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<p>Here, 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 modelling the predator–prey community with the so-called prey-taxis. By that, we mean the interactions of two species of particles or cells or anything else through which the species called predators is capable of directional movement in response to the density gradient of the other one called prey. In contrast with the widely studied models, we suppose the predators possess the same kind of sensitivity with respect to the intensity of an external field, which is independent from the community state. Such a field can be due to the spatiotemporal inhomogeneity of the environment arising from natural or artificial reasons. This is what we call an external signal. We assume that the external signal takes a general short-wave form and construct the complete asymptotic expansions of the short-wave solutions. This result generalizes the prior one by Morgulis and Malal (<CitationRef CitationID="CR3">2025</CitationRef>) 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 travelling wave, that makes our result novel even in one dimension. Our 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. 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 concrete systems, we get several examples of suppressing the diffusive and tactical transports, of robustness to the signal or, oppositely, blurring the borderline in the parametric space between the areas of stability and instability of the predator–prey coexistence equilibrium.</p>

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PREY-TAXIS VS A SHORTWAVE EXTERNAL SIGNAL IN MULTIPLE DIMENSIONS

  • 期刊论文

  • 发表日期：

    2025年12月6日

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  • Andrey Morgulis,
  • Karrar Malal

摘要

Here, 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 modelling the predator–prey community with the so-called prey-taxis. By that, we mean the interactions of two species of particles or cells or anything else through which the species called predators is capable of directional movement in response to the density gradient of the other one called prey. In contrast with the widely studied models, we suppose the predators possess the same kind of sensitivity with respect to the intensity of an external field, which is independent from the community state. Such a field can be due to the spatiotemporal inhomogeneity of the environment arising from natural or artificial reasons. This is what we call an external signal. We assume that the external signal takes a general short-wave form and construct the complete asymptotic expansions of the short-wave solutions. This result generalizes the prior one by Morgulis and 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 travelling wave, that makes our result novel even in one dimension. Our 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. 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 concrete systems, we get several examples of suppressing the diffusive and tactical transports, of robustness to the signal or, oppositely, blurring the borderline in the parametric space between the areas of stability and instability of the predator–prey coexistence equilibrium.