Existence, uniqueness, and Ulam–Hyers stability of periodic solutions for a discrete Nicholson’s blowflies equation with proportional delay

This paper investigates the existence, uniqueness, and stability of periodic solutions for a discrete Nicholson’s blowflies equation with proportional delay. Proportional delay, in which the delay is defined as a fraction of the current time, naturally arises in many discrete dynamical systems, including neural networks and biological population models. The original problem is transformed into an equivalent fixed-point equation by introducing a suitable operator on a Banach space of periodic functions. Sufficient conditions for the existence of N-periodic solutions are established using the Leray–Schauder nonlinear alternative, while uniqueness is obtained through Banach’s fixed-point theorem. In addition, the Ulam–Hyers stability of the periodic solution is investigated, showing that small perturbations in the governing equation lead to only small deviations from the exact solution. To validate the theoretical findings, numerical simulations are carried out using Python. Several representative examples are considered by varying the proportionaldelay parameter, delay structure, birth coefficient, and nonlinear coefficient. The numerical results confirm the analytical predictions and clearly illustrate the influence of these parameters on the existence, stability, and qualitative behavior of the periodic solutions, thereby demonstrating the applicability and effectiveness of the proposed theoretical approach.