Fractal Dynamics Exploration and Hybrid Control of a Time-Delayed Fractional-Order Plankton System

The present paper puts forward a time-delayed fractional-order phytoplankton-zooplankton system and systematically investigates its fractal dynamics, stability, bifurcation mechanism, and control strategy. Firstly, the existence, uniqueness, positivity and boundedness of solutions are derived. The equilibrium’s stability is analyzed, and the critical condition for delay-induced Hopf bifurcation is obtained. Subsequently, a nonlinear time-delayed feedback controller is introduced with the objective of suppressing unstable oscillations and bifurcation behaviors. The discussion encompasses the stability and bifurcation of the controlled system, demonstrating the efficacy of the controller in expanding the stable region and constraining fractal-like oscillatory patterns. Finally, the results of the theoretical model are verified by means of numerical simulations. This study contributes to the fractal and fractional dynamics of ecological models and provides a theoretical basis for the stability regulation of plankton systems.