  Reaction-diffusion-advection media on semi-infinite domains are important in
chemical, biological and ecological applications, yet remain a challenge for
pattern formation theory. To demonstrate the rich emergence of nonlinear
traveling waves and stationary periodic states, we review results obtained
using a membrane reactor as a case model. Such solutions coexist in overlapping
parameter regimes and their temporal stability is determined by the boundary
conditions (periodic vs. mixed) which either preserve or destroy the
translational symmetry, i.e., selection mechanisms under realistic Danckwerts
boundary conditions. A brief outlook is given at the end.
