Stochastic asymptotical stability of the predator-free and positive equilibrium for food chain model with vigilance under stochastic perturbations

Abstract

Most stability analyzes of stochastic differential equations (SDEs) are based on the construction of Lyapunov functions, but it is difficult to find suitable Lyapunov functions in some cases. This paper studies a class of stochastic food chain models with vigilance. First, the asymptotical mean square (MS) stability of the corresponding linearization problem was analyzed using Kronecker product and semi-vectorization dimensionality reduction methods, and the sufficient conditions and stable regions of the local stochastic asymptotic stability of the stochastic nonlinear food chain model were derived. Afterwards, the effects of noise intensity and dual parameters changes on the stochastic asymptotical stability (SAS) of the system were analyzed through numerical simulation, and the following conclusions were drawn: mid-level predators respond most strongly to disturbance, while bottom prey are the least sensitive to disturbance. Overall, the increase in perturbation reduces the geometric volume of the stochastically asymptotically stable domain.