Dynamics of a stochastic one-prey two-predator model with Lévy jumps

•Lévy noise of one species increases extinction risk of this species and its predator.•Lévy noise of one species has no impact on the extinction or not of its prey.•In some cases, Lévy noises do not effect the stability in distribution of the model. This paper is concerned with a one-prey two-predator model with both white noises and Lévy noises. We first carry out the almost complete parameters analysis for the model. In each case we show that each species is either persistent in the mean or extinct, depending on some critical values. Then we establish the sufficient criteria for stability in distribution of the model. Finally, we use some numerical examples to demonstrate the analytical findings.