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Unique stationary distribution and ergodicity of a stochastic Logistic model with distributed delay

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  • Sun, Xinguo
  • Zuo, Wenjie
  • Jiang, Daqing
  • Hayat, Tasawar

Abstract

Dynamics of a stochastic Logistic model with distributed delay are considered. We first transfer a scalar stochastic Logistic model with strong kernel or weak kernel into an equivalent stochastic system through the linear chain technique. Then we obtain the sufficient and necessary conditions for extinction and persistence of the species with probability one. Moreover, in the case of persistence, we prove that there exists a unique stationary distribution by the Markov semigroups theory. The results show that, the stronger white noise results in the extinction of the species and the weaker white noise guarantees the existence of a unique stationary distribution, though for the deterministic model with strong kernel or weak kernel, the average delay may induce the existence of a group of small amplitude periodic solutions.

Suggested Citation

  • Sun, Xinguo & Zuo, Wenjie & Jiang, Daqing & Hayat, Tasawar, 2018. "Unique stationary distribution and ergodicity of a stochastic Logistic model with distributed delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 864-881.
    • Handle: RePEc:eee:phsmap:v:512:y:2018:i:c:p:864-881
    DOI: 10.1016/j.physa.2018.08.048
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    References listed on IDEAS

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    1. Lin, Yuguo & Jiang, Daqing & Wang, Shuai, 2014. "Stationary distribution of a stochastic SIS epidemic model with vaccination," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 394(C), pages 187-197.
    2. Li, Dingshi, 2013. "The stationary distribution and ergodicity of a stochastic generalized logistic system," Statistics & Probability Letters, Elsevier, vol. 83(2), pages 580-583.
    3. Jiang, Daqing & Zuo, Wenjie & Hayat, Tasawar & Alsaedi, Ahmed, 2016. "Stationary distribution and periodic solutions for stochastic Holling–Leslie predator–prey systems," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 460(C), pages 16-28.
    4. Guodong Liu & Xiaohong Wang & Xinzhu Meng & Shujing Gao, 2017. "Extinction and Persistence in Mean of a Novel Delay Impulsive Stochastic Infected Predator-Prey System with Jumps," Complexity, Hindawi, vol. 2017, pages 1-15, June.
    5. Zuo, Wenjie & Jiang, Daqing & Sun, Xinguo & Hayat, Tasawar & Alsaedi, Ahmed, 2018. "Long-time behaviors of a stochastic cooperative Lotka–Volterra system with distributed delay," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 506(C), pages 542-559.
    6. Rudnicki, Ryszard, 2003. "Long-time behaviour of a stochastic prey-predator model," Stochastic Processes and their Applications, Elsevier, vol. 108(1), pages 93-107, November.
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    Cited by:

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    2. Wenxu Ning & Zhijun Liu & Lianwen Wang & Ronghua Tan, 2021. "Analysis of a Stochastic Competitive Model with Saturation Effect and Distributed Delay," Methodology and Computing in Applied Probability, Springer, vol. 23(4), pages 1435-1459, December.
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    4. Xu, Dongsheng & Liu, Ming & Xu, Xiaofeng, 2020. "Analysis of a stochastic predator–prey system with modified Leslie–Gower and Holling-type IV schemes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).
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    6. Lu, Chun, 2021. "Dynamics of a stochastic Markovian switching predator–prey model with infinite memory and general Lévy jumps," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 181(C), pages 316-332.
    7. Zhang, Xiaofeng & Yuan, Rong, 2022. "Stochastic bifurcation and density function analysis of a stochastic logistic equation with distributed delay and weak kernel," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 195(C), pages 56-70.

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