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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. |
