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

Huijie Ji, Ping Yu, Hongxia Sun and Yuhang Zhen ()
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Huijie Ji: School of Mathematics Science, Shanxi Normal University, Taiyuan 030031, China
Ping Yu: School of Mathematics Science, Shanxi Normal University, Taiyuan 030031, China
Hongxia Sun: School of Mathematics Science, Shanxi Normal University, Taiyuan 030031, China
Yuhang Zhen: School of Computer Science and Technology, Zhejiang University of Water Resources and Electric Power, Hangzhou 310018, China

Mathematics, 2025, vol. 13, issue 21, 1-23

Abstract: 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.

Keywords: countable switching states; state-dependent switching; Lévy jump; stochastic Lotka–Volterra system; extinction (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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