Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T19:48:53.120Z Has data issue: false hasContentIssue false
  • English
  • Français

Time adaptive numerical solution of a highly non-linear degenerate cross-diffusion system arising in multi-species biofilm modelling

Part of: Parabolic equations and systems Numerical analysis: Ordinary differential equations

Published online by Cambridge University Press:  10 September 2018

MARYAM GHASEMI ,
STEFANIE SONNER  and
HERMANN J. EBERL
MARYAM GHASEMI
Affiliation:
Department Applied Mathematics, University of Waterloo, Waterloo Ontario, Canada email: m23ghase@uwaterloo.ca
STEFANIE SONNER
Affiliation:
Department Mathematics, Radboud University, Nijmegen, the Netherlands email: stefanie.sonner@uni-graz.at
HERMANN J. EBERL
Affiliation:
Department Mathematics and Statistics, University of Guelph, Guelph Ontario, Canada email: heberl@uoguelph.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We propose a numerical method for the simulation of a quasi-linear parabolic biofilm model that exhibits three non-linear diffusion effects: (i) a power law degeneracy, (ii) a super diffusion singularity and (iii) non-linear cross-diffusion. The method is based on a spatial Finite Volume discretisation in which cross-diffusion terms are formally treated as convection terms. Time-integration of the resulting semi-discretised system is carried out using an error-controlled, time-adaptive, embedded Rosenbrock–Wanner method. We compare several variants of the method and two variants of the model to investigate how details such as the choice cross-diffusion coefficients, and specific variants of the time integrator affect simulation time.

Keywords

biofilmcross-diffusiontime-adaptivity

MSC classification

Primary: 35K65: Degenerate parabolic equations
Secondary: 65L05: Initial value problems 68U20: Simulation 92D25: Population dynamics (general)
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Special Issue 6: Biofilms , December 2018 , pp. 1035 - 1061
Copyright
Copyright © Cambridge University Press 2018 

Footnotes

†This studies was financially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) with a PGS-D scholarship awarded to MG, as well as a Discovery Grant and a Research Tools and Infrastructure Grant awarded to HJE.

References

[1] Anaya, V., Bendahmane, M., Langlais, M. & Sepülveda, M. (2015) A convergent finite volume method for a model of indirectly transmitted diseases with nonlocal cross-diffusion. J. Comput. Math. Appl. 70, 132–157.Google Scholar
[2] Anderl, J. N., Franklin, M. J. & Stewart, P. S. (2000) Role of antibiotic penetration limitation in Klebsiella pneumoniae biofilm resistance to ampicillin and ciprofloxacin. Antimicrob. Agents Chemother. 44, 1818–1824.Google Scholar
[3] Andreianov, B., Bendahmane, M. & Baier, R. R. (2011) Analysis of a finite volume method for a cross-diffusion model in population dynamics. Math. Models Methods Appl. Sci. (ISSN: 0218–2025) 21 (2), 307–344, world scientific publishing.Google Scholar
[4] Barrett, J. W. & Blowey, J. F. (2004) Finite element approximation of a nonlinear cross-diffusion population model. Numer. Math. 98, 195221.Google Scholar
[5] Costerton, J. W., Stewart, P. S. & Greenberg, E. P. (2017) Bacterial biofilms: A common cause of persistent infections. Science 248 (5418), 1318–1322.Google Scholar
[6] Eberl, H. J., Parker, D. F. & Van Loosdrecht, C. M. (2001) A new deterministic spatio-temporal continuum model for biofilm development. J. Theor. Med. 3, 161–175.Google Scholar
[7] Eberl, H. J. & Collinson, S. (2009) A modelling and simulation study of siderophore mediated antagonsim in dual-species biofilms. Theor. Biol. Med. Mod. 6, 30.Google Scholar
[8] Efendiev, M. A., Zelik, S. V. & Eberl, H. J. (2009) Existence and longtime behaviour of a biofilm model. Comm. Pure Appl. Anal. 8 (2), 509–531.Google Scholar
[9] Ghasemi, M. & Eberl, H. J. (2017) Extension of a regularization based time-adaptive numerical method for a degenerate diffusion-reaction biofilm growth model to systems involving quorum sensing. Proc. Comp. Sci. 108, 1893–1902.Google Scholar
[10] Ghasemi, M. & Eberl, H. J. (2018) Time adaptive numerical solution of a highly degenerate diffusion-reaction biofilm model based on regularisatio. J. Sci. Comp. 74, 1060–1090.Google Scholar
[11] Galiano, G., Garzon, M. L. & Jungel, A. (2001) Analysis and numerical solution of a nonlinear cross-diffusion system arising in population dynamics. Rev. R. Acad. Cien. Ser. A Mat. 95, 281295.Google Scholar
[12] Galiano, G., Garzon, M. L. & Jungel, A. (2003) Semi-discretization in time and numerical convergence of solutions of a nonlinear cross-diffusion population model. Numer. Math. 93, 655673.Google Scholar
[13] Hall-Stoodley, L., Costerton, J. W., & Stoodley, P. (2004) Bacterial biofilms: From the natural environment to infectious diseases. Nat. Rev. Microbiol. 2 (2), 95–108.Google Scholar
[14] Khassehkhan, H., Efendiev, M. A. & Eberl, H. J. (2008) Existence and simulations of solutions to a degenerate diffusion-reaction model of an amensalistic biofilm control system, Discrete Cont. Dyn. Syst. B. 12 (2), 371–388.Google Scholar
[15] Khassehkhan, H., Hillen, T. & Eberl, H. J. (2009). A nonlinear master equation for a degenerate diffusion model of biofilm growth, LNCS. 5544, 735–744Google Scholar
[16] Lang, J. & Teleaga, D. (2008) Towards a fully space-time adaptive FEM for magnetoquasistatics. IEEE Trans. Magn. 44, 1238–1241.Google Scholar
[17] Lear, G. & Lewis, G. D. (2012) Microbial Biofilms: Current Research and Applications, Caister Academic Press, Poole, U.K. ISBN 978-1-904455-96-7.Google Scholar
[18] van Loosdrecht, M. C. M., Heijnen, J. J., Eberl, H., Kreft, J. & Picioreanu, C. (2002) Mathematical modelling of biofilm structures. Antonie van Leeuwenhoek 81 (1), 245–256Google Scholar
[19] Muhammad, N. & Eberl, H. J. (2010) OpenMP parallelization of a Mickens time-integration scheme for a mixed-culture biofilm model and its performance on multi-core and multi-processor computers. LNCS 5976, 180–195.Google Scholar
[20] Muhammad, N. & Eberl, H. J. (2011) Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones. Math. Biosci. 233 (1), 1–18.Google Scholar
[21] Rahman, K. A. & Eberl, H. J. (2014) Numerical treatment of a cross-diffusion model of biofilm exposure to antimicrobials. LNCS 8384, 134–144.Google Scholar
[22] Rahman, K. A., Sudarsan, R. & Eberl, H. J. (2015) A mixed culture biofilm model with cross-diffusion. Bull. Math. Biol. 77 (11), 2086–2124.Google Scholar
[23] Rang, J. & Angermann, L. (2005) New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1. BIT Numer. Math. 45, 761–787.Google Scholar
[24] Rang, J. (2014) An analysis of the Prothero-Robinson example for constructing new DIRK and ROW methods. J. Comput. Appl. Math. 262, 105–114.Google Scholar
[25] Rang, J. (2015) Improved traditional Rosenbrock-Wanner methods for stiff ODEs and DAEs. J. Comp. Appl. Math. 286, 128–144.Google Scholar
[26] Rittmann, B. E. & McCarty, P. L. (2001) Environmental Biotechnology: Principles and Applications, McGraw-Hill, United States.Google Scholar
[27] Saad, Y. (1994) SPARSKIT: A basic tool for sparse matrix computations. URL: http://www.users.cs.umn.edu/saad/software/SPARSKIT/sparskit.html. (accessed 2014).Google Scholar
[28] Schwermer, C. U. et al. (2008) Impact of nitrate on the structure and function of bacterial biofilm communities in pipelines used for injection of seawater into oil fields. Appl. Environ. Microbiol. 74 (9), 2841–51.Google Scholar
[29] Sirca, S. & Horvat, M. (2012) Computational Methods for Physicists: Compendium for Students, Graduate Texts in Physics (Berlin: Springer).Google Scholar
[30] Sneddon, I. A. (1961) Special Functions of Mathematical Physics and Chemistry. Oliver and Boyd, Edinburgh.Google Scholar
[31] Stewart, P. S. & Costerton, J. W. (2001) Antibiotic resistance of bacteria in biofilms. Lancet 358 (9276), 135–8.Google Scholar
[32] Stewart, P. S. (2003) Diffusion in biofilms. J. Bacteriol. 185, 1485–1491.Google Scholar
[33] Van der Vorst, H. A. (1992) Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 13 (2), 631–644.Google Scholar
[34] Walter, W. (1927) Ordinary Differential Equations, Springer, New York.Google Scholar
[35] Watnick, P. & Kolter, R. (2000) Biofilm – city of microbes (Minireview) J. Bacteriol. 182 (10), 2675–2679.Google Scholar
[36] Walters, E., Hille, A., He, M., Ochmann, C. & Horn, H. (2009) Simultaneous nitrification/denitrification in a biofilm airlift suspension (BAS) reactor with biodegradable carrier material. Water Res. 43 (18), 4461–4468.Google Scholar