Narrow Results

Non-autonomous single-species growth chemostat models with general response functions are considered. In the models, the dilution rate and removal rate are allowed to be different from each other. A series of new criteria on the boundedness, permanence, persistence, average persistence and extinction of the population is established. In particular, when models degenerate into the almost periodic case, the equivalences of the permanence, persistence and average persistence of species are obtained. These results improve and extend some well-known corresponding results obtained in Refs. 1, 14 and 17.

In this paper, we consider a predator–prey chemostat model with ratio-dependent Monod type functional response and periodic input and washout at different fixed times. We obtain an exact periodic solution with substrate and prey. The stability analysis for this periodic solutions yields an invasion threshold for the period of pulses of the predator. When the impulsive period is more than the threshold, there are periodic oscillations in the substrate, prey, and predator. If the impulsive period further increases, the system undergoes a complex dynamic process. By analyzing bifurcation diagrams, we can see that the impulsive system shows two kinds of bifurcation, which are period-doubling and period-halving.