There exist many experimental examples of spatial pattern formation in electrochemical systems. Using a recently proposed model, we present numerical simulations of the spatio-temporal behavior in the bistable and oscillatory regime. The simulations reveal that for some parameter regimes only spatially inhomogeneous solutions exist. The impact of the system parameters on the spatial coupling as well as on the existence of patterns is elaborated. The predictions coincide with experimental results for a simple electrochemical reaction, the reduction of peroxodisulfate.