Stabilization of solutions of a semilinear parabolic equation modelling quorum sensing in a biofilm

Article type: Research Article

Authors: Fila, Marek | Winkler, Michael

Affiliations: Department of Applied Mathematics and Statistics, Comenius University, 84248 Bratislava, Slovakia | Fachbereich Mathematik, Universität Duisburg-Essen, 45117 Essen, Germany

Abstract: We analyze a mathematical model introduced by Anguige, Ward and King (J. Math. Biol. 51 (2005), 557–594) to describe a quorum sensing mechanism in spatially-structured populations of the bacteria Pseudomonas aeruginosa. In the biologically relevant limit case when the spatial distribution of bacteria is constant in time, this model reduces to a single semilinear parabolic equation for the concentration of the signal substance N-(3-oxododecanoyl)-homoserine lactone (AHL). We show that under some mild technical assumptions on the nonlinear AHL production rate, the AHL concentration approaches a uniquely determined steady state, and that this convergence takes place at an exponential rate.

Keywords: semilinear parabolic equation, quorum sensing, steady state, stabilization

DOI: 10.3233/ASY-2008-0920

Journal: Asymptotic Analysis, vol. 62, no. 1-2, pp. 89-106, 2009