Dynamic properties for a stochastic food chain model

Abstract

In this paper, Lyapunov exponents of ergodic invariant measures are used to study dynamic properties for a stochastic food chain model, which consists of two competing predators and one prey. Ayala’s experimental result, or rather, competitive coexistence is showed to be possible in this random case. Furthermore, the considered stochastic model have five points of dynamical bifurcation, which happen to be the thresholds (between survival and extinction) of the system or each species. In addition, the necessity of introducing environmental noise is verified by the fact that environmental driving force can drive the system towards extinction from partial extinction or coexistence. Moreover, all the theoretical results are well verified by numerical simulations. It is worth mentioning that we make a first attempt at using meshing method and statistical data to test Lyapunov exponents for two-dimensional boundary measures, and this is an innovation in the numerical methods.