Approximating the dynamics of communicating cells in a diffusive medium by ODEs-homogenization with localization

Bacteria may change their behavior depending on the population density. Here we study a dynamical model in which cells of radius within a diffusive medium communicate with each other via diffusion of a signalling substance produced by the cells. The model consists of an initial boundary value problem for a parabolic PDE describing the exterior concentration of the signalling substance, coupled with ODEs for the masses of the substance within each cell. We show that for small the model can be approximated by a hierarchy of models, namely first a system of coupled delay ODEs, and in a second step by coupled ODEs. We give some illustrations of the dynamics of the approximate model.