# Population Evolution and Interacting Particle Systems

**Head :** Sylvie Méléard, Professor at Ecole Polytechnique.

**Confirmed researchers :**

Vincent Bansaye, Professeur Chargé de cours

Thierry Bodineau, CNRS Senior Researcher

Giovanni Conforti, Assistant professor

Lucas Gerin, Professeur Chargé de cours

Carl Graham, CNRS Junior Researcher

Igor Kortchemski, CNRS Junior Researcher

Gaël Raoul, CNRS Junior Researcher

Amandine Véber, CNRS Junior Researcher

**Ph.D. students and Post-Docs :**

Céline Bonnet, Ph.D. student

Dorian Collot, Ph.D.

Benoît Dagalier, Ph.D. student

Alejandro Fernandez-Montero, Ph.D. student

José Mendez, Ph.D. student at CMAP and UPMC

Pierre Montagnon, Ph.D. student at CMAP and INRA

Felipe Munoz, Ph.D. student at CMAP and University of Chile

Adrien Prodhomme, Ph.D. student at CMAP and University of Tours

Loïc Richier, Post-doc

Tristan Roget, Ph.D. student

Michele Salvi, Post-doc

Josué Tchouanti, Ph.D. student

Paul Thevenin, Ph.D. student

Julie Tourniaire, Ph.D. student

Anne van Gorp, Ph.D. student

**Former Members :**

Etienne Adam, CPGE teacher

Florent Barret, Assistant Professor (Nanterre)

Valère Bitseki Penda, Assistant Professor (Dijon)

Juliette Bouhours, Teacher

Fabien Condamine, CNRS (Montpellier)

Camille Coron, Assistant Professor (Orsay)

Manon Costa, Assistant Professor (Toulouse)

Joseba Dalmau, Post-doc (NYU Shanghai)

Clément Erignoux, Post-doc (IMPA, Rio)

Raphaël Forien, INRA (Avignon)

Sarah Kaakai, Assistant Professor (Le Mans)

Hélène Leman, INRIA (Lyon)

Aline Marguet, Post-doc (Grenoble)

Sepideh Mirrahimi, CNRS (Toulouse)

Daniel Moen, Assistant Professor (Oklahoma State University)

Hélène Morlon, CNRS Senior Researcher (ENS Paris)

Matthieu Richard, CPGE teacher

Jonathan Rolland, Post-doc (Lausanne)

Charline Smadi-Lasserre, Researcher at IRSTEA

Denis Villemonais, Assistant Professor (Nancy)

**Projects associated to the group : ** Chaire MMB, ANR MODELOV, ANR Large Scale Dynamics, ANR ABIM, ANR CADENCE, ANR GRAAL

**Research Activities**

Our group develops and studies probabilistic models of complex dynamics/systems, describing the interactions between populations, individuals or particles. In particular, we aim at characterising the evolution in time of these models and their convergence towards stationary or quasi-stationary distributions. The group PEIPS is structured around two major topics.

**Stochastic Analysis and Partial Differential Equations for the Evolution of the Living**

We develop apropriate stochastic and deterministic models to capture some major phenomena related to biodiversity, ecology and evolution. More specifically, we consider some complex systems that are essentially based on individual behaviours (cells, bacteria, species, populations, metapopulations) and take into account the biology of the problem as much as possible. A stochastic modelling enables us to quantify the fluctuations emanating from diverse sources: randomness in the size of a small population related to the births and deaths of individuals (genetic drift), random mutations appearing at reproduction (during the replication of DNA), randomly changing environments (climate changes), random displacements of individuals (impact of fragmenting their habitat). A deterministic modelling offers a more macroscopic point of view, in which individual behaviours are integrated into the evolution of the global system.

Our approach aims at constructing "good" models, in concertation with biologists, that are meant to be as close as possible to the phenomenon studied, but also sufficiently simple to be able to provide quantitative answers to the questions considered. These models are multi-scale, and depend on numerous parameters which quantify the relations between the different scales of time and space within the genetic, ecological and phenotypic parameters. The biological questions are mainly about the evolution (invasion and fixation of mutations, spatially structured genealogies, evolutionary branching, speciation, benefits of sexual and asexual reproduction) and the dynamics of populations (extinction, competition, scaling limits, quasi-stationary states, behaviour in random environments).

The tools that we use are essentially based on stochastic calculus, PDEs, measure-valued processes, coalescents and processes in random environment.

Our group holds the chaire program ''Modélisation Mathématique et Biodiversité'' with the Museum National d’Histoire Naturelle.

*Evolutionary branching and Darwin's finches :*

**Particle Systems, Statistical Mechanics, Random Discrete Models**

This research topic primarily deals with mathematical problems coming from out of equilibrium statistical mechanics. The goal is to use probabilistic methods to describe physical systems comprising a large number of interacting particles. The out of equilibrium statistical mechanics offers some very rich research perspectives, as it is much less understood than statistical mechanics at equilibrium (not only from the mathematical point of view, but also from the physics viewpoint). Indeed, there are no analogues to the Gibbs theory which may describe the large variety of observed macroscopic phenomena through microscopic models.

In this context, where the theoretical concepts are still to be defined, the study of specific microscopic models is essential to make progress in the analysis of transportation mechanisms and stationary measures. A model very well studied by our group is the asymetric exclusion process, which plays an analogous role to that of the Ising model in out of equilibrium statistical mechanics. More generally, we are interested in the out of equilibrium stationary distributions and their long-range correlations, the fluctuations in flows and in the dynamic phase transitions. Similar questions on the stationary distributions appear also in the theory of cellular automata, which are defined in a purely dynamic way.

*A simulation of out-of-equilibrium asymmetric exclusion :*

This topic is also tightly linked to some models of directed percolation and polymers, and more generally to certain properties of large discrete combinatorial objects: random trees and graphs, random planar maps.