Metastability in Chemotaxis Models

We consider pattern formation in a chemotaxis model with a vanishing chemotaxis coefficient at high population densities. This model was developed in Hillen and Painter (2001, Adv. Appli. Math. 26(4), 280-301.) to model volume effects. The solutions show spatio-temporal patterns which allow for ultr...

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Bibliographic Details
Published in:Journal of dynamics and differential equations 2005-04, Vol.17 (2), p.293-330
Main Authors: Potapov, A. B., Hillen, T.
Format: Article
Language:English
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Summary:We consider pattern formation in a chemotaxis model with a vanishing chemotaxis coefficient at high population densities. This model was developed in Hillen and Painter (2001, Adv. Appli. Math. 26(4), 280-301.) to model volume effects. The solutions show spatio-temporal patterns which allow for ultra-long transients and merging or coarsening. We study the underlying bifurcation structure and show that the existence time for the pseudo- structures exponentially grows with the size of the system. We give approximations for one-step steady state solutions. We show that patterns with two or more steps are metastable and we approximate the two-step interaction using asymptotic expansions. This covers the basic effects of coarsening/merging and dissolving of local maxima. These effects are similar to pattern dynamics in other chemotaxis models, in spinodal decomposition of Cahn-Hilliard models, or to metastable patterns in microwave heating models.
ISSN:1040-7294
1572-9222
DOI:10.1007/s10884-005-2938-3