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On a comparison method to reaction-diffusion systems and its applications to chemotaxis
Numerical simulation of chemotaxis models on stationary surfaces
1. | Institut für Angewandte Mathematik, TU Dortmund, 44227 Dortmund, Germany, Germany, Germany |
References:
[1] |
M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc., 74 (2006), 453-474.
doi: 10.1112/S0024610706023015. |
[2] |
M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system, Scientiae Mathematicae Japonicae, 59 (2004), 577-590. |
[3] |
D. Ambrosi, F. Bussolino and L. Preziosi, A review of vasculogenesis models, J. Theor. Med., 6 (2005), 1-19.
doi: 10.1080/1027366042000327098. |
[4] |
A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, Journal of Theoretical Medicine, 2 (2000), 129-154.
doi: 10.1080/10273660008833042. |
[5] |
M. Bergdorf, I. F. Sbalzarini and P. Koumoutsakos, A Lagrangian particle method for reaction-diffusion systems on deforming surfaces, J. Math. Biol., 61 (2010), 649-663.
doi: 10.1007/s00285-009-0315-2. |
[6] |
M. A. J. Chaplain, The mathematical modelling of tumour angiogenesis and invasion, ACTA Biotheoretica, 43 (1995), 387-402.
doi: 10.1007/BF00713561. |
[7] |
M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bulletin of Mathematical Biology, 60 (1998), 857-900. |
[8] |
M. A. J. Chaplain, Mathematical modelling of angiogenesis, Journal of Neuro-Oncology, 50 (2000), 37-51. |
[9] |
M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA Journal of Mathematics Applied in Medicine and Biology, 10 (1993), 149-168.
doi: 10.1093/imammb/10.3.149. |
[10] |
A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), 169-205.
doi: 10.1007/s00211-008-0188-0. |
[11] |
L. Corriasa, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[12] |
L. Corriasa and B. Perthame, Asymptotic decay for the solutions of the parabolic Keller-Segel chemotaxis system in critical spaces, Mathematical and Computer Modelling, 47 (2008), 755-764.
doi: 10.1016/j.mcm.2007.06.005. |
[13] |
G. Dziuk and C. M. Elliott, Eulerian finite element method for parabolic PDEs on implicit surfaces, Interfaces and Free Boundaries, 10 (2008), 119-138.
doi: 10.4171/IFB/182. |
[14] |
Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models, J. Comput. Appl. Math., 224 (2009), 168-181.
doi: 10.1016/j.cam.2008.04.030. |
[15] |
Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model, SIAM J. Numer. Anal., 47 (2008/09), 386-408.
doi: 10.1137/07070423X. |
[16] |
F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.
doi: 10.1007/s00211-006-0024-3. |
[17] |
H. Gajewski, W. Jäger and A. Koshelev, About loss of regularity and 'blow up' of solutions for quasilinear parabolic systems, R-Report, Institut fuer Angewandte Analysis und Stochastik (IAAS), Berlin, 1993. |
[18] |
H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[19] |
A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation, Phys. Rev. Lett., 90 118101 (2003) [4 pages].
doi: 10.1103/PhysRevLett.90.118101. |
[20] |
D. Horstmann, Generalizing the Keller-Segel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in the Presence of Attraction and Repulsion Between Competitive Interacting Species, J. Nonlinear Sci., 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[21] |
D. Horstmann and M. Lucia, Uniqueness and symmetry of equilibria in a chemotaxis model, Journal für die Reine und angewandte Mathematik (Crelle's Journal), 654 (2011), 83-124.
doi: 10.1515/CRELLE.2011.030. |
[22] |
D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European Journal of Applied Mathematics, 12, (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[23] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, Journal of Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[24] |
E. Keller and L. 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. |
[25] |
E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[26] |
D. Kuzmin, R. Löhner and S. Turek, Flux-Corrected Transport, Springer, 2nd edition, 2012. |
[27] |
D. Kuzmin and M. Möller, Algebraic flux correction I. Scalar conservation laws, in Flux-Corrected Transport: Principles, Algorithms, and Applications (Eds. D. Kuzmin, R. Löhner, S. Turek), Springer, Berlin (2005), 155-206.
doi: 10.1007/3-540-27206-2_6. |
[28] |
D. Kuzmin and S. Turek, Flux correction tools for finite elements, J. Comput. Phys., 175 (2002), 525-558.
doi: 10.1006/jcph.2001.6955. |
[29] |
D. Kuzmin, Explicit and implicit FEM-TVD algorithms with flux linearization, J. Comput. Phys., 228 (2009), 2517-2534.
doi: 10.1016/j.jcp.2008.12.011. |
[30] |
D. Kuzmin, Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes, Journal of Computational and Applied Mathematics, 236 (2012), 2317-2337.
doi: 10.1016/j.cam.2011.11.019. |
[31] |
R. J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys., 131 (1997), 327-353. |
[32] |
M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. |
[33] |
T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[34] |
K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469. |
[35] |
N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365.
doi: 10.1093/imanum/drl018. |
[36] |
T. Senba and T. Suzuki, Parabolic System of Chemotaxis: Blowup in a Finite and the Infinite Time, IMS Workshop on Reaction-Diffusion Systems (Shatin, 1999). Methods Appl. Anal., 8 (2001), 349-367. |
[37] |
G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly, The EMBO Journal, 22 (2003), 1771-1779.
doi: 10.1093/emboj/cdg176. |
[38] |
R. Strehl, A. Sokolov, D. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems, Computational methods in applied mathematics, 10 (2010), 219-232.
doi: 10.2478/cmam-2010-0013. |
[39] |
R. Strehl, A. Sokolov, D. Kuzmin, D. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D, Journal of Computational and Applied Mathematics, 239 (2013), 290-303.
doi: 10.1016/j.cam.2012.09.041. |
[40] |
R. Strehl, A. Sokolov and S. Turek, Efficient, accurate and flexible Finite Element solvers for Chemotaxis problems, Computers and Mathematics with Applications, 64 (2012), 175-189.
doi: 10.1016/j.camwa.2011.12.040. |
[41] |
S. Turek, Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach, Springer, Berlin, 1999.
doi: 10.1007/3-540-48092-7. |
[42] |
R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation, Proc. R. Soc. Lond. B, 266 (1999), 299-304.
doi: 10.1098/rspb.1999.0637. |
[43] |
R. Tyson, S. R. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), 359-375.
doi: 10.1007/s002850050153. |
[44] |
R. Tyson, L. G. Stern and R. J. LeVeque, Fractional step methods applied to a chemotaxis model, J. Math. Biol., 41 (2000), 455-475.
doi: 10.1007/s002850000038. |
[45] |
C. M. Elliott, B. Stinner and C. Venkataraman, Modelling cell motility and chemotaxis with evolving surface finite elements, J. R. Soc. Interface, 9 (2012), 3027-3044.
doi: 10.1098/rsif.2012.0276. |
[46] |
C. Landsberg, F. Stenger, A. Deutsch, M. Gelinsky, A. Rösen-Wolff and A. Voigt, Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds-a modeling approach, J. Biomech, 44 (2011), 359-364. |
[47] |
S. T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31 (1979), 335-362.
doi: 10.1016/0021-9991(79)90051-2. |
show all references
References:
[1] |
M. Aida, T. Tsujikawa, M. Efendiev, A. Yagi and M. Mimura, Lower estimate of the attractor dimension for a chemotaxis growth system, J. London Math. Soc., 74 (2006), 453-474.
doi: 10.1112/S0024610706023015. |
[2] |
M. Aida and A. Yagi, Target pattern solutions for chemotaxis-growth system, Scientiae Mathematicae Japonicae, 59 (2004), 577-590. |
[3] |
D. Ambrosi, F. Bussolino and L. Preziosi, A review of vasculogenesis models, J. Theor. Med., 6 (2005), 1-19.
doi: 10.1080/1027366042000327098. |
[4] |
A. R. A. Anderson, M. A. J. Chaplain, E. L. Newman, R. J. C. Steele and A. M. Thompson, Mathematical modelling of tumour invasion and metastasis, Journal of Theoretical Medicine, 2 (2000), 129-154.
doi: 10.1080/10273660008833042. |
[5] |
M. Bergdorf, I. F. Sbalzarini and P. Koumoutsakos, A Lagrangian particle method for reaction-diffusion systems on deforming surfaces, J. Math. Biol., 61 (2010), 649-663.
doi: 10.1007/s00285-009-0315-2. |
[6] |
M. A. J. Chaplain, The mathematical modelling of tumour angiogenesis and invasion, ACTA Biotheoretica, 43 (1995), 387-402.
doi: 10.1007/BF00713561. |
[7] |
M. A. J. Chaplain, Continuous and discrete mathematical models of tumor-induced angiogenesis, Bulletin of Mathematical Biology, 60 (1998), 857-900. |
[8] |
M. A. J. Chaplain, Mathematical modelling of angiogenesis, Journal of Neuro-Oncology, 50 (2000), 37-51. |
[9] |
M. A. J. Chaplain and A. M. Stuart, A model mechanism for the chemotactic response of endothelial cells to tumour angiogenesis factor, IMA Journal of Mathematics Applied in Medicine and Biology, 10 (1993), 149-168.
doi: 10.1093/imammb/10.3.149. |
[10] |
A. Chertock and A. Kurganov, A second-order positivity preserving central-upwind scheme for chemotaxis and haptotaxis models, Numer. Math., 111 (2008), 169-205.
doi: 10.1007/s00211-008-0188-0. |
[11] |
L. Corriasa, B. Perthame and H. Zaag, Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[12] |
L. Corriasa and B. Perthame, Asymptotic decay for the solutions of the parabolic Keller-Segel chemotaxis system in critical spaces, Mathematical and Computer Modelling, 47 (2008), 755-764.
doi: 10.1016/j.mcm.2007.06.005. |
[13] |
G. Dziuk and C. M. Elliott, Eulerian finite element method for parabolic PDEs on implicit surfaces, Interfaces and Free Boundaries, 10 (2008), 119-138.
doi: 10.4171/IFB/182. |
[14] |
Y. Epshteyn, Discontinuous Galerkin methods for the chemotaxis and haptotaxis models, J. Comput. Appl. Math., 224 (2009), 168-181.
doi: 10.1016/j.cam.2008.04.030. |
[15] |
Y. Epshteyn and A. Kurganov, New interior penalty discontinuous Galerkin methods for the Keller-Segel chemotaxis model, SIAM J. Numer. Anal., 47 (2008/09), 386-408.
doi: 10.1137/07070423X. |
[16] |
F. Filbet, A finite volume scheme for the Patlak-Keller-Segel chemotaxis model, Numer. Math., 104 (2006), 457-488.
doi: 10.1007/s00211-006-0024-3. |
[17] |
H. Gajewski, W. Jäger and A. Koshelev, About loss of regularity and 'blow up' of solutions for quasilinear parabolic systems, R-Report, Institut fuer Angewandte Analysis und Stochastik (IAAS), Berlin, 1993. |
[18] |
H. Gajewski and K. Zacharias, Global behavior of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[19] |
A. Gamba, D. Ambrosi, A. Coniglio, A. de Candia, S. Di Talia, E. Giraudo, G. Serini, L. Preziosi and F. Bussolino, Percolation, morphogenesis, and Burgers dynamics in blood vessels formation, Phys. Rev. Lett., 90 118101 (2003) [4 pages].
doi: 10.1103/PhysRevLett.90.118101. |
[20] |
D. Horstmann, Generalizing the Keller-Segel Model: Lyapunov Functionals, Steady State Analysis, and Blow-Up Results for Multi-species Chemotaxis Models in the Presence of Attraction and Repulsion Between Competitive Interacting Species, J. Nonlinear Sci., 21 (2011), 231-270.
doi: 10.1007/s00332-010-9082-x. |
[21] |
D. Horstmann and M. Lucia, Uniqueness and symmetry of equilibria in a chemotaxis model, Journal für die Reine und angewandte Mathematik (Crelle's Journal), 654 (2011), 83-124.
doi: 10.1515/CRELLE.2011.030. |
[22] |
D. Horstmann and G. Wang, Blow-up in a chemotaxis model without symmetry assumptions, European Journal of Applied Mathematics, 12, (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[23] |
D. Horstmann and M. Winkler, Boundedness vs. blow-up in a chemotaxis system, Journal of Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[24] |
E. Keller and L. 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. |
[25] |
E. Keller and L. Segel, Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[26] |
D. Kuzmin, R. Löhner and S. Turek, Flux-Corrected Transport, Springer, 2nd edition, 2012. |
[27] |
D. Kuzmin and M. Möller, Algebraic flux correction I. Scalar conservation laws, in Flux-Corrected Transport: Principles, Algorithms, and Applications (Eds. D. Kuzmin, R. Löhner, S. Turek), Springer, Berlin (2005), 155-206.
doi: 10.1007/3-540-27206-2_6. |
[28] |
D. Kuzmin and S. Turek, Flux correction tools for finite elements, J. Comput. Phys., 175 (2002), 525-558.
doi: 10.1006/jcph.2001.6955. |
[29] |
D. Kuzmin, Explicit and implicit FEM-TVD algorithms with flux linearization, J. Comput. Phys., 228 (2009), 2517-2534.
doi: 10.1016/j.jcp.2008.12.011. |
[30] |
D. Kuzmin, Linearity-preserving flux correction and convergence acceleration for constrained Galerkin schemes, Journal of Computational and Applied Mathematics, 236 (2012), 2317-2337.
doi: 10.1016/j.cam.2011.11.019. |
[31] |
R. J. LeVeque, Wave propagation algorithms for multi-dimensional hyperbolic systems, J. Comput. Phys., 131 (1997), 327-353. |
[32] |
M. Mimura and T. Tsujikawa, Aggregating pattern dynamics in a chemotaxis model including growth, Physica A, 230 (1996), 499-543. |
[33] |
T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[34] |
K. Osaki and A. Yagi, Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcialaj Ekvacioj, 44 (2001), 441-469. |
[35] |
N. Saito, Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis, IMA J. Numer. Anal., 27 (2007), 332-365.
doi: 10.1093/imanum/drl018. |
[36] |
T. Senba and T. Suzuki, Parabolic System of Chemotaxis: Blowup in a Finite and the Infinite Time, IMS Workshop on Reaction-Diffusion Systems (Shatin, 1999). Methods Appl. Anal., 8 (2001), 349-367. |
[37] |
G. Serini, D. Ambrosi, E. Giraudo, A. Gamba, L. Preziosi and F. Bussolino, Modeling the early stages of vascular network assembly, The EMBO Journal, 22 (2003), 1771-1779.
doi: 10.1093/emboj/cdg176. |
[38] |
R. Strehl, A. Sokolov, D. Kuzmin and S. Turek, A flux-corrected finite element method for chemotaxis problems, Computational methods in applied mathematics, 10 (2010), 219-232.
doi: 10.2478/cmam-2010-0013. |
[39] |
R. Strehl, A. Sokolov, D. Kuzmin, D. Horstmann and S. Turek, A positivity-preserving finite element method for chemotaxis problems in 3D, Journal of Computational and Applied Mathematics, 239 (2013), 290-303.
doi: 10.1016/j.cam.2012.09.041. |
[40] |
R. Strehl, A. Sokolov and S. Turek, Efficient, accurate and flexible Finite Element solvers for Chemotaxis problems, Computers and Mathematics with Applications, 64 (2012), 175-189.
doi: 10.1016/j.camwa.2011.12.040. |
[41] |
S. Turek, Efficient Solvers for Incompressible Flow Problems: An Algorithmic and Computational Approach, Springer, Berlin, 1999.
doi: 10.1007/3-540-48092-7. |
[42] |
R. Tyson, S. R. Lubkin and J. D. Murray, A minimal mechanism for bacterial pattern formation, Proc. R. Soc. Lond. B, 266 (1999), 299-304.
doi: 10.1098/rspb.1999.0637. |
[43] |
R. Tyson, S. R. Lubkin and J. D. Murray, Model and analysis of chemotactic bacterial patterns in a liquid medium, J. Math. Biol., 38 (1999), 359-375.
doi: 10.1007/s002850050153. |
[44] |
R. Tyson, L. G. Stern and R. J. LeVeque, Fractional step methods applied to a chemotaxis model, J. Math. Biol., 41 (2000), 455-475.
doi: 10.1007/s002850000038. |
[45] |
C. M. Elliott, B. Stinner and C. Venkataraman, Modelling cell motility and chemotaxis with evolving surface finite elements, J. R. Soc. Interface, 9 (2012), 3027-3044.
doi: 10.1098/rsif.2012.0276. |
[46] |
C. Landsberg, F. Stenger, A. Deutsch, M. Gelinsky, A. Rösen-Wolff and A. Voigt, Chemotaxis of mesenchymal stem cells within 3D biomimetic scaffolds-a modeling approach, J. Biomech, 44 (2011), 359-364. |
[47] |
S. T. Zalesak, Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31 (1979), 335-362.
doi: 10.1016/0021-9991(79)90051-2. |
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