×

A finite volume scheme for the Patlak-Keller-Segel chemotaxis model. (English) Zbl 1098.92006

Summary: A finite volume method is presented to discretize the Patlak-Keller-Segel (PKS) model for chemosensitive movements [C. S. Patlak, Random walk with persistence and external bias. Bull. Math. Biol. Biophys. 15, 311–338 (1953); E. F. Keller and L. A. Segel, Traveling band of chemotactic bacteria: a theoretical analysis. J. Theor. Biol. 30, 235–248 (1971; Zbl 1170.92308)]. First, we prove existence and uniqueness of a numerical solution to the proposed scheme. Then we give a priori estimates and establish a threshold on the initial mass, for which we show that the numerical approximation converges to the solution to the PKS system when the initial mass is lower than this threshold. Numerical simulations are performed to verify accuracy and the properties of the scheme. Finally, in the last section we investigate blow-up of the solution for large mass.

MSC:

92C17 Cell movement (chemotaxis, etc.)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65C20 Probabilistic models, generic numerical methods in probability and statistics

Citations:

Zbl 1170.92308

Software:

Chemotaxis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brenner M.P., Levitov L., Budrene E.O. (1995) Physical mechanisms for chemotactic pattern formation by bacteria. Biophy’s’. J. 74,1677–1693 · doi:10.1016/S0006-3495(98)77880-4
[2] Brezis H. (1987) Analyse Fonctionelle: Théorie et Applications. Masson, Paris
[3] Chainais-Hillairet C., Liu J.-G., Peng Y.-J. (2003) Finite volume scheme for multi-dimensional drift-diffusion equations and convergence analysis. M2AN Math. Model Numer. Anal. 37, 319–338 · Zbl 1032.82038 · doi:10.1051/m2an:2003028
[4] Childress S., Percus J.K. (1981) Nonlinear aspects of chemotaxis. Math. Biosci; 56, 217–237 · Zbl 0481.92010 · doi:10.1016/0025-5564(81)90055-9
[5] Coudière Y., Gallouët Th., Herbin R. (2001) Discrete Sobolev inequalities and L p error estimates for finite volume solutions of convection diffusion equations. M2AN Math. Model Numer. Anal. 35, 767–778 · Zbl 0990.65122
[6] DeVore R., Sharpley R.: Maximal functions easuring smoothness. Mem. Amer. Math. Soc. 293, viii + 115 (1984) · Zbl 0529.42005
[7] Eymard, R., Gallouet, Th., Herbin, R.: Finite volume methods.In: Handbook of Numerical Analysis, vol. VII, North-Holland, Amsterdam
[8] Eymard R., Gallouet Th., Herbin R., Michel A. (2002) Convergence of a finite volume scheme for nonlinear degenerate parabolic equations. Numer. Math. 92, 41–82 · Zbl 1005.65099 · doi:10.1007/s002110100342
[9] Filbet F., Laurençcot Ph., Perthame B. (2005) Derivation of hyperbolic models for chemosensitive movement. J. Math Biol. 50, 189–207 · Zbl 1080.92014 · doi:10.1007/s00285-004-0286-2
[10] Filbet F., Shu C.-W. (2005) Approximation of hyperbolic models for chemosensitive movement. SIAM J. Sci. Comput. 27(3): 850–872 · Zbl 1141.35396 · doi:10.1137/040604054
[11] Gajewski H., Zacharias K. (1998) Global behavior of a reaction diffusion system modelling chemotaxis. Math. Nachr. 195, 77–114 · Zbl 0918.35064 · doi:10.1002/mana.19981950106
[12] Herrero M.A., Medina E., Velázquez J.J.L. (1997) Finite-time aggregation into a single point in a reaction-diffusion system. Nonlinearity 10, 1739–1754 · Zbl 0909.35071 · doi:10.1088/0951-7715/10/6/016
[13] Herrero M.A., Velazquez J.L.L. (1997) A blow up mechanism for a chemotaxis model. Ann. Scuola Normale Superiore 24, 633–683 · Zbl 0904.35037
[14] Horstmann D. (2003) From 1970 until now: The Keller–Segel model in chemotaxis and its consequences I. Jahresber. DMV 105, 103–165 · Zbl 1071.35001
[15] Horstmann D. (2004) From 1970 until now: The Keller-Segel model in chemotaxis and its consequences II. Jahresber. DMV 106, 51–69 · Zbl 1072.35007
[16] Tyson R., Stern L.J., LeVeque R.J. (2000) Fractional step methods applied to a chemotaxis model. J. Math. Biol. 41, 455–475 · Zbl 1002.92003 · doi:10.1007/s002850000038
[17] Keller E.F., Segel L.A.(1971) Traveling band of chemotactic bacteria: a theoritical analysis. J. Theor. Biol. 30, 235–248 · Zbl 1170.92308 · doi:10.1016/0022-5193(71)90051-8
[18] Maini P.K. (2001) Application of mathematical modelling to biological pattern formation. Coherent structures in complex systems. Lecture Notes in Physics, vol 567. Springer, Berlin Heidelberg New York · Zbl 0985.92011
[19] Marrocco A. (2003) 2D simulation of chemotaxis bacteria aggregation. ESAIM:M2AN 37(4): 617–630 · Zbl 1065.92006 · doi:10.1051/m2an:2003048
[20] Murray J.D. (2003) Mathematical Biology, 3rd edn. vol. 2. Springer, Berlin Heidelberg New York · Zbl 1006.92002
[21] Nanjundiah V. (1973) Chemotaxis, signal relaying and aggregation morphology. J. Theor. Biol. 42, 63–105 · doi:10.1016/0022-5193(73)90149-5
[22] Patlak C.S. (1953) Random walk with persistense and external bias. Bull. Math. Biol. Biophys. 15, 311–338 · Zbl 1296.82044 · doi:10.1007/BF02476407
[23] Perthame B.(2004) PDE models for chemotactic movements: parabolic, hyperbolic and kinetic. Appl. Math 49, 539–564 · Zbl 1099.35157 · doi:10.1007/s10492-004-6431-9
[24] Simon J. (1987) Compact sets in the space L p (0,T; B). Ann. Math. Appl. 146, 65–96 · Zbl 0629.46031 · doi:10.1007/BF01762360
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.