analysis of the simpler stochastic process driving the type frequencies in the population. Kingman’s coalescence rates are recovered from the contribution of the spinal immigration to a small deviation of this frequency process, and the underneath spatial structure only influences the effective population size, hence the genetic diversity of the population. This is joint work with Félix Foutel-Rodier (MAP5) and Emmanuel Schertzer (University of Vienna).

Orateur: Mathilde André (IBENS, PSL, INSERM & Université de Vienne)

3 A generalised spatial branching process with ancestral branching to model the growth of a filamentous fungus¶

We introduce a spatial branching process to model the growth of the mycelial network of a filamentous fungus. In this model, each filament is described by the position of its tip, the trajectory of which is the solution to a stochastic differential equation with a drift term which depends on all the other trajectories. Each filament can branch either at its tip or along its length, that is to say at some past position of its tip, at some time- and space-dependent rates. It can stop growing at some rate which also depends on the positions of the other tips. We study the large population limit of this process and we characterise the limiting process as the weak solution to a system of partial differential equations.

Orateur: Lena Kuwata (MAP5, Université Paris Cité)

16:30 Pause café (courte)

4 Long-time behaviour of a multidimensional age-dependent branching process with a singular jump kernel¶

In this article, we investigate the ergodic behaviour of a multidimensional age-dependent branching process with a singular jump kernel, motivated by studying the phenomenon of telomere shortening in cell populations. Our model tracks individuals evolving within a continuous-time framework indexed by a binary tree, characterised by age and a multidimensional trait. Branching events occur with rates dependent on age, where offspring inherit traits from their parent with random increase or decrease in some coordinates, while the most of them are left unchanged. Exponential ergodicity is obtained at the cost of an exponential normalisation, despite the fact that we have an unbounded age-dependent birth rate that may depend on the multidimensional trait, and a non-compact transition kernel. These two difficulties are respectively treated by stochastically comparing our model to Bellman-Harris processes, and by using a weak form of a Harnack inequality. We conclude this study by giving examples where the assumptions of our main result are verified. This talk is based on joint work with Milica Tomasevic.

Orateur: Jules Olayé (CMAP, École polytechnique)