Triply Coupled Systems of Differential Equations with Time-Dependent Delay and Application to Three-Species Food-Chain Dynamics

We introduce a class of triply coupled systems of differential equations with fractal–fractional Caputo derivatives and time-dependent delays. This framework captures long-memory effects and complex structural patterns while allowing delays to evolve over time, offering greater realism than constant-delay models. The existence and uniqueness of solutions are established using fixed point theory, and Hyers–Ulam stability is analyzed. A numerical scheme based on the Adams–Bashforth method is implemented to approximate solutions. The approach is illustrated through a numerical example and applied to a three-species food-chain model, comparing scenarios with and without time-dependent delays to demonstrate their impact on system dynamics.