Acoustic Wind in a Hyperbolic Predator—Prey System
We address a hyperbolic model for prey-sensitive predators interacting with purely diffusive prey. We adopt the Cattaneo formulation for describing the predators’ transport. Given the hyperbolicity, the long-lived short-wave patterns occur for sufficiently weak prey diffusivities. The main result is...
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| Format: | Article |
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MDPI AG
2023-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/5/1265 |
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| author | Andrey Morgulis |
| author_facet | Andrey Morgulis |
| author_sort | Andrey Morgulis |
| collection | DOAJ |
| description | We address a hyperbolic model for prey-sensitive predators interacting with purely diffusive prey. We adopt the Cattaneo formulation for describing the predators’ transport. Given the hyperbolicity, the long-lived short-wave patterns occur for sufficiently weak prey diffusivities. The main result is that the non-linear interplay of the short waves generically excites the slowly growing amplitude modulation for wide ranges of the model parameters. We have observed such a feature in the numerical experiments and support our conclusions with a short-wave asymptotic solution in the limit of vanishing prey diffusivity. Our reasoning relies on the so-called homogenized system that governs slow evolutions of the amplitudes of the short-wave parcels. It includes a term (called wind) which is absent in the original model and only comes from averaging over the short waves. It is the wind that (unlike any of the other terms!) is capable of exciting the instability and pumping the growth of solutions. There is quite a definite relationship between the predators’ transport coefficients to be held for getting rid of the wind. Interestingly, this relationship had been introduced in prior studies of small-scale mosaics in the spatial distributions of some real-life populations. |
| first_indexed | 2024-03-11T07:17:47Z |
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| language | English |
| last_indexed | 2024-03-11T07:17:47Z |
| publishDate | 2023-03-01 |
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| spelling | doaj.art-90058da10d824aa389b30edd3c974e712023-11-17T08:10:29ZengMDPI AGMathematics2227-73902023-03-01115126510.3390/math11051265Acoustic Wind in a Hyperbolic Predator—Prey SystemAndrey Morgulis0I.I.Vorovich Institute for Mathematic, Mechanics and Computer Science, Southern Federal University, 344090 Rostov-na-Donu, RussiaWe address a hyperbolic model for prey-sensitive predators interacting with purely diffusive prey. We adopt the Cattaneo formulation for describing the predators’ transport. Given the hyperbolicity, the long-lived short-wave patterns occur for sufficiently weak prey diffusivities. The main result is that the non-linear interplay of the short waves generically excites the slowly growing amplitude modulation for wide ranges of the model parameters. We have observed such a feature in the numerical experiments and support our conclusions with a short-wave asymptotic solution in the limit of vanishing prey diffusivity. Our reasoning relies on the so-called homogenized system that governs slow evolutions of the amplitudes of the short-wave parcels. It includes a term (called wind) which is absent in the original model and only comes from averaging over the short waves. It is the wind that (unlike any of the other terms!) is capable of exciting the instability and pumping the growth of solutions. There is quite a definite relationship between the predators’ transport coefficients to be held for getting rid of the wind. Interestingly, this relationship had been introduced in prior studies of small-scale mosaics in the spatial distributions of some real-life populations.https://www.mdpi.com/2227-7390/11/5/1265Patlak–Keller–Segel systemsCattaneo model for a chemosensitive motionhyperbolic modelspattern formationaveraginghomogenization |
| spellingShingle | Andrey Morgulis Acoustic Wind in a Hyperbolic Predator—Prey System Mathematics Patlak–Keller–Segel systems Cattaneo model for a chemosensitive motion hyperbolic models pattern formation averaging homogenization |
| title | Acoustic Wind in a Hyperbolic Predator—Prey System |
| title_full | Acoustic Wind in a Hyperbolic Predator—Prey System |
| title_fullStr | Acoustic Wind in a Hyperbolic Predator—Prey System |
| title_full_unstemmed | Acoustic Wind in a Hyperbolic Predator—Prey System |
| title_short | Acoustic Wind in a Hyperbolic Predator—Prey System |
| title_sort | acoustic wind in a hyperbolic predator prey system |
| topic | Patlak–Keller–Segel systems Cattaneo model for a chemosensitive motion hyperbolic models pattern formation averaging homogenization |
| url | https://www.mdpi.com/2227-7390/11/5/1265 |
| work_keys_str_mv | AT andreymorgulis acousticwindinahyperbolicpredatorpreysystem |