  We study pattern formation in a reaction-diffusion system for a benthic
bacteria-nutrient model in a marine sediment, which originally contains some
spatially varying coefficients and with these shows some layering of patterns.
Using the Landau reduction for the system with homogeneous coefficients, we
locally analyze Turing bifurcations in 1D and 2D. Moreover, we use the software
pde2path to compute the corresponding branches globally and find a number of
snaking branches of patterns over patterns. This shows that spatially varying
patterns are not necessarily due to spatially varying coefficients. We find a
type of hexagon patches on a homogeneous background with no prior mention in
the literature. We show the first numerically calculated solution-branch, which
connects two different types of hexagons in parameter space. We call states on
this branch rectangles. We check numerically, if the stability changes for
hexagons and stripes, which are continued homogeneously into the third
dimension. We find that stripes and one type of hexagons have the same stable
range over bounded 2D and 3D domains, while the other type of hexagons becomes
unstable earlier. Here we find a snaking branch of solutions, which are spatial
connects between hexagonal prisms and a genuine 3D pattern (balls).
