Rich dynamics in a stochastic predator-prey model with protection zone for the prey and multiplicative noise applied on both species

In this manuscript, a new approach of a stochastic predator-prey interaction with protection zone for the prey is developed and studied. The considered mathematical model consists of a system of two stochastic differential equations, SDEs, describing the interaction between the prey and predator populations where the prey exhibits a social behavior called also by "herd behavior." First, according to the theory of the SDEs, some properties of the solution are obtained, including: the existence and uniqueness of the global positive solution and the stochastic boundedness of the solutions. Then, the sufficient conditions for the persistence in the mean and the extinction of the species are established, where the extinction criteria are discussed in two different cases, namely, the first case is the survival of the prey population, while the predator population goes extinct; the second case is the extinction of all prey and predator populations. Next, by constructing a suitable stochastic Lyapunov function and under certain parametric restrictions, it has been proved that the system has a unique stationary distribution which is ergodic. Finally, some numerical simulations based on the Milstein's higher-order scheme are performed to illustrate the theoretical predictions.