# Acoustic Wind in a Hyperbolic Predator--Prey System
## 3 Motivating Examples  
Throughout this section, we will be discussing three numerical solutions to the Cauchy problem for Equations (1)-(3) with small diffusivity of prey and identically constant coefficients \(\mu\) and \(\chi\). The initial data and other settings are as follows  
\[p|_{t=0}=p_{e}\Big{(}1+\epsilon\mathrm{e}^{-x^{2}}+a\sin^{N}(2\pi nx)\Big{)}, \quad q|_{t=0}=\sqrt{\mu}p_{e}a\sin^{N}(2\pi nx),\quad s|_{t=0}=s_{e}, \tag{5}\]  
\[F=p(\gamma s-\beta),\;G=s(\alpha-p-s),\;s_{e}=\beta/\gamma,\;p_{e}=\alpha-s_{e },\;\beta/(\gamma a)=0.4,\;\alpha=5/4,\;\gamma=2/3, \tag{6}\]  
\[\mu\equiv\mathrm{const}=4,\;\
u=1,\;\delta=0.01,\;\chi\equiv\mathrm{const}= \mu\
u/(p_{e}s_{e}),\;\kappa=1. \tag{7}\]  
Quantities \(s_{e}\) and \(p_{e}\) are the equilibrium species densities in the sense that  
\[F(p_{e},s_{e})=0,\quad G(p_{e},s_{e})=0,\]  
i.e., the system (1)-(3) allows the homogeneous equilibrium in which \(p\equiv p_{e}\), \(s\equiv s_{e}\) and \(q\equiv 0\). The value assigned to the parameter \(\chi\) is a threshold for the linear instability of such an equilibrium in the case of \(\delta=0\) (see Section 6).  
Figure 1 illustrates the numerical solutions, which cover the spatial range of \((-35,35)\). For more details of the implementation, see Section 7. Each row of snapshots shows the instantaneous graphs of the deviation of the species densities from their equilibrium values for one concrete set of initial data (5). The top row shows those patterns that arise from a short-wave modulation of the homogeneous equilibrium. The middle row demonstrates an interplay of the same modulation with a small smooth localized perturbation. The bottom row corresponds to the same perturbation but with no underlying short wave. Table 1 states the correspondence between the rows and the versions of initial data (5).  
Overall, the top snapshots row displays a short-wave profile of the predator density amid almost homogeneous prey distribution. The former resembles a linear travelling wave subject to slow amplitude damping. It is not entirely unexpected since the short-wave limit of the governing equations leads to the linear acoustic system with the sound speed equal to \(\sqrt{\mu}\), namely,  
\[p_{t}+q_{x}=0,\quad q_{t}+\mu p_{x}=0.\]  
The amplitude damping is due to the resistance term, \(\
u q\), which is not entirely negligible for the wavelengths adopted for computing.  
The middle row shows that the small smooth localized initial perturbation persists and even gets amplified despite the existing resistance. Notice the asymmetry too. It cannot be explained using the above acoustic approximation only. These features sharply contrast with the bottom row, which displays a symmetrical picture of decaying perturbation. This symmetry is due to the mirror invariance of Equations (1)-(3) and initial data (5) (the latter takes place for \(a=0\) only).  
The listed observations suggest that short waves are capable of pumping the perturbations. In the rest of the article, we will be addressing this conjecture with the use of homogenization.  
\begin{table}
\begin{tabular}{c c} \hline
**Row** & **Settings of the Initial Data** \\ \hline Top & \(N=5,n=3,\epsilon=0,a=1/2\) \\ \hline Middle & \(N=5,n=3,\epsilon=0.1,a=1/2\) \\ \hline Bottom & \(N=5,n=3,\epsilon=0.1,a=0\) \\ \hline \end{tabular}
\end{table}
Table 1: This table states the correspondence between patterns shown in Figure 1 and concrete versions of the initial data (5).