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

Yousef Rohanizadegan; Stefanie Sonner; Hermann J. Eberl; Yousef Rohanizadegan; Stefanie Sonner; Hermann J. Eberl

doi:10.3934/mbe.2020119

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 3: 2236-2271. doi: 10.3934/mbe.2020119

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Research article

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

1.
Department of Mathematics and Statistics, University of Guelph, 50 Stone Rd E, Guelph ON, N1G 2W1, Canada
2.
Radboud University, Department of Mathematics, Postbus 9010, 6500 GL Nijmegen, The Netherlands

Received: 28 October 2019 Accepted: 02 January 2020 Published: 15 January 2020

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.
- attachment,
- cellulolytic biofilm,
- cellulosic biofilm,
- mathematical model,
- nonlinear diffusion,
- numerical simulation,
- stochastic differential equation
Citation: Yousef Rohanizadegan, Stefanie Sonner, Hermann J. Eberl. Discrete attachment to a cellulolytic biofilm modeled by an Itô stochastic differential equation[J]. Mathematical Biosciences and Engineering, 2020, 17(3): 2236-2271. doi: 10.3934/mbe.2020119

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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.

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