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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 |
References:
[1] |
A. Adler, Chemotaxis in bacteria,, Ann. Rev. Biochem., 44 (1975), 341.
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.
doi: 10.1007/BF00275919. |
[3] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.
|
[4] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, 133 (1993), 9.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (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.
|
[16] |
I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods,, J. Sci. Comput., 39 (2009), 115.
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.
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.
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.
|
[20] |
D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. DMV, 106 (2004), 51.
|
[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. |
[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. |
[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. |
[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).
|
[25] |
G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (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.
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.
|
[28] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9.
|
[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. |
[30] |
C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311.
doi: 10.1007/BF02476407. |
[31] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).
|
[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. |
[33] |
B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics, (2007).
|
[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. |
[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. |
[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. |
[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. |
[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. |
show all references
References:
[1] |
A. Adler, Chemotaxis in bacteria,, Ann. Rev. Biochem., 44 (1975), 341.
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.
doi: 10.1007/BF00275919. |
[3] |
H. Amann, Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems,, Differential Integral Equations, 3 (1990), 13.
|
[4] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,, in, 133 (1993), 9.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[14] |
D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lecture Notes in Mathematics, 840 (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.
|
[16] |
I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods,, J. Sci. Comput., 39 (2009), 115.
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.
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.
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.
|
[20] |
D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences. II,, Jahresber. DMV, 106 (2004), 51.
|
[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. |
[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. |
[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. |
[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).
|
[25] |
G. M. Lieberman, "Second Order Parabolic Differential Equations,", World Scientific Publishing Co., (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.
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.
|
[28] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9.
|
[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. |
[30] |
C. S. Patlak, Random walk with persistence and external bias,, Bull. Math. Biophys., 15 (1953), 311.
doi: 10.1007/BF02476407. |
[31] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).
|
[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. |
[33] |
B. Perthame, "Transport Equations in Biology,", Frontiers in Mathematics, (2007).
|
[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. |
[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. |
[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. |
[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. |
[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. |
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