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 [R] 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 [u] of the signalling substance, coupled with [N] ODEs for the masses [ai] of the substance within each cell. We show that for small [R] the model can be approximated by a hierarchy of models, namely first a system of [N] coupled delay ODEs, and in a second step by [N] coupled ODEs. We give some illustrations of the dynamics of the approximate model.