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January  2014, 34(1): 99-119. doi: 10.3934/dcds.2014.34.99

## Analysis of a degenerate biofilm model with a nutrient taxis term

 1 Department of Mathematics and Statistics, University of Guelph, Guelph, On, N1G 2W1, Canada 2 Helmholtz Zentrum München, Institute of Computational Biology, Ingolstädter Landstrasse1, D-85764 Neuherberg,, Germany 3 Inst. Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097 Warszawa, Poland 4 Centre for Mathematical Sciences, Technical University of Munich, Boltzmannstr. 3, 85748 Garching, Germany

Received  October 2012 Revised  March 2013 Published  June 2013

We introduce and analyze a prototype model for chemotactic effects in biofilm formation. The model is a system of quasilinear parabolic equations into which two thresholds are built in. One occurs at zero cell density level, the second one is related to the maximal density which the cells cannot exceed. Accordingly, both diffusion and taxis terms have degenerate or singular parts. This model extends a previously introduced degenerate biofilm model by combining it with a chemotaxis equation. We give conditions for existence and uniqueness of weak solutions and illustrate the model behavior in numerical simulations.
Citation: Hermann J. Eberl, Messoud A. Efendiev, Dariusz Wrzosek, Anna Zhigun. Analysis of a degenerate biofilm model with a nutrient taxis term. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 99-119. doi: 10.3934/dcds.2014.34.99
##### References:
 [1] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis'' (eds. H. Triebel and H. J. Schmeisser), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126. [3] D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Mathematics, 1224, Springer, Berlin, 1986. doi: 10.1007/BFb0072687. [4] R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9. [5] J. Dockery and I. Klapper, Finger formation in biofilm layers,, SIAM J. Appl. Math., 62 (): 853.  doi: 10.1137/S0036139900371709. [6] H. J. Eberl and L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, in "Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations," El. J. Diff Equs Conf., 15, Southwest Texas State Univ., San Marcos, TX, (2007), 77-96. [7] H. J. Eberl, D. F. Parker and M. C. M. Van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3 (2001), 161-176. doi: 10.1080/10273660108833072. [8] M. A. Efendiev and S. Sonner, Verifying mathematical models with diffusion, transport and interaction, in "Current Advances in Nonlinear Analysis and Related Topics," GAKUTO Internat. Ser. Math. Sci. Appl., 32, Gakkōtosho, Tokyo, (2010), 41-67. [9] M. A. Efendiev, R. Lasser and S. Sonner, Necessary and sufficient conditions for an infinite system of parabolic equations preserving the positive cone, Int. J. Biomath. & Biostats, 1 (2010), 47-52. [10] M. A. Efendiev, S. V. Zelik and H. J. Eberl, Existence and long time behaviour of a biofilm model, Comm. Pure and Appl. Analysis, 8 (2009), 509-531. doi: 10.3934/cpaa.2009.8.509. [11] M. A. Efendiev and T. Senba, On the well-posedness of a class of PDEs including porous medium and chemotaxis effect, Adv. Differ. Equ., 16 (2011), 937-954. [12] M. A. Efendiev and A. Zhigun, On a 'balance' condition for a class of PDEs including porous medium and chemotaxis effect: Non-autonomous case, Adv. Math. Sci. Appl., 21 (2011), 285-304. [13] H. Fgaier, B. Feher, R. C. McKellar and H. J. Eberl, Predictive modeling of siderphore production by Pseudomonas fluorscens under iron limitation, J. Theor. Biol., 251 (2008), 348-362. doi: 10.1016/j.jtbi.2007.11.026. [14] H. Fgaier and H. J. Eberl, Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation, Disc. Cont. Dyn. Sys. Suppl., (2009), 230-239. [15] M. R. Frederick, C. Kuttler, B. A. Hense, J. Müller and H. J. Eberl, A mathematical model of quorum sensing in patchy biofilm communities with slow background flow, Can. Appl. Math. Quart., 18 (2011), 267-298. [16] R. Kowalczyk, A. Gamba and L. Preciosi, On the stability of homogeneous solutions to some aggregation models, Discrete Contin. Dynam. Systems-Series B, 4 (2004), 203-220. [17] V. Gordon, Personal email communication, July 4, (2011). [18] Ph. Laurençot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion, in "Nonlinear Elliptic and Parabolic Problems," Progr. Nonlinear Differential Equations Appl., 64, Birkhäuser, Basel, (2005), 273-290. doi: 10.1007/3-7643-7385-7_16. [19] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod; Gauthier-Villars, Paris, 1969. [20] P. M. Lushnikov, N. Chen and M. Alber, Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact, Phys. Rev. E., 78 (2008), 061904, 12 pp. doi: 10.1103/PhysRevE.78.061904. [21] M. A. Molina, J.-L. Ramos and M. Espinosa-Urgel, Plant-associated biofilms, Rev. Environ. Sci. Biotech., 2 (2003), 99-108. doi: 10.1023/B:RESB.0000040458.35960.25. [22] N. Muhammad and H. J. Eberl, Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones, Math. Biosci., 233 (2011), 1-18. doi: 10.1016/j.mbs.2011.05.006. [23] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002), 501-543. [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [25] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360. [26] R. Singh, D. Paul and R. K. Jain, Biofilms: Implications in bioremediation, TRENDS in Microbiol., 14 (2006), 389-397. doi: 10.1016/j.tim.2006.07.001. [27] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 1994. [28] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Diff. Equ., 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003. [29] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978. [30] Z. A Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297. doi: 10.1088/0951-7715/24/12/001. [31] Z. A Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972. [32] O. Wanner, H. Eberl, E. Morgenroth, D. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, "Mathematical Modeling of Biofilms," IWA Publishing, London, 2006. [33] D. Wrzosek, Chemotaxis models with a threshold cell density, in "Parabolic and Navier-Stokes Equations. Part 2," Banach Center Publ., 81, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 553-566. doi: 10.4064/bc81-0-35. [34] D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonl. Analysis TMA, 73 (2010), 338-349. doi: 10.1016/j.na.2010.02.047. [35] D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Model. Nat. Phenom., 5 (2010), 123-147. doi: 10.1051/mmnp/20105106. [36] A. Yagi, "Abstract Parabolic Evolution Equations and their Applications," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5. [37] P. M. Yaryura, M. León, O. S. Correa, N. L. Kerber, N. L. Pucheu and A. F. García, Assessment of the role of chemotaxis and biofilm formation as requirement for colonization of roots and seed of soybean plants by Bacillus amyloliqufaciens BNM339, Curr. Microbiol., 56 (2008), 625-632. doi: 10.1007/s00284-008-9137-5.

show all references

##### References:
 [1] R. Adams, "Sobolev Spaces," Pure and Applied Mathematics, Vol. 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis'' (eds. H. Triebel and H. J. Schmeisser), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126. [3] D. G. Aronson, The porous medium equation, in "Nonlinear Diffusion Problems" (Montecatini Terme, 1985), Lecture Notes in Mathematics, 1224, Springer, Berlin, 1986. doi: 10.1007/BFb0072687. [4] R. Denk, M. Hieber and J. Prüss, Optimal $L^p-L^q$-estimates for parabolic boundary value problems with inhomogeneous data, Math. Z., 257 (2007), 193-224. doi: 10.1007/s00209-007-0120-9. [5] J. Dockery and I. Klapper, Finger formation in biofilm layers,, SIAM J. Appl. Math., 62 (): 853.  doi: 10.1137/S0036139900371709. [6] H. J. Eberl and L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology, in "Proceedings of the Sixth Mississippi State-UBA Conference on Differential Equations and Computational Simulations," El. J. Diff Equs Conf., 15, Southwest Texas State Univ., San Marcos, TX, (2007), 77-96. [7] H. J. Eberl, D. F. Parker and M. C. M. Van Loosdrecht, A new deterministic spatio-temporal continuum model for biofilm development, J. Theor. Med., 3 (2001), 161-176. doi: 10.1080/10273660108833072. [8] M. A. Efendiev and S. Sonner, Verifying mathematical models with diffusion, transport and interaction, in "Current Advances in Nonlinear Analysis and Related Topics," GAKUTO Internat. Ser. Math. Sci. Appl., 32, Gakkōtosho, Tokyo, (2010), 41-67. [9] M. A. Efendiev, R. Lasser and S. Sonner, Necessary and sufficient conditions for an infinite system of parabolic equations preserving the positive cone, Int. J. Biomath. & Biostats, 1 (2010), 47-52. [10] M. A. Efendiev, S. V. Zelik and H. J. Eberl, Existence and long time behaviour of a biofilm model, Comm. Pure and Appl. Analysis, 8 (2009), 509-531. doi: 10.3934/cpaa.2009.8.509. [11] M. A. Efendiev and T. Senba, On the well-posedness of a class of PDEs including porous medium and chemotaxis effect, Adv. Differ. Equ., 16 (2011), 937-954. [12] M. A. Efendiev and A. Zhigun, On a 'balance' condition for a class of PDEs including porous medium and chemotaxis effect: Non-autonomous case, Adv. Math. Sci. Appl., 21 (2011), 285-304. [13] H. Fgaier, B. Feher, R. C. McKellar and H. J. Eberl, Predictive modeling of siderphore production by Pseudomonas fluorscens under iron limitation, J. Theor. Biol., 251 (2008), 348-362. doi: 10.1016/j.jtbi.2007.11.026. [14] H. Fgaier and H. J. Eberl, Parameter identification and quantitative comparison of differential equations that describe physiological adaptation of a bacterial population under iron limitation, Disc. Cont. Dyn. Sys. Suppl., (2009), 230-239. [15] M. R. Frederick, C. Kuttler, B. A. Hense, J. Müller and H. J. Eberl, A mathematical model of quorum sensing in patchy biofilm communities with slow background flow, Can. Appl. Math. Quart., 18 (2011), 267-298. [16] R. Kowalczyk, A. Gamba and L. Preciosi, On the stability of homogeneous solutions to some aggregation models, Discrete Contin. Dynam. Systems-Series B, 4 (2004), 203-220. [17] V. Gordon, Personal email communication, July 4, (2011). [18] Ph. Laurençot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion, in "Nonlinear Elliptic and Parabolic Problems," Progr. Nonlinear Differential Equations Appl., 64, Birkhäuser, Basel, (2005), 273-290. doi: 10.1007/3-7643-7385-7_16. [19] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires," Dunod; Gauthier-Villars, Paris, 1969. [20] P. M. Lushnikov, N. Chen and M. Alber, Macroscopic dynamics of biological cells interacting via chemotaxis and direct contact, Phys. Rev. E., 78 (2008), 061904, 12 pp. doi: 10.1103/PhysRevE.78.061904. [21] M. A. Molina, J.-L. Ramos and M. Espinosa-Urgel, Plant-associated biofilms, Rev. Environ. Sci. Biotech., 2 (2003), 99-108. doi: 10.1023/B:RESB.0000040458.35960.25. [22] N. Muhammad and H. J. Eberl, Model parameter uncertainties in a dual-species biofilm competition model affect ecological output parameters much stronger than morphological ones, Math. Biosci., 233 (2011), 1-18. doi: 10.1016/j.mbs.2011.05.006. [23] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Quart., 10 (2002), 501-543. [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [25] J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl. (4), 146 (1987), 65-96. doi: 10.1007/BF01762360. [26] R. Singh, D. Paul and R. K. Jain, Biofilms: Implications in bioremediation, TRENDS in Microbiol., 14 (2006), 389-397. doi: 10.1016/j.tim.2006.07.001. [27] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," Second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 258, Springer-Verlag, New York, 1994. [28] Y. Sugiyama and H. Kunii, Global existence and decay properties for a degenerate Keller-Segel model with a power factor in drift term, J. Diff. Equ., 227 (2006), 333-364. doi: 10.1016/j.jde.2006.03.003. [29] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978. [30] Z. A Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect, Nonlinearity, 24 (2011), 3279-3297. doi: 10.1088/0951-7715/24/12/001. [31] Z. A Wang, M. Winkler and D. Wrzosek, Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume filling effect and degenerate diffusion, SIAM J. Math. Anal., 44 (2012), 3502-3525. doi: 10.1137/110853972. [32] O. Wanner, H. Eberl, E. Morgenroth, D. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, "Mathematical Modeling of Biofilms," IWA Publishing, London, 2006. [33] D. Wrzosek, Chemotaxis models with a threshold cell density, in "Parabolic and Navier-Stokes Equations. Part 2," Banach Center Publ., 81, Polish Acad. Sci. Inst. Math., Warsaw, (2008), 553-566. doi: 10.4064/bc81-0-35. [34] D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion, Nonl. Analysis TMA, 73 (2010), 338-349. doi: 10.1016/j.na.2010.02.047. [35] D. Wrzosek, Volume filling effect in modelling chemotaxis, Math. Model. Nat. Phenom., 5 (2010), 123-147. doi: 10.1051/mmnp/20105106. [36] A. Yagi, "Abstract Parabolic Evolution Equations and their Applications," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5. [37] P. M. Yaryura, M. León, O. S. Correa, N. L. Kerber, N. L. Pucheu and A. F. García, Assessment of the role of chemotaxis and biofilm formation as requirement for colonization of roots and seed of soybean plants by Bacillus amyloliqufaciens BNM339, Curr. Microbiol., 56 (2008), 625-632. doi: 10.1007/s00284-008-9137-5.
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