# Waves in a Hyperbolic Predator-Prey System
## 6 Numerical Experiments  
In this section, we present the numerical solutions to system (7)-(9) that we formulate with the use of the Lotka-Volterra kinetics (40) and predators' diffusivity \(\mu=(1+\kappa s)^{-1}\). The diffusivity determines the sensitivity, \(\chi(s)\), by the simplified integrability condition (27). For the numerical implementation, we eliminate the flux, \(q\), from this system with the use of ansatz by [PERSON] (see references provided above). As a result, we arrive at the following equations:  
\[p_{tt}+\
u(p_{t}-F(p,s)) = (\mu(s)p)_{xx}+(F(p,s))_{t},\] (46) \[s_{t} = G(p,s)+\delta s_{xx}.\] (47  
Implementation of the numerical solution should cover a spatial interval wide enough to put the artificial boundaries far out of the domain in which the phenomena of interest occurs. We had controlled all the results below by widening the spatial area and found them reproducing themselves sustainably for values of \(L\) higher than 8. In particular, \(L=10\) for all the figures presented in this section. In a similar way, we checked the influence of the mesh refining and varying the level of smoothing used for preparing the initial data and found the results of this inspection quite satisfactory. The first set of numerical experiments is for answering the question of whether the shock waves persist for the positive values of the resistivity, \(\
u\) and the prey diffusivity, \(\delta\). To this end, we have been taking the profiles of the species' densities, \(s\) and \(p\) from the waves (43)-(45) and putting them as the initial profiles, \(s_{0}\) and \(p_{0}\), which enter the boundary conditions (48). At that, we have been smoothing the shocks slightly. Further, we have been putting \(q_{0}=c_{*}p_{0}\), where \(c_{*}=\sqrt{\mu(s_{e})}\) is the shock wave speed. This choice is consistent with the definition of the shock waves. Appendix A provides the concrete details of setting initial conditions, the control parameters values, etc. The answer to the question formulated above is fully affirmative. The shocks become a bit smoother, but keep propagating at an almost constant speed that is nearly equal to the value of \(c_{*}\). Figure 2 shows the typical behavior of the slightly smoothed counterpart of the patchy shock wave. This figure tells us that the smoothed patch spreads as a kind of soliton, which is shaped rather sharply for the small resistivity and prey diffusivity. An increase in the resistivity produces scattering of the predators behind the rear front of the wave. The next piece of computing addresses the interactions of the observed solitary waves with some perturbations applied initially. These are:  
1. a small displacement of the species density profiles one relative to the other one with no deformation;
2. a small deformation of the species density profiles;
3. a small droplet of predators localized behind the main core;
4. a small droplet of predators localized ahead of the main core. In cases (a) and (b), the smallness means that the magnitudes of mutual displacements (deformations) are approximately ten percent of the magnitude of the main patch. The smallness of the droplet means that its mass is ten times smaller than the mass of the main patch, while they both are localized in the intervals of nearly equal lengths. In Appendix A, there is the exact formulation of the initial conditions and the concrete values of the control parameters. In cases (a) and (b), the effects of the initial perturbations manifest themselves mostly by the predators scattering, which goes almost the same way as shown in the bottom row of frames in Figure 2, with no any qualitative distinctions. So, we do not illustrate these cases. Case (c) is similar to the above, but shedding the predators is rather intensive. We illustrate this case in the top row of frames in Figure 3. Case (d) demonstrates a very peculiar interplay of solitary waves. Namely, the main patch and the droplet attract one to the other one until they clash.