Short-wave signal versus the indirect prey-taxis

Andrey Morgulis; Karrar Malal; Andrey Morgulis; Karrar Malal

doi:10.3934/math.20261409

AIMS Mathematics

2026, Volume 11, Issue 11: 36409-36443. doi: 10.3934/math.20261409

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Short-wave signal versus the indirect prey-taxis

Andrey Morgulis ^{1,2
,
,},
Karrar Malal ¹

1.
I.I. Vorovich Institute of Mathematics, Mechanucs & Computer Science, Southern Federal University, Rostov-on-Don, Russia
2.
Southern Mathematical Institute of Vladikavkaz Scientific Center of Russian Academy of Science, Vladikavkaz, Russia

Received: 25 June 2026 Revised: 19 October 2026 Accepted: 21 October 2026 Published: 04 November 2026
MSC : 35B27, 35B35, 35C20, 35Q92

We address short-wave asymptotics for one class of quasi-linear second-order PDE's 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, which is the interaction of two species of particles or cells through which the predator species is capable of moving directionally while searching for the prey species. However, we suppose the predators to be sensitive not to the prey density but to another cue field produced by the prey. Additionally, producing the cue is assumed to be sensitive to the intensity of an external field, which is independent from the community state and mimics the environmental inhomogeneity due to natural or artificial reasons. This is what we call the "external signal". We assume that the external signal takes a general short-wave form and construct a formal asymptotic expansion for the short-wave solutions with no restrictions on the spatial dimension or kinetics of inter- or intra-specific reactions modulo solving a hierarchy of the so-called slow systems. With no special assumption, we explicitly write down both the leading term in the asymptotics and the leading slow system, which governs the slow evolution of the amplitude of the leading term. Further, we apply the leading approximation to study 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 construct examples of suppressing the taxical transport, examples of the robustness of the species equilibrium to the signal up to a very strong stabilization, and, oppositely, destabilization and blurring the borderline in the parametric space between the areas of stability and instability of this equilibrium.
- Patlak–Keller–Segel systems,
- prey–taxis,
- averaging,
- homogenization,
- short-wave asymptotic,
- induced stability
Citation: Andrey Morgulis, Karrar Malal. Short-wave signal versus the indirect prey-taxis[J]. AIMS Mathematics, 2026, 11(11): 36409-36443. doi: 10.3934/math.20261409

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Abstract

We address short-wave asymptotics for one class of quasi-linear second-order PDE's 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, which is the interaction of two species of particles or cells through which the predator species is capable of moving directionally while searching for the prey species. However, we suppose the predators to be sensitive not to the prey density but to another cue field produced by the prey. Additionally, producing the cue is assumed to be sensitive to the intensity of an external field, which is independent from the community state and mimics the environmental inhomogeneity due to natural or artificial reasons. This is what we call the "external signal". We assume that the external signal takes a general short-wave form and construct a formal asymptotic expansion for the short-wave solutions with no restrictions on the spatial dimension or kinetics of inter- or intra-specific reactions modulo solving a hierarchy of the so-called slow systems. With no special assumption, we explicitly write down both the leading term in the asymptotics and the leading slow system, which governs the slow evolution of the amplitude of the leading term. Further, we apply the leading approximation to study 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 construct examples of suppressing the taxical transport, examples of the robustness of the species equilibrium to the signal up to a very strong stabilization, and, oppositely, destabilization and blurring the borderline in the parametric space between the areas of stability and instability of this equilibrium.

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