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

Abstract

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.

Keywords:

• Differential equation models
• regulation
• external stimulus
• steady state
• dynamical systems

Notes

1 The symbols ω and ζ are commonly used in physics to denote frequency and damping ratio. Please note that this set of symbols is different with many other articles in the field (e.g., Boker & Graham, 1998), where the frequency parameter is denoted by η which equals our –ω², and the damping parameter is denote by ζ which equals our –2ωζ.