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Striped patterns are known to bifurcate in reaction-diffusion systems withdifferential isotropic diffusions at a supercritical Turing instability. Inthis paper we study the impact of weak anisotropy by directional advection onthe stability of stripes with respect to various lattice modes, and the role ofquadratic terms therein. We focus on the generic form of planarreaction-diffusion systems with two components near such a bifurcation. Usingcentre manifold reduction we derive a rigorous parameter expansion for thecritical eigenvalues for lattice mode perturbations, specifically nearly squareand nearly hexagonal ones. This provides detailed formulae for the loci ofstability boundaries under the influence of the advection and quadratic terms.In particular, the well known destabilising effect of quadratic terms can becounterbalanced by advection, which leads to intriguing arrangements ofstability boundaries. We illustrate these results numerically by a specificexample. Finally, we show numerical computations of these stability boundariesin the extended Klausmeier model for vegetation patterns and show stripesbifurcate stably in the presence of sufficiently strong advection.

The impact of advection on the stability of stripes on lattices near planar Turing instabilities