Prey-taxis, short waves and pattern formation

It is common to see the prey-taxis as an interaction of two species, in which particles of one of them ("predators") are capable of deterministic movements in search of particles of the other environment ("prey"). At the macroscopic level, this means advecting the predators controlled by the concentration gradient of prey. We’ll be discussing three models of this interaction. First, we’ll be talking about a parabolic model based on the expression of the local flux of a predator directly from the Patlak-Keller-Segel (PKS) law. Next, we’ll be considering two less common models, which in one or another way take into account the predators inertia. Namely, the prey gradient drives either the local acceleration of the predators or the local time derivative of their flux. In all three cases the diffusion of predators and prey is taken into account. We’ll be deriving the asymptotics of short-wave solutions to all three outlined systems. In particular, we’ll be getting the closed systems, called homogenized, that govern the main terms of these asymptotics. Following Kapitza’s theory of the inverted pendulum, we’ll be using the homogenized systems to estimate the effect of the short-wave excitation on the stability of the homogeneous quasi-equilibria – that is, the simplest short-wave structures that represent counterparts of the homogeneous equilibria of the original systems.