Appears in collection : 2022 - T1 - WS3 - Mathematical models in ecology and evolution

Joint work with Stephen Cantrell, Yuan Lou and Benoît Perthame.

To investigate the evolution of dispersal in spatially heterogeneous environments, Dockery et al. in 1998 formulated a di˙usion-competition system of N species which are identical except for their di˙usion rates, and conjectured that the slowest di˙using species al-ways competitively exclude all its counterparts. A continuum version of the problem was formulated by Perthame and Souganidis in 2016. In this selection-mutation model, the population is structured by both space and the di˙usion rate, and where mutation acts on the latter phenotypic variable. The rare mutation limit of the time-dependent solution is believed to be well described by certain Hamilton-Jacobi equation with a constraint, but rigorous results are limited to the case without spatial structure. In this talk, we will describe some recent progress on both problems by introducing the concept of principal Floquet bundle for parabolic equations.