In this paper we deal with a semilinear hyperbolic chemotaxis model in one space dimension evolving on a network, with suitable transmission conditions at nodes. This framework is motivated by tissue-engineering scaffolds used for improving wound healing. We introduce a numerical scheme, which guarantees global mass densities conservation. Moreover our scheme is able to yield a correct approximation of the effects of the source term at equilibrium. Several numerical tests are presented to show the behavior of solutions and to discuss the stability and the accuracy of our approximation.
Mots-clés : hyperbolic system on network, initial-boundary value problem, transmission conditions, asymptotic behavior, finite difference schemes, chemotaxis
@article{M2AN_2014__48_1_231_0,
author = {Bretti, G. and Natalini, R. and Ribot, M.},
title = {A hyperbolic model of chemotaxis on a network: a numerical study},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {231--258},
publisher = {EDP-Sciences},
volume = {48},
number = {1},
year = {2014},
doi = {10.1051/m2an/2013098},
mrnumber = {3177843},
zbl = {1285.92004},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2013098/}
}
TY - JOUR AU - Bretti, G. AU - Natalini, R. AU - Ribot, M. TI - A hyperbolic model of chemotaxis on a network: a numerical study JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 231 EP - 258 VL - 48 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2013098/ DO - 10.1051/m2an/2013098 LA - en ID - M2AN_2014__48_1_231_0 ER -
%0 Journal Article %A Bretti, G. %A Natalini, R. %A Ribot, M. %T A hyperbolic model of chemotaxis on a network: a numerical study %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 231-258 %V 48 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2013098/ %R 10.1051/m2an/2013098 %G en %F M2AN_2014__48_1_231_0
Bretti, G.; Natalini, R.; Ribot, M. A hyperbolic model of chemotaxis on a network: a numerical study. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 1, pp. 231-258. doi : 10.1051/m2an/2013098. http://www.numdam.org/articles/10.1051/m2an/2013098/
[1] , and , Asymptotic high-order schemes for 2 × 2 dissipative hyperbolic systems. SIAM J. Numer. Anal. 46 (2008) 869-894. | MR | Zbl
[2] and , An anisotropic biphasic theory of tissue-equivalent mechanics: The interplay among cell traction, fibrillar network deformation, fibril alignment, and cell contact guidance. J. Biomech. Eng. 119 (1997) 137-145.
[3] , , , , and , Stability of switched hyperbolic systems: the example of SMB chromatography” submitted IEEE-CDC (2013).
[4] , and , A numerical scheme for a hyperbolic relaxation model on networks. Numerical Analysis and Applied Mathematics ICNAAM 2011, AIP Conf. Proc. 1389 (2011) 1412-1415.
[5] , , Mathematical framework to model migration of cell population in extracellular matrix, in Cell mechanics. From single scale-based models to multiscale modeling. A. Chauviere, L. Preziosi and C. Verdier Eds., Taylor & Francis Group, CRC Press publisher, (2010) 285-318. | MR
[6] and , Wave propagation, observation and control in 1 - d flexible multi-structures, vol. 50 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Berlin (2006). | MR | Zbl
[7] and , Cattaneo models for chemosensitive movement. Numerical solution and pattern formation. J. Math. Biol. 46 (2003) 153-170; corrected version after misprinted p. 160 in J. Math. Biol. 46 (2003) 461-478. | MR | Zbl
[8] , and , Derivation of hyperbolic models for chemosensitive movement. J. Math. Biol. 50 (2005) 189-207. | MR | Zbl
[9] , , , , , , , and , Percolation, morphogenesis, and Burgers dynamics in blood vessels formation. Phys. Rev. Lett. 90 (2003) 118101.1-118101.4.
[10] and , Traffic flow on networks. Conservation laws models, vol. 1 of AIMS Series on Applied Mathematics. American Institute of Mathematical Sciences (AIMS), Springfield, MO (2006). | MR | Zbl
[11] , Asymptotic-preserving and well-balanced schemes for the 1D Cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes. J. Math. Anal. Appl. 388 (2012) 964-983. | MR | Zbl
[12] , Well-balanced numerical approximations display asymptotic decay toward Maxwellian distributions for a model of chemotaxis in a bounded interval. SIAM J. Sci. Comput. 34 (2012) A520-A545. | MR
[13] and , Stability results for a diffusion equation with functional drift approximating a chemotaxis model. Trans. Amer. Math. Soc. 300 (1987) 235-258. | MR | Zbl
[14] , Some analytical results for hyperbolic chemotaxis model on networks Ph.D. thesis, Università di Roma “La Sapienza” (2012).
[15] and , Nonlinear transmission problems for quasilinear diffusion problems. Networks and Heterogeneous media 2 (2007) 359-381. | MR | Zbl
[16] , , and , Stability of constant states and qualitative behavior of solutions to a one dimensional hyperbolic model of chemotaxis. Discrete Contin. Dyn. Syst. Ser. B 12 (2009) 39-76. | MR | Zbl
[17] , A mathematical model for mesenchymal and chemosensitive cell dynamics. J. Math. Biol. 64 (2012) 361-401. | MR | Zbl
[18] and , A second order model of road junctions in fluid models of traffic networks. Netw. Heterog. Media 2 (2007) 227-253. | MR | Zbl
[19] , Hyperbolic models for chemosensitive movement. Special issue on kinetic theory. Math. Mod. Methods Appl. Sci. 12 (2002) 1007-1034. | MR | Zbl
[20] , and , Existence of weak solutions for a hyperbolic model of chemosensitive movement. J. Math. Anal. Appl. 26 (2001) 173-199. | MR | Zbl
[21] and , Hyperbolic models for chemotaxis in 1-D. Nonlinear Anal. Real World Appl. 1 (2000) 409-433. | MR | Zbl
[22] and , Thermodynamic analysis of the permeability of biological membrane to non-electrolytes. Biochimica et Biophysica Acta 27 (1958) 229-246.
[23] and , Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970) 399-415. | Zbl
[24] , , , , and , Microarchitecture of Three-Dimensional Scaffolds Inuences Cell Migration Behavior via Junction Interactions. Biophys. J. 29 (2008) 4013-4024.
[25] , From 1970 until present: the Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Deutsch. Math.-Verein. 105 (2003) 103-165. | MR | Zbl
[26] and , Cell proliferation and migration in silk broin 3D scaffolds. Biomaterials 30 (2009) 2956-2965.
[27] , Mathematical biology. I. An introduction, 3rd edn., vol. 17 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York (2002); II. Spatial models and biomedical applications, 3rd edn., vol. 18 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York (2003). | MR | Zbl
[28] , Convergence to equilibrium for the relaxation approximation of conservation laws. Commun. Pure Appl. Math. 49 (1996) 795-823. | MR | Zbl
[29] and , An asymptotic high order mass-preserving scheme for a hyperbolic model of chemotaxis. SIAM J. Num. Anal. 50 (2012) 883-905. | MR | Zbl
[30] , Transport equations in biology, Frontiers in Mathematics. Birkhäuser (2007). | MR | Zbl
[31] and , Multiphase modelling of tumour growth and extracellular matrix interaction: mathematical tools and applications. J. Math. Biol. 58 (2009) 625-656. | MR
[32] , A theoretical study of receptor mechanisms in bacterial chemotaxis. SIAM J. Appl. Math. 32 (1977) 653-665. | Zbl
[33] , , , , , , , , A G-CSF functionalized PLLA scaffold for wound repair: an in vitro preliminary study. Conf. Proc. IEEE Eng. Med. Biol. Soc. (2010).
[34] and , Stabilization of the wave equation on 1-D networks. SIAM J. Control Optim. 48 (2009) 2771-2797. | MR | Zbl
Cité par Sources :
