Collective Behavior: Macroscopic versus Kinetic Descriptions

Travelling waves for kinetic equations issued from biology

Vincent Calvez

École Normale Supérieure de Lyon

Recently, kinetic-type models have raised a lot of interest in multiscale modelling of collective motion and dispersal evolution. For instance, kinetic models are a very good option for modelling concentration waves of chemotactic bacteria in a micro-channel [Saragosti et al. Proc. Natl. Acad. Sci. USA 2011]. Another example is the propagation of some invasive species with a high heterogeneity in dispersal capability among individuals. A minimal selection-mutation-diusion model, analogous to a kinetic equation, has been proposed [Benichou et al. Phys. Rev. E 2012], following previous works by Champagnat-Meleard and Arnold-Desvillettes-Prevost. It has been shown to reproduce the qualitative features of the propagating front [Bouin et al. C. R. Math. Acad. Sci. Paris 2012].

I will present some recent progresses about the existence of travelling waves for two analogous models: (1) a kinetic reaction-transport model, which coincides with the Fisher-KPP equation in the diusive limit, (2) a selection-mutation-diusion model where the phenotypical heterogeneity in the population aects the diusion of individuals.

I will present some quantitative results about the travelling front (when it exists), and some qualitative results about the spreading of the population (in the case where travelling waves do not exist). The last situation occurs when variables v or  take values in unbounded sets.