Short-wave signal versus indirect prey-taxis

Abstract

We address a short-wave asymptotic for one class of quasi-linear second-order PDE systems involving the cross-diffusion described by the so-called Patlak-Keller-Segel law. It is common to employ these equations for modeling the predator-prey community with the prey-taxis that means the interactions of two species of particles or cells or anything else through which the species called "predators" is capable of moving directionally while searching for the other species called "prey." However, we suppose the predators to be sensitive not to the prey density but to a driving signal produced by the prey. Additionally, the production of the driving signal is assumed to be sensitive to the intensity of an external field, which is independent from the community state. This is what we call the external signal. It can be due to the spatiotemporal inhomogeneity of the environment arising from natural or artificial reasons. We assume that the external signal takes a general short-wave form and construct a complete asymptotic expansion for the short-wave solutions with no restrictions on the spatial dimension or kinetics of inter- or intra-specific reactions. Further, we apply the short wave asymptotic to studying the stability or instability induced by the external signal following Kapitza's theory for the upside-down pendulum. Applying the general results to some special classes of external signals, we get examples of suppressing the taxical transport, examples of robustness of the species equilibrium to the signal up to a very strong stabilization or, oppositely, destabilization and somewhat like blurring the borderline in the parametric space between the areas of stability and instability of this equilibrium. These results contribute to filling the gap in the literature, since the theory and techniques for the asymptotic integration of systems described above represent a weakly charted area.