Formation of stationary spatially multiperiodic or even chaotic patterns is analyzed for a simple model of a cross-flow reactor with two consecutive reactions and realistically high Le and Pe. Spatial patterns emerge much like dynamic temporal patterns in a mixed system of the same kinetics. The sequence of period doubling bifurcations is determined for the corresponding ODE system and is completely confirmed by direct numerical simulations of the full PDE model. The incorporation of a slow nondiffusing inhibitor led to chaotic spatiotemporal patterns.