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Approximating the dynamics of communicating cells in a diffusive medium by ODEs – homogenization with localization. (English) Zbl 1277.35036
Summary: 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 \(a_i\) 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.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
35C20 Asymptotic expansions of solutions to PDEs
35Q92 PDEs in connection with biology, chemistry and other natural sciences
35K10 Second-order parabolic equations
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