Prey-taxis vs an external signal: short-wave asymptotic and stability analysis

TL;DR

This paper analyzes predator-prey models with prey-taxis and external signals, deriving asymptotic solutions to understand how short-wave external signals influence pattern formation and stability in these ecological systems.

Contribution

The study develops a complete asymptotic expansion for short-wave solutions in predator-prey models with prey-taxis, incorporating external signals and comparing stability with and without signals.

Findings

External signals do not create new instability domains.

Signals can significantly widen existing instability regions.

Pattern stability depends on signal propagation speed and system kinetics.

Abstract

We consider two models of predator-prey community with prey-taxis, one relies on Patlak-Keller-Segel law (Lee et al, 2009), the other one employs the Cattaneo model of heat transfer following Dolak and Hillen (2003). Thus, the former one uses the prey density gradient for directing the predators flux, and the latter one -- for directing the vector density of sources of the predators flux. We assume the predators to be also capable of responding to an external signal in the same manner. Additionally, we assume that some scaling makes the dimensionless prey diffusivity and the wavelength of the external signal small quantities of the same order. In the case of Cattaneo's model, we additionally assume the resistivity to varying the predators flux to be as high as the reciprocal of the prey diffusivity. With these assumptions we construct the complete asymptotic expansion of the short-wave solution. We use this asymptotic to examine the effect of the short-wave signal on the formation of spatiotemporal patterns. We do so by comparing the stability of equilibria with no signal to that of the quasi-equilibria, by which we mean the simplest patterns directly forced by the external signal. Historically, such an approach goes back to Kapitza's theory for upside-down pendulum, and it had been already applied to examining other predator-prey systems and/or other limits (Morgulis and Ilin, 2020, Morgulis, 2023). This time the overall conclusion is that the external signal is likely not capable of creating the instability domain in the parametric space from nothing but it can substantially widen the one that is non-empty with no signal. The details, however, essentially depend on the speed at which the external signal propagates and on the system kinetics, at least when the signal amplitude is small