Dynamics and stability of a fractional-order renewable resource model with generalized Mittag-Leffler kernel and fractal dimension

Fractional-order derivatives have a wide range of applications in real-world problems. In this work, we focus on a renewable resource in biomass management with population effects using a generalized Mittag-Leffler kernel. We modify the system into a time-fractional-order biomass model with population impacts on resource biomass while accounting for the suffocation effect, and analyze its dynamics under the Atangana--Baleanu Caputo (ABC) fractional operator. We establish positivity, boundedness, existence, and uniqueness of the fractional-order model using fixed-point theory. We confirm global stability by means of the fractional derivative of a Lyapunov function and compute Ulam--Hyers stability using the spectral radius of the associated matrix. The proposed model has a unique solution according to the Banach contraction principle. The numerical scheme for the fractal-fractional Atangana--Baleanu Riemann--Liouville (ABR) extension of the ABC model is constructed using a two-step Lagrange polynomial in the generalized form of the Mittag-Leffler kernel. Simulations are carried out at various parameter values, fractional orders, and fractal dimensions. The numerical results reveal that the dynamics vary significantly with fractional order, leading to either convergence to a steady state or oscillatory behavior. The study offers a framework for capturing population-resource interactions and demonstrates how fractional order can inform control of biomass consumption under population pressure. These insights are useful for planning and decision-making regarding the social impact of biomass.