Fractional-order differential equations have recently been utilized in phytoplankton-zooplankton models to represent genetics influences. These influences are frequently seen in aquatic ecosystems but cannot be completely explained by standard integer-order models. The focus of this work was to analyze the dynamics of a fractional-order phytoplankton-zooplankton model that incorporates a Holling type-Ⅱ functional response. This model was discretized using the Caputo fractional derivative. We carried out an extensive stability analysis to determine the equilibrium states of the system and establish the requirements under which these points are stable or unstable. We used the bifurcation theory to investigate the development of bifurcations when the parameters of the considered model varied, which showed intricate dynamical features including chaos and oscillations. We analyzed Neimark-Sacker and flip bifurcations using a discretization parameter $ \omega. $ The chaos control was presented. Computational simulations were executed in order to independently confirm the theoretical results and point out the diverse dynamics that the model displays. Our findings provide further clarification on the interactions between phytoplankton and zooplankton species, demonstrating significant environmental implications for marine ecosystems.
Citation: M. B. Almatrafi. Stability and bifurcation of a fractional-order phytoplankton-zooplankton model with Holling type-Ⅱ response[J]. AIMS Mathematics, 2025, 10(11): 25545-25567. doi: 10.3934/math.20251131
Fractional-order differential equations have recently been utilized in phytoplankton-zooplankton models to represent genetics influences. These influences are frequently seen in aquatic ecosystems but cannot be completely explained by standard integer-order models. The focus of this work was to analyze the dynamics of a fractional-order phytoplankton-zooplankton model that incorporates a Holling type-Ⅱ functional response. This model was discretized using the Caputo fractional derivative. We carried out an extensive stability analysis to determine the equilibrium states of the system and establish the requirements under which these points are stable or unstable. We used the bifurcation theory to investigate the development of bifurcations when the parameters of the considered model varied, which showed intricate dynamical features including chaos and oscillations. We analyzed Neimark-Sacker and flip bifurcations using a discretization parameter $ \omega. $ The chaos control was presented. Computational simulations were executed in order to independently confirm the theoretical results and point out the diverse dynamics that the model displays. Our findings provide further clarification on the interactions between phytoplankton and zooplankton species, demonstrating significant environmental implications for marine ecosystems.
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M. B. Almatrafi. Stability and bifurcation of a fractional-order phytoplankton-zooplankton model with Holling type-Ⅱ response[J]. AIMS Mathematics, 2025, 10(11): 25545-25567. doi: 10.3934/math.20251131
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