With the development of micdrofluidics technology and associated flow reactors it is of interest to investigate the possibility for self-replicating patterns in such reactors. The focus of this research is wether the rich spatio-temproal complexity observed in the Gray-Scott dynamics in reactors driven by a diffusive inflow from a reservoir can be reproduced in the much simpler flow reactor setting. Here we introduce an extended Gray-Scott model for a reaction-diffusion-advection system, and we demonstrate that the system can exhibit complex spatio-temporal dynamics, including self-replicating spots.