Discrete attachment to a cellulolytic biofilm modeled by an Itô stochastic differential equation

Abstract

We propose a mathematical framework for introducing random attachment of bacterial cells in a deterministic continuum model of cellulosic biofilms. The underlying growth model is a highly nonlinear coupled PDE-ODE system. It is regularised and discretised in space. Attachment is described then via an auxiliary stochastic process that induces impulses in the biomass equation. The resulting system is an Itô stochastic differential equation. Unlike the more direct approach of modeling attachment by additive noise, the proposed model preserves non-negativity of solutions. Our numerical simulations are able to reproduce characteristic features of cellulolytic biofilms with cell attachment from the aqueous phase. Grid refinement studies show convergence for the expected values of spatially integrated biomass density and carbon concentration. We also examine the sensitivity of the model with respect to the parameters that control random attachment.

Keywords: attachment; cellulolytic biofilm; cellulosic biofilm; mathematical model; nonlinear diffusion; numerical simulation; stochastic differential equation.