Traveling waves and their spectral instability in volume–filling chemotaxis model

Abstract

In this paper, I consider a volume-filling chemotaxis model with a small cell diffusion coefficient and chemotactic sensitivity. By the geometric singular perturbation theory together with the center-stable and center unstable manifolds, one gets the existence of a positive traveling wave connecting the two constant steady states (0,0) and (b,αbβ) with a small wave speed ϵc. In addition, the traveling wave is monotone for b≥1 and is not monotone for 0<b<1. Moreover, by the spectral analysis it shows that the above traveling wave is spectrally unstable in some exponentially weighted spaces.