  It is well known that for reaction-diffusion systems with differential
isotropic diffusions, a Turing instability yields striped solutions. In this
paper we study the impact of weak anisotropy by directional advection on such
solutions, and the role of quadratic terms. We focus on the generic form of
planar reaction-diffusion systems with two components near such a bifurcation.
Using Lyapunov-Schmidt reduction and Floquet-Bloch decomposition we derive a
rigorous parameter expansion for existence and stability against large
wavelength perturbations. This provides detailed formulae for the loci of
bifurcations and so-called Eckhaus as well as zigzag stability boundaries under
the influence of the advection and quadratic terms. In particular, while
destabilisation of the background state is through modes perpendicular to the
advection (Squire-theorem), we show that stripes can bifurcate zigzag unstably.
We illustrate these results numerically by an example. Finally, we show
numerical computations of these stability boundaries in the extended Klausmeier
model for vegetation patterns and show stripes bifurcate stably in the presence
of advection.
