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Stochastic functional Kolmogorov equations, I: Persistence

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  • Nguyen, Dang H.
  • Nguyen, Nhu N.
  • Yin, George

Abstract

This work (Part (I)) together with its companion (Part (II)) develops a new framework for stochastic functional Kolmogorov equations, which are nonlinear stochastic differential equations depending on the current as well as the past states. Because of the complexity of the results, it seems to be instructive to divide our contributions to two parts. In contrast to the existing literature, our effort is to advance the knowledge by allowing delay and past dependence, yielding essential utility to a wide range of applications. A long-standing question of fundamental importance pertaining to biology and ecology is: What are the minimal necessary and sufficient conditions for long-term persistence and extinction (or for long-term coexistence of interacting species) of a population? Regardless of the particular applications encountered, persistence and extinction are properties shared by Kolmogorov systems. While there are many excellent treaties of stochastic-differential-equation-based Kolmogorov equations, the work on stochastic Kolmogorov equations with past dependence is still scarce. Our aim here is to answer the aforementioned basic question. This work, Part (I), is devoted to characterization of persistence, whereas its companion, Part (II) is devoted to extinction. The main techniques used in this paper include the newly developed functional Itô formula and asymptotic coupling and Harris-like theory for infinite dimensional systems specialized to functional equations. General theorems for stochastic functional Kolmogorov equations are developed first. Then a number of applications are examined covering, improving, and substantially extending the existing literature. Furthermore, our results reduce to that in the existing literature of Kolmogorov systems when there is no past dependence.

Suggested Citation

  • Nguyen, Dang H. & Nguyen, Nhu N. & Yin, George, 2021. "Stochastic functional Kolmogorov equations, I: Persistence," Stochastic Processes and their Applications, Elsevier, vol. 142(C), pages 319-364.
    • Handle: RePEc:eee:spapps:v:142:y:2021:i:c:p:319-364
    DOI: 10.1016/j.spa.2021.09.007
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    References listed on IDEAS

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    1. Liu, Qun & Jiang, Daqing & Shi, Ningzhong & Hayat, Tasawar & Alsaedi, Ahmed, 2016. "Asymptotic behavior of a stochastic delayed SEIR epidemic model with nonlinear incidence," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 462(C), pages 870-882.
    2. Reiß, M. & Riedle, M. & van Gaans, O., 2006. "Delay differential equations driven by Lévy processes: Stationarity and Feller properties," Stochastic Processes and their Applications, Elsevier, vol. 116(10), pages 1409-1432, October.
    3. Jian Wu, 2019. "Analysis of a Three-Species Stochastic Delay Predator-Prey System with Imprecise Parameters," Methodology and Computing in Applied Probability, Springer, vol. 21(1), pages 43-67, March.
    4. repec:hhs:iuiwop:487 is not listed on IDEAS
    5. Jorgen W. Weibull, 1997. "Evolutionary Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262731215, December.
    6. Nguyen, Dang H. & Nguyen, Nhu N. & Yin, George, 2020. "General nonlinear stochastic systems motivated by chemostat models: Complete characterization of long-time behavior, optimal controls, and applications to wastewater treatment," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4608-4642.
    7. Liu, Qun & Jiang, Daqing, 2016. "The threshold of a stochastic delayed SIR epidemic model with vaccination," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 461(C), pages 140-147.
    8. Sun, Shulin & Zhang, Xiaofeng, 2018. "Asymptotic behavior of a stochastic delayed chemostat model with nonmonotone uptake function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 512(C), pages 38-56.
    9. Liu, Qun & Chen, Qingmei & Jiang, Daqing, 2016. "The threshold of a stochastic delayed SIR epidemic model with temporary immunity," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 450(C), pages 115-125.
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    Cited by:

    1. Cao, Nan & Fu, Xianlong, 2023. "Stationary distribution and extinction of a Lotka–Volterra model with distribute delay and nonlinear stochastic perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 169(C).
    2. Hongjiang Qian & Zhexin Wen & George Yin, 2022. "Numerical solutions for optimal control of stochastic Kolmogorov systems with regime-switching and random jumps," Statistical Inference for Stochastic Processes, Springer, vol. 25(1), pages 105-125, April.
    3. Ky Q. Tran & Bich T. N. Le & George Yin, 2022. "Harvesting of a Stochastic Population Under a Mixed Regular-Singular Control Formulation," Journal of Optimization Theory and Applications, Springer, vol. 195(3), pages 1106-1132, December.

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