Stationary distribution of a stochastic three species predator–prey model with anti-predator behavior

A stochastic three species predator–prey model with intermediate predator and anti-predator behavior is established and studied in this paper. By constructing suitable Lyapunov functions and combining knowledge of stochastic differential equations, the behavior of global positive solutions to the model are investigated. Firstly, we prove that there is unique global positive solution to the model, and establish the stochastic boundedness of the positive solution as well. Secondly, the sufficient condition for the existence of a unique ergodic stationary distribution is provided. Thirdly, the persistence and extinction of the populations are discussed. Finally, some numerical simulations demonstrate that the results obtained in this paper are true, moreover, the influence of white noise on the populations is revealed.