LAUSR

In this paper, we study the dynamics of stochastic time-delayed predator–prey model with modified Leslie–Gower and ratio-dependent schemes including additional food for predator and prey with Michaelis–Menten type harvest. We introduce the effects of time delay, Michaelis–Menten type harvest and stochastic perturbation under the structure of the original model to make the model more consistent with the actual system. We first prove that the system has a globally unique positive solution. Secondly we obtain conditions for the persistence in mean and extinction of the system. Besides, we verify that the system is stochastic permanence under certain conditions. In addition to that, we prove that the system has an ergodic stationary distribution when the parameters satisfy certain conditions. Finally, some numerical simulations were performed to verify the correctness and validity of the theoretical results.