Spatial patterns and coexistence mechanisms in systems with unidirectional flow

River ecosystems are the prime example of environments where unidirectional flow influences the dispersal of individuals. Spatial patterns of community composition and species replacement emerge from complex interplays of hydrological, geochemical, biological, and ecological factors. Local processes affecting algal dynamics are well understood, but a mechanistic basis for large scale emerging patterns is lacking. To understand how these patterns could emerge in rivers, we analyze a reaction-advection-diffusion model for two competitors in heterogeneous environments. The model supports waves that invade upstream up to a well-defined "upstream invasion limit". We discuss how these waves are produced and present their key properties. We suggest that patterns of species replacement and coexistence along spatial axes reflect stalled waves, produced from diffusion, advection, and species interactions. Emergent spatial scales are plausible given parameter estimates for periphyton. Our results apply to other systems with unidirectional flow such as prevailing winds or climate-change scenarios.