Pattern analysis in a benthic bacteria-nutrient system
  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). 
Daniel Wetzel
math.DS nlin.PS