Long-Term Behavior of Lotka–Volterra Model with Lévy Jump in Countable State-Dependent Environments

In this study, we analyze a multi-species mutualistic Lotka–Volterra model with Lévy jumps and regime-switching. A defining feature of the work lies in modeling the random environment through state-dependent switching in an infinite countable state space. Our main objective is to establish the sufficient conditions of the extinction and stochastic permanence of the model. First, we analyze the existence and uniqueness of the model’s solution, followed by an examination of the solution’s stochastic ultimate boundedness. Moreover, the challenges arising from state-dependent switching are addressed using the stochastic comparison method. Due to the presence of the jump component, more complex conditions are required to achieve a finite partition of the countably infinite space. Furthermore, the M-matrix theory is also used to obtain the stochastic permanence property. Finally, two specific examples are provided to illustrate the conclusions in this paper.