Mathematical analysis of tri-trophic food webs with carrying capacity and Holling-type predation using fractal-fractional Caputo derivatives

This research investigated a tri-trophic food chain model, incorporating carrying capacity and Holling-type predation, formulated using the fractal-fractional Caputo derivative. The four equilibrium states---trivial, prey-only, prey-predator, and coexistence---presented and their stability was discussed. Existence and uniqueness of solutions were established using Schaefer's and Banach's fixed point theorems. Also, stability requirements in the sense of Hyers-Ulam (H-U) were investigated. Numerical simulations were performed using the extended numerical method of Adams-Bashforth-Moulton (ABM), and comparative results were graphically presented to demonstrate the impact of varying fractal-fractional orders. A sensitivity analysis revealed how perturbations in individual parameters influence the model's outcome. The model accounts for memory and hereditary effects in ecological interactions. The proposed method enhances accuracy, stability, and convergence for long-time simulations compared to classical models.