# Waves in a Hyperbolic Predator-Prey System
## 1 Introduction  
The Patlak-Keller-Segel (PKS) law provides a simple macroscopic model for a perceptual motion (taxis) of the particles ensemble in response to the spatial gradient of a stimulus (or signal) field. The commonly recognized formulation of the PKS flux reads as: \(\chi p\
abla s\), where the notations of \(p\), \(s\), and \(\chi\) stand for the density of the medium that moves in response to the signal, the intensity of the signal itself and the sensitivity coefficient, correspondingly. A multitude of parabolic PKS systems resulting from the summation of the diffusive and PKS fluxes stand as the subject of intensive research for last five decades. A considerable number of reviews expose the advances in this area, e.g., [1, 2, 3]. However, usage of the parabolic framework is not a unique way of treating the PKS law. For example, [PERSON] and [PERSON] [4] proposed a different formulation known as the Cattaneo model for chemosensitive movement. In contrast with the parabolic models, this one takes into account the inertia of the response and becomes hyperbolic. At that, the flux has a contribution from the local time derivative of itself in addition to those mentioned above. From the articles [5, 6], it follows that the common parabolic model, the Cattaneomodel and several other hyperbolic models represent the approximations of the kinetics equations under different hypotheses. There is a concise review by [PERSON] [7] of this subject that covers deriving the models and the issues of exact solutions, stability and bifurcations, etc. It makes clear that the hyperbolic models are not less natural than their parabolic counterparts, e.g., while modelling the aggregation processes in the active media. Nevertheless, the former receive much less attention than the latter. The mentionedreview by [PERSON] reports the deficiency of results for understanding the generic pattern emerging from the dynamics of the local hyperbolic models even in the case of one spatial dimension (see Table 9.1, Chapter 9) despite a considerable piece of work exposed therein. Anyway, the hyperbolic models are the subject of continuing research. Among the recent results, those published in [8; 9] have much to do with the present study. These articles address propagating the travelling shock waves and the gradual formation of the shocks from the initially smooth solutions for an infinite time extent. The analysis covers the rigorous proof of both features for a non-local hyperbolic model of a media, the density of which governs its velocity via the action of a given first-order pseudo-differential operator. Interestingly, the travelling shock waves coexist with the smooth ones. The latter are the widely studied features in the parabolic case, see, e.g., [10; 11; 12] and the references therein. The studies of their hyperbolic counterparts likely traced back to [PERSON] [13; 14; 15; 16]. In the present paper, we address similar issues but in a different context. We consider the Cattaneo model for a predator-prey community with the Lotka-Volterra kinetics. Regarding the predator flux, we assume that the diffusivity coefficient, \(\mu=\mu(p,s)\), and the sensitivity coefficient, \(\chi=\chi(p,s)\), are given functions in the species densities, \(p\) and \(s\). In addition, we assume that 1-form \(-\mu(p,s)dp+p\chi(p,s)ds\) is exact. Despite being restrictive, this assumption relies on certain biological grounds [17; 18]. Formally, it entails the conservatism of the flux equation. Thanks to this circumstance, we take advantage of considering the travelling shock waves and arrive at very simple exact solutions for the limit case of sedentary prey and a highly-inertial predator. In the aforementioned articles, [PERSON] addressed the travelling waves in a semi-linear version of the Cattaneo model. However, the model dealt with here is not semi- but quasi-linear. Besides, the conservatism allows for the elimination of the flux from the governing equations using the ansatz by [PERSON]; see the reference above again.