Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises

Jiangtao Yang

doi:10.3934/dcdsb.2020371

Article Contents

2021, Volume 26, Issue 10: 5641-5660. Doi: 10.3934/dcdsb.2020371

This issue Previous Article Stable transition layers in an unbalanced bistable equation Next Article Parameter identification on Abelian integrals to achieve Chebyshev property

Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises

Jiangtao Yang^,

School of Mathematics and Statistics, Yangtze Normal University, Chongqing, 408100, China

^* Corresponding author: Jiangtao Yang
^* Corresponding author: Jiangtao Yang

Received: June 30, 2020

Early access: December 2020

Published: October 2021

The author is supported by Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant No. KJQN202001401)

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Abstract

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.

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Mathematics Subject Classification: Primary: 92B05, 60G51, 60G57, 60H35.

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Figure 1. Dynamical behaviors of model (46) with $ c(t) = 0.85+0.75\sin t-\int_{\mathbb{Z}}h(t,z)\pi(dz) $: (a) $ h = 0.4 $; (b) $ h = 2 $. Other parameters are given by (47)

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Figure 2. Dynamical behaviors of model (46) with $ c(t) = 0.85+0.75\sin t $: (a) $ h = 0.4 $; (b) $ h = -0.6 $. Other parameters are given by (47)

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