IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v39y2009i3p1395-1409.html
   My bibliography  Save this article

Stability and bifurcation in a two harmful phytoplankton–zooplankton system

Author

Listed:
  • Zhao, Jiantao
  • Wei, Junjie

Abstract

In this paper, a mathematical model consisting of two harmful phytoplankton and zooplankton with discrete time delays is considered. We prove that a sequence of Hopf bifurcations occur at the interior equilibrium as the delay increases. Meanwhile, the phenomenon of stability switches is found under certain conditions. The direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are determined by using the theory of normal form and center manifold. Numerical simulations are given to support the theoretical results.

Suggested Citation

  • Zhao, Jiantao & Wei, Junjie, 2009. "Stability and bifurcation in a two harmful phytoplankton–zooplankton system," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1395-1409.
  • Handle: RePEc:eee:chsofr:v:39:y:2009:i:3:p:1395-1409
    DOI: 10.1016/j.chaos.2007.05.019
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077907003815
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.chaos.2007.05.019?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. Zhang, Shuwen & Tan, Dejun & Chen, Lansun, 2006. "Chaos in periodically forced Holling type IV predator–prey system with impulsive perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 27(4), pages 980-990.
    2. Jing, Zhujun & Yang, Jianping, 2006. "Bifurcation and chaos in discrete-time predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 259-277.
    3. Huang, Dongwei & Wang, Hongli & Feng, Jianfeng & Zhu, Zhi-wen, 2006. "Hopf bifurcation of the stochastic model on HAB nonlinear stochastic dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 27(4), pages 1072-1079.
    4. Zhang, Shuwen & Chen, Lansun, 2006. "A study of predator–prey models with the Beddington–DeAnglis functional response and impulsive effect," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 237-248.
    5. Zhang, Yujuan & Xiu, Zhilong & Chen, Lansun, 2005. "Chaos in a food chain chemostat with pulsed input and washout," Chaos, Solitons & Fractals, Elsevier, vol. 26(1), pages 159-166.
    6. Zhang, Shuwen & Tan, Dejun & Chen, Lansun, 2006. "Chaos in periodically forced Holling type II predator–prey system with impulsive perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 28(2), pages 367-376.
    7. Xiang, Zhongyi & Song, Xinyu, 2006. "Extinction and permanence of a two-prey two-predator system with impulsive on the predator," Chaos, Solitons & Fractals, Elsevier, vol. 29(5), pages 1121-1136.
    8. Li, Zhenqing & Wang, Weiming & Wang, Hailing, 2006. "The dynamics of a Beddington-type system with impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 29(5), pages 1229-1239.
    9. Hui, Jing & Chen, Lansun, 2006. "Dynamic complexities in a periodically pulsed ratio-dependent predator–prey ecosystem modeled on a chemostat," Chaos, Solitons & Fractals, Elsevier, vol. 29(2), pages 407-416.
    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. Agnihotri, Kulbhushan & Kaur, Harpreet, 2019. "The dynamics of viral infection in toxin producing phytoplankton and zooplankton system with time delay," Chaos, Solitons & Fractals, Elsevier, vol. 118(C), pages 122-133.
    2. Jiang, Zhichao & Zhang, Tongqian, 2017. "Dynamical analysis of a reaction-diffusion phytoplankton-zooplankton system with delay," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 693-704.
    3. Raw, Sharada Nandan & Sahu, Sevak Ram, 2023. "Strong stability with impact of maturation delay and diffusion on a toxin producing phytoplankton–zooplankton model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 210(C), pages 547-570.
    4. Thakur, Nilesh Kumar & Ojha, Archana & Tiwari, Pankaj Kumar & Upadhyay, Ranjit Kumar, 2021. "An investigation of delay induced stability transition in nutrient-plankton systems," Chaos, Solitons & Fractals, Elsevier, vol. 142(C).
    5. Shi, Renxiang & Yu, Jiang, 2017. "Hopf bifurcation analysis of two zooplankton-phytoplankton model with two delays," Chaos, Solitons & Fractals, Elsevier, vol. 100(C), pages 62-73.
    6. Li, Kai & Wei, Junjie, 2009. "Stability and Hopf bifurcation analysis of a prey–predator system with two delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(5), pages 2606-2613.
    7. Poria, Swarup & Poria, Anindita Tarai & Chatterjee, Prasanta, 2009. "Synchronization threshold of a coupled n-dimensional time-delay system," Chaos, Solitons & Fractals, Elsevier, vol. 41(3), pages 1123-1124.

    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. Wang, Weiming & Wang, Xiaoqin & Lin, Yezhi, 2008. "Complicated dynamics of a predator–prey system with Watt-type functional response and impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 37(5), pages 1427-1441.
    2. Wang, Weiming & Wang, Hailing & Li, Zhenqing, 2008. "Chaotic behavior of a three-species Beddington-type system with impulsive perturbations," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 438-443.
    3. Li, Dong & Wang, Shilong & Zhang, Xiaohong & Yang, Dan, 2009. "Impulsive control of uncertain Lotka–Volterra predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1572-1577.
    4. Zhao, Min & Lv, Songjuan, 2009. "Chaos in a three-species food chain model with a Beddington–DeAngelis functional response," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2305-2316.
    5. Kim, Hye Kyung & Baek, Hunki, 2013. "The dynamical complexity of a predator–prey system with Hassell–Varley functional response and impulsive effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 94(C), pages 1-14.
    6. Elettreby, M.F., 2009. "Two-prey one-predator model," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2018-2027.
    7. Chen, Yiping & Liu, Zhijun, 2009. "Modelling and analysis of an impulsive SI model with Monod-Haldane functional response," Chaos, Solitons & Fractals, Elsevier, vol. 39(4), pages 1698-1714.
    8. Guo, Hongjian & Chen, Lansun & Song, Xinyu, 2009. "Dynamic analysis of a kind of species control model concerning impulsively releasing pathogen and infective predator," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1326-1336.
    9. Liu, Zhijun & Tan, Ronghua, 2007. "Impulsive harvesting and stocking in a Monod–Haldane functional response predator–prey system," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 454-464.
    10. Wang, Xiaoqin & Wang, Weiming & Lin, Xiaolin, 2008. "Chaotic behavior of a Watt-type predator–prey system with impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 37(3), pages 706-718.
    11. Wang, Xiaoqin & Wang, Weiming & Lin, Xiaolin, 2009. "Dynamics of a two-prey one-predator system with Watt-type functional response and impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 40(5), pages 2392-2404.
    12. Xiang, Zhongyi & Song, Xinyu, 2009. "The dynamical behaviors of a food chain model with impulsive effect and Ivlev functional response," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2282-2293.
    13. Zhao, Qiuyue & Liu, Shutang & Niu, Xinglong, 2019. "Dynamic behavior analysis of a diffusive plankton model with defensive and offensive effects," Chaos, Solitons & Fractals, Elsevier, vol. 129(C), pages 94-102.
    14. Yu, Hengguo & Zhao, Min & Lv, Songjuan & Zhu, Lili, 2009. "Dynamic complexities in a parasitoid-host-parasitoid ecological model," Chaos, Solitons & Fractals, Elsevier, vol. 39(1), pages 39-48.
    15. Wang, Jiang & Chen, Liangquan & Fei, Xianyang, 2007. "Bifurcation control of the Hodgkin–Huxley equations," Chaos, Solitons & Fractals, Elsevier, vol. 33(1), pages 217-224.
    16. Huo, Liang’an & Jiang, Jiehui & Gong, Sixing & He, Bing, 2016. "Dynamical behavior of a rumor transmission model with Holling-type II functional response in emergency event," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 450(C), pages 228-240.
    17. Wang, Xiaoqin & Wang, Weiming & Lin, Xiaolin, 2009. "Dynamics of a periodic Watt-type predator–prey system with impulsive effect," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1270-1282.
    18. Zhong, Shihong & Xia, Juandi & Liu, Biao, 2021. "Spatiotemporal dynamics analysis of a semi-discrete reaction-diffusion Mussel-Algae system with advection," Chaos, Solitons & Fractals, Elsevier, vol. 151(C).
    19. Wang, Weiming & Wang, Hailing & Li, Zhenqing, 2007. "The dynamic complexity of a three-species Beddington-type food chain with impulsive control strategy," Chaos, Solitons & Fractals, Elsevier, vol. 32(5), pages 1772-1785.
    20. Wang, Xiaoqin & Wang, Weiming & Lin, Yezhi & Lin, Xiaolin, 2009. "The dynamical complexity of an impulsive Watt-type prey–predator system," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 731-744.

    More about this item

    Statistics

    Access and download statistics

    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:chsofr:v:39:y:2009:i:3:p:1395-1409. 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: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    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.