IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v130y2020i8p4608-4642.html
   My bibliography  Save this article

General nonlinear stochastic systems motivated by chemostat models: Complete characterization of long-time behavior, optimal controls, and applications to wastewater treatment

Author

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

Abstract

This paper focuses on a general class of systems of nonlinear stochastic differential equations, inspired by stochastic chemostat models. In the first part, the system is formulated as a hybrid switching diffusion. A complete characterization of the asymptotic behavior of the system under consideration is provided. It is shown that the long-term properties of the system can be classified by using a real-valued parameter λ. If λ≤0, the bacteria will die out, which means that the process does not operate. If λ>0, the system has an invariant probability measure and the transition probability of the solution process converges to that of the invariant measure. The rate of convergence is also obtained. One of the distinct features of this paper is that the critical case λ=0 is also considered. Moreover, numerical examples are given to illustrate our results. In the second part of the paper, controlled diffusions with a long-run average objective function are treated. The associated Hamilton–Jacobi–Bellman (HJB) equation is derived and the existence of an optimal Markov control is established. The techniques and methods of analysis in this paper can be applied to many other stochastic Kolmogorov systems.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:8:p:4608-4642
    DOI: 10.1016/j.spa.2020.01.010
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304414919303126
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.spa.2020.01.010?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Lutz Becks & Frank M. Hilker & Horst Malchow & Klaus Jürgens & Hartmut Arndt, 2005. "Experimental demonstration of chaos in a microbial food web," Nature, Nature, vol. 435(7046), pages 1226-1229, June.
    2. Geiß, Christel & Manthey, Ralf, 1994. "Comparison theorems for stochastic differential equations in finite and infinite dimensions," Stochastic Processes and their Applications, Elsevier, vol. 53(1), pages 23-35, September.
    3. Fritsch, Coralie & Harmand, Jérôme & Campillo, Fabien, 2015. "A modeling approach of the chemostat," Ecological Modelling, Elsevier, vol. 299(C), pages 1-13.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. 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.
    2. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wade, M.J. & Harmand, J. & Benyahia, B. & Bouchez, T. & Chaillou, S. & Cloez, B. & Godon, J.-J. & Moussa Boudjemaa, B. & Rapaport, A. & Sari, T. & Arditi, R. & Lobry, C., 2016. "Perspectives in mathematical modelling for microbial ecology," Ecological Modelling, Elsevier, vol. 321(C), pages 64-74.
    2. Occhipinti, Guido & Solidoro, Cosimo & Grimaudo, Roberto & Valenti, Davide & Lazzari, Paolo, 2023. "Marine ecosystem models of realistic complexity rarely exhibits significant endogenous non-stationary dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 175(P1).
    3. Damir Filipović & Sander Willems, 2020. "A term structure model for dividends and interest rates," Mathematical Finance, Wiley Blackwell, vol. 30(4), pages 1461-1496, October.
    4. Rajasekar, S.P. & Pitchaimani, M., 2019. "Qualitative analysis of stochastically perturbed SIRS epidemic model with two viruses," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 207-221.
    5. C. A. Tapia Cortez & J. Coulton & C. Sammut & S. Saydam, 2018. "Determining the chaotic behaviour of copper prices in the long-term using annual price data," Palgrave Communications, Palgrave Macmillan, vol. 4(1), pages 1-13, December.
    6. Ding, Xiaodong & Wu, Rangquan, 1998. "A new proof for comparison theorems for stochastic differential inequalities with respect to semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 78(2), pages 155-171, November.
    7. Wang, Tianxiao & Yong, Jiongmin, 2015. "Comparison theorems for some backward stochastic Volterra integral equations," Stochastic Processes and their Applications, Elsevier, vol. 125(5), pages 1756-1798.
    8. Mukama, Denis Sospeter & Ghani, Mohammad & Mbalawata, Isambi Sailon, 2023. "Persistence, Extinction, and boundedness in pth moment of hybrid stochastic logistic systems by delay feedback control based on discrete-time observation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 661-677.
    9. Richard Ottermanns & Kerstin Szonn & Thomas G Preuß & Martina Roß-Nickoll, 2014. "Non-Linear Analysis Indicates Chaotic Dynamics and Reduced Resilience in Model-Based Daphnia Populations Exposed to Environmental Stress," PLOS ONE, Public Library of Science, vol. 9(5), pages 1-13, May.
    10. Bildirici, Melike E. & Sonustun, Bahri, 2021. "Chaotic behavior in gold, silver, copper and bitcoin prices," Resources Policy, Elsevier, vol. 74(C).
    11. Bouderbala, Ilhem & El Saadi, Nadjia & Bah, Alassane & Auger, Pierre, 2019. "A simulation study on how the resource competition and anti-predator cooperation impact the motile-phytoplankton groups’ formation under predation stress," Ecological Modelling, Elsevier, vol. 391(C), pages 16-28.
    12. Tapia, Carlos & Coulton, Jeff & Saydam, Serkan, 2020. "Using entropy to assess dynamic behaviour of long-term copper price," Resources Policy, Elsevier, vol. 66(C).
    13. Fritsch, Coralie & Campillo, Fabien & Ovaskainen, Otso, 2017. "A numerical approach to determine mutant invasion fitness and evolutionary singular strategies," Theoretical Population Biology, Elsevier, vol. 115(C), pages 89-99.
    14. Sakai, Kenshi & Noguchi, Yuko & Asada, Shin-ichi, 2008. "Detecting chaos in a citrus orchard: Reconstruction of nonlinear dynamics from very short ecological time series," Chaos, Solitons & Fractals, Elsevier, vol. 38(5), pages 1274-1282.
    15. Damir Filipovi'c & Sander Willems, 2018. "A Term Structure Model for Dividends and Interest Rates," Papers 1803.02249, arXiv.org, revised May 2020.
    16. Liu Baisen & Wang Liangliang & Cao Jiguo, 2018. "Bayesian estimation of ordinary differential equation models when the likelihood has multiple local modes," Monte Carlo Methods and Applications, De Gruyter, vol. 24(2), pages 117-127, June.
    17. David G. Green, 2023. "Emergence in complex networks of simple agents," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 18(3), pages 419-462, July.
    18. Wu, Zhen & Xu, Mingyu, 2009. "Comparison theorems for forward backward SDEs," Statistics & Probability Letters, Elsevier, vol. 79(4), pages 426-435, February.
    19. Sun, Shulin & Sun, Yaru & Zhang, Guang & Liu, Xinzhi, 2017. "Dynamical behavior of a stochastic two-species Monod competition chemostat model," Applied Mathematics and Computation, Elsevier, vol. 298(C), pages 153-170.
    20. Shang, Zuchong & Qiao, Yuanhua, 2024. "Complex dynamics of a four-species food web model with nonlinear top predator harvesting and fear effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 223(C), pages 458-484.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:130:y:2020:i:8:p:4608-4642. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.