For a model of microbial flow reactor with two competing populations, we show the coexistence of weakly coupled traveling waves in the sense that one organism undergoes a population growth while another organism remains in a very low population density in the first half interval of the space line; the population densities then exchange the position in the next half interval. This type of traveling waves can occur only if the input nutrient slightly exceeds the maximum carrying capacity for two populations. This means, in the lack of adequate nutrient, two competing organisms will manage to survive in a more economical way.