Traveling waves in a chemotaxis model with logistic growth

Tong Li; Jeungeun Park

doi:10.3934/dcdsb.2019147

Article Contents

2019, Volume 24, Issue 12: 6465-6480. Doi: 10.3934/dcdsb.2019147

This issue Previous Article Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation Next Article Non-oscillation principle for eventually competitive and cooperative systems

Traveling waves in a chemotaxis model with logistic growth

Tong Li and
Jeungeun Park^,

Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA

^* Corresponding author: Jeungeun Park
^* Corresponding author: Jeungeun Park

Received: October 31, 2018

Revised: December 31, 2018

Early access: July 2019

Published: September 2019

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Abstract

Traveling wave solutions of a chemotaxis model with a reaction term are studied. We investigate the existence and non-existence of traveling wave solutions in certain ranges of parameters. Particularly for a positive rate of chemical growth, we prove the existence of a heteroclinic orbit by constructing a positively invariant set in the three dimensional space. The monotonicity of traveling waves is also analyzed in terms of chemotaxis, reaction and diffusion parameters. Finally, the traveling wave solutions are shown to be linearly unstable.

Keywords:

Mathematics Subject Classification: Primary: 35B35, 35C07, 35K57, 92C17; Secondary: 35K55.

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Figure 1. Sketch of the compact sets: $ \mathcal{P} $ in (A) and $ \mathcal{T} $ in (B)

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Figure 2. Numerical simulations of traveling wave solutions $ (U(\xi), V(\xi)) $ of the system (1). For (A), $ D = 1, s = 2, \beta = 0.2, \mu = 1 $ and $ \chi(v) = \cos(10 v) - \sin(20 v) +2 $. For (B), $ D = 1, s = 2, \beta = 1, \mu = \frac{1}{2} $ and $ \chi(v) = \frac{1}{(1+v)^{2}}. $

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