  We study general linear transport-reaction systems on an arbitrary
dimensional hypercube with periodic boundary conditions. Transport-reaction
systems are often used to model the finite speed movement and interaction of
particles, bacteria or animals. We first show a weak spectral mapping theorem
and demonstrate its' application. Secondly, we introduce a certain class of
so-called hyperbolic instabilities, which provide a natural framework for
transport-driven instabilities on one-dimensional domains: They are either
Turing patterns or increasingly oscillating hyperbolic instabilities. A new
algebraic condition for the existence of Turing patterns is obtained as a
side-product.
