Using analytical equations to represent nonlinear relationships

Table functions, also referred to as graphical functions, provide a powerful and user-friendly way to represent nonlinear relationships between variables in system dynamics (SD) models. However, in many cases modelers may benefit from using analytical equations to represent nonlinear relationships for model sensitivity testing and also for communicating with researchers in other fields and disciplines. We propose six analytical equations that can be used to represent many of the nonlinear relationships commonly formulated using table functions in SD models. Specifically, this article provides guidance on using the generalized logistic function, the exponential function, the modified exponential function, the quadratic function, the logarithmic function and the power function to replace existing table functions. Importantly, we also present a version of each equation that includes an interior reference point. We demonstrate how to apply these analytical equations in SD models by replacing the table functions in the original World Dynamics model. We also provide a Python script to help implement our recommended procedure for incorporating the six analytical equations into models and a Vensim macro for each analytical equation.