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

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.