MS-stability of the predator-free and positive equilibria for food chain model with vigilance under stochastic perturbations

Abstract Most stability analyzes of SDE models are based on the construction of Lyapunov functions, but it is difficult to find suitable Lyapunov functions in some cases. This paper investigates a vigilant food chain model assuming that the model is subject to random perturbations proportional to deviations from the system state and equilibrium point. Using Kronecker product and semi-vectorization dimensionality reduction method, the explicit sufficient condition and stable region of asymptotically mean square(MS) stability of stochastic system are obtained. The influence of noise intensity and bi-parameters changes on the mean square stable region are analyzed. From the analysis, the following conclusions are obtained: mid-level predators are the most responsive to disturbance, while bottom prey are least sensitive to disturbance. Overall, the increase in perturbation reduces the MS-stable geometry volume and maximum resilience.