  Turing instabilities for a two species reaction-diffusion systems is studied
under anisotropic diffusion. More specifically, the diffusion constants which
characterize the ability of the species to relocate in space are direction
sensitive. Under this working hypothesis, the conditions for the onset of the
instability are mathematically derived and numerically validated. Patterns
which closely resemble those obtained in the classical context of isotropic
diffusion, develop when the usual Turing condition is violated, along one of
the two accessible directions of migration. Remarkably, the instability can
also set in when the activator diffuses faster than the inhibitor, along the
direction for which the usual Turing conditions are not matched.
