Prey-taxis VS a Shortwave External Signal in Multiple Dimensions

TL;DR

This paper models predator-prey interactions with prey-taxis and external signals in multiple dimensions, deriving asymptotic solutions for short-wave signals and analyzing stability without assuming traveling wave forms.

Contribution

It extends previous work by addressing multiple dimensions and removing the traveling wave assumption, providing new asymptotic analysis of external signals in predator-prey models.

Findings

Derived complete asymptotic expansions for short-wave solutions in multiple dimensions.

Generalized prior results to more complex spatial settings.

Analyzed stability and instability influenced by external signals using Kapitza's theory.

Abstract

We consider a model of the predator--prey community with prey-taxis. By that we mean the capability of the predators to get moving in a certain direction on the macroscopic level in response to the prey density gradients. Additionally, we suppose the same kind of sensitivity with respect to one more signal, called external, the production of which goes on independently of the community state. Such a signal can be due to the spatiotemporal inhomogeneity of the environment that results from the natural or artificial reasons. The model employs the Patlak--Keller--Segel law for responses to both ones. We assume that the external signal takes a general short-wave form, and we construct the complete asymptotic expansions of the short-wave solutions. This result generalizes the prior one by Morgulis \& Malal (2025) in two respects. First, we have addressed the case of multiple dimensions. Second, we have got rid of assuming the signal and corresponding solutions to take the form of a traveling wave, that makes our result novel even in one dimension. Further, we apply the short wave asymptotic to studying the stability or instability imposed by the external signal following Kapitza' theory for upside-down pendulum.