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, e.g., to study the growth of living tissues, colonies of microorganisms and more highly organized populations. The interaction between particles of two species, one of which (predator) pursues the other (prey) is investigated. Predator movement is described by the Cattaneo heat equation, and the prey is only capable of diffusing. Due to the hyperbolicity of the Cattaneo model, the presence of long-lived short-wave patterns can be expected in the case of sufficiently low diffusion of preys. However, the mechanism of instability and failure of such patterns is found. Explicit relations for the predator transport coefficients are derived that block this mechanism.