Homogenization, Drift, Stabilization and Destabilization for the Patlak-Keller-Segel Systems Driven by the Indirect Taxis and the Short-Wavelength External Signals

We study a model of the active continuous media driven by the interspecific taxis and also by an external signal. We employ the Patlak-Keller-Segel law for modelling the tactical motions. We address the short-wavelength signals with the use the homogenization. We show that such a signal exerts the effect on the large-scale dynamics by the drift that arises from the homogenization. We analyse in detail the signals which have the form of travelling waves. We find out that they are capable of producing the quasi-equilibria—that is, the short-wavelength patterns which stay in equilibrium on average. We examine the stability of the quasi-equilibria and compare the results to the case when the signal is off. The effect of the signal turns out to be not single-valued but depending on the speed at which the signal-producing wave propagates. 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 signal in both cases.