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Modelling and analysis of a phytoplankton–zooplankton system with continuous and discrete time

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  • S. Kartal
  • M. Kar
  • N. Kartal
  • F. Gurcan

Abstract

In this study, a phytoplankton–zooplankton system has been modelled using a system of differential equations with piecewise constant arguments, which represents a new approach to modelling phytoplankton–zooplankton interaction. To analyse the dynamic behaviour of the model, we consider the solution of the system in a certain subinterval, which yields a system of difference equations. Some theoretical results on the boundedness character and local stability properties for the discrete dynamical system are obtained. In addition, we explain the biological dynamics of the bloom in the plankton model through Neimark–Sacker bifurcation and obtain the threshold values for different parameters that govern the periodic nature of the bloom. We conclude that, while other studies explained that the bloom depended on only one parameter, this study explains that the bloom depended on three different parameters, namely $$\theta $$θ (rate of toxin production per phytoplankton), $$\beta $$β (zooplankton growth efficiency) and $$K$$K (environmental carrying capacity of phytoplankton).

Suggested Citation

  • S. Kartal & M. Kar & N. Kartal & F. Gurcan, 2016. "Modelling and analysis of a phytoplankton–zooplankton system with continuous and discrete time," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 22(6), pages 539-554, November.
    • Handle: RePEc:taf:nmcmxx:v:22:y:2016:i:6:p:539-554
    DOI: 10.1080/13873954.2016.1204323
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    References listed on IDEAS

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    1. Gurcan, Fuat & Kartal, Senol & Ozturk, Ilhan & Bozkurt, Fatma, 2014. "Stability and bifurcation analysis of a mathematical model for tumor–immune interaction with piecewise constant arguments of delay," Chaos, Solitons & Fractals, Elsevier, vol. 68(C), pages 169-179.
    2. Jef Huisman & Franz J. Weissing, 1999. "Biodiversity of plankton by species oscillations and chaos," Nature, Nature, vol. 402(6760), pages 407-410, November.
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    1. Bozkurt, Fatma & Yousef, Ali & Bilgil, Halis & Baleanu, Dumitru, 2023. "A mathematical model with piecewise constant arguments of colorectal cancer with chemo-immunotherapy," Chaos, Solitons & Fractals, Elsevier, vol. 168(C).
    2. Mutasa, Farikayi K. & Jones, Brian & Musekwa-Hove, Senelani D., 2020. "Modelling and analysis of Limnothrissa miodon population in a Lake," Chaos, Solitons & Fractals, Elsevier, vol. 136(C).

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