Dynamic Regulation Responding to an External Stimulus: A Differential Equation Model

This study examines the dynamic regulation process responding to an external stimulus. The damped oscillator model has been used to describe this process. However, the model does not allow a nonzero steady state, even though the oscillations may continue and do not necessarily damp toward zero. This study introduces the driven damped oscillator model which has an additional parameter to identify different patterns of the steady state. Three methods, generalized local linear approximation, continuous time structural equation modeling, and analytic solutions of differential equations are provided to estimate model parameters. A simulation study indicates that parameters in the driven damped oscillator model are well recovered. The model is then illustrated using a data set on the daily reports of sales after a sale promotion. Potential applications and possible expansions of this model are also discussed.