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## PEIPS: Evolution de Populations et Systèmes de Particules en Interaction

**Head:** Sylvie Méléard, Professeur, Ecole Polytechnique.

Vincent Bansaye, Professeur chargé de cours

Thierry Bodineau, Directeur de recherche CNRS

Lucas Gerin, Professeur chargé de cours

Carl Graham, Chargé de recherche CNRS

Igor Kortchemski , Chargé de Recherche CNRS

Sylvie Méléard, Professeur

Gael Raoul, Chargé de Recherche CNRS

Amandine Véber, Chargée de recherche CNRS

**Affiliated confirmed researcher :**

Jean-René Chazottes, Directeur de recherche CNRS, CPHT

Céline Bonnet, PhD Student

Juliette Bouhours, Post-doc

Raphaël Forien, PhD Student

Aline Marguet, PhD Student

Pierre Montagnon, PhD Student

Tristan Roget, PhD Student

Etienne Adam

Clément Erignoux

Valère Bitseki Penda

Camille Coron

Manon Costa

Hélène Leman

Hélène Morlon

Matthieu Richard

**Related projects: ** ANR Manège, Chaire MMB, ANR GRAAL

Our team develops probabilistic models describing complex dynamics or systems, which take into account the interaction between populations, individuals, or particles. We notably seek to determine the time evolution of these models, as well as their convergence to stationary or quasi-stationary distributions. The PEIPS team is organized around two major themes:

**A Probability Theory and Partial Differential Equations for the Evolution of Living Beings**

We develop pertinent stochastic and deterministic models for phenomena related to biodiversity, ecology, and evolution. More specifically, we consider complex systems based essentially on individual behaviors (cells, bacteria, species, populations, metapopulations) and which take into account biological facts as much as possible. Stochastic models allow to quantify fluctuations due to various sources: randomness in the size of a small population due to the deaths and births of individuals (genetic drift), randomness of mutations appearing at the level of reproduction (in DNA replication), randomness in environmental change (climatic change), randomness in the movement of individuals (impact of habitat fragmentation). Deterministic models yield a more macroscopic perspective in which individual behavior is integrated into an evolution describing the global behavior of the system.

Our approach aims to construct, in relation with biologists, “good” models, in the sense that they are as close as possible to the phenomena under study, but also sufficiently simple to yield quantitative answers to the problem under consideration. These are multi-scale models, depending on numerous parameters which will quantify the relations between these different time-scales, space-scales, and between the genetic, ecological, and phenotypic parameters. Biological questions bear essentially on the evolution - mutant invasion and fixation, spatially structured genealogies, evolutionary branching, speciation, scope of sexual and asexual reproduction - and on the dynamics of populations - extinction, competition, scale limits, quasi-stationary states, behavior in random media.

The tools we use are essentially related to stochastic calculus, PDEs, measured-valued processes, coalescents, and processes in random media.

Our team carries the *chaire Modélisation Mathématique et Biodiversité* together with *Museum National d’Histoire Naturelle*.

*Evolutionary branching and Darwin Finch :*

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

This research axis is concerned firstly about mathematical problems stemming from non-equilibrium statistical mechanics. The goal is to describe by probabilistic methods some systems constituted of a large number of interacting particles. Non-equilibrium statistical mechanics offers very rich research perspectives, since it is much less well understood than equilibrium statistical mechanics, not only from a mathematical but also from a physical viewpoint. Indeed, there is no theory analogue to Gibbs theory which allows to understand starting from microscopic models the variety of macroscopic phenomena that are observed.

In this context, in which theoretical concepts are yet to be defined, the study of specific microscopic models is essential in order to advance in the analysis of transport phenomena and of stationary measures. A model under much scrutiny by our team is the asymmetric exclusion process, which plays a role analogue to the role of the Ising model for non-equilibrium statistical mechanics. More generally, we are interested in out of equilibrium stationary measures and their long-range correlations, current fluctuations, and dynamical phase transitions. Similar questions on stationary measures also arise in the theory of cellular automata which are defined in purely dynamical terms.

*A aimulation of asymmetric exclusion out of equilibrium:*

This theme is also closely related to directed percolation and polymer models, and more generally to certain properties of large discrete combinatorial objects: random trees and graphs, random planar maps.