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Free boundary approach for the attachment in the initial phase of multispecies biofilm growth

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Abstract

In this work, a free boundary problem is presented for the attachment process in the initial phase of multispecies biofilm formation. The free boundary is represented by the biofilm thickness and it is assumed to be initially zero. The growth of attached species is governed by nonlinear hyperbolic PDEs. The free boundary evolution is governed by a first-order differential equation depending on the attachment, detachment, biomass velocity and substrates. The quasi-static diffusion of substrates is modelled by a system of semi-linear elliptic PDEs. The qualitative analysis of solutions leads to prove existence, uniqueness and some properties of solutions. We highlight that the free boundary velocity is greater than the characteristic velocity during the first instants of biofilm formation and the free boundary is a space-like line. It is proved that the attachment function depends linearly on the concentrations of all the attaching species. The first phase of biofilm growth is shown to be completely determined by environmental conditions and characterized by a specific mathematical inequality. The opposite inequality describes the further phase where the bulk liquid stops to directly affect the biofilm life. The mentioned inequalities could be assumed as rigorous definitions of non-mature and mature biofilms, respectively.

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Acknowledgements

This study has been performed under the auspices of the G.N.F.M. of Indam. The authors acknowledge the Progetto Giovani G.N.F.M. 2017 Analisi di sistemi biologici complessi and the project VOLAC—Valorization of OLive oil wastes for sustainable production of biocide-free Antibiofilm Compounds of Cariplo Foundation (Grant Number 2017-0977) for financial support.

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Authors and Affiliations

  1. Department of Mathematics and Applications, University of Naples Federico II, Complesso Montesantangelo, via Cintia, 80126, Napoli, Italy

    B. D’Acunto, L. Frunzo, V. Luongo & M. R. Mattei

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  1. B. D’Acunto
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  2. L. Frunzo
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  3. V. Luongo
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  4. M. R. Mattei
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Correspondence to M. R. Mattei.

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D’Acunto, B., Frunzo, L., Luongo, V. et al. Free boundary approach for the attachment in the initial phase of multispecies biofilm growth. Z. Angew. Math. Phys. 70, 91 (2019). https://doi.org/10.1007/s00033-019-1134-y

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  • DOI: https://doi.org/10.1007/s00033-019-1134-y

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