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Travelling-wave analysis of a model describing tissue degradation by bacteria

Published online by Cambridge University Press:  01 October 2007

D. HILHORST ,
J. R. KING  and
M. RÖGER
D. HILHORST
Affiliation:
CNRS and Laboratoire de Mathématiques, Université de Paris-Sud, 91405 Orsay Cedex, France (email: Danielle.Hilhorst@math.u-psud.fr)
J. R. KING
Affiliation:
Centre for Mathematical Medicine, Theoretical Mechanics Section, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK (email: John.King@nottingham.ac.uk)
M. RÖGER
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, D-04103 Leipzig, Germany (email: roeger@mis.mpg.de)
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Abstract

We study travelling-wave solutions for a reaction-diffusion system arising as a model for host-tissue degradation by bacteria. This system consists of a parabolic equation coupled with an ordinary differential equation. For large values of the ‘degradation-rate parameter’ solutions are well approximated by solutions of a Stefan-like free boundary problem, for which travelling-wave solutions can be found explicitly. Our aim is to prove the existence of travelling waves for all sufficiently large wave speeds for the original reaction-diffusion system and to determine the minimal speed. We prove that for all sufficiently large degradation rates, the minimal speed is identical to the minimal speed of the limit problem. In particular, in this parameter range, non-linear selection of the minimal speed occurs.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 5 , October 2007 , pp. 583 - 605
Copyright
Copyright © Cambridge University Press 2007

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