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Numerical study of two-species chemotaxis models
1. | Mathematics Department, Tulane University, 6823 St. Charles Ave, New Orleans, LA 70118, United States |
2. | Institute of Mathematics, University of Mainz, Staudingerweg 9, 55099 Mainz, Germany |
References:
[1] |
A. Chertock, Y. Epshteyn and A. Kurganov, High-order finite-difference and finite-volume methods for chemotaxis models,, in preparation., ().
|
[2] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[3] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosc., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[4] |
C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math., 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[5] |
E. E. Espejo, A. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential Integral Equations, 25 (2012), 251-288. |
[6] |
E. E. Espejo, A. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential Integral Equations, 23 (2010), 451-462. |
[7] |
E. E. Espejo, K. Vilches and C. Conca, Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math., 24 (2013), 297-313.
doi: 10.1017/S0956792512000411. |
[8] |
E. E. Espejo Arenas, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[9] |
A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Math. Models Methods Appl. Sci., 14 (2004), 503-533.
doi: 10.1142/S0218202504003337. |
[10] |
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. |
[11] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683. |
[12] |
I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods, J. Sci. Comput., 39 (2009), 115-128.
doi: 10.1007/s10915-008-9252-2. |
[13] |
T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[14] |
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. |
[15] |
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.
doi: 10.3934/dcdsb.2007.7.125. |
[16] |
D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165. |
[17] |
D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. DMV, 106 (2004), 51-69. |
[18] |
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. |
[19] |
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. |
[20] |
M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Diff. Integral Eqns., 4 (2003), 427-452. |
[21] |
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, 23, American Mathematical Society, Providence, R.I., (1968). |
[22] |
K.-A. Lie and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput., 24 (2003), 1157-1174.
doi: 10.1137/S1064827501392880. |
[23] |
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. |
[24] |
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. |
[25] |
H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408-463.
doi: 10.1016/0021-9991(90)90260-8. |
[26] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[27] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math: Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[28] |
B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. |
[29] |
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.
doi: 10.1137/S0036139902415117. |
[30] |
P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21 (1984), 995-1011.
doi: 10.1137/0721062. |
[31] |
J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1198-1223.
doi: 10.1137/S0036139903433888. |
[32] |
J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1224-1248.
doi: 10.1137/S003613990343389X. |
[33] |
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.
doi: 10.1137/S0036141098339897. |
[34] |
G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
show all references
References:
[1] |
A. Chertock, Y. Epshteyn and A. Kurganov, High-order finite-difference and finite-volume methods for chemotaxis models,, in preparation., ().
|
[2] |
A. Chertock, A. Kurganov, X. Wang and Y. Wu, On a chemotaxis model with saturated chemotactic flux, Kinet. Relat. Models, 5 (2012), 51-95.
doi: 10.3934/krm.2012.5.51. |
[3] |
S. Childress and J. K. Percus, Nonlinear aspects of chemotaxis, Math. Biosc., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[4] |
C. Conca, E. Espejo and K. Vilches, Remarks on the blowup and global existence for a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math., 22 (2011), 553-580.
doi: 10.1017/S0956792511000258. |
[5] |
E. E. Espejo, A. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential Integral Equations, 25 (2012), 251-288. |
[6] |
E. E. Espejo, A. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential Integral Equations, 23 (2010), 451-462. |
[7] |
E. E. Espejo, K. Vilches and C. Conca, Sharp condition for blow-up and global existence in a two species chemotactic Keller-Segel system in $\mathbbR^2$, European J. Appl. Math., 24 (2013), 297-313.
doi: 10.1017/S0956792512000411. |
[8] |
E. E. Espejo Arenas, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis (Munich), 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[9] |
A. Fasano, A. Mancini and M. Primicerio, Equilibrium of two populations subject to chemotaxis, Math. Models Methods Appl. Sci., 14 (2004), 503-533.
doi: 10.1142/S0218202504003337. |
[10] |
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. |
[11] |
M. A. Herrero and J. J. L. Velázquez, A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683. |
[12] |
I. Higueras, Characterizing strong stability preserving additive Runge-Kutta methods, J. Sci. Comput., 39 (2009), 115-128.
doi: 10.1007/s10915-008-9252-2. |
[13] |
T. Hillen and K. Painter, Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[14] |
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. |
[15] |
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.
doi: 10.3934/dcdsb.2007.7.125. |
[16] |
D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences I, Jahresber. DMV, 105 (2003), 103-165. |
[17] |
D. Horstmann, From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II, Jahresber. DMV, 106 (2004), 51-69. |
[18] |
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. |
[19] |
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. |
[20] |
M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Diff. Integral Eqns., 4 (2003), 427-452. |
[21] |
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, 23, American Mathematical Society, Providence, R.I., (1968). |
[22] |
K.-A. Lie and S. Noelle, On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput., 24 (2003), 1157-1174.
doi: 10.1137/S1064827501392880. |
[23] |
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. |
[24] |
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. |
[25] |
H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., 87 (1990), 408-463.
doi: 10.1016/0021-9991(90)90260-8. |
[26] |
W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[27] |
C. S. Patlak, Random walk with persistence and external bias, Bull. Math: Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[28] |
B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. |
[29] |
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.
doi: 10.1137/S0036139902415117. |
[30] |
P. K. Sweby, High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., 21 (1984), 995-1011.
doi: 10.1137/0721062. |
[31] |
J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. I. Motion of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1198-1223.
doi: 10.1137/S0036139903433888. |
[32] |
J. J. L. Velázquez, Point dynamics in a singular limit of the Keller-Segel model. II. Formation of the concentration regions, SIAM J. Appl. Math., 64 (2004), 1224-1248.
doi: 10.1137/S003613990343389X. |
[33] |
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.
doi: 10.1137/S0036141098339897. |
[34] |
G. Wolansky, Multi-components chemotactic system in the absence of conflicts, European J. Appl. Math., 13 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
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