<p>In this paper, we study a predator-prey model with additional food for predator. By using white noise to perturb the natural growth rates and introduce a jump process, we model the corresponding stochastic differential equations. The effect of fear and prey refuge on population dynamics is also considered. First, we use Itô’s formula to prove the existence and uniqueness of a global positive solution and its boundedness. Next, sufficient conditions for the extinction and persistence of both species have been given. Then the stochastic permanence of our system is investigated under some conditions. Our main results demonstrate that sufficiently large white noise could drive both species to extinction. However, Lévy noise enhances the survival of both prey and predator species. Our analytical derivations are justified through numerical simulations which show the reliability of the model from the ecological point of view. In addition, we have investigated the impact of fear effect, prey refuge and the additional food biomass on this model by numerical simulation.</p>