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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 & Continuous Dynamical Systems - A, 2014, 34 (1) : 99-119. doi: 10.3934/dcds.2014.34.99
##### References:
 [1] R. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, 133 (1993), 9.   Google Scholar [3] D. G. Aronson, The porous medium equation,, in, 1224 (1985).  doi: 10.1007/BFb0072687.  Google Scholar [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.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [5] J. Dockery and I. Klapper, Finger formation in biofilm layers,, SIAM J. Appl. Math., 62 (): 853.  doi: 10.1137/S0036139900371709.  Google Scholar [6] H. J. Eberl and L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology,, in, 15 (2007), 77.   Google Scholar [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.  doi: 10.1080/10273660108833072.  Google Scholar [8] M. A. Efendiev and S. Sonner, Verifying mathematical models with diffusion, transport and interaction,, in, 32 (2010), 41.   Google Scholar [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.   Google Scholar [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.  doi: 10.3934/cpaa.2009.8.509.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  doi: 10.1016/j.jtbi.2007.11.026.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [17] V. Gordon, Personal email communication,, July 4, (2011).   Google Scholar [18] Ph. Laurençot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion,, in, 64 (2005), 273.  doi: 10.1007/3-7643-7385-7_16.  Google Scholar [19] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod; Gauthier-Villars, (1969).   Google Scholar [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).  doi: 10.1103/PhysRevE.78.061904.  Google Scholar [21] M. A. Molina, J.-L. Ramos and M. Espinosa-Urgel, Plant-associated biofilms,, Rev. Environ. Sci. Biotech., 2 (2003), 99.  doi: 10.1023/B:RESB.0000040458.35960.25.  Google Scholar [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.  doi: 10.1016/j.mbs.2011.05.006.  Google Scholar [23] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Quart., 10 (2002), 501.   Google Scholar [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [25] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar [26] R. Singh, D. Paul and R. K. Jain, Biofilms: Implications in bioremediation,, TRENDS in Microbiol., 14 (2006), 389.  doi: 10.1016/j.tim.2006.07.001.  Google Scholar [27] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Second edition, 258 (1994).   Google Scholar [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.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar [29] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library, 18 (1978).   Google Scholar [30] Z. A Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect,, Nonlinearity, 24 (2011), 3279.  doi: 10.1088/0951-7715/24/12/001.  Google Scholar [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.  doi: 10.1137/110853972.  Google Scholar [32] O. Wanner, H. Eberl, E. Morgenroth, D. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, "Mathematical Modeling of Biofilms,", IWA Publishing, (2006).   Google Scholar [33] D. Wrzosek, Chemotaxis models with a threshold cell density,, in, 81 (2008), 553.  doi: 10.4064/bc81-0-35.  Google Scholar [34] D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion,, Nonl. Analysis TMA, 73 (2010), 338.  doi: 10.1016/j.na.2010.02.047.  Google Scholar [35] D. Wrzosek, Volume filling effect in modelling chemotaxis,, Math. Model. Nat. Phenom., 5 (2010), 123.  doi: 10.1051/mmnp/20105106.  Google Scholar [36] A. Yagi, "Abstract Parabolic Evolution Equations and their Applications,", Springer Monographs in Mathematics, (2010).  doi: 10.1007/978-3-642-04631-5.  Google Scholar [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.  doi: 10.1007/s00284-008-9137-5.  Google Scholar

show all references

##### References:
 [1] R. Adams, "Sobolev Spaces,", Pure and Applied Mathematics, (1975).   Google Scholar [2] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, 133 (1993), 9.   Google Scholar [3] D. G. Aronson, The porous medium equation,, in, 1224 (1985).  doi: 10.1007/BFb0072687.  Google Scholar [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.  doi: 10.1007/s00209-007-0120-9.  Google Scholar [5] J. Dockery and I. Klapper, Finger formation in biofilm layers,, SIAM J. Appl. Math., 62 (): 853.  doi: 10.1137/S0036139900371709.  Google Scholar [6] H. J. Eberl and L. Demaret, A finite difference scheme for a degenerated diffusion equation arising in microbial ecology,, in, 15 (2007), 77.   Google Scholar [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.  doi: 10.1080/10273660108833072.  Google Scholar [8] M. A. Efendiev and S. Sonner, Verifying mathematical models with diffusion, transport and interaction,, in, 32 (2010), 41.   Google Scholar [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.   Google Scholar [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.  doi: 10.3934/cpaa.2009.8.509.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.  doi: 10.1016/j.jtbi.2007.11.026.  Google Scholar [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.   Google Scholar [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.   Google Scholar [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.   Google Scholar [17] V. Gordon, Personal email communication,, July 4, (2011).   Google Scholar [18] Ph. Laurençot and D. Wrzosek, A chemotaxis model with threshold density and degenerate diffusion,, in, 64 (2005), 273.  doi: 10.1007/3-7643-7385-7_16.  Google Scholar [19] J.-L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", Dunod; Gauthier-Villars, (1969).   Google Scholar [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).  doi: 10.1103/PhysRevE.78.061904.  Google Scholar [21] M. A. Molina, J.-L. Ramos and M. Espinosa-Urgel, Plant-associated biofilms,, Rev. Environ. Sci. Biotech., 2 (2003), 99.  doi: 10.1023/B:RESB.0000040458.35960.25.  Google Scholar [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.  doi: 10.1016/j.mbs.2011.05.006.  Google Scholar [23] K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Can. Appl. Math. Quart., 10 (2002), 501.   Google Scholar [24] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).  doi: 10.1007/978-1-4612-5561-1.  Google Scholar [25] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl. (4), 146 (1987), 65.  doi: 10.1007/BF01762360.  Google Scholar [26] R. Singh, D. Paul and R. K. Jain, Biofilms: Implications in bioremediation,, TRENDS in Microbiol., 14 (2006), 389.  doi: 10.1016/j.tim.2006.07.001.  Google Scholar [27] J. Smoller, "Shock Waves and Reaction-Diffusion Equations,", Second edition, 258 (1994).   Google Scholar [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.  doi: 10.1016/j.jde.2006.03.003.  Google Scholar [29] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library, 18 (1978).   Google Scholar [30] Z. A Wang, M. Winkler and D. Wrzosek, Singularity formation in chemotaxis systems with volume-filling effect,, Nonlinearity, 24 (2011), 3279.  doi: 10.1088/0951-7715/24/12/001.  Google Scholar [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.  doi: 10.1137/110853972.  Google Scholar [32] O. Wanner, H. Eberl, E. Morgenroth, D. Noguera, C. Picioreanu, B. Rittmann and M. van Loosdrecht, "Mathematical Modeling of Biofilms,", IWA Publishing, (2006).   Google Scholar [33] D. Wrzosek, Chemotaxis models with a threshold cell density,, in, 81 (2008), 553.  doi: 10.4064/bc81-0-35.  Google Scholar [34] D. Wrzosek, Model of chemotaxis with threshold density and singular diffusion,, Nonl. Analysis TMA, 73 (2010), 338.  doi: 10.1016/j.na.2010.02.047.  Google Scholar [35] D. Wrzosek, Volume filling effect in modelling chemotaxis,, Math. Model. Nat. Phenom., 5 (2010), 123.  doi: 10.1051/mmnp/20105106.  Google Scholar [36] A. Yagi, "Abstract Parabolic Evolution Equations and their Applications,", Springer Monographs in Mathematics, (2010).  doi: 10.1007/978-3-642-04631-5.  Google Scholar [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.  doi: 10.1007/s00284-008-9137-5.  Google Scholar
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