# Waves in a Hyperbolic Predator-Prey System
## 7 Discussion  
We have addressed a Cattaneo-type dynamics of the predator-prey system with the Lotka-Volterra kinetics term in one spatial dimension. By assumption, only the predators are capable of the perceptual motions, and the flux of them generally reads as \(p\chi(p,s)s_{x}-\mu(p,s)p_{x}\), where \(\mu(p,s)\) and \(\chi(p,s)\) are some prescribed functions. To start with, we have considered the sedentary prey. In this approximation, the governing equations turn to form a strictly hyperbolic system, for which we have formulated the criterion for reducing to [PERSON]'s invariants in terms of sensitivity, \(\chi(p,s)\), and diffusivity, \(\mu(p,s)\), explicitly. It has turned out, however, that the class of systems obeying this criterion looks rather artificial. At the same time, reducing the governing equations to the conservation laws happened to be less restrictive. In particular, such a reduction is possible provided that \(\mu=\mu(s)\), \(\kappa\chi(s)=-d\mu/ds\). Since there are biological rationales for this structural relation (see the reference above), we have accepted it. For the systems of conservation laws, the shock waves are natural, and we have derived a system of conditions on the shocks. In the inertial limit--that is, for \(\
u=0\)--we have discovered a very simple exact solutions that describe the travelling shock waves. The wave pattern represents a semi-infinite or even finite patch of predators that propagates at a constant speed. There are no predators outside the patch while both species coexist in the equilibrium state inside, see Figure 1. The wave speed is equal to \(\pm\sqrt{\mu(s_{e})}\), where \(s_{e}\) is the prey density at the equilibrium state. The waves carrying semi-infinite patches describes the transitions between two equilibria states. The community goes either from the extinction of both species to the coexistence at the equilibrium or from the coexistence at the equilibrium to the extinction of the predators and restoring the prey up to the carrying capacity. In this sense they resemble the KPP-Fisher waves. Here, however, the extinction of predators means that the wave either has not brought them to the areal extent under consideration yet or has moved them out already. The interesting feature is that the front of the predators' invasion always moves towards the smaller concentration of prey. This is not as paradoxical as it can seem given the locality of the predator-prey interactions. Thus, two patterns of behavior are emergent from propagating these waves. The first is restoring the resource up to the carrying capacity after the migrating predators' withdrawal. The second is the transition from the prey dying out to the mutual equilibrium with the spreading predators. The travelling shock waves  
Figure 6: The figure illustrates propagating two waves due to the explosion of a smoothly localized unit mass of predators amid the uniform distribution of prey, the density of which is equal to the carrying capacity. The blue lines are for the instantaneous graphs of the predators’ density. The green lines are for the deviation of the prey density from the carrying capacity. The detailed description of the initial states and other settings used for computing this figure is in Appendix A.  
carrying a finite mass of the predators combine both features. Indeed, the profile of such a wave involves the transitions from the mutual extinction via the coexistence to the extinction of predators. Besides, the patch of predators moves towards the smaller concentration of prey again. So, the corresponding pattern of behavior looks like preventing the prey from dying out and even restoring the prey population due to migrating a compact core of the predators. We have extended the travelling shock waves to the positive values of the prey diffusivity, \(\delta\), and the resistivity, \(\
u\), numerically. It turns out that they withstand such an extension, at least while the values of \(\delta\) and \(\
u\) remain small enough.