Analysis of variable-order fractional enzyme kinetics model with time delay

In enzymatic reactions, studying reaction rates and mechanisms helps us understand how concentration, temperature, and catalysts influence the speed of chemical transformations. This field is critical for optimizing processes in biotechnology, pharmaceuticals, and food industries. Traditional enzyme kinetics models may overlook the influence of past system states. In this paper, we propose a variable-order Caputo fractional derivative enzyme kinetics model that incorporates constant time delays to capture memory effects and nonlocal behavior more accurately. We establish the existence and uniqueness of solutions using fixed-point theory. The proposed model stability is analyzed through Ulam-Hyers and generalized Ulam-Hyers concepts. A robust and an effective numerical approach is employed to reveal the intricate dynamics of the model and demonstrate the significance of the variable-order Caputo fractional derivative with time delay. Incorporating a delay term and employing the variable-order Caputo fractional derivative, this model refines conventional enzyme kinetics, leading to a more precise characterization of biological catalytic processes.