# Acoustic Wind in a Hyperbolic Predator—Prey System

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## Abstract

**:**

## 1. Introduction

## 2. The Governing Equations and Scaling

## 3. Motivating Examples

## 4. Homogenization and Wind

## 5. The Case of Constant Diffusivity and Sensitivity

## 6. Linear Stability

- The neutral submanifold is the graph of function $\chi ={\chi}_{nt}(\gamma ,p,s,{k}^{2})$, where$${\chi}_{nt}=\frac{\mu \phantom{\rule{0.166667em}{0ex}}\nu}{{p}_{e}\phantom{\rule{0.166667em}{0ex}}{s}_{e}}+\frac{\gamma \phantom{\rule{0.166667em}{0ex}}{p}_{e}\phantom{\rule{0.166667em}{0ex}}{{s}_{e}}^{2}+{s}_{e}\phantom{\rule{0.166667em}{0ex}}{\nu}^{2}+{{s}_{e}}^{2}\nu}{{p}_{e}\phantom{\rule{0.166667em}{0ex}}{s}_{e}\phantom{\rule{0.166667em}{0ex}}{k}^{2}}.$$For a specific equilibrium and a specific wave number k, the normal mode is stable for $\chi <{\chi}_{nt}$ and becomes unstable otherwise.
- The critical submanifold is the graph of function $\chi ={\chi}_{cr}=\mu \nu /{p}_{e}{s}_{e}$. A specific equilibrium is stable provided that $\chi <{\chi}_{cr}$ and becomes unstable otherwise. This instability is always oscillatory and occurs to shorter waves; namely, for every $\chi >{\chi}_{cr}$ there exists ${k}_{*}>0$ such that every eigenmode is unstable (stable) provided that $\left|k\right|>{k}_{*}$ ($\left|k\right|<{k}_{*}$).
- The above conclusions mainly withstand allowing the prey diffusion, except for the shortwave instability; namely, the neutral submanifold is the graph of function$$\chi ={\stackrel{\u02c7}{\chi}}_{nt}={\delta}^{2}{k}^{2}A+\delta B+{\chi}_{nt},$$
- At that, the critical submanifold is the graph of function$$\chi ={\stackrel{\u02c7}{\chi}}_{cr}=\underset{k}{min}{\stackrel{\u02c7}{\chi}}_{nt}=\frac{\delta ({\nu}^{2}+2\nu {s}_{e}+\gamma {p}_{e}{s}_{e}+\nu {s}_{e}\sqrt{{\nu}^{2}+\nu {s}_{e}+\gamma {p}_{e}{s}_{e}})+\mu \nu}{{p}_{e}{s}_{e}}>{\chi}_{cr}.$$

## 7. Numerical Results for the Homogenized System

## 8. Conclusions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**This figure shows the patterns produced by three versions of the initial data (5), see Table 1. Each frame shows instantaneous graphs of the deviation of the species densities from their equilibrium values. At that, blue (green) colour corresponds to the predator (prey) and the horizontal axis corresponds to the spatial coordinate, x. The upper (middle) row demonstrates the short-wave pattern produced by initial data (5) where $\u03f5=0$ ($\u03f5\ne 0$). The bottom row displays the decaying pattern produced by initial data (5) for $a=0$. The snapshot timing, left to right, is $t\approx 0.0,2.0,3.9,6.0$ for the upper row, $t\approx 0.0,2.0,41,5.38$ for the middle row and $t\approx 0.0,3.1,6.9,10.0$ for the bottom row.

**Figure 2.**The typical neutral curves of $\mu $ versus squared wavenumber for five values of $\alpha =0.5,0.87,1.12,1.43,1.8,2.0$ (

**left**panel) and graphs of the increments of the eigenmodes ($\mathrm{Re}\phantom{\rule{0.166667em}{0ex}}\lambda $) vs. the squared wavenumbers for $\mu =0.1,0.18,0.25,1.25,2.25,3.0$ and $\alpha =3/4$ (

**right**). Each increase in the diffusivity coefficient increases the increment so that its graph moves upward and crosses the horizontal axis closer to zero. On the left panel, the area of stability is confined within the coordinate axes and the neutral curves. Each increase in $\alpha $ widens the stability areas so that the neutral curves moves right and upward. The values of other parameters are as follows: $\nu =1,\gamma =2/3,\beta =0.4\gamma \alpha $, $\chi ={\chi}_{cr}$ and $\theta $ is equal to its maximum.

**Figure 3.**The typical pattern of the smooth localized perturbation with or without the wind, see Section 7 for specific settings. Each frame shows instantaneous graphs of the deviation of the species densities from their equilibrium values. At that, blue (green) colour corresponds to the predator (prey) and the horizontal axis corresponds to the spatial coordinate, x. The upper row displays growth and sharpening of the perturbation due to the wind-induced pumping. Parameter $\theta <0$ is maximal in the absolute value. The snapshot timing, left to right, is $t\approx 0.0,3.4,9.3$. The bottom row displays a decaying pattern with no wind so that $\theta =0$. The snapshot timing, left to right, is $t\approx 0.0,5.2,10$.

Row | Settings of the Initial Data |
---|---|

Top | $N=5,n=3,\u03f5=0,a=1/2$ |

Middle | $N=5,n=3,\u03f5=0.1,a=1/2$ |

Bottom | $N=5,n=3,\u03f5=0.1,a=0$ |

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**MDPI and ACS Style**

Morgulis, A.
Acoustic Wind in a Hyperbolic Predator—Prey System. *Mathematics* **2023**, *11*, 1265.
https://doi.org/10.3390/math11051265

**AMA Style**

Morgulis A.
Acoustic Wind in a Hyperbolic Predator—Prey System. *Mathematics*. 2023; 11(5):1265.
https://doi.org/10.3390/math11051265

**Chicago/Turabian Style**

Morgulis, Andrey.
2023. "Acoustic Wind in a Hyperbolic Predator—Prey System" *Mathematics* 11, no. 5: 1265.
https://doi.org/10.3390/math11051265