Stability in chemostat equations with delayed nutrient recycling

The growth of a species feeding on a limiting nutrient supplied at a constant rate is modelled by chemostat-type equations with a general nutrient uptake function and delayed nutrient recycling. Conditions for boundedness of the solutions and the existence of non-negative equilibria are given for the integrodifferential equations with distributed time lags. When the time lags are neglected conditions for the global stability of the positive equilibrium and for the extinction of the species are provided. The positive equilibrium continues to be locally stable when the time lag in recycling is considered and this is proved for a wide class of memory functions. Computer simulations suggest that even in this case the region of stability is very large, but the solutions tend to the equilibrium through oscillations.