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In this paper, we study a three-species predator-prey model with modified LG-Holling type II with Lévy jumps, and we take the competition among predators into consideration. First, We use an Ornstein-Uhlenbeck process to describe the environmental stochasticity and prove that there is a unique positive solution to the system by mathematical analysis skills such as comparison theorem. Futhermore, the extinction or persistence in the mean of each species under different conditions is obtained. Finally, some numerical simulations are carried out to support our main results. [ABSTRACT FROM AUTHOR]

In this paper, we use a mean-reverting Ornstein-Uhlenbeck process to simulate the stochastic perturbations in the environment, and then a modified Leslie–Gower Holling-type II predator-prey stochastic model in a polluted environment with interspecific competition and pulse toxicant input is proposed. Through constructing V-function and applying It o ^ ′ s formula, the sharp sufficient conditions including strongly persistent in the mean, persistent in the mean and extinction are established. In addition, the theoretical results are verified by numerical simulation. [ABSTRACT FROM AUTHOR]

This paper is concerned with a modified Leslie–Gower and Holling-type II two-predator one-prey model with Lévy jumps. First, we use an Ornstein–Uhlenbeck process to describe the environmental stochasticity and prove that there is a unique positive solution to the system. Moreover, sufficient conditions for persistence in the mean and extinction of each species are established. Finally, we give some numerical simulations to support the main results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a three-species competitive model with both white noises and Lévy noises. We first carry out the almost complete parameters analysis for the model and establish the critical value between persistence in the mean and extinction for each species. The sufficient criteria for stability in distribution of solutions are obtained. Finally, numerical simulations are carried out to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with a two-dimensional stochastic Lotka-Volterra mutualistic parasite-host system with Lévy jumps. First, the study reveals that each species is either extinct or persistent in the mean, relying on some critical values. The result shows that both the environmental noises and Lévy jumps play important roles in determining the persistent in the mean or extinction of each species. Then, we establish the sufficient criteria for stability on distribution of the system. Finally, we give some numerical simulations to support the main results. Some recent works are extended and improved greatly. [ABSTRACT FROM AUTHOR]