We perform the nonlinear stability analysis of a one-dimensional chemotaxis model of bacterial self-organization, assuming that bacteria respond sharply to chemical signals. The resulting discontinuous advection speed represents the key challenge for the stability analysis. We follow a perturbative approach, where the shape of the cellular profile is clearly separated from its global motion, allowing us to circumvent the discontinuity issue. Further, the homogeneity of the problem leads to two conservation laws, which express themselves in differently weighted functional spaces. This discrepancy between the weights represents another key methodological challenge. We derive an improved Poincaré inequality that allows to transfer the information encoded in the conservation laws to the appropriately weighted spaces. As a result, we obtain exponential relaxation to equilibrium with an explicit rate. A numerical investigation illustrates our results.
Ce travail est consacré à l’analyse de stabilité non-linéaire d’un modèle de chimiotactisme pour l’auto-organisation de colonies de bactéries. Une hypothèse centrale de ce modèle est que les bactéries répondent de manière très abrupte aux variations du signal chimique. Le champ d’advection discontinue qui en résulte est la principale difficulté dans l’analyse de stabilité. Nous adoptons une approche perturbative, dans laquelle le profil de densité cellulaire est détaché du mouvement global de la population, ce qui nous permet de contourner cette difficulté. De l’homogénéité du problème résultent deux lois de conservation qui font intervenir des poids différents. Cette disparité représente une autre difficulté à surmonter. Nous étalissons une inégalité de Poincaré améliorée qui permet de transférer l’information d’un poids à l’autre. En conclusion, nous déduisons de cette analyse la convergence exponentielle vers l’équilibre avec un taux explicite. Cette étude est complétée par des exemples numériques.
Accepted:
Published online:
Vincent Calvez 1; Franca Hoffmann 2
CC-BY 4.0
@article{AFST_2023_6_32_2_287_0,
author = {Vincent Calvez and Franca Hoffmann},
title = {Nonlinear stability of chemotactic clustering with discontinuous advection},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {287--318},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 32},
number = {2},
year = {2023},
doi = {10.5802/afst.1737},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1737/}
}
TY - JOUR AU - Vincent Calvez AU - Franca Hoffmann TI - Nonlinear stability of chemotactic clustering with discontinuous advection JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2023 SP - 287 EP - 318 VL - 32 IS - 2 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1737/ DO - 10.5802/afst.1737 LA - en ID - AFST_2023_6_32_2_287_0 ER -
%0 Journal Article %A Vincent Calvez %A Franca Hoffmann %T Nonlinear stability of chemotactic clustering with discontinuous advection %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2023 %P 287-318 %V 32 %N 2 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1737/ %R 10.5802/afst.1737 %G en %F AFST_2023_6_32_2_287_0
Vincent Calvez; Franca Hoffmann. Nonlinear stability of chemotactic clustering with discontinuous advection. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 2, pp. 287-318. doi : 10.5802/afst.1737. https://afst.centre-mersenne.org/articles/10.5802/afst.1737/
[1] Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., Volume 25 (2015) no. 09, pp. 1663-1763 (Accessed 2016-10-21) | DOI | MR | Zbl
[2] On the parabolic-elliptic Patlak–Keller–Segel system in dimension 2 and higher, Sémin. Laurent Schwartz, EDP Appl. (2011), pp. 1-26 (Accessed 2020-05-26) | DOI | Numdam | Zbl
[3] Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal., Volume 163 (1999) no. 1, pp. 1-28 (Accessed 2020-05-26) | DOI | MR | Zbl
[4] Hypocoercivity and sub-exponential local equilibria (2019) (https://arxiv.org/abs/1911.10961)
[5] Chemotactic waves of bacteria at the mesoscale, J. Eur. Math. Soc., Volume 22 (2019) no. 2, pp. 593-668 (Accessed 2020-01-28) | DOI | MR | Zbl
[6] Particle approximation of the one dimensional Keller–Segel equation, stability and rigidity of the blow-up, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 3, pp. 1175-1208 | DOI | MR | Zbl
[7] Confinement by biased velocity jumps: Aggregation of Escherichia coli, Kinet. Relat. Models, Volume 8 (2015) no. 4, pp. 651-666 (Accessed 2015-08-12) | DOI | MR | Zbl
[8] Model hierarchies for cell aggregation by chemotaxis, Math. Models Methods Appl. Sci., Volume 16 (2006) no. 7, suppl., pp. 1173-1197 | DOI | MR | Zbl
[9] Derivation of hyperbolic models for chemosensitive movement, J. Math. Biol., Volume 50 (2005) no. 2, pp. 189-207 | DOI | MR | Zbl
[10] Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients, Bull. Math. Biol., Volume 53 (1991) no. 5, pp. 721-749 | DOI | Zbl
[11] Lagrangian numerical approximations to one-dimensional convolution-diffusion equations, SIAM J. Sci. Comput., Volume 28 (2006) no. 4, pp. 1203-1227 | DOI | MR | Zbl
[12] The diffusion limit of transport equations derived from velocity-jump processes, SIAM J. Appl. Math., Volume 61 (2000) no. 3, pp. 751-775 | DOI | MR | Zbl
[13] A user’s guide to PDE models for chemotaxis, J. Math. Biol., Volume 58 (2008) no. 1, p. 183 (Accessed 2019-12-30) | DOI | MR | Zbl
[14] Quand est-ce que des bornes de Hardy permettent de calculer une constante de Poincaré exacte sur la droite ?, Ann. Fac. Sci. Toulouse, Math., Volume 17 (2008) no. 1, pp. 121-192 (Accessed 2020-05-26) | DOI | MR | Zbl
[15] On a linear runs and tumbles equation, Kinet. Relat. Models, Volume 10 (2017) no. 3, pp. 799-822 | DOI | MR | Zbl
[16] Motility of Escherichia coli cells in clusters formed by chemotactic aggregation, Proc. Natl. Acad. Sci. USA, Volume 100 (2003) no. 23, pp. 13259-13263 (Accessed 2014-03-21) | DOI
[17] stability of travelling waves with applications to convective porous media flow, Commun. Pure Appl. Math., Volume 35 (1982) no. 6, pp. 737-749 | DOI | MR | Zbl
[18] Mathematical models for chemotaxis and their applications in self-organisation phenomena, J. Theor. Biol., Volume 481 (2019), pp. 162-182 (Accessed 2020-05-25) | DOI | MR | Zbl
[19] Transport models for chemotactic cell populations based on individual cell behavior, Chem. Eng. Sci., Volume 44 (1989) no. 12, pp. 2881-2897 (Accessed 2020-05-25) | DOI
[20] Mathematical description of bacterial traveling pulses, PLoS Comput. Biol., Volume 6 (2010) no. 8, e1000890 (Accessed 2016-07-01) | DOI | MR
[21] Directional persistence of chemotactic bacteria in a traveling concentration wave, Proc. Natl. Acad. Sci. USA, Volume 108 (2011) no. 39, pp. 16235-16240 (Accessed 2014-02-25) | DOI
[22] -stability of nonlinear waves in scalar conservation laws, Evolutionary equations. Vol. I (Handbook of Differential Equations), North-Holland, 2004, pp. 473-553 | MR | Zbl
[23] Overview of mathematical approaches used to model bacterial chemotaxis II: Bacterial populations, Bull. Math. Biol., Volume 70 (2008) no. 6, pp. 1570-1607 (Accessed 2014-03-11) | DOI | MR | Zbl
Cited by Sources: