This paper considers a stochastic single-species model with Lévy noises and time periodic coefficients. By Lyapunov functions and stochastic estimates, the threshold conditions between the time-average persistence in probability and extinction for the model are derived where Lévy noises play an important role in persistence and extinction of populations. It is shown that the time-average persistence in probability of the model implies the existence and uniqueness of positive periodic solution and the existence and uniqueness of periodic measure of the model. An example and its numerical simulations are given to verify the effectiveness of the theoretical results.

中文翻译：

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具有 Lévy 噪声的随机单物种模型的持久性、消光性和周期解

本文考虑具有 Lévy 噪声和时间周期系数的随机单物种模型。通过李雅普诺夫函数和随机估计，推导出模型的时间平均持续概率与灭绝之间的阈值条件，其中 Lévy 噪声在种群的持续和灭绝中起重要作用。结果表明，模型概率的时间平均持久性意味着正周期解的存在唯一性和模型周期测度的存在唯一性。通过算例及其数值模拟验证了理论结果的有效性。