The effect of L$ \acute{E} $vy noise and white noise on a Leslie-Gower predator-prey system with prey refuge

A stochastic Leslie-Gower predator-prey system with prey refuge is discussed, Le ' vy noise and white noise are adopted to simulate the random change of environment. The existence of global positive solution and the stochastic boundedness is proved, the threshold conditions of extinction and stable in the mean are given, the long time behavior of solution for the predator-prey system is also considered. In the end, some examples of numerical simulation are given to verify these conclusions. Compared to the deterministic system, the prey population and the predator population will be extinct, when the intensity of noise is large enough, however, it does not happen in the deterministic system, where the positive equilibrium is always globally stable. These results indicate that the stochastic change of environment has great influence on the population dynamic system.