Nonlinear flux-limited models for chemotaxis on networks

Raul Borsche; Axel Klar; T. N. Ha Pham

doi:10.3934/nhm.2017017

Article Contents

2017, Volume 12, Issue 3: 381-401. Doi: 10.3934/nhm.2017017

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Nonlinear flux-limited models for chemotaxis on networks

Technische Universität Kaiserslautern, Department of Mathematics, Erwin-Schrödinger-Straße, 67663 Kaiserslautern, Germany

* Corresponding author: klar@mathematik.uni-kl.de
* Corresponding author: klar@mathematik.uni-kl.de

Received: September 2016

Revised: May 2017

Published: October 2017

The first author is supported by DFG grant BO 4768/1. The second author is supported by DFG grant 1105/27 and by DFG, RTG 1932.

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Abstract

In this paper we consider macroscopic nonlinear moment models for the approximation of kinetic chemotaxis equations on a network. Coupling conditions at the nodes of the network for these models are derived from the coupling conditions of kinetic equations. The results of the different models are compared and relations to a Keller-Segel model on networks are discussed. For a numerical approximation of the governing equations an asymptotic well-balanced schemes is extended to directed graphs. Kinetic and macroscopic equations are investigated numerically and their solutions are compared for tripod and more general networks.

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Mathematics Subject Classification: Primary: 92C17, 92C42; Secondary: 65M06.

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Figure 1. Sketch of a tripod network

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Figure 2. Numerical solutions of the four models on an interval at time $t=0.2$ with $\epsilon = 1$ (top) and $\epsilon = 0.5$ (bottom)

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Figure 3. Numerical solutions of the four models on an interval at time $t=0.2$ with $\epsilon = 0.1$ (top) and $\epsilon = 10^{-6}$ (bottom)

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Figure 4. Linear models ($P_1$) with negative full or half range densities and positivity preserving nonlinear models ($M_1$)

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Figure 5. Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 1$

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Figure 6. Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 0.5$

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Figure 7. Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 0.1$

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Figure 8. Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 10^-6$

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Figure 9. Comparison of the numerical solutions on a larger network at $t=5$

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Figure 10. Total mass over time in the large network

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