Stability analysis of a stochastic Gilpin–Ayala model driven by Lévy noise

• We consider a stochastic one-dimensional Gilpin–Ayala model driven by Lévy noise.
• We show that the solution of this system is global and positive under certain conditions.
• Almost sure exponential stability and moment exponential stability of the trivial solution are discussed.
• Numerical example is introduced to illustrate the results.

A stochastic one-dimensional Gilpin–Ayala model driven by Lévy noise is presented in this paper. Firstly, we show that this model has a unique global positive solution under certain conditions. Then sufficient conditions for the almost sure exponential stability and moment exponential stability of the trivial solution are established. Results show that the jump noise can make the trivial solution stable under some conditions. Numerical example is introduced to illustrate the results.