Numerical simulation of chemotactic bacteria aggregation via mixed finite elements
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 4, p. 617-630

We start from a mathematical model which describes the collective motion of bacteria taking into account the underlying biochemistry. This model was first introduced by Keller-Segel [13]. A new formulation of the system of partial differential equations is obtained by the introduction of a new variable (this new variable is similar to the quasi-Fermi level in the framework of semiconductor modelling). This new system of P.D.E. is approximated via a mixed finite element technique. The solution algorithm is then described and finally we give some preliminary numerical results. Especially our method is well adapted to compute the concentration of bacteria.

DOI : https://doi.org/10.1051/m2an:2003048
Classification:  35Q,  65M,  92B,  92C
Keywords: biophysics, chemotaxis, numerical simulation, mixed finite element
@article{M2AN_2003__37_4_617_0,
     author = {Marrocco, Americo},
     title = {Numerical simulation of chemotactic bacteria aggregation via mixed finite elements},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {4},
     year = {2003},
     pages = {617-630},
     doi = {10.1051/m2an:2003048},
     zbl = {1065.92006},
     mrnumber = {2018433},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_4_617_0}
}
Marrocco, Americo. Numerical simulation of chemotactic bacteria aggregation via mixed finite elements. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 4, pp. 617-630. doi : 10.1051/m2an:2003048. http://www.numdam.org/item/M2AN_2003__37_4_617_0/

[1] M.D. Betterton and M.P. Brenner, Collapsing bacterial cylinders. Phys. Rev. E 64 (2001) 061904.

[2] M.P. Brenner, L.S. Levitov and E.O. Budrene, Physical mechanisms for chemotactic pattern formation bybacteria. Biophys. J. 74 (1998) 1677-1693.

[3] M.P. Brenner, P. Constantin, L.P. Kadanof, A. Schenkel and S.C. Venhataramani, Diffusion, attraction and collapse. Nonlinearity 12 (1999) 1071-1098. | Zbl 0942.35018

[4] L. Corrias, B. Perthame and H. Zaag, A model motivated by angiogenesis. C. Rendus Acad. Sc. Paris, to appear.

[5] A El Boukili and A. Marrocco, Arclength continuation methods and applications to 2d drift-diffusion semiconductor equations. Rapport de recherche 2546, INRIA (mai 1995).

[6] A. El Boukili, Analyse mathématique et simulation numérique bidimensionnelle des dispositifs semi-conducteurs à hétérojonctions par l'approche éléments finis mixtes. Ph.D. thesis, Univ. Pierre et Marie Curie, Paris (décembre 1995).

[7] R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics, Studies in Applied Mathematics. SIAM, Philadelphia (1989). | MR 1060954 | Zbl 0698.73001

[8] M.A. Herrero and J.J.L. Velázquez, Chemotactic collapse for the keller-segel model. J. Math. Biol. 35 (1996) 177-194. | Zbl 0866.92009

[9] M.A. Herrero, E. Medina and J.J.L. Velázquez, Finite time aggregation into a single point in a reaction-diffusion system. Nonlinearity 10 (1997) 1739-1754. | Zbl 0909.35071

[10] F. Hecht and A. Marrocco, Numerical simulation of heterojunction structures using mixed finite elements and operator splitting, in 10th International Conference on Computing Methods in Applied Sciences and Engineering, R. Glowinski Ed., Nova Science Publishers, Le Vésinet (February 1992) 271-286.

[11] F. Hecht and A. Marrocco, Mixed finite element simulation of heterojunction structures including a boundary layer model for the quasi-fermi levels. COMPEL 13 (1994) 757-770. | Zbl 0824.65136

[12] W. Jäger and S. Luckhaus, On explosion of solution to a system of partial differential equations modelling chemotaxis. Trans. Amer. Math. Soc. 239 (1992) 819-824. | Zbl 0746.35002

[13] E.F. Keller and L.A. Segel, Model for chemotaxis. J. Theor. Biol. 30 (1971) 225-234.

[14] A. Marrocco and Ph. Montarnal, Simulation des modèles energy-transport à l'aide des éléments finis mixtes. C.R. Acad. Sci. Paris I 323 (1996) 535-541. | Zbl 0858.65111

[15] Ph. Montarnal, Modèles de transport d'énergie des semi-conducteurs, études asymptotiques et résolution par des éléments finis mixtes. Ph.D. thesis, Université Paris VI (octobre 1997).

[16] A. Marrocco, 2d simulation of chemotactic bacteria aggregation. Rapport de recherche 4667, INRIA (décembre 2002).