Mathematical Biosciences and Engineering (MBE)

Traveling bands for the Keller-Segel model with population growth

Pages: 717 - 737, Volume 12, Issue 4, August 2015      doi:10.3934/mbe.2015.12.717

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Shangbing Ai - Department of Mathematical Sciences, University of Alabama in Huntsville, Huntsville, AL 35899, United States (email)
Zhian Wang - Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China (email)

Abstract: This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that collective interaction of cell growth and chemical consumption rate plays an essential role in the generation of traveling bands. The research in the paper provides new insights into the mechanisms underlying the chemotactic pattern formation of wave bands.

Keywords:  Chemotaxis, Keller-Segel model, cell kinetics, traveling waves, minimal wave speed.
Mathematics Subject Classification:  35C07, 35K55, 46N60, 62P10, 92C17.

Received: April 2014;      Accepted: November 2014;      Available Online: April 2015.