Mathematical modeling of allelopathic stimulatory phytoplankton species using fractal–fractional derivatives

In the current study, we employ the novel fractal–fractional operator in the Atangana–Baleanu sense to investigate the dynamics of an interacting phytoplankton species model. Initially, we utilize the Picard-Lindelöf theorem to validate the uniqueness and existence of solutions for the model. We then explore equilibrium points within the phytoplankton model and conduct Hyers–Ulam stability analysis. Additionally, we present a numerical scheme utilizing the Newton polynomial to validate our analytical findings. Numerical simulations illustrate the dynamical behavior of the model across various fractal and fractional parameter values, visualized through graphical representations. Our simulations reveal that the stability of equilibrium points is not significantly impacted with the long-term memory effect, which is characterized by fractal–fractional order values. However, an increase in fractal–fractional parameters accelerates the convergence of solutions to their intended equilibrium states.