Instabilities in hyperbolic reaction–diffusion system with cross diffusion and species-dependent inertia

The hyperbolic reaction–diffusion (HRD) equation may overcome the physical shortcomings of the parabolic reaction–diffusion (PRD) equation, where the initially localized disturbance propagates infinitely fast through space. Instead, species often exhibit inertia, resulting in delayed effect in their spatial movement. Incorporating such response time for the onset of species flow due to a concentration gradient leads to an HRD equation with inertia. In this paper, we develop the general theory for pattern-forming instabilities in a two-species HRD system, which becomes a PRD system in the limiting case, with cross-diffusion and species-dependent inertia to explore how they play a role in the pattern forming instabilities. In particular, we determine various criteria for diffusion-induced instabilities (like Turing, wave, wave–Turing) and Hopf-induced instabilities (like pure Hopf, Hopf–wave, Hopf–Turing, and Hopf–wave–Turing) arise due to the cross-diffusion and inertial time. The theoretical results are demonstrated with an example where the Brusselator system represents the local interaction.