Abstract
The Keller-Segel system is a linear parabolic-elliptic system, which describes the aggregation of slime molds resulting from their chemotactic features. By chemotaxis we understand the movement of an organism (like bacteria) in response to chemical stimulus, for example attraction by certain chemicals in the environment.
In this paper, we use the results of a paper by Zhou and Saito to validate our finite volume method with respect to blow-up analysis and equilibrium solutions. Based on these results, we study model variations and their blow-up behavior numerically.
We will discuss the question whether or not conservative numerical methods are able to model a blow-up behavior in the case of non-global existence of solutions.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ascher, U.M.: Numerical methods for evolutionary differential equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)
Blanchet, A., Dolbeault, J., Perthame, B.: Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions. Electron. J. Differ. Equ. 2006, 33 (2006)
Cho, C.-H.: A numerical algorithm for blow-up problems revisited. Numer. Algorithms 75(3), 675–697 (2017)
Dolbeault, J., Perthame, B.: Optimal critical mass in the two dimensional Keller-Segel model in \(R^2\). C. R. Math. Acad. Sci. Paris 339(9), 611–616 (2004)
Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In: Handbook of numerical analysis, Vol. 7: Solution of equations in \(\mathbb{R}^n\) (Part 3). Techniques of scientific computing, pp. 713–1020. Elsevier, Amsterdam (2000)
Gajewski, H., Zacharias, K.: Global behaviour of a reaction-diffusion system modelling chemotaxis. Math. Nachr. 195, 77–114 (1998)
Hillen, T., Painter, K.J.: A user’s guide to PDE models for chemotaxis. J. Math. Biol. 58(1–2), 183–217 (2009)
Horstmann, D.: Aspekte positiver Chemotaxis. Univ. Köln, Köln (1999)
Jäger, W., Luckhaus, S.: On explosions of solutions to a system of partial differential equations modelling chemotaxis. Trans. Am. Math. Soc. 329(2), 819–824 (1992)
Keller, E.F., Segel, L.A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26(3), 399–415 (1970)
Keller, E.F., Segel, L.A.: Model for chemotaxis. J. Theor. Biol. 30(2), 225–234 (1971)
Müller, J., Kuttler, C.: Methods and models in mathematical biology. Springer, Deterministic and stochastic approaches. Berlin (2015)
Nagai, T.: Blow-up of radially symmetric solutions to a chemotaxis system. Adv. Math. Sci. Appl. 5(2), 581–601 (1995)
Perthame, B.: Transport equations in biology. Birkhäuser, Basel (2007)
Saito, N.: Conservative upwind finite-element method for a simplified Keller-Segel system modelling chemotaxis. IMA J. Numer. Anal. 27(2), 332–365 (2007)
Walentiny, D.: Mathematical modeling and numerical study of the blow-up behaviour of a Keller-Segel chemotaxis system using a finite volume method. Master’s thesis, TU Berlin (2017)
Zhou, G., Saito, N.: Finite volume methods for a Keller-Segel system: discrete energy, error estimates and numerical blow-up analysis. Numer. Math. 135(1), 265–311 (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Bärwolff, G., Walentiny, D. (2018). Numerical and Analytical Investigation of Chemotaxis Models. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2018. ICCSA 2018. Lecture Notes in Computer Science(), vol 10961. Springer, Cham. https://doi.org/10.1007/978-3-319-95165-2_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-95165-2_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-95164-5
Online ISBN: 978-3-319-95165-2
eBook Packages: Computer ScienceComputer Science (R0)