Modeling bacterial attachment to surfaces as an early stage of biofilm development

Fadoua El Moustaid; Amina Eladdadi; Lafras Uys

doi:10.3934/mbe.2013.10.821

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2013, 10(3): 821-842. doi: 10.3934/mbe.2013.10.821

Modeling bacterial attachment to surfaces as an early stage of biofilm development

Fadoua El Moustaid ^1, , Amina Eladdadi ^2, and Lafras Uys ^1,

1.	African Institute for Mathematical Sciences, 6 Melrose road, Muizenberg, 7945, South Africa, South Africa
2.	The College of Saint Rose, Department of Mathematics, 432 Western Avenue, Albany, NY 12203, United States

Received June 2012 Revised January 2013 Published April 2013

Abstract
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Biofilms are present in all natural, medical and industrial surroundings where bacteria live. Biofilm formation is a key factor in the growth and transport of both beneficial and harmful bacteria. While much is known about the later stages of biofilm formation, less is known about its initiation which is an important first step in the biofilm formation. In this paper, we develop a non-linear system of partial differential equations of Keller-Segel type model in one-dimensional space, which couples the dynamics of bacterial movement to that of the sensing molecules. In this case, bacteria perform a biased random walk towards the sensing molecules. We derive the boundary conditions of the adhesion of bacteria to a surface using zero-Dirichlet boundary conditions, while the equation describing sensing molecules at the interface needed particular conditions to be set. The numerical results show the profile of bacteria within the space and the time evolution of the density within the free-space and on the surface. Testing different parameter values indicate that significant amount of sensing molecules present on the surface leads to a faster bacterial movement toward the surface which is the first step of biofilm initiation. Our work gives rise to results that agree with the biological description of the early stages of biofilm formation.

Keywords: bacterial biofilm, Keller-Segel model, Chemotaxis, sensing molecules..

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Fadoua El Moustaid, Amina Eladdadi, Lafras Uys. Modeling bacterial attachment to surfaces as an early stage of biofilm development. Mathematical Biosciences & Engineering, 2013, 10 (3) : 821-842. doi: 10.3934/mbe.2013.10.821

References:

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[29]	J. W. Costerton, Overview of microbial biofilms,, Journal of Industrial Microbiology and Biotechnology, 15 (1995), 137. doi: 10.1007/BF01569816. Google Scholar
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[31]	J. W. Costerton, G. G. Geesey and G. K. Cheng, How bacteria stick,, Sci. Am., 238 (1978), 86. doi: 10.1038/scientificamerican0178-86. Google Scholar
[32]	KJ. Engel, R. Nagel., "One Parameter Semigroups for Linear Evolution Equation,", S. Axler editor, (2000). Google Scholar
[33]	K. K. Jefferson, What drives bacteria to produce a biofilm?,, FEMS Microbiology Letters, 236 (2004), 163. Google Scholar
[34]	K. Kang, T. Kolokolnikov and J. Ward, The stability and dynamics of a spike in the 1D Keller Segel model,, IMA Journal of Applied Mathematics, (2007). doi: 10.1093/imamat/hxl028. Google Scholar
[35]	K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda and N. Shigesada, Modeling spatio-temporal patterns generated by bacillus subtilis,, Journal of Theoretical Biology, 188 (1997), 177. Google Scholar
[36]	K. Sauer, A. K. Camper, G. D. Ehrlich, J. W. Costerton and D. G. Davies, Pseudomonas aeruginosa displays multiple phenotypes during development as a biofilm,, Journal of Bacteriology, 184 (2002), 1140. doi: 10.1128/jb.184.4.1140-1154.2002. Google Scholar
[37]	L. R. Johnson, Microcolony and biofilm formation as a survival strategy for bacteria,, Journal of Theoretical Biology, 251 (2008), 24. doi: 10.1016/j.jtbi.2007.10.039. Google Scholar
[38]	M. Ballyk and H. Smith, A model of microbial growth in a plug flow reactor with wall attachment,, Mathematical Biosciences, 158 (1999), 95. doi: 10.1016/S0025-5564(99)00006-1. Google Scholar
[39]	M. Burmolle, T. Rolighed Thomsen, M. Fazli, I. Dige, L. Christensen, P. Homoe, M. Tvede, B. Nyvad, T. Tolker-Nielsen, M. Givskov, C. Moser, K. Kirketerp-Moller, H. Krogh Johansen, N. Hoiby, P. Ostrup Jensen, S. J. Sorensen and T. Bjarnsholt, Biofilms in chronic infections a matter of opportunity monospecies biofilms in multispecies infections,, FEMS Immunol. Med. Microbiol, 59 (2010), 324. doi: 10.1111/j.1574-695X.2010.00714.x. Google Scholar
[40]	M. G. Fagerlind, J. S. Webb, N. Barraud, D. McDougald, A. Jansson, P. Nilsson, M. Harln, S. Kjelleberg and S. A. Rice, Dynamic modelling of cell death during biofilm development,, Journal of Theoretical Biology, 259 (2012), 23. doi: 10.1016/j.jtbi.2011.10.007. Google Scholar
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[43]	M. R. Rahbar, I. Rasooli, S. Latif, M. Gargari, J. Amani and Y. Fattahian, In silico analysis of antibody triggering biofilm associated protein in Acinetobacter baumannii,, Journal of Theoretical Biology, 266 (2010), 275. doi: 10.1016/j.jtbi.2010.06.014. Google Scholar
[44]	M. Tindall, P. Maini, S. Porter and J. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations,, Bulletin of Mathematical Biology, 70 (2008), 1570. doi: 10.1007/s11538-008-9322-5. Google Scholar
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[46]	N. Hoiby, T. Bjarnsholt, M. Givskov, S. Molin and O. Ciofu, Antibiotic resistance of bacterial biofilms,, International Journal of Antimicrobial Agents, 35 (2010), 322. doi: 10.1016/j.ijantimicag.2009.12.011. Google Scholar
[47]	N. Hoiby, T. Bjarnsholt, M. L. Givskov, S. Molin and O. Ciofu, Antibiotic resistance of bacterial biofilms,, International Journal of Antimicrobial Agents, 35 (2010), 322. doi: 10.1016/j.ijantimicag.2009.12.011. Google Scholar
[48]	O. Wanner and W. Gujer, A multispecies biofilm model,, Biotechnology and Bioengineering, 28 (1986), 314. doi: 10.1002/bit.260280304. Google Scholar
[49]	P. Carol, Microbiology: Biofilms invade microbiology,, Journal of Science, 273 (1996), 1795. Google Scholar
[50]	P. Watnick and R. Kolter, Biofilm, city of microbes,, Journal of Bacteriology, 182 (2000), 2675. doi: 10.1128/JB.182.10.2675-2679.2000. Google Scholar
[51]	Q. Wang and T. Zhang, Review of mathematical models for biofilms,, Solid State Communications, 150 (2010), 21. doi: 10.1016/j.ssc.2010.01.021. Google Scholar
[52]	R. Erban and G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, multiscale model,, Journal of Simul., 3 (2005), 362. doi: 10.1137/040603565. Google Scholar
[53]	R. J. Leveque, "Finite Volume Methods for Hyperbolic Problems,", Cambridge University Press, (2002). doi: 10.1017/CBO9780511791253. Google Scholar
[54]	R. M. Donlan and J. W. Costerton, Survival mechanisms of clinically relevant microorganisms,, Clinical Microbiology Reviews, 15 (2002). doi: 10.1128/CMR.15.2.167-193.2002. Google Scholar
[55]	T. R. de Kievit, Quorum sensing in Pseudomonas aeruginosa biofilms,, Environmental Microbiology, 11 (2009), 279. Google Scholar
[56]	T. Tolker-Nielsen, U. C. Brinch, P. C. Ragas, J. B. Andersen, C. S. Jacobsen and S. Molin, Development and dynamics of Pseudomonas sp. biofilms,, Journal of Bacteriology, 182 (2000). Google Scholar
[57]	S. Abdul Rani, B. Pitts, H. Beyenal, R. A. Veluchamy, Z. Lewandowski, W. M. Davison, K. Buckingham-Meyer and P. S. Stewart, Spatial patterns of DNA replication, protein synthesis, and oxygen concentration within bacterial biofilms reveal diverse physiological states,, Journal of Bacteriology, 189 (2007), 4223. doi: 10.1128/JB.00107-07. Google Scholar
[58]	Z. Lewandowski and H. Beyenal, Mechanisms of microbially influenced corrosion,, Springer Berlin Heidelberg, 4 (2009), 35. Google Scholar
[59]	, http://grants.nih.gov/grants/guide/pa-files/PA-03-047.html, Last Accessed on June 11, (2012). Google Scholar

show all references

References:

[1]	A. Bejan, "Convection Heat Transfer,", John Wiley and Sons Inc., (1984). Google Scholar
[2]	A. Coghlan, Slime city,, New Scientist, 151 (1996), 32. Google Scholar
[3]	A. P. Petroff, TD. Wu, B. Liang, J. Mui, JL. Guerquin-Kern, H. Vali, D. H. Rothman and T. Bosak, Reaction diffusion model of nutrient uptake in a biofilm: Theory and experiment,, Journal of Theoretical Biology, 289 (2001), 90. doi: 10.1016/j.jtbi.2011.08.004. Google Scholar
[4]	C. D. Nadell, J. B. Xavier and K. R. Foster, The sociobiologyof biofilms,, FEMS Microbiol Review, (2009), 1. Google Scholar
[5]	C. Picioreanu, M. C. M. Van Loosdrecht and J. J. Heijnen, Mathematical modeling of biofilm structure with a hybrid differential-discrete cellular automaton approach,, Biotechnology and Bioengineering, 58 (1997). doi: 10.1002/(SICI)1097-0290(19980405)58:1<101::AID-BIT11>3.0.CO;2-M. Google Scholar
[6]	C. S. Laspidou, A. Kungolos, P. Samaras C. S. Laspidou, A. Kungolos and P. Samaras, Cellular-automata and individual-based approaches for the modeling of biofilm structures: Pros and cons,, Journal of Desalination, 250 (2010), 390. doi: 10.1016/j.desal.2009.09.062. Google Scholar
[7]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences,, 2003., (). Google Scholar
[8]	D. Jones, V. K. Hristo and D. Le and H. Smith, Bacterial wall attachment in a flow reactor,, SIAM Journal, 62 (2002), 1728. doi: 10.1137/S0036139901390416. Google Scholar
[9]	D. Priscilla, Biofilms: The environmental playground of Legionella pneumophila,, Journal of Environmental Microbiology, 12 (2010), 557. Google Scholar
[10]	D. V. Nicolau Jr., J. P. Armitage and P. K. Maini, Directional persistence and the optimality of run-and-tumble chemotaxis,, Computational Biology and Chemistry, 33 (2009), 269. doi: 10.1016/j.compbiolchem.2009.06.003. Google Scholar
[11]	E. A. J. F. Peters and Th. M. A. O. M. Barenbrug, Efficient brownian dynamics simulation of particles near walls. I. Reflecting and adsorbing walls,, Physical Review E, 66 (2002). doi: 10.1103/PhysRevE.66.056701. Google Scholar
[12]	E. A. J. F. Peters and Th. M. A. O. M. Barenbrug, Efficient brownian dynamics simulation of particles near walls. II. sticky walls,, Physical Review E, 66 (2002). doi: 10.1103/PhysRevE.66.056702. Google Scholar
[13]	E. Ben Jacob, O. Schochet, A. Tenenbaum, I. Cohen, A. Czirok and T. Vicsek, Generic modelling of cooperative growth patterns in bacterial colonies,, Nature Journal, 368 (1994), 46. Google Scholar
[14]	E. F. Keller, Science as a medium for friendship: How the Keller-Segel models came about,, Bull. Math. Biol., 68 (2009), 1033. doi: 10.1007/s11538-006-9097-5. Google Scholar
[15]	E. F. Keller and L. A. Segel, Initiation of slide mold aggregation viewed as an instability,, Journal of Theoretical Biology, 26 (1970), 399. Google Scholar
[16]	E. F. Keller and L. A. Segel, Model for chemotaxis,, Journal of Theoretical Biology, 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6. Google Scholar
[17]	E. F. Keller and L. A. Segel, Traveling bands of chemotactic bacteria: A theoretical analysis,, Journal of Theoretical Biology, 30 (1971), 235. doi: 10.1016/0022-5193(71)90051-8. Google Scholar
[18]	G. D. Zacarias, C. P. Ferreira and J. X. Velasco-Hernandez, Porosity and tortuosity relations as revealed by a mathematical model of biofilm structure,, Journal of Theoretical Biology, 233 (2005), 245. doi: 10.1016/j.jtbi.2004.10.006. Google Scholar
[19]	H. Donnelly, Uniqueness of positive solutions of the heat equation,, American Mathematical Society, 99 (1987). doi: 10.1090/S0002-9939-1987-0870800-6. Google Scholar
[20]	H. F. Jenkinson and H. M. Lappin-Scott, Biofilms adhere to stay,, Journal of Trends in Microbiology, 9 (2001), 9. doi: 10.1016/S0966-842X(00)01891-6. Google Scholar
[21]	H. Stoodley, Luanne, W. Costerton and P. Stoodley, Bacterial biofilms: From the natural environment to infectious diseases,, Review of Microbiology, 2 (2004), 1740. Google Scholar
[22]	H. Tamboto, K. Vickery and A. K. Deva, Subclinical (Biofilm) infection causes capsular contracture in a porcine model following augmentation mammaplasty,, Plastic & Reconstructive Surgery, 126 (2010), 835. doi: 10.1097/PRS.0b013e3181e3b456. Google Scholar
[23]	I. Klapper and J. Dockery, Mathematical description of microbial biofilms,, SIAM Journal, 52 (2010), 221. doi: 10.1137/080739720. Google Scholar
[24]	J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications,", S. S. Antman editor, (2003). Google Scholar
[25]	J. D. Murray, "Mathematical Biology I: An Introduction,", S. S. Antman editor, (2002). Google Scholar
[26]	J. E. Guyer, D. Wheeler and J. A. Warren, FiPy: Partial differential equations with python,, Journal of Computer Science and Engineering, 11 (2009), 6. doi: 10.1109/MCSE.2009.52. Google Scholar
[27]	J. L. Goldberg and A. J. Schwartz, "Systems of Ordinary Differential Equations: An Introduction,", I. N. Herstein and Gian-Carlo Rota editor Harper and Row publisher., (). Google Scholar
[28]	J. S. Poindexter and E. R. Leadbetter, " Bacteria in Nature 2: Methods and Special Applications in Bacterial Ecology,", Spring Street Editor, (1986). Google Scholar
[29]	J. W. Costerton, Overview of microbial biofilms,, Journal of Industrial Microbiology and Biotechnology, 15 (1995), 137. doi: 10.1007/BF01569816. Google Scholar
[30]	J. W. Costerton, Introduction to biofilm,, International Journal of Antimicrobial Agents, 11 (1999), 217. doi: 10.1016/S0924-8579(99)00018-7. Google Scholar
[31]	J. W. Costerton, G. G. Geesey and G. K. Cheng, How bacteria stick,, Sci. Am., 238 (1978), 86. doi: 10.1038/scientificamerican0178-86. Google Scholar
[32]	KJ. Engel, R. Nagel., "One Parameter Semigroups for Linear Evolution Equation,", S. Axler editor, (2000). Google Scholar
[33]	K. K. Jefferson, What drives bacteria to produce a biofilm?,, FEMS Microbiology Letters, 236 (2004), 163. Google Scholar
[34]	K. Kang, T. Kolokolnikov and J. Ward, The stability and dynamics of a spike in the 1D Keller Segel model,, IMA Journal of Applied Mathematics, (2007). doi: 10.1093/imamat/hxl028. Google Scholar
[35]	K. Kawasaki, A. Mochizuki, M. Matsushita, T. Umeda and N. Shigesada, Modeling spatio-temporal patterns generated by bacillus subtilis,, Journal of Theoretical Biology, 188 (1997), 177. Google Scholar
[36]	K. Sauer, A. K. Camper, G. D. Ehrlich, J. W. Costerton and D. G. Davies, Pseudomonas aeruginosa displays multiple phenotypes during development as a biofilm,, Journal of Bacteriology, 184 (2002), 1140. doi: 10.1128/jb.184.4.1140-1154.2002. Google Scholar
[37]	L. R. Johnson, Microcolony and biofilm formation as a survival strategy for bacteria,, Journal of Theoretical Biology, 251 (2008), 24. doi: 10.1016/j.jtbi.2007.10.039. Google Scholar
[38]	M. Ballyk and H. Smith, A model of microbial growth in a plug flow reactor with wall attachment,, Mathematical Biosciences, 158 (1999), 95. doi: 10.1016/S0025-5564(99)00006-1. Google Scholar
[39]	M. Burmolle, T. Rolighed Thomsen, M. Fazli, I. Dige, L. Christensen, P. Homoe, M. Tvede, B. Nyvad, T. Tolker-Nielsen, M. Givskov, C. Moser, K. Kirketerp-Moller, H. Krogh Johansen, N. Hoiby, P. Ostrup Jensen, S. J. Sorensen and T. Bjarnsholt, Biofilms in chronic infections a matter of opportunity monospecies biofilms in multispecies infections,, FEMS Immunol. Med. Microbiol, 59 (2010), 324. doi: 10.1111/j.1574-695X.2010.00714.x. Google Scholar
[40]	M. G. Fagerlind, J. S. Webb, N. Barraud, D. McDougald, A. Jansson, P. Nilsson, M. Harln, S. Kjelleberg and S. A. Rice, Dynamic modelling of cell death during biofilm development,, Journal of Theoretical Biology, 259 (2012), 23. doi: 10.1016/j.jtbi.2011.10.007. Google Scholar
[41]	M. M. Ballyk, D. A. Jones and H. L. Smith, Microbial competition in reactors with wall attachment,, Microbial Ecology, 41 (2001), 210. Google Scholar
[42]	M. Mimura, H. Sakaguchi and M. Matsushita, Reaction diffusion modeling of bacterial colony patterns,, Physica A: Statistical Mechanics and its Applications, 282 (2000), 283. doi: 10.1016/S0378-4371(00)00085-6. Google Scholar
[43]	M. R. Rahbar, I. Rasooli, S. Latif, M. Gargari, J. Amani and Y. Fattahian, In silico analysis of antibody triggering biofilm associated protein in Acinetobacter baumannii,, Journal of Theoretical Biology, 266 (2010), 275. doi: 10.1016/j.jtbi.2010.06.014. Google Scholar
[44]	M. Tindall, P. Maini, S. Porter and J. Armitage, Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations,, Bulletin of Mathematical Biology, 70 (2008), 1570. doi: 10.1007/s11538-008-9322-5. Google Scholar
[45]	N. Balaban, "Control of Biofilom Infections by Signal Manipulation,", J. William Costerton Editor Springer Publisher, (2008). doi: 10.1007/978-3-540-73853-4. Google Scholar
[46]	N. Hoiby, T. Bjarnsholt, M. Givskov, S. Molin and O. Ciofu, Antibiotic resistance of bacterial biofilms,, International Journal of Antimicrobial Agents, 35 (2010), 322. doi: 10.1016/j.ijantimicag.2009.12.011. Google Scholar
[47]	N. Hoiby, T. Bjarnsholt, M. L. Givskov, S. Molin and O. Ciofu, Antibiotic resistance of bacterial biofilms,, International Journal of Antimicrobial Agents, 35 (2010), 322. doi: 10.1016/j.ijantimicag.2009.12.011. Google Scholar
[48]	O. Wanner and W. Gujer, A multispecies biofilm model,, Biotechnology and Bioengineering, 28 (1986), 314. doi: 10.1002/bit.260280304. Google Scholar
[49]	P. Carol, Microbiology: Biofilms invade microbiology,, Journal of Science, 273 (1996), 1795. Google Scholar
[50]	P. Watnick and R. Kolter, Biofilm, city of microbes,, Journal of Bacteriology, 182 (2000), 2675. doi: 10.1128/JB.182.10.2675-2679.2000. Google Scholar
[51]	Q. Wang and T. Zhang, Review of mathematical models for biofilms,, Solid State Communications, 150 (2010), 21. doi: 10.1016/j.ssc.2010.01.021. Google Scholar
[52]	R. Erban and G. Othmer, From signal transduction to spatial pattern formation in E. coli: A paradigm for multiscale modeling in biology, multiscale model,, Journal of Simul., 3 (2005), 362. doi: 10.1137/040603565. Google Scholar
[53]	R. J. Leveque, "Finite Volume Methods for Hyperbolic Problems,", Cambridge University Press, (2002). doi: 10.1017/CBO9780511791253. Google Scholar
[54]	R. M. Donlan and J. W. Costerton, Survival mechanisms of clinically relevant microorganisms,, Clinical Microbiology Reviews, 15 (2002). doi: 10.1128/CMR.15.2.167-193.2002. Google Scholar
[55]	T. R. de Kievit, Quorum sensing in Pseudomonas aeruginosa biofilms,, Environmental Microbiology, 11 (2009), 279. Google Scholar
[56]	T. Tolker-Nielsen, U. C. Brinch, P. C. Ragas, J. B. Andersen, C. S. Jacobsen and S. Molin, Development and dynamics of Pseudomonas sp. biofilms,, Journal of Bacteriology, 182 (2000). Google Scholar
[57]	S. Abdul Rani, B. Pitts, H. Beyenal, R. A. Veluchamy, Z. Lewandowski, W. M. Davison, K. Buckingham-Meyer and P. S. Stewart, Spatial patterns of DNA replication, protein synthesis, and oxygen concentration within bacterial biofilms reveal diverse physiological states,, Journal of Bacteriology, 189 (2007), 4223. doi: 10.1128/JB.00107-07. Google Scholar
[58]	Z. Lewandowski and H. Beyenal, Mechanisms of microbially influenced corrosion,, Springer Berlin Heidelberg, 4 (2009), 35. Google Scholar
[59]	, http://grants.nih.gov/grants/guide/pa-files/PA-03-047.html, Last Accessed on June 11, (2012). Google Scholar

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[3]	Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231
[4]	Qi Wang, Jingyue Yang, Lu Zhang. Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3547-3574. doi: 10.3934/dcdsb.2017179
[5]	Shangbing Ai, Zhian Wang. Traveling bands for the Keller-Segel model with population growth. Mathematical Biosciences & Engineering, 2015, 12 (4) : 717-737. doi: 10.3934/mbe.2015.12.717
[6]	Vincent Calvez, Benoȋt Perthame, Shugo Yasuda. Traveling wave and aggregation in a flux-limited Keller-Segel model. Kinetic & Related Models, 2018, 11 (4) : 891-909. doi: 10.3934/krm.2018035
[7]	Norikazu Saito. Error analysis of a conservative finite-element approximation for the Keller-Segel system of chemotaxis. Communications on Pure & Applied Analysis, 2012, 11 (1) : 339-364. doi: 10.3934/cpaa.2012.11.339
[8]	Marco Di Francesco, Donatella Donatelli. Singular convergence of nonlinear hyperbolic chemotaxis systems to Keller-Segel type models. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 79-100. doi: 10.3934/dcdsb.2010.13.79
[9]	Zhichun Zhai. Well-posedness for two types of generalized Keller-Segel system of chemotaxis in critical Besov spaces. Communications on Pure & Applied Analysis, 2011, 10 (1) : 287-308. doi: 10.3934/cpaa.2011.10.287
[10]	Tian Xiang. On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4911-4946. doi: 10.3934/dcds.2014.34.4911
[11]	Yadong Shang, Jianjun Paul Tian, Bixiang Wang. Asymptotic behavior of the stochastic Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1367-1391. doi: 10.3934/dcdsb.2019020
[12]	Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara. Stationary solutions for some shadow system of the Keller-Segel model with logistic growth. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1023-1034. doi: 10.3934/dcdss.2015.8.1023
[13]	Jean Dolbeault, Christian Schmeiser. The two-dimensional Keller-Segel model after blow-up. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 109-121. doi: 10.3934/dcds.2009.25.109
[14]	Shen Bian, Jian-Guo Liu, Chen Zou. Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$. Kinetic & Related Models, 2014, 7 (1) : 9-28. doi: 10.3934/krm.2014.7.9
[15]	Wenting Cong, Jian-Guo Liu. Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 307-338. doi: 10.3934/dcdsb.2017015
[16]	Xinru Cao. Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3369-3378. doi: 10.3934/dcdsb.2017141
[17]	José Antonio Carrillo, Stefano Lisini, Edoardo Mainini. Uniqueness for Keller-Segel-type chemotaxis models. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1319-1338. doi: 10.3934/dcds.2014.34.1319
[18]	Qi Wang, Lu Zhang, Jingyue Yang, Jia Hu. Global existence and steady states of a two competing species Keller--Segel chemotaxis model. Kinetic & Related Models, 2015, 8 (4) : 777-807. doi: 10.3934/krm.2015.8.777
[19]	Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061
[20]	Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597

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