Drift, stabilizing and destabilizing for a Patlak-Keller-Segel system with the short-wavelength external signal

TL;DR

This paper investigates how short-wavelength external signals, modeled as traveling waves, can either stabilize or destabilize patterns in a Patlak-Keller-Segel system, with effects depending on wave speed and amplitude.

Contribution

It introduces a homogenization approach to analyze external signal effects on stability in Patlak-Keller-Segel models, revealing the non-single-valued influence of traveling waves.

Findings

Traveling wave speed determines whether the external signal stabilizes or destabilizes the system.

Destabilization occurs when wave amplitude exceeds a threshold at high wave speeds.

The effect of the external wave is exponential in its amplitude.

Abstract

This article aims at exploring the short-wavelength stabilization and destabilization of the advection-diffusion systems formulated using the Patlak-Keller-Segel cross-diffusion. We study a model of the taxis partly driven by an external signal. We address the general short-wavelength signal using the homogenization technique, and then we give a detailed analysis of the signals emitted as the travelling waves. It turns out that homogenizing produces the drift of species, which is the main translator of the external signal effects, in particular, on the stability issues. We examine the stability of the quasi-equilibria - that is, the simplest short-wavelength patterns fully imposed by the external signal. Comparing the results to the case of switching the signal off allows us to estimate the effect of it. For instance, the effect of the travelling wave turns out to be not single-valued but depending on the wave speed. Namely, there is an independent threshold value such that increasing the amplitude of the wave destabilizes the quasi-equilibria provided that the wave speed is above this value. Otherwise, the same action exerts the opposite effect. It is worth to note that the effect is exponential in the amplitude of the wave in both cases.