Fractal–Fractional Coupled Systems with Constant and State- Dependent Delays: Existence Theory and Ecological Applications

This study introduces a new class of coupled differential systems described by fractal–fractional Caputo derivatives with both constant and state-dependent delays. In contrast to traditional delay differential equations, the proposed framework integrates memory effects and geometric complexity while capturing adaptive feedback delays that vary with the system’s state. Such a formulation provides a closer representation of biological and physical processes in which delays are not fixed but evolve dynamically. Sufficient conditions for the existence and uniqueness of solutions are established using fixed-point theory, while the stability of the solution is investigated via the Hyers–Ulam (HU) stability approach. To demonstrate applicability, the approach is applied to two illustrative examples, including a predator–prey interaction model. The findings advance the theory of fractional-order systems with mixed delays and offer a rigorous foundation for developing realistic, application-driven dynamical models.