Thresholds and critical states for a stochastic predator–prey model with mixed functional responses

In this paper, we are committed to the study of thresholds (between persistence and extinction) for all the species in a stochastic predator–prey model, which takes both Beddington–DeAngelis and Holling-II functional responses. One interesting thing we find is that the ith Lyapunov exponent defined for an ergodic invariant measure may just happen to be the threshold for the ith species. Furthermore, we discuss the priority levels among multiple thresholds for the same species, which is a novel feature of this paper. A brief summary of priority level is that, Lyapunov exponent for a high-dimensional measure has higher priority than that of a low-dimensional measure. At the end of the paper, we analyze dynamic properties for some critical states.