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Stability and bifurcation in a two harmful phytoplankton–zooplankton system

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  • 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.

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  • 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
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    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.
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    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. 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.
    4. 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.
    5. 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.
    6. 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.
    7. 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).

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