On a chemotaxis model with saturated chemotactic flux

Previous Article
Ghost effect for a vapor-vapor mixture
KRM Home
This Issue
Next Article
A smooth 3D model for fiber lay-down in nonwoven production processes

March 2012, 5(1): 51-95. doi: 10.3934/krm.2012.5.51

On a chemotaxis model with saturated chemotactic flux

Alina Chertock ^1, , Alexander Kurganov ^2, , Xuefeng Wang ^2, and Yaping Wu ^3,

1.	Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States
2.	Mathematics Department, Tulane University, New Orleans, LA 70118
3.	Department of Mathematics, Capital Normal University, Beijing 100048, China

Received May 2011 Revised August 2011 Published January 2012

Abstract
Related Papers

We propose a PDE chemotaxis model, which can be viewed as a regularization of the Patlak-Keller-Segel (PKS) system. Our modification is based on a fundamental physical property of the chemotactic flux function---its boundedness. This means that the cell velocity is proportional to the magnitude of the chemoattractant gradient only when the latter is small, while when the chemoattractant gradient tends to infinity the cell velocity saturates. Unlike the original PKS system, the solutions of the modified model do not blow up in either finite or infinite time in any number of spatial dimensions, thus making it possible to use bounded spiky steady states to model cell aggregation. After obtaining local and global existence results, we use the local and global bifurcation theories to show the existence of one-dimensional spiky steady states; we also study the stability of bifurcating steady states. Finally, we numerically verify these analytical results, and then demonstrate that solutions of the two-dimensional model with nonlinear saturated chemotactic flux function typically develop very complicated spiky structures.

Keywords: spiky steady states, local and global existence, Chemotaxis, bifurcation, saturation of flux, second-order positivity preserving upwind scheme., hybrid finite-volume-finite-difference method.

Mathematics Subject Classification: Primary: 92C17, 35K50, 35B36, 76M12, 76M20; Secondary: 35B3.

Citation: Alina Chertock, Alexander Kurganov, Xuefeng Wang, Yaping Wu. On a chemotaxis model with saturated chemotactic flux. Kinetic & Related Models, 2012, 5 (1) : 51-95. doi: 10.3934/krm.2012.5.51

References:

[1]	A. Adler, Chemotaxis in bacteria,, Ann. Rev. Biochem., 44 (1975), 341. doi: 10.1146/annurev.bi.44.070175.002013. Google Scholar
[2]	W. Alt, Biased random walk models for chemotaxis and related diffusion approximations,, J. Math. Biol., 9 (1980), 147. doi: 10.1007/BF00275919. Google Scholar
[3]	H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13. Google Scholar
[4]	H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, 133 (1993), 9. Google Scholar
[5]	J. T. Bonner, "The Cellular Slime Molds,", 2^nd ed., (1967). Google Scholar
[6]	E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of escherichia coli,, Nature, 349 (1991), 630. doi: 10.1038/349630a0. Google Scholar
[7]	E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 376 (1995), 49. doi: 10.1038/376049a0. Google Scholar
[8]	A. Chertock, Y. Epshteyn and A. Kurganov, High-order finite-difference and finite-volume methods for chemotaxis models,, in preparartion., (). Google Scholar
[9]	S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Biosc., 56 (1981), 217. doi: 10.1016/0025-5564(81)90055-9. Google Scholar
[10]	M. H. Cohen and A. Robertson, Wave propagation in the early stages of aggregation of cellular slime molds,, J. Theor. Biol., 31 (1971), 101. doi: 10.1016/0022-5193(71)90124-X. Google Scholar
[11]	M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar
[12]	M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161. doi: 10.1007/BF00282325. Google Scholar
[13]	S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods,, SIAM Rev., 43 (2001), 89. doi: 10.1137/S003614450036757X. Google Scholar
[14]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981). Google Scholar
[15]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Normale Superiore Pisa Cl. Sci. (4), 24 (1997), 633. Google Scholar
[16]	I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods,, J. Sci. Comput., 39 (2009), 115. doi: 10.1007/s10915-008-9252-2. Google Scholar
[17]	T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125. doi: 10.3934/dcdsb.2007.7.125. Google Scholar
[18]	T. Hillen and K. J. 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
[19]	D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. DMV, 105 (2003), 103. Google Scholar
[20]	D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. DMV, 106 (2004), 51. Google Scholar
[21]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar
[22]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar
[23]	E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6. Google Scholar
[24]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translated from the Russian by S. Smith, (1967). Google Scholar
[25]	G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996). Google Scholar
[26]	C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7. Google Scholar
[27]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411. Google Scholar
[28]	W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9. Google Scholar
[29]	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
[30]	C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar
[31]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar
[32]	J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$,, J. Anal. Math., 76 (1998), 289. doi: 10.1007/BF02786939. Google Scholar
[33]	B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics, (2007). Google Scholar
[34]	L. M. Prescott, J. P. Harley and D. A. Klein, "Microbiology,", 3^rd ed., (1996). Google Scholar
[35]	M. A. Rivero, R. T. Tranquillo, H. M. Buettner and D. A. Lauffenburger, Transport models for chemotactic cell populations based on individual cell behavior,, Chem. Eng. Sci., 44 (1989), 1. doi: 10.1016/0009-2509(89)85098-5. Google Scholar
[36]	J. Shi and X. Wang, On the global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009. Google Scholar
[37]	B. D. Sleeman, M. J. Ward and J. C. Wei, The existence and stability of spike patterns in a chemotaxis model,, SIAM J. Appl. Math., 65 (2005), 790. doi: 10.1137/S0036139902415117. Google Scholar
[38]	X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics,, SIAM J. Math. Anal., 31 (2000), 535. doi: 10.1137/S0036141098339897. Google Scholar
[39]	D. Woodward, R. Tyson, M. Myerscough, J. Murray, E. Budrene and H. Berg, Spatio-temporal patterns generated by S. typhimurium,, Biophys. J., 68 (1995), 2181. doi: 10.1016/S0006-3495(95)80400-5. Google Scholar

show all references

References:

[1]	A. Adler, Chemotaxis in bacteria,, Ann. Rev. Biochem., 44 (1975), 341. doi: 10.1146/annurev.bi.44.070175.002013. Google Scholar
[2]	W. Alt, Biased random walk models for chemotaxis and related diffusion approximations,, J. Math. Biol., 9 (1980), 147. doi: 10.1007/BF00275919. Google Scholar
[3]	H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13. Google Scholar
[4]	H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, 133 (1993), 9. Google Scholar
[5]	J. T. Bonner, "The Cellular Slime Molds,", 2^nd ed., (1967). Google Scholar
[6]	E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of escherichia coli,, Nature, 349 (1991), 630. doi: 10.1038/349630a0. Google Scholar
[7]	E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria,, Nature, 376 (1995), 49. doi: 10.1038/376049a0. Google Scholar
[8]	A. Chertock, Y. Epshteyn and A. Kurganov, High-order finite-difference and finite-volume methods for chemotaxis models,, in preparartion., (). Google Scholar
[9]	S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis,, Math. Biosc., 56 (1981), 217. doi: 10.1016/0025-5564(81)90055-9. Google Scholar
[10]	M. H. Cohen and A. Robertson, Wave propagation in the early stages of aggregation of cellular slime molds,, J. Theor. Biol., 31 (1971), 101. doi: 10.1016/0022-5193(71)90124-X. Google Scholar
[11]	M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues,, J. Functional Analysis, 8 (1971), 321. doi: 10.1016/0022-1236(71)90015-2. Google Scholar
[12]	M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability,, Arch. Rational Mech. Anal., 52 (1973), 161. doi: 10.1007/BF00282325. Google Scholar
[13]	S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods,, SIAM Rev., 43 (2001), 89. doi: 10.1137/S003614450036757X. Google Scholar
[14]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (1981). Google Scholar
[15]	M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model,, Ann. Scuola Normale Superiore Pisa Cl. Sci. (4), 24 (1997), 633. Google Scholar
[16]	I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods,, J. Sci. Comput., 39 (2009), 115. doi: 10.1007/s10915-008-9252-2. Google Scholar
[17]	T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125. doi: 10.3934/dcdsb.2007.7.125. Google Scholar
[18]	T. Hillen and K. J. 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
[19]	D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. I,, Jahresber. DMV, 105 (2003), 103. Google Scholar
[20]	D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. DMV, 106 (2004), 51. Google Scholar
[21]	D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system,, J. Differential Equations, 215 (2005), 52. doi: 10.1016/j.jde.2004.10.022. Google Scholar
[22]	E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar
[23]	E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6. Google Scholar
[24]	O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, "Linear and Quasilinear Equations of Parabolic Type,", Translated from the Russian by S. Smith, (1967). Google Scholar
[25]	G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (1996). Google Scholar
[26]	C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1. doi: 10.1016/0022-0396(88)90147-7. Google Scholar
[27]	T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis,, Funkcial. Ekvac., 40 (1997), 411. Google Scholar
[28]	W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9. Google Scholar
[29]	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
[30]	C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311. doi: 10.1007/BF02476407. Google Scholar
[31]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983). Google Scholar
[32]	J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$,, J. Anal. Math., 76 (1998), 289. doi: 10.1007/BF02786939. Google Scholar
[33]	B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics, (2007). Google Scholar
[34]	L. M. Prescott, J. P. Harley and D. A. Klein, "Microbiology,", 3^rd ed., (1996). Google Scholar
[35]	M. A. Rivero, R. T. Tranquillo, H. M. Buettner and D. A. Lauffenburger, Transport models for chemotactic cell populations based on individual cell behavior,, Chem. Eng. Sci., 44 (1989), 1. doi: 10.1016/0009-2509(89)85098-5. Google Scholar
[36]	J. Shi and X. Wang, On the global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788. doi: 10.1016/j.jde.2008.09.009. Google Scholar
[37]	B. D. Sleeman, M. J. Ward and J. C. Wei, The existence and stability of spike patterns in a chemotaxis model,, SIAM J. Appl. Math., 65 (2005), 790. doi: 10.1137/S0036139902415117. Google Scholar
[38]	X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics,, SIAM J. Math. Anal., 31 (2000), 535. doi: 10.1137/S0036141098339897. Google Scholar
[39]	D. Woodward, R. Tyson, M. Myerscough, J. Murray, E. Budrene and H. Berg, Spatio-temporal patterns generated by S. typhimurium,, Biophys. J., 68 (1995), 2181. doi: 10.1016/S0006-3495(95)80400-5. Google Scholar

[1]	Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-20. doi: 10.3934/dcdsb.2019146
[2]	Yingwen Guo, Yinnian He. Fully discrete finite element method based on second-order Crank-Nicolson/Adams-Bashforth scheme for the equations of motion of Oldroyd fluids of order one. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2583-2609. doi: 10.3934/dcdsb.2015.20.2583
[3]	Wei Qu, Siu-Long Lei, Seak-Weng Vong. A note on the stability of a second order finite difference scheme for space fractional diffusion equations. Numerical Algebra, Control & Optimization, 2014, 4 (4) : 317-325. doi: 10.3934/naco.2014.4.317
[4]	Cheng Wang, Jian-Guo Liu. Positivity property of second-order flux-splitting schemes for the compressible Euler equations. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 201-228. doi: 10.3934/dcdsb.2003.3.201
[5]	T. Hillen, K. Painter, Christian Schmeiser. Global existence for chemotaxis with finite sampling radius. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 125-144. doi: 10.3934/dcdsb.2007.7.125
[6]	Caterina Calgaro, Meriem Ezzoug, Ezzeddine Zahrouni. Stability and convergence of an hybrid finite volume-finite element method for a multiphasic incompressible fluid model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 429-448. doi: 10.3934/cpaa.2018024
[7]	Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644
[8]	Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768
[9]	Yaping Wu, Qian Xu. The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 367-385. doi: 10.3934/dcds.2011.29.367
[10]	Nora Aïssiouene, Marie-Odile Bristeau, Edwige Godlewski, Jacques Sainte-Marie. A combined finite volume - finite element scheme for a dispersive shallow water system. Networks & Heterogeneous Media, 2016, 11 (1) : 1-27. doi: 10.3934/nhm.2016.11.1
[11]	Francis Filbet, Roland Duclous, Bruno Dubroca. Analysis of a high order finite volume scheme for the 1D Vlasov-Poisson system. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 283-305. doi: 10.3934/dcdss.2012.5.283
[12]	Nan Li, Song Wang, Shuhua Zhang. Pricing options on investment project contraction and ownership transfer using a finite volume scheme and an interior penalty method. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-20. doi: 10.3934/jimo.2019006
[13]	Rui Li, Yingjing Shi. Finite-time optimal consensus control for second-order multi-agent systems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 929-943. doi: 10.3934/jimo.2014.10.929
[14]	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
[15]	Pavol Kútik, Karol Mikula. Diamond--cell finite volume scheme for the Heston model. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 913-931. doi: 10.3934/dcdss.2015.8.913
[16]	Ronald E. Mickens. A nonstandard finite difference scheme for the drift-diffusion system. Conference Publications, 2009, 2009 (Special) : 558-563. doi: 10.3934/proc.2009.2009.558
[17]	Renaud Leplaideur. From local to global equilibrium states: Thermodynamic formalism via an inducing scheme. Electronic Research Announcements, 2014, 21: 72-79. doi: 10.3934/era.2014.21.72
[18]	Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[19]	So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343
[20]	Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

2018 Impact Factor: 1.38

American Institute of Mathematical Sciences