Traveling waves in a Keller–Segel model with logistic growth

Bacterial diffusion, proliferation and chemotactic aggregation play an important role in forming a traveling wave in a model for chemotaxis. In this paper, we investigate the existence and non-existence of traveling wave solutions of a Keller–Segel type model for chemotaxis, where logistic cell growth is considered and chemotactic sensitivity function is a general $C^1$ function that represents positive or negative chemotaxis. To show the existence of traveling waves, we use techniques from dynamical system theory. By applying the techniques, we determine the range of parameter values of the bacterial chemotaxis and the kinetics of cell and chemical for which traveling wave solutions exist. Furthermore, we examine the monotonicity of the traveling wave solutions. Finally, we conclude that the traveling waves are spectrally unstable.

Keywords

traveling waves, chemotaxis, Keller–Segel model, cell growth, reaction diffusion system

2010 Mathematics Subject Classification

Primary 34C37, 35C07, 35K57, 35Q92, 92C17. Secondary 34A34, 35B35.

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Received 8 September 2020

Received revised 26 September 2021

Accepted 29 September 2021

Published 21 March 2022