Persistence, extinction and practical exponential stability of impulsive stochastic competition models with varying delays

Yuxiao Zhao; Hong Lin; Xiaoyan Qiao; Yuxiao Zhao; Hong Lin; Xiaoyan Qiao

doi:10.3934/math.20231152

AIMS Mathematics

2023, Volume 8, Issue 10: 22643-22661. doi: 10.3934/math.20231152

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Research article

Persistence, extinction and practical exponential stability of impulsive stochastic competition models with varying delays

1.
School of Mathematics and Information Science, Shandong Technology and Business University, Yantai, 264005, China
2.
School of Mathematical and Computational Science, Hunan University of Science and Technology, Xiangtan, 411201, China
3.
Institute of Intelligence Science and Engineering, Shenzhen Polytechnic, Shenzhen, 518055, China

Received: 30 April 2023 Revised: 28 June 2023 Accepted: 04 July 2023 Published: 17 July 2023
MSC : 34A37

This paper studies the persistence, extinction and practical exponential stability of impulsive stochastic competition models with time-varying delays. The existence of the global positive solutions is investigated by the relationship between the solutions of the original system and the equivalent system, and the sufficient conditions of system persistence and extinction are given. Moreover, our study shows the following facts: (1) The impulsive perturbation does not affect the practical exponential stability under the condition of bounded pulse intensity. (2) In solving the stability of non-Markovian processes, it can be transformed into solving the stability of Markovian processes by applying Razumikhin inequality. (3) In some cases, a non-Markovian process can produce Markovian effects. Finally, numerical simulations obtained the importance and validity of the theoretical results for the existence of practical exponential stability through the relationship between parameters, pulse intensity and noise intensity.
- impulsive stochastic model,
- time-varying delays,
- practical exponential stability,
- persistence
Citation: Yuxiao Zhao, Hong Lin, Xiaoyan Qiao. Persistence, extinction and practical exponential stability of impulsive stochastic competition models with varying delays[J]. AIMS Mathematics, 2023, 8(10): 22643-22661. doi: 10.3934/math.20231152

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Abstract

This paper studies the persistence, extinction and practical exponential stability of impulsive stochastic competition models with time-varying delays. The existence of the global positive solutions is investigated by the relationship between the solutions of the original system and the equivalent system, and the sufficient conditions of system persistence and extinction are given. Moreover, our study shows the following facts: (1) The impulsive perturbation does not affect the practical exponential stability under the condition of bounded pulse intensity. (2) In solving the stability of non-Markovian processes, it can be transformed into solving the stability of Markovian processes by applying Razumikhin inequality. (3) In some cases, a non-Markovian process can produce Markovian effects. Finally, numerical simulations obtained the importance and validity of the theoretical results for the existence of practical exponential stability through the relationship between parameters, pulse intensity and noise intensity.

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