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Effects of toxin delay on the dynamics of a phytoplankton–zooplankton model

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  • Zhang, Jia-Fang
  • Wang, Shaoli
  • Kong, Xiangjun

Abstract

This paper is concerned with a phytoplankton–zooplankton model with toxin delay. It is shown that the positive equilibrium of the system is stable when the toxin delay is not included. However, we find that the incorporation of a discrete toxin delay not only can destabilize the positive equilibrium of the system but also can cause a Hopf bifurcation at the positive equilibrium as it crosses some critical values. In particular, the normal form of the Hopf bifurcation arising in the system is determined to investigate the direction and the stability of periodic solutions bifurcating from these Hopf bifurcations. To verify the obtained conditions, a special numerical example is also included. These results in this work demonstrate that the toxin delay plays an important role on deriving the complex dynamics.

Suggested Citation

  • Zhang, Jia-Fang & Wang, Shaoli & Kong, Xiangjun, 2018. "Effects of toxin delay on the dynamics of a phytoplankton–zooplankton model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 1150-1162.
    • Handle: RePEc:eee:phsmap:v:505:y:2018:i:c:p:1150-1162
    DOI: 10.1016/j.physa.2018.04.049
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    References listed on IDEAS

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    1. Das, Kalyan & Ray, Santanu, 2008. "Effect of delay on nutrient cycling in phytoplankton–zooplankton interactions in estuarine system," Ecological Modelling, Elsevier, vol. 215(1), pages 69-76.
    2. Roy, Shovonlal, 2009. "The coevolution of two phytoplankton species on a single resource: Allelopathy as a pseudo-mixotrophy," Theoretical Population Biology, Elsevier, vol. 75(1), pages 68-75.
    3. Shi, Hong-Bo & Li, Yan, 2015. "Global asymptotic stability of a diffusive predator–prey model with ratio-dependent functional response," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 71-77.
    4. Chakraborty, Subhendu & Roy, Shovonlal & Chattopadhyay, J., 2008. "Nutrient-limited toxin production and the dynamics of two phytoplankton in culture media: A mathematical model," Ecological Modelling, Elsevier, vol. 213(2), pages 191-201.
    5. Huo, Hai-Feng & Yang, Peng & Xiang, Hong, 2018. "Stability and bifurcation for an SEIS epidemic model with the impact of media," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 490(C), pages 702-720.
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    Cited by:

    1. Gökçe, Aytül & Yazar, Samire & Sekerci, Yadigar, 2020. "Delay induced nonlinear dynamics of oxygen-plankton interactions," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    2. Kejun Zhuang & Gao Jia & Dezhi Liu, 2019. "Stability and Hopf Bifurcation in a Three-Component Planktonic Model with Spatial Diffusion and Time Delay," Complexity, Hindawi, vol. 2019, pages 1-17, July.
    3. Chen, Zhewen & Tian, Zhuyan & Zhang, Shuwen & Wei, Chunjin, 2020. "The stationary distribution and ergodicity of a stochastic phytoplankton–zooplankton model with toxin-producing phytoplankton under regime switching," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 537(C).

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