American Institute of Mathematical Sciences

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

On a chemotaxis model with saturated chemotactic flux

 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

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.
Citation: Alina Chertock, Alexander Kurganov, Xuefeng Wang, Yaping Wu. On a chemotaxis model with saturated chemotactic flux. Kinetic and 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), 341-356. doi: 10.1146/annurev.bi.44.070175.002013. [2] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177. doi: 10.1007/BF00275919. [3] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [4] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126. [5] J. T. Bonner, "The Cellular Slime Molds," 2nd ed., Princeton University Press, Princeton, New Jersey, 1967. [6] E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of escherichia coli, Nature, 349 (1991), 630-633. doi: 10.1038/349630a0. [7] E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53. doi: 10.1038/376049a0. [8] A. Chertock, Y. Epshteyn and A. Kurganov, High-order finite-difference and finite-volume methods for chemotaxis models,, in preparartion., (). [9] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosc., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. [10] M. H. Cohen and A. Robertson, Wave propagation in the early stages of aggregation of cellular slime molds, J. Theor. Biol., 31 (1971), 101-118. doi: 10.1016/0022-5193(71)90124-X. [11] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [12] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325. [13] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112. doi: 10.1137/S003614450036757X. [14] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [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), 633-683. [16] I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods, J. Sci. Comput., 39 (2009), 115-128. doi: 10.1007/s10915-008-9252-2. [17] T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125-144 (electronic). doi: 10.3934/dcdsb.2007.7.125. [18] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [19] D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. DMV, 105 (2003), 103-165. [20] D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. DMV, 106 (2004), 51-69. [21] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. [22] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [23] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. [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, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. [25] G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [26] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [27] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [28] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [29] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. doi: 10.1007/BF00277392. [30] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. [31] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. [32] J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$, J. Anal. Math., 76 (1998), 289-319. doi: 10.1007/BF02786939. [33] B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. [34] L. M. Prescott, J. P. Harley and D. A. Klein, "Microbiology," 3rd ed., Wm. C. Brown Publishers, Chicago-London, 1996. [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), 1-17. doi: 10.1016/0009-2509(89)85098-5. [36] J. Shi and X. Wang, On the global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. [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), 790-817 (electronic). doi: 10.1137/S0036139902415117. [38] X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560 (electronic). doi: 10.1137/S0036141098339897. [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), 2181-2189. doi: 10.1016/S0006-3495(95)80400-5.

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References:
 [1] A. Adler, Chemotaxis in bacteria, Ann. Rev. Biochem., 44 (1975), 341-356. doi: 10.1146/annurev.bi.44.070175.002013. [2] W. Alt, Biased random walk models for chemotaxis and related diffusion approximations, J. Math. Biol., 9 (1980), 147-177. doi: 10.1007/BF00275919. [3] H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems, Differential Integral Equations, 3 (1990), 13-75. [4] H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in "Function Spaces, Differential Operators and Nonlinear Analysis" (Friedrichroda, 1992), Teubner-Texte Math., 133, Teubner, Stuttgart, (1993), 9-126. [5] J. T. Bonner, "The Cellular Slime Molds," 2nd ed., Princeton University Press, Princeton, New Jersey, 1967. [6] E. O. Budrene and H. C. Berg, Complex patterns formed by motile cells of escherichia coli, Nature, 349 (1991), 630-633. doi: 10.1038/349630a0. [7] E. O. Budrene and H. C. Berg, Dynamics of formation of symmetrical patterns by chemotactic bacteria, Nature, 376 (1995), 49-53. doi: 10.1038/376049a0. [8] A. Chertock, Y. Epshteyn and A. Kurganov, High-order finite-difference and finite-volume methods for chemotaxis models,, in preparartion., (). [9] S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosc., 56 (1981), 217-237. doi: 10.1016/0025-5564(81)90055-9. [10] M. H. Cohen and A. Robertson, Wave propagation in the early stages of aggregation of cellular slime molds, J. Theor. Biol., 31 (1971), 101-118. doi: 10.1016/0022-5193(71)90124-X. [11] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Functional Analysis, 8 (1971), 321-340. doi: 10.1016/0022-1236(71)90015-2. [12] M. G. Crandall and P. H. Rabinowitz, Bifurcation, perturbation of simple eigenvalues and linearized stability, Arch. Rational Mech. Anal., 52 (1973), 161-180. doi: 10.1007/BF00282325. [13] S. Gottlieb, C.-W. Shu and E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev., 43 (2001), 89-112. doi: 10.1137/S003614450036757X. [14] D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. [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), 633-683. [16] I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods, J. Sci. Comput., 39 (2009), 115-128. doi: 10.1007/s10915-008-9252-2. [17] T. Hillen, K. Painter and C. Schmeiser, Global existence for chemotaxis with finite sampling radius, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 125-144 (electronic). doi: 10.3934/dcdsb.2007.7.125. [18] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [19] D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. DMV, 105 (2003), 103-165. [20] D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. II, Jahresber. DMV, 106 (2004), 51-69. [21] D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107. doi: 10.1016/j.jde.2004.10.022. [22] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415. doi: 10.1016/0022-5193(70)90092-5. [23] E. F. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. [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, Translations of Mathematical Monographs, Vol. 23, American Mathematical Society, Providence, R.I., 1967. [25] G. M. Lieberman, "Second Order Parabolic Differential Equations," World Scientific Publishing Co., Inc., River Edge, NJ, 1996. [26] C.-S. Lin, W.-M. Ni and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1988), 1-27. doi: 10.1016/0022-0396(88)90147-7. [27] T. Nagai, T. Senba and K. Yoshida, Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433. [28] W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. [29] H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298. doi: 10.1007/BF00277392. [30] C. S. Patlak, Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338. doi: 10.1007/BF02476407. [31] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. [32] J. Pejsachowicz and P. J. Rabier, Degree theory for $C^1$ Fredholm mappings of index $0$, J. Anal. Math., 76 (1998), 289-319. doi: 10.1007/BF02786939. [33] B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. [34] L. M. Prescott, J. P. Harley and D. A. Klein, "Microbiology," 3rd ed., Wm. C. Brown Publishers, Chicago-London, 1996. [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), 1-17. doi: 10.1016/0009-2509(89)85098-5. [36] J. Shi and X. Wang, On the global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations, 246 (2009), 2788-2812. doi: 10.1016/j.jde.2008.09.009. [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), 790-817 (electronic). doi: 10.1137/S0036139902415117. [38] X. Wang, Qualitative behavior of solutions of chemotactic diffusion systems: Effects of motility and chemotaxis and dynamics, SIAM J. Math. Anal., 31 (2000), 535-560 (electronic). doi: 10.1137/S0036141098339897. [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), 2181-2189. doi: 10.1016/S0006-3495(95)80400-5.
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