Analysis of a Beddington-DeAngelis food chain chemostat with periodically varying dilution rate

A model of a Beddington-DeAngelis type food chain chemostat with periodically varying dilution rate is considered. The system consists of one predator, one prey and limiting substrate. The periodic solutions of the subsystem with substrate and prey, which are the boundary periodic solutions of the system, are obtained. The stability analysis of the boundary periodic solution yields a predator invasion threshold. Above this threshold, there are periodic oscillations in the system. Furthermore, a model with sinusoidal dilution rate is numerically simulated. By bifurcation diagrams with different bifurcation parameters, the periodic system shows two kinds of bifurcations, whose are period-doubling and period-halving.