# American Institute of Mathematical Sciences

September  2017, 12(3): 381-401. doi: 10.3934/nhm.2017017

## 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

Received  September 2016 Revised  May 2017 Published  September 2017

Fund Project: 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

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.

Citation: Raul Borsche, Axel Klar, T. N. Ha Pham. Nonlinear flux-limited models for chemotaxis on networks. Networks & Heterogeneous Media, 2017, 12 (3) : 381-401. doi: 10.3934/nhm.2017017
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##### References:
Sketch of a tripod network
Numerical solutions of the four models on an interval at time $t=0.2$ with $\epsilon = 1$ (top) and $\epsilon = 0.5$ (bottom)
Numerical solutions of the four models on an interval at time $t=0.2$ with $\epsilon = 0.1$ (top) and $\epsilon = 10^{-6}$ (bottom)
Linear models ($P_1$) with negative full or half range densities and positivity preserving nonlinear models ($M_1$)
Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 1$
Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 0.5$
Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 0.1$
Numerical solutions on a tripod network at time $t=0.2$, $\Delta x=0.02$, $\epsilon = 10^-6$
Comparison of the numerical solutions on a larger network at $t=5$
Total mass over time in the large network
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