  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.
