# 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 & 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,", 2nd 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,", 3rd 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

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##### 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,", 2nd 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,", 3rd 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
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