Section: Research Program

Multiscale propagation phenomena in biology

Project team positioning

The originality of our work is the quantitative description of propagation phenomena accounting for several time and spatial scales. Here, propagation has to be understood in a broad sense. This includes propagation of invasive species, chemotactic waves of bacteira, evoluation of age structures populations ... Our main objectives are the quantitative calculation of macroscopic quantities as the rate of propagation, and microscopic distributions at the edge and the back of the front. These are essential features of propagation which are intimately linked in the long time dynamics.

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, evolution in age-structured population 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 biological situations. 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 (e.g. PDMP), for which we gave important contributions too in collaboration with E. Bouin (CEREMADE Dauphine), E. Grenier (Inria NUMED) and G. Nadin (Univ. Paris 6).

These advances are linked to the work of other Inria teams (MAMBA, 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. As a main result, he could establish the existence of concentration waves of chemotactic bacteria E. coli in a fully coupled kinetic/reaction-diffusion system previously validated on experimental data.

In collaboration with a group of theoretical biologists at ISEM Montpellier (O. Ronce and O. Cotto), and J. Garnier (Univ. Savoie), Th. Lepoutre (Inria DRACULA), Th. Bourgeron (Inria NUMED) he has investigated quantitatively the maladaptation of an age-structured population in a changing environment. He has unravelled a striking phenomenon of severe maladaptation specific to age structure. This was observed on numerical simulations by biologists, but it has now a systematic mathematical comprehension.

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, in collaboration with P. Gabriel (UVSQ) and S. Gaubert (Inria MAXPLUS).

Alvaro Mateos Gonzalez has started his PhD on September 2014 under the supervision of Vincent Calvez, and Hugues Berry (BEAGLE), . He has already collaborated fruitfully with Thomas Lepoutre (DRACULA) and Hugues 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. In collaboration with P. Gabriel (UVSQ) and V. Calvez (Inria NUMED) he has investigated large deviations of heterogenous continuous time random walks.

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.

• 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 anomalous diffusion processes, and evolution

  • H. Berry (Inria BEAGLE), V. Calvez (ENS de Lyon, Inria NUMED), Th. Lepoutre (Inria DRACULA), P. Gabriel (Univ. UVSQ), O. Ronce (ISEM Montpellier), O. Cotto (ISEM Montpellier), J. Garnier (Univ. Savoie).