# Acoustic Wind in a Hyperbolic Predator--Prey System
## 6 Linear Stability
We call a submanifold in the parametric space _neutral_ if the transversal intersection of it by such a path generically entails the transversal intersection of the imaginary axis by a branch of eigenvalues of the corresponding stability problems. The unstable eigenmodes disappearing or arising upon crossing the neutral submanifold is what we call _an occurrence of instability_. We say that the instability is _oscillatory_ (monotonic) if the corresponding branch of the eigenvalues is not real (is real). In the case dealt with here, the stability test for an individual normal mode reduces to counting the eigenvalues of some complex matrix \(3\times 3\) that belong to the right semi-plane. We allude to widely recognized textbook [9] for more details, though the procedure is routine. Nevertheless, it is worth noting that the characteristic polynomials of the mentioned matrices depend not on the parameter \(\theta\) itself but on \(\theta^{2}\) only--that is, the direction of the wind is nothing to do with the stability. This feature also agrees with the invariance of the homogenized system. Here is the summary of our outcomes concerning the linear stability of equilibrium (40) for \(\theta=0\)--that is, for the original system. * The neutral submanifold is the graph of function \(\chi=\chi_{nt}(\gamma,p,s,k^{2})\), where \[\chi_{nt}=\frac{\mu\,\
u}{p_{e}\,s_{e}}+\frac{\gamma\,p_{e}\,s_{e}^{2}+s_{e} \,\
u^{2}+s_{e}\,\
u}{p_{e}\,s_{e}\,k^{2}}.\] (42) 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 \(\chi=\chi_{cr}=\mu\
u/p_{e}s_{e}\). 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 \(|k|>k_{*}\) (\(|k|<k_{*}\)). * The above conclusions mainly withstand allowing the prey diffusion, except for the shortwave instability; namely, the neutral submanifold is the graph of function \[\chi=\tilde{\chi}_{nt}=\delta^{2}k^{2}A+\delta B+\chi_{nt},\] where \(\chi_{nt}\) is function (42) while factors \(A,B\) do not depend on the parameters that enter the right-hand side explicitly. * At that, the critical submanifold is the graph of function \[\chi=\tilde{\chi}_{cr}=\min_{k}\tilde{\chi}_{nt}=\frac{\delta(\
u^{2}+2 vs_{e} +\gamma p_{e}s_{e}+\
u s_{e}\sqrt{\
u^{2}+vs_{e}+\gamma p_{e}s_{e}})+\mu\
u}{ p_{e}s_{e}}>\chi_{cr}.\]  
Again, for every \(\delta>0\) a specific equilibrium is stable provided that \(\chi<\tilde{\chi}_{cr}\) and becomes unstable otherwise. This instability is always oscillatory and there exist real numbers \(k_{1}>k_{0}>0\) such that every eigenmode's mode is unstable provided that its wavenumber obeys the inequality \(k_{0}<|k|<k_{1}\). Interestingly, function \(\tilde{\chi}_{cr}\) is linear in \(\mu\) and in \(\delta\) too. Let us pass to the homogenized system (37)-(39) and test the linear stability for \(\chi=\chi_{cr}\) and \(\theta\) equal to its maximal absolute value (35). It turns out that the equilibrium (40) is unstable while it is stable for \(\theta=0\) in accordance with what we have been saying above. Hence, the wind can make the equilibria unstable. In more detail, let \(\chi\) and \(\theta\) be set as above. Then for every admissible setting of the other parameters there exist numbers \(k_{2}>k_{1}>0\) such that there are unstable normal modes for every wavelength \(k:\ |k|>k_{2}\) and all the normal modes are stable for every \(k:\ |k|<k_{1}\). Figure 2 (the right frame) is to illustrate this conclusion. Note that generically there are three different eigenvalues \(\lambda\) for every \(k\). Their expressions in the problem parameters are commonly called the eigenvalues branches. These branches remains smooth unless the eigenvalues collide, while their collisions lead to singularities.