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Traveling waves in a chemotaxis model with logistic growth

  • * Corresponding author: Jeungeun Park

    * Corresponding author: Jeungeun Park
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  • 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.

    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)

    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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