Abstract
Hyperbolic reaction–diffusion equations have recently attracted attention both for their application to a variety of biological and chemical phenomena, and for their distinct features in terms of propagation speed and novel instabilities not present in classical two-species reaction–diffusion systems. We explore the onset of diffusive instabilities and resulting pattern formation for such systems. Starting with a rather general formulation of the problem, we obtain necessary and sufficient conditions for the Turing and wave instabilities in such systems, thereby classifying parameter spaces for which these diffusive instabilities occur. We find that the additional temporal terms do not strongly modify the Turing patterns which form or parameters which admit them, but only their regions of existence. This is in contrast to the case of additional space derivatives, where past work has shown that resulting patterned structures are sensitive to second-order cross-diffusion and first-order advection. We also show that additional temporal terms are necessary for the emergence of spatiotemporal patterns under the wave instability. We find that such wave instabilities exist for parameters which are mutually exclusive to those parameters leading to stationary Turing patterns. This implies that wave instabilities may occur in cases where the activator diffuses faster than the inhibitor, leading to routes to spatial symmetry breaking in reaction–diffusion systems which are distinct from the well studied Turing case.
•Turing and wave instabilities are classified in hyperbolic reaction–diffusion systems.•The Turing instability mechanism is unchanged due to second-order time derivatives.•Cross-diffusion modifies the Turing instability and resulting Turing patterns.•Wave instabilities are shown to be possible in systems of two equations.•Second-order time derivatives strongly influence the wave instability.