The Long Time Behavior of Equilibrium Status of a Predator-Prey System with Delayed Fear in Deterministic and Stochastic Scenarios

In view of a time lag between the time that the prey perceive predator signals and make some changes or behavior responses, we establish a predator-prey system with direct and indirect predation in this paper. First, we investigate the existence, boundedness, and global asymptotical stability of the positive equilibrium status. Next, by perturbing the mortality rates of prey species and predator species, we stretch the deterministic system to the stochastic scenario and investigate the existence of stochastic process and the global asymptotical stability of the equilibrium status. The analytical findings show that fear affects the value of the equilibrium status, and stochastic disturbance affects its stability, but time delay has no effect if some conditions are satisfied, which are verified by some examples and numerical simulations.