Stability of the quasi-equilibria of Keller–Segel systems in strictly inhomogeneous environment

It is well known that a local bifurcation of the equilibrium of a system of Patlak–Keller–Segels’ type (PKS) often turns out to be the first link in the chain of dynamical transitions leading to rather complex regimes of motion. However, as far as we are aware, the studies of the first transition cover only the homogeneous equilibria of homogeneous (i.e. translationally invariant) systems. In this article, we consider the effect of inhomogeneity. For this purpose, we have been introducing a PKS system, modeling two species, one of which (predator) is capable of searching the other one (prey). In addition to the prey-taxis, the predator has been endowed with taxis driven by environmental characteristics, such as temperature, salinity, terrain relief, etc. In other words, the predator can perceive an external signal. When the external signal is off, then we get a very simple homogeneous PKS-type system, which is, nevertheless, capable of transiting from the homogeneous equilibrium to self-oscillatory wave motions via a local bifurcation. Notably, this transition does not involve the predator's kinetics, but the taxis only. We have been examining the short wavelength signals using the homogenization technique. It turns out that a short wavelength signal typically causes an exponential reduction of the predators' motility in comparison with the homogeneous system in response to the increase in the external signal level. The loss of motility to a great extent prevents the occurrence of the waves and dramatically stabilizes the primitive quasi-equilibria fully imposed by the external signal. It can be said that intense small-scale environmental fluctuations disorient and distract predators, and prevent them from effectively pursuing their prey.