Section: Research Program

Multiscale propagation phenomena in biology

Project team positioning

The originality of our work is the quantitative description of propagation phenomena for some models including several scales. We are able to compute the macroscopic speed of propagation and the distribution with respect to the microscopic variable at relevant locations (e.g. the edge and the back of the front) in a wide variety of models.

Multiscale modeling of propagation phenomena raises a lot of interest in several fields of application. This ranges from shock waves in kinetic equations (Boltzmann, BGK, etc...), bacterial chemotactic waves, selection-mutation models with spatial heterogeneities, age-structured models for epidemiology or subdiffusive processes.

Earlier works generally focused on numerical simulations, hydrodynamic limits to average over the microscopic variable, or specific models with only local features, not suitable for most of the relevant models. Our contribution enables to derive the relevant features of propagation analytically, and far from the hydrodynamic regime for a wide range of models including nonlocal interaction terms.

Our recent understanding is closely related to the analysis of large deviations in multiscale dispersion equations, for which we give important contributions too.

These advances are linked to the work of other Inria teams (BANG, DRACULA, BEAGLE), and collaborators in mathematics, physics and theoretical biology in France, Austria and UK.

Recent results

Vincent Calvez has focused on the modelling and analysis of propagation phenomena in structured populations. This includes chemotactic concentration waves, transport-reaction equations, coupling between ecological processes (reaction-diffusion) and evolutionary processes (selection of the fittest trait, adaptation), evolution of age structured poulations, and anomalous diffusion.

He has also continued his work on the optimal control of monotone linear dynamical systems, using the Hamilton-Jacobi framework, and the weak KAM theory.

Emeric Bouin has defended his PhD on December 2nd, 2014. He has accomplished his work under the supervision of Vincent Calvez and Emmanuel Grenier. He has studied propagation phenomena in multiscale models. He has focused on some specific behaviours arising from the multiscale nature of the problem, which are not described by classical reaction-diffusion models. For example, he has discovered unexpected acceleration behaviour in kinetic reaction-transport equations (Bouin, Calvez and Nadin, Arch. Ration. Mech. Anal. 2014).

Laetitia Giraldi was a post-doctoral fellow funded by the ANR grant MODPOL under the supervision of Vincent Calvez. She studied thoroughly a biomechanical model for the growth of plant or yeats cells. This new model couples standard equations for the displacement of the cell wall under internal pressure, and a reaction-diffusion equation set on the membrane accounting for the growth pattern has a function of the cell geometry. A rigorous linear stability analysis of the growing spherical shape, together with the development of a stable numerical scheme opens the way to future research in the coupling between growth and geometry.

Alvaro Mateos Gonzalez has started a PhD on September 2014 under the supervision of Vincent Calvez, and Hugues Berry (BEAGLE). He has already collaborated fruitfully with Thomas Lepoutre (DRACULA) and Hughes Berry to investigate the long-time asymptotics of a degenerate renewal equation. This is a first step towards the mathematical analysis of anomalous diffusion processes.

Collaborations

• Mathematical description of bacterial chemotactic waves:

  • N. Bournaveas (Univ. Edinburgh), V. Calvez (ENS de Lyon, Inria NUMED) B. Perthame (Univ. Paris 6, Inria BANG), Ch. Schmeiser (Univ. Vienna), N. Vauchelet: design of the model, analysis of traveling waves, analysis of optimal strategies for bacterial foraging.

  • J. Saragosti, V. Calvez (ENS de Lyon, Inria NUMED), A. Buguin, P. Silberzan (Institut Curie, Paris): experiments, design of the model, identification of parameters.

  • F. Filbet, C. Yang (Univ. Lyon 1): numerical simulations in 2D in curved geometries.

• Transport-reaction waves and large deviations:

  • E. Bouin, V. Calvez (ENS de Lyon, Inria NUMED), E. Grenier (ENS de Lyon, Inria NUMED), G. Nadin (Univ. Paris 6)

• Selection-mutation models of invasive species:

  • E. Bouin (ENS de Lyon, Inria NUMED), V. Calvez (ENS de Lyon, Inria NUMED), S. Mirrahimi (Inst. Math. Toulouse): construction of traveling waves, asymptotic propagation of fronts,

  • E. Bouin (ENS de Lyon, Inria NUMED), V. Calvez (ENS de Lyon, Inria NUMED), N. Meunier, (Univ. Paris 5), B. Perthame (Univ. Paris 6, Inria Bang), G. Raoul (CEFE, Montpellier), R. Voituriez (Univ. Paris 6): formal analysis, derivation of various asymptotic regimes.

• Age-structured equations for subdiffusive processes (just starting)

  • H. Berry (Inria BEAGLE), V. Calvez (ENS de Lyon, Inria NUMED), Th. Lepoutre (Inria DRACULA), P. Gabriel (Univ. UVSQ)

This work is also supportet by a PEPS project (CNRS) "Physique Théorique et ses Interfaces", led by N. Vauchelet (Univ. Paris 6).