Reaction, diffusion and chemotaxis in wave propagation

Shangbing Ai; Wenzhang Huang; Zhi-An Wang

doi:10.3934/dcdsb.2015.20.1

Article Contents

2015, Volume 20, Issue 1: 1-21. Doi: 10.3934/dcdsb.2015.20.1

This issue Previous Article Next Article An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence

Reaction, diffusion and chemotaxis in wave propagation

1.
Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899
2.
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

Received: June 2014

Published: January 2015

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Abstract

By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusion-chemotaxis model describing biological processes such as the bacterial chemotactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribution to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.

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

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