A Mathematical Model of Discrete Attachment to a Cellulolytic Biofilm using Random DEs

Abstract

We propose a new mathematical framework for the addition of stochastic attachment to biofilm models via the use of random ordinary differential equations. We focus our approach on a spatially explicit model of cellulolytic biofilm growth and formation that comprises a PDE-ODE coupled system to describe the biomass and carbon respectively. The model equations are discretized in space using a standard finite volume method. We introduce discrete attachment events into the discretized model via an impulse function with a standard stochastic process as input. We solve our model with an implicit ODE solver. We provide basic simulations to investigate the qualitative features of our model. We then perform a grid refinement study to investigate the spatial convergence of our model. We investigate model behaviour while varying key attachment parameters. Lastly, we use our attachment model to provide evidence for a stable travelling wave solution to the original PDE-ODE coupled system.