Analysis of Impulsive and Proportional Delay Problems: Theory and Application to a Housefly Population Model

This paper addresses a class of multi-point initial value problems with impulses and proportional delays. The framework is based on the Atangana–Baleanu–Caputo (ABC) fractional derivative, which allows the model to incorporate hereditary memory effects absent in standard integer-order systems. Using suitable fixed-point arguments, conditions ensuring the existence and uniqueness of solutions are derived. The reliability and robustness of the obtained solutions is analyzed through the Hyers–Ulam (H-U) method and generalized H-U stability. To demonstrate the theoretical findings, a general numerical example model and a fractional-order housefly population model are considered that incorporate impulsive effects and delay terms reflecting real ecological feedback. Numerical simulations illustrate how variations in the fractional order and impulse intensity influence the dynamic behavior of the adult population. The results reveal that impulsive interventions can effectively regulate population oscillations, while the fractional order governs the rate of decay and long-term stability of the system.