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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
##### Keywords:
dimension reduction; quorum sensing
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##### References:
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