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

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.

Automated summary

This paper investigates the persistence, extinction, and practical exponential stability of impulsive stochastic competition models with time-varying delays. The existence of global positive solutions is examined through the relationship between the original system and the equivalent system, and sufficient conditions for system persistence and extinction are provided. The study also reveals that impulsive perturbation has no impact on practical exponential stability under bounded pulse intensity, non-Markovian processes can be transformed into solving the stability of Markovian processes using Razumikhin inequality, and non-Markovian processes can sometimes produce Markovian effects. Numerical simulations further validate the importance and validity of the theoretical results for practical exponential stability.