# American Institute of Mathematical Sciences

September  2021, 41(9): 4145-4183. doi: 10.3934/dcds.2021032

## Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching

 1 School of Mathematical Sciences, Anhui University, Hefei, 230601, China 2 School of Mathematical Science, Huaiyin Normal University, Huai'an, 223300, China 3 Jiangsu Key Laboratory for NSLSCS, School of Mathematical Sciences, Nanjing Normal University, Nanjing, 210023, China

* Corresponding author: Hui Wan

Received  June 2020 Revised  November 2020 Published  February 2021

This work investigates the dynamics of competitive Kolmogorov systems formulated in a semi-Markov regime-switching framework. The conditional holding time of each environmental regime is allowed to follow arbitrary probability distribution on the nonnegative half-line in the sense of approximations. Sharp sufficient conditions of the coexistence and competitive exclusion of species are established, and in the case of species coexistence, the convergence rate of the transition probability to the unique stationary measure is estimated. In weaker conditions, these results extend the existing results to the semi-Markov regime-switching environment. Particularly, the method of proving the exponential convergence of the transition probability to the invariant measure for the population models formulated as random differential equations driven by a semi-Markov process is proposed.

Citation: Dan Li, Hui Wan. Coexistence and exclusion of competitive Kolmogorov systems with semi-Markovian switching. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4145-4183. doi: 10.3934/dcds.2021032
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