Instability and short waves in a hyperbolic predator–prey system

This paper presents a mathematical model of a medium consisting of active particles capable of adjusting their movement depending on so-called signals or stimuli. Such models are used, for example, in studying the growth of living tissues, colonies of microorganisms and more highly organized populations. The interaction between two types of particles, one of which (predator) pursues the other (prey) is investigated. The predator's movement is described by the Cattaneo heat equation, and the prey is only capable of diffusing. In view of the hyperbolicity of the Cattaneo model, in the case of sufficiently weak diffusion of preys, the presence of long-lived short-wave structures can be assumed. However, the mechanism of instability and failure of such structures is found. The relations for the transport coefficients of the predator that block this mechanism are derived explicitly.