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I am interested in the study, the simulation and the microscopic and macroscopic modeling of populations subject to evolution, from the viewpoint of ecology (interactions in the population are precisely modeled), in the biological theoretical framework of adaptive dynamics, of population genetics, and more generally in population dynamics.
One of the problem I am studying is the mathematical understanding of the phenomenon of evolutionary branching, where a population initially concentrated around a single phenotype is driven by the evolutionary forces to states where two (or more) distinct phenotypes coexist, without geographical separation. This phenomenon is at the origin of diversity in ecosystems, and is believed to be an important mechanism for sympatric speciation, i.e. separation of a single species into two without geographical separation. I have developed and participated in two approaches to this question, the first one based on stochastic individual-based models, and the second one based on the asymptotic analysis of nonlocal PDEs.
I am also interested in the study of spatial organisation (clustering) of spatially structured populations, of robustness and of evolution by punctuated equilibria. In this context, I also study the spatial structuration of clonal plants via stochastic individual-based models. Another line of research I am currently developing in collaboration with C. Fritsch during her post-doc concerns the growth of bacterial populations structured by mass within a chemostat.
I am also interested in the population genetics of branching populations, where I try to extend classical properties of constant population size models (Wright-Fisher, Cannings or Moran models) to populations growing according to general branching processes. In particular, I am studying the frequency spectrum in such a population subject to neutral mutations.
Finally, stochastic models of growth of tumoral populations taking into account their pH regulation mechanisms were developed in collaboration with J. Pouyssegur, N. Mazure, D. Talay during the PhD thesis of J. Claisse.
The basic natural models in this biological setting are measure-valued interacting particles systems with birth, death, competition, mutation, resource consumption, dispersal or spatial diffusion.
I am interested in various biologistically realistic scalings of this microscopic model, yielding the convergence to, on the one hand, PDEs, SPDEs, deterministic or stochastic integro-differential equations and superprocesses; and, on the other hand, to jump processes, ODEs or diffusion processes, by multi-scale analysis (timescales separation). I am also interested in probabilistic interpretations of some other natural PDE models arising in this setting, that can be used to study evolutionary branching from the analytical side.
I also study the long time behaviour of these limiting models, linked to the biological phenomenon of evolutionary branching. In the case of the diffusion processes, this problem is linked to the large deviations theory and to the problem of exit from a domain. The mathematical study of evolutionary branching also requires fine properties of long time behaviour of competitive Lotka-Volterra systems.
In population genetics, out work on branching populations relies strongly on special properties of appropriate contour processes, which allow to recover a Markovian property (Lévy process) in models where life-lengths need not to be exponential. This is the subject of the current PhD thesis of B. Henry.
Finally, an important part of my work is devoted to numerical simulations of the microscopic models and their scaling limits, specifically for spatially structured populations.
This part of my work is developed in three main directions:
This point is developed mainly for the Poisson-Boltzmann equation mentioned above as an application to molecular dynamics. My interest lies in the theoretical development of probabilistic interpretations of linear and non-linear equations with stochastic differential equations, branching diffusions and backward stochastic differential equations, and in the practical implementation of these methods. Apart from the applications to Poisson-Boltzmann equation, I am currently working on the use of branching diffusions and large deviations to obtain a new interpretation of the Hamilton-Jacobi equation with constraint arising from the PDE approach to adaptive dynamics and evolutionary branching. I am also working on variance reduction techniques to solve some of the drawbacks of Feynman-Kac formulas based on branching diffusions.
For general absorbed Markov processes, quasi-stationary distributions (QSDs) are distributions which are invariant for the process conditionally on non-absorption. I developed recently with D. Villemonais general necessary and sufficient criteria to ensure the existence, uniqueness, and exponential convergence w.r.t. the total variation norm to QSDs. These results allow us to extend and generalize many results obtained in the past years to a large variety of processes (birth and deat processes with catastrophes, neutron transport processes, individual-based models). We are currently working on the implications of our results on one-dimensional diffusion with killing, and to multi-dimensional diffusions with K. Coulibaly-Pasquier. My works with P. Diaconis and L. Miclo and with S. Roelly also concern this topic.
SDE are involved in many parts of my work as a tool for modeling of to solve PDEs using Monte-Carlo techniques. I also worked on strong existence and pathwise uniqueness for SDEs with non-smooth coefficients in collaboration with P.-E. Jabin. One of the extensions of these results I am currently working on concerns the construction and the strong error analysis of Euler schemes for SDEs with non-smooth coefficients.
This point was already mentionned above concerning the mathematical study of evolutionary branching (see my work with P.-E. Jabin and G. Raoul). I am also interested in the question of well-posedness for dynamical systems. Jointly with P.-E. Jabin, we improved the existing conditions on the force fields in Hamiltonian dynamical systems to ensure existence and uniqueness when the force field has less than one derivative (H^3/4).
My interest in this area comes from three applications: