Modeling chemotaxis from L^2--closure moments in kinetic theory of active particles

Previous Article
Preface: Special issue on cancer modeling, analysis and control
DCDS-B Home
This Issue
Next Article
Designing proliferating cell population models with functional targets for control by anti-cancer drugs

June 2013, 18(4): 847-863. doi: 10.3934/dcdsb.2013.18.847

Modeling chemotaxis from $L^2$--closure moments in kinetic theory of active particles

Nicola Bellomo ^1, , Abdelghani Bellouquid ^2, , Juanjo Nieto ^3, and Juan Soler ^3,

1.	Department of Mathematica Sciences, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy
2.	University Cadi Ayyad, Ecole Nationale des Sciences Appliquées, Safi
3.	Departamento de Matemática Aplicada, Universidad de Granada, Spain, Spain

Received March 2012 Revised May 2012 Published February 2013

Abstract
Related Papers

This paper deals with the derivation of macroscopic tissue models from the underlying description delivered by a class of equations modeling binary mixtures of multi-cellular systems by methods of the kinetic theory for active particles. Cellular interactions generate both modification of biological functions and proliferative-destructive events. The analysis refers to a suitable hyperbolic approximation to show how the macroscopic tissue behavior can be described from the underlying cellular description. The approach is specifically focused on the modeling of chemotaxis phenomena by the Keller--Segel approximation.

Keywords: chemotaxis, cancer cells, closure method, Living systems, Keller--Segel., kinetic theory, multi-cellular systems.

Mathematics Subject Classification: 35Q92, 92C17, 35K5.

Citation: Nicola Bellomo, Abdelghani Bellouquid, Juanjo Nieto, Juan Soler. Modeling chemotaxis from $L^2$--closure moments in kinetic theory of active particles. Discrete & Continuous Dynamical Systems - B, 2013, 18 (4) : 847-863. doi: 10.3934/dcdsb.2013.18.847

References:

[1]	A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nature Reviews - Cancer, 8 (2008), 227. Google Scholar
[2]	N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Complexity and mathematical tools toward the modelling of multicellular growing systems,, Math. Comput. Modelling., 51 (2010), 441. doi: 10.1016/j.mcm.2009.12.002. Google Scholar
[3]	N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems,, Math. Models Methods Appl. Sci., 20 (2010), 1. doi: 10.1142/S0218202510004568. Google Scholar
[4]	N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512005885. Google Scholar
[5]	N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives,, Physics of Life Reviews, 8 (2011), 1. Google Scholar
[6]	A. Bellouquid, On the asymptotic analysis of kinetic models towards the compressible Euler and acoustic equations,, Math. Models Methods Appl. Sci., 14 (2004), 853. doi: 10.1142/S0218202504003477. Google Scholar
[7]	A. Bellouquid and E. De Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models,, Nonlinear Analysis: Real World. Appl., 12 (2011), 1111. doi: 10.1016/j.nonrwa.2010.09.005. Google Scholar
[8]	A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology,, Math. Models Methods Appl. Sci., 23 (2013). doi: 10.1142/S0218202512500650. Google Scholar
[9]	A. Bellouquid and M. Delitala, Kinetic (cellular) models of cell progression and competition with the immune system,, Z. Agnew. Math. Phys., 55 (2004), 295. doi: 10.1007/s00033-003-3057-9. Google Scholar
[10]	A. Bellouquid and M. Delitala, Mathematical models and tools of kinetic theory towards modelling complex biological systems,, Math. Models Methods Appl. Sci., 15 (2005), 1639. doi: 10.1142/S0218202505000923. Google Scholar
[11]	A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,", Birkäuser, (2006). Google Scholar
[12]	C. Cattani and A. Ciancio, Separable transition density in the hybrid model for tumor-immune system competition,, Comp. Math. Meth. in Medicine, (2012). Google Scholar
[13]	F. Cerreti, B. Perthame, C. Schmeiser, M. Tang and V. Vauchelet, Waves for the hyperbolic Keller-Segel model,, Math. Models Methods Appl. Sci., 21 (2011), 825. Google Scholar
[14]	F. A. Chalub, P. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits,, Monatsh. Math., 142 (2004), 123. doi: 10.1007/s00605-004-0234-7. Google Scholar
[15]	F. A. Chalub, Y. Dolak-Struss, P. Markowich, D. Oeltz, C. Schmeiser and A. Soref, Model hierarchies for cell aggregation by chemotaxis,, Math. Models Methods Appl. Sci., 16 (2006), 1173. doi: 10.1142/S0218202506001509. Google Scholar
[16]	M. A. J. Chaplain, M. Lachowicz, Z. Szyman'ska and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., 21 (2011), 719. doi: 10.1142/S0218202511005192. Google Scholar
[17]	K. C. Chen, R. M. Ford and P. T. Cummings, Perturbation expansion of Alt's cell balance equations reduces to Segel's 1d equation for shallow chemoattractant gradients,, SIAM J. Appl. Math., 59 (1999), 35. doi: 10.1137/S0036139996301283. Google Scholar
[18]	A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinetic and Related Models., 5 (2012), 51. doi: 10.3934/krm.2012.5.51. Google Scholar
[19]	Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation,, J. Math. Biol., 2 (2003), 153. doi: 10.1007/s00285-002-0173-7. Google Scholar
[20]	A. D'Onofrio, P. Cerrai and A. Gandolfi, "New Challenges for Cancer Systems Biomedicine,'', SIMAI-Springer Series, (2012). Google Scholar
[21]	H. Du, Z. Xu, J. D. Shrout and M. Alber, Multiscale modeling of Pesudomonas Aeruginosa swarming,, Math. Models Methods Appl. Sci., 21 (2011), 939. Google Scholar
[22]	F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar
[23]	J. Folkman, Role of angiogenesis in tumor growth and methastasis,, Seminar Oncology, 29 (2002), 15. Google Scholar
[24]	H. L. Hartwell, J. J. Hopfield, S. Leibner and A. W. Murray, From molecular to modular cell biology,, Nature, 402 (1999). Google Scholar
[25]	J. Haskovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system,, Comm. Part. Diff. Eqs., 36 (2011), 940. doi: 10.1080/03605302.2010.538783. Google Scholar
[26]	M. A. Herrero, A. Köhn and J. M. Pérez-Pomares, Modelling vascular morphogenesis: Current views on blood vessels development,, Math. Models Methods Appl. Sci., 19 (2009), 1483. doi: 10.1142/S021820250900384X. Google Scholar
[27]	M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system,, Nonlinearity, 10 (1997), 1739. doi: 10.1088/0951-7715/10/6/016. Google Scholar
[28]	T. Hillen, On the $L^2$-moment closure of transport equation: The Cattaneo approximation,, Disc. Cont. Dyn. Syst. B, 4 (2004), 961. doi: 10.3934/dcdsb.2004.4.961. Google Scholar
[29]	T. Hillen and J. K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar
[30]	T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump processes,, SIAM J. Appl. Math., 61 (2000), 751. doi: 10.1137/S0036139999358167. Google Scholar
[31]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: Jahresber I,, Deutsch. Math.-Verein, 105 (2003), 103. Google Scholar
[32]	E. F. Keller and L. A. Segel, Initiation od slide mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. Google Scholar
[33]	E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. Google Scholar
[34]	E. F. Keller and L. A. Segel, Traveling Bands of chemotactic Bacteria: A Theoretical Analysis,, J. Theor. Biol., 30 (1971), 235. Google Scholar
[35]	E. F. Keller, Assessing the Keller-Segel model: How has it fared? In biological growth and spread,, Proc. Conf. Math. Biol. Heidelberg, (1980), 379. Google Scholar
[36]	P. Koumoutsakos, B. Bayati, F. Milde and G. Tauriello, Particle simulations of morphogenesis,, Math. Models Methods Appl. Sci., 21 (2011), 955. Google Scholar
[37]	R. Kowalczyck, Preventing blow-up in a chemotaxis model,, J. Math. Anal. Appl., 305 (2005), 566. doi: 10.1016/j.jmaa.2004.12.009. Google Scholar
[38]	M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumors,, Math. Models Methods Appl. Sci., 15 (2005), 1667. doi: 10.1142/S0218202505000935. Google Scholar
[39]	M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems,, Probab. Eng. Mechanics, 26 (2011), 54. Google Scholar
[40]	M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology,, Nonlinear Analysis: Real World Applications, 12 (2011), 2396. doi: 10.1016/j.nonrwa.2011.02.014. Google Scholar
[41]	C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552. Google Scholar
[42]	A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques,, SIAM J. Math. Analysis, 40 (2008), 215. doi: 10.1137/050645269. Google Scholar
[43]	A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model,, Math. Models Methods Appl. Sci., 20 (2010), 440. doi: 10.1142/S0218202510004301. Google Scholar
[44]	I. Muller and T. Ruggeri, "Rational Extended Thermodynamics,", Springer, (1998). doi: 10.1007/978-1-4612-2210-1. Google Scholar
[45]	J. D. Murray, "Mathematical Biology,", Springer, (1989). Google Scholar
[46]	H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar
[47]	H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar
[48]	K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Canadian Appl. Math. Quart., 10 (2002), 501. Google Scholar
[49]	C. S. Patlak, Random walk with persistant and external bias,, Bull. Math. Biol., 15 (1953), 311. Google Scholar
[50]	L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model,, Studies Appl. Math, 123 (2009), 337. doi: 10.1111/j.1467-9590.2009.00457.x. Google Scholar
[51]	C. Ringhofer, C. Schmeiser and A. Zwirchmayr, Moment methods for the semiconductor Boltzmann equation on bounded position domains,, SIAM J. Num. Anal., 39 (2001), 1078. doi: 10.1137/S0036142998335984. Google Scholar
[52]	L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653. Google Scholar
[53]	M. A. Stolarska, K. I. M. Yangjin and H. G. Othmer, Multi-scale models of cell and tissue dynamics,, Phil. Trans. Royal Society A: Math. Phys. Eng. Sci., 367 (2009), 3525. doi: 10.1098/rsta.2009.0095. Google Scholar
[54]	Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling,, Nonlinear Analysis: RWA, 12 (2011), 418. doi: 10.1016/j.nonrwa.2010.06.027. Google Scholar
[55]	R. A. Weinberg, "The Biology of Cancer,", Garland Sciences - Taylor and Francis, (2007). Google Scholar

show all references

References:

[1]	A. R. A. Anderson and V. Quaranta, Integrative mathematical oncology,, Nature Reviews - Cancer, 8 (2008), 227. Google Scholar
[2]	N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Complexity and mathematical tools toward the modelling of multicellular growing systems,, Math. Comput. Modelling., 51 (2010), 441. doi: 10.1016/j.mcm.2009.12.002. Google Scholar
[3]	N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multiscale biological tissue models and flux-limited chemotaxis for multicellular growing systems,, Math. Models Methods Appl. Sci., 20 (2010), 1. doi: 10.1142/S0218202510004568. Google Scholar
[4]	N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, On the asymptotic theory from microscopic to macroscopic growing tissue models: An overview with perspectives,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512005885. Google Scholar
[5]	N. Bellomo and B. Carbonaro, Towards a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives,, Physics of Life Reviews, 8 (2011), 1. Google Scholar
[6]	A. Bellouquid, On the asymptotic analysis of kinetic models towards the compressible Euler and acoustic equations,, Math. Models Methods Appl. Sci., 14 (2004), 853. doi: 10.1142/S0218202504003477. Google Scholar
[7]	A. Bellouquid and E. De Angelis, From kinetic models of multicellular growing systems to macroscopic biological tissue models,, Nonlinear Analysis: Real World. Appl., 12 (2011), 1111. doi: 10.1016/j.nonrwa.2010.09.005. Google Scholar
[8]	A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology,, Math. Models Methods Appl. Sci., 23 (2013). doi: 10.1142/S0218202512500650. Google Scholar
[9]	A. Bellouquid and M. Delitala, Kinetic (cellular) models of cell progression and competition with the immune system,, Z. Agnew. Math. Phys., 55 (2004), 295. doi: 10.1007/s00033-003-3057-9. Google Scholar
[10]	A. Bellouquid and M. Delitala, Mathematical models and tools of kinetic theory towards modelling complex biological systems,, Math. Models Methods Appl. Sci., 15 (2005), 1639. doi: 10.1142/S0218202505000923. Google Scholar
[11]	A. Bellouquid and M. Delitala, "Mathematical Modeling of Complex Biological Systems. A Kinetic Theory Approach,", Birkäuser, (2006). Google Scholar
[12]	C. Cattani and A. Ciancio, Separable transition density in the hybrid model for tumor-immune system competition,, Comp. Math. Meth. in Medicine, (2012). Google Scholar
[13]	F. Cerreti, B. Perthame, C. Schmeiser, M. Tang and V. Vauchelet, Waves for the hyperbolic Keller-Segel model,, Math. Models Methods Appl. Sci., 21 (2011), 825. Google Scholar
[14]	F. A. Chalub, P. Markovich, B. Perthame and C. Schmeiser, Kinetic models for chemotaxis and their drift-diffusion limits,, Monatsh. Math., 142 (2004), 123. doi: 10.1007/s00605-004-0234-7. Google Scholar
[15]	F. A. Chalub, Y. Dolak-Struss, P. Markowich, D. Oeltz, C. Schmeiser and A. Soref, Model hierarchies for cell aggregation by chemotaxis,, Math. Models Methods Appl. Sci., 16 (2006), 1173. doi: 10.1142/S0218202506001509. Google Scholar
[16]	M. A. J. Chaplain, M. Lachowicz, Z. Szyman'ska and D. Wrzosek, Mathematical modelling of cancer invasion: The importance of cell-cell adhesion and cell-matrix adhesion, Math. Models Methods Appl. Sci., 21 (2011), 719. doi: 10.1142/S0218202511005192. Google Scholar
[17]	K. C. Chen, R. M. Ford and P. T. Cummings, Perturbation expansion of Alt's cell balance equations reduces to Segel's 1d equation for shallow chemoattractant gradients,, SIAM J. Appl. Math., 59 (1999), 35. doi: 10.1137/S0036139996301283. Google Scholar
[18]	A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux,, Kinetic and Related Models., 5 (2012), 51. doi: 10.3934/krm.2012.5.51. Google Scholar
[19]	Y. Dolak and T. Hillen, Cattaneo models for chemotaxis, numerical solution and pattern formation,, J. Math. Biol., 2 (2003), 153. doi: 10.1007/s00285-002-0173-7. Google Scholar
[20]	A. D'Onofrio, P. Cerrai and A. Gandolfi, "New Challenges for Cancer Systems Biomedicine,'', SIMAI-Springer Series, (2012). Google Scholar
[21]	H. Du, Z. Xu, J. D. Shrout and M. Alber, Multiscale modeling of Pesudomonas Aeruginosa swarming,, Math. Models Methods Appl. Sci., 21 (2011), 939. Google Scholar
[22]	F. Filbet, P. Laurençot and B. Perthame, Derivation of hyperbolic models for chemosensitive movement,, J. Math. Biol., 50 (2005), 189. doi: 10.1007/s00285-004-0286-2. Google Scholar
[23]	J. Folkman, Role of angiogenesis in tumor growth and methastasis,, Seminar Oncology, 29 (2002), 15. Google Scholar
[24]	H. L. Hartwell, J. J. Hopfield, S. Leibner and A. W. Murray, From molecular to modular cell biology,, Nature, 402 (1999). Google Scholar
[25]	J. Haskovec and C. Schmeiser, Convergence of a stochastic particle approximation for measure solutions of the 2D Keller-Segel system,, Comm. Part. Diff. Eqs., 36 (2011), 940. doi: 10.1080/03605302.2010.538783. Google Scholar
[26]	M. A. Herrero, A. Köhn and J. M. Pérez-Pomares, Modelling vascular morphogenesis: Current views on blood vessels development,, Math. Models Methods Appl. Sci., 19 (2009), 1483. doi: 10.1142/S021820250900384X. Google Scholar
[27]	M. A. Herrero, E. Medina and J. J. L. Velázquez, Finite-time aggregation into a single point in a reaction-diffusion system,, Nonlinearity, 10 (1997), 1739. doi: 10.1088/0951-7715/10/6/016. Google Scholar
[28]	T. Hillen, On the $L^2$-moment closure of transport equation: The Cattaneo approximation,, Disc. Cont. Dyn. Syst. B, 4 (2004), 961. doi: 10.3934/dcdsb.2004.4.961. Google Scholar
[29]	T. Hillen and J. K. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar
[30]	T. Hillen and H. G. Othmer, The diffusion limit of transport equations derived from velocity jump processes,, SIAM J. Appl. Math., 61 (2000), 751. doi: 10.1137/S0036139999358167. Google Scholar
[31]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences: Jahresber I,, Deutsch. Math.-Verein, 105 (2003), 103. Google Scholar
[32]	E. F. Keller and L. A. Segel, Initiation od slide mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. Google Scholar
[33]	E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. Google Scholar
[34]	E. F. Keller and L. A. Segel, Traveling Bands of chemotactic Bacteria: A Theoretical Analysis,, J. Theor. Biol., 30 (1971), 235. Google Scholar
[35]	E. F. Keller, Assessing the Keller-Segel model: How has it fared? In biological growth and spread,, Proc. Conf. Math. Biol. Heidelberg, (1980), 379. Google Scholar
[36]	P. Koumoutsakos, B. Bayati, F. Milde and G. Tauriello, Particle simulations of morphogenesis,, Math. Models Methods Appl. Sci., 21 (2011), 955. Google Scholar
[37]	R. Kowalczyck, Preventing blow-up in a chemotaxis model,, J. Math. Anal. Appl., 305 (2005), 566. doi: 10.1016/j.jmaa.2004.12.009. Google Scholar
[38]	M. Lachowicz, Micro and meso scales of description corresponding to a model of tissue invasion by solid tumors,, Math. Models Methods Appl. Sci., 15 (2005), 1667. doi: 10.1142/S0218202505000935. Google Scholar
[39]	M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems,, Probab. Eng. Mechanics, 26 (2011), 54. Google Scholar
[40]	M. Lachowicz, Individually-based Markov processes modeling nonlinear systems in mathematical biology,, Nonlinear Analysis: Real World Applications, 12 (2011), 2396. doi: 10.1016/j.nonrwa.2011.02.014. Google Scholar
[41]	C. D. Levermore, Moment closure hierarchies for kinetic theories,, J. Stat. Phys., 83 (1996), 1021. doi: 10.1007/BF02179552. Google Scholar
[42]	A. Marciniak-Czochra and M. Ptashnyk, Derivation of a macroscopic receptor-based model using homogenization techniques,, SIAM J. Math. Analysis, 40 (2008), 215. doi: 10.1137/050645269. Google Scholar
[43]	A. Marciniak-Czochra and M. Ptashnyk, Boundedness of solutions of a haptotaxis model,, Math. Models Methods Appl. Sci., 20 (2010), 440. doi: 10.1142/S0218202510004301. Google Scholar
[44]	I. Muller and T. Ruggeri, "Rational Extended Thermodynamics,", Springer, (1998). doi: 10.1007/978-1-4612-2210-1. Google Scholar
[45]	J. D. Murray, "Mathematical Biology,", Springer, (1989). Google Scholar
[46]	H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar
[47]	H. G. Othmer and T. Hillen, The diffusion limit of transport equations II: Chemotaxis equations,, SIAM J. Appl. Math., 62 (2002), 1222. doi: 10.1137/S0036139900382772. Google Scholar
[48]	K. Painter and T. Hillen, Volume-filling and quorum-sensing in models for chemosensitive movement,, Canadian Appl. Math. Quart., 10 (2002), 501. Google Scholar
[49]	C. S. Patlak, Random walk with persistant and external bias,, Bull. Math. Biol., 15 (1953), 311. Google Scholar
[50]	L. E. Payne and B. Straughan, Decay for a Keller-Segel chemotaxis model,, Studies Appl. Math, 123 (2009), 337. doi: 10.1111/j.1467-9590.2009.00457.x. Google Scholar
[51]	C. Ringhofer, C. Schmeiser and A. Zwirchmayr, Moment methods for the semiconductor Boltzmann equation on bounded position domains,, SIAM J. Num. Anal., 39 (2001), 1078. doi: 10.1137/S0036142998335984. Google Scholar
[52]	L. A. Segel, A theoretical study of receptor mechanisms in bacterial chemotaxis,, SIAM J. Appl. Math., 32 (1977), 653. Google Scholar
[53]	M. A. Stolarska, K. I. M. Yangjin and H. G. Othmer, Multi-scale models of cell and tissue dynamics,, Phil. Trans. Royal Society A: Math. Phys. Eng. Sci., 367 (2009), 3525. doi: 10.1098/rsta.2009.0095. Google Scholar
[54]	Y. Tao, Global existence for a haptotaxis model of cancer invasion with tissue remodeling,, Nonlinear Analysis: RWA, 12 (2011), 418. doi: 10.1016/j.nonrwa.2010.06.027. Google Scholar
[55]	R. A. Weinberg, "The Biology of Cancer,", Garland Sciences - Taylor and Francis, (2007). Google Scholar

[1]	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
[2]	Alexandra Fronville, Abdoulaye Sarr, Vincent Rodin. Modelling multi-cellular growth using morphological analysis. Discrete & Continuous Dynamical Systems - B, 2017, 22 (1) : 83-99. doi: 10.3934/dcdsb.2017004
[3]	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
[4]	Sachiko Ishida. An iterative approach to $L^\infty$-boundedness in quasilinear Keller-Segel systems. Conference Publications, 2015, 2015 (special) : 635-643. doi: 10.3934/proc.2015.0635
[5]	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
[6]	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
[7]	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
[8]	Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046
[9]	Robert A. Gatenby, B. Roy Frieden. The Role of Non-Genomic Information in Maintaining Thermodynamic Stability in Living Systems. Mathematical Biosciences & Engineering, 2005, 2 (1) : 43-51. doi: 10.3934/mbe.2005.2.43
[10]	Patrizia Pucci, Maria Cesarina Salvatori. On an initial value problem modeling evolution and selection in living systems. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 807-821. doi: 10.3934/dcdss.2014.7.807
[11]	Sachiko Ishida, Tomomi Yokota. Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 345-354. doi: 10.3934/proc.2013.2013.345
[12]	Sachiko Ishida, Tomomi Yokota. Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2569-2596. doi: 10.3934/dcdsb.2013.18.2569
[13]	Sachiko Ishida. $L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 335-344. doi: 10.3934/proc.2013.2013.335
[14]	Sachiko Ishida, Yusuke Maeda, Tomomi Yokota. Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2537-2568. doi: 10.3934/dcdsb.2013.18.2537
[15]	Tong Li, Anthony Suen. Existence of intermediate weak solution to the equations of multi-dimensional chemotaxis systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 861-875. doi: 10.3934/dcds.2016.36.861
[16]	Mihaela Negreanu, J. Ignacio Tello. On a comparison method to reaction-diffusion systems and its applications to chemotaxis. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2669-2688. doi: 10.3934/dcdsb.2013.18.2669
[17]	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
[18]	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
[19]	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
[20]	Hui Huang, Jian-Guo Liu. Well-posedness for the Keller-Segel equation with fractional Laplacian and the theory of propagation of chaos. Kinetic & Related Models, 2016, 9 (4) : 715-748. doi: 10.3934/krm.2016013

2018 Impact Factor: 1.008

American Institute of Mathematical Sciences