Mathematics of traveling waves in chemotaxis --Review paper--

Zhi-An Wang

doi:10.3934/dcdsb.2013.18.601

Article Contents

2013, Volume 18, Issue 3: 601-641. Doi: 10.3934/dcdsb.2013.18.601

This issue Previous Article Next Article Random attractors for stochastic FitzHugh-Nagumo systems driven by deterministic non-autonomous forcing

Mathematics of traveling waves in chemotaxis --Review paper--

Zhi-An Wang^1,

1.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received: October 31, 2012

Revised: October 31, 2012

Published: April 2013

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Abstract

This article surveys the mathematical aspects of traveling waves of a class of chemotaxis models with logarithmic sensitivity, which describe a variety of biological or medical phenomena including bacterial chemotactic motion, initiation of angiogenesis and reinforced random walks. The survey is focused on the existence, wave speed, asymptotic decay rates, stability and chemical diffusion limits of traveling wave solutions. The main approaches are reviewed and related analytical results are given with sketchy proofs. We also develop some new results with detailed proofs to fill the gap existing in the literature. The numerical simulations of steadily propagating waves will be presented along the study. Open problems are proposed for interested readers to pursue.

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Mathematics Subject Classification: 35C07, 35K55, 46N60, 62P10, 92C17.

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