Dynamic complexities in a periodically pulsed ratio-dependent predator–prey ecosystem modeled on a chemostat

Abstract

This paper contains a three dimensional ratio-dependent predator–prey system modeling for a chemostat with predator, prey and periodically pulsed substrate. Using the discrete dynamical system determined by the stroboscopic map, we obtain an exact periodic solution with positive concentrations of substrate and prey in the absence of the protozoan predator. A stability analysis for this solution yields an invasion threshold (the smallest value of the predator’s predation constant consistent with invasion of the chemostat). Above this threshold, there are periodic oscillations in substrate, prey and predator. Increasing the predator’s predation constant yields a series of period-doubling bifurcations, leading to chaotic dynamics, which implies that the dynamical behaviors of the periodically pulsed ratio-dependent predator–prey ecosystem are very complex, including small-amplitude oscillations, large-amplitude cycles, and chaos. This suggests that limiting substrate pulse, in effect, provides a natural period or cyclicity that allows for a period-doubling route to chaos.