A mathematical model of discrete attachment to a cellulolytic biofilm using random DEs

Jack M. Hughes; Hermann J. Eberl; Stefanie Sonner; Jack M. Hughes; Hermann J. Eberl; Stefanie Sonner

doi:10.3934/mbe.2022310

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 7: 6582-6619. doi: 10.3934/mbe.2022310

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A mathematical model of discrete attachment to a cellulolytic biofilm using random DEs

1.
Department of Mathematics, University of British Columbia, 1984 Mathematics Rd., Vancouver, BC V6T 1Z2, Canada
2.
Department of Mathematics and Statistics, University of Guelph, 50 Stone Rd E., Guelph, ON N1G 2W1, Canada
3.
Radboud University, IMAPP – Mathematics, Postbus 9010, 6500 GL Nijmegen, The Netherlands

Received: 15 January 2022 Revised: 02 April 2022 Accepted: 19 April 2022 Published: 26 April 2022

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.
- attachment,
- cellulolytic biofilm,
- cellulolsic biofilm,
- mathematical model,
- nonlinear diffusion,
- numerical simulation,
- random ordinary differential equation
Citation: Jack M. Hughes, Hermann J. Eberl, Stefanie Sonner. A mathematical model of discrete attachment to a cellulolytic biofilm using random DEs[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 6582-6619. doi: 10.3934/mbe.2022310

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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.

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