bang-bang controlled chemostat culture in which two microbial populations compete for the same nutrient was studied from the standpoint of bifurcation theory. The dilution rate was periodically switched between two fixed values. The bifurcations of periodic states due to changes in the control parameters, the period and the switching ratio were investigated and the region of co-existence in the control parameter plane was determined. The stability of each periodic state was determined by seeking the values of the corresponding characteristic multipliers. The bifurcation8 without any topological contradiction were determined by applying the index theory. lNTRODUCIION The co-existence. problem of two or more microbial populations which compete with each other under various growth limiting conditions is of great importance not only from the ecological viewpoint but also from the standpoint of practical applications such as SCP (single cell protein) production, waste water treatment and so on. It is well known that two populations which compete with one another for a single limiting nutrient cannot co-exist under the condition of a spatially uniform environment and time-invariant external influences. This suggests that the co-existence state of both populations might be realizable by using either any spatially inhomogeneous environment or any time varying external influences. The former has been applied to SCP production