Long Time Behavior of a Stochastic Prey–Predator System With Prey Refuge and Predation Fear

The primary goal of this paper is to investigate the long-term behavior of a prey–predator system that incorporates the effects of refuge, fear, and degenerate diffusion. Due to the degenerate nature of the diffusion matrix, the uniform ellipticity condition is not satisfied. Consequently, the Markov semigroup theory is utilized to establish the existence of a unique stable stationary distribution, which indicates the distribution density of any positive solution converges to a unique invariant density as (Formula presented.). Additionally, numerical simulations using the Euler's method are presented to demonstrate the influence of refuge, fear, and environmental noise on the dynamics of the system.