A new approach to modelling sigmoidal curves

Growth and diffusion phenomena have become of great interest to investigators in many disciplines, such as Biology, Demography, Economy, Agriculture, etc. These processes are generally analyzed by means of growth curves. As, in nature, it is not possible for any variable to continue growing indefinitely, we can consider any growth process to have an upper limit or saturation level. Thus, should a model represent a growth phenomenon, it will be described by a sigmoidal or S-shaped curve. There are a wide variety of growth models in general and specific literature. Of these, the logistic model is without doubt one of the most studied in practice, as well as some modifications of it, including recent investigations directed to the decomposition of a growth curve into various logistic components [Technol. Forecast. Soc. Change 47 (1994) 89; Technol. Forecast. Soc. Change 61 (1999) 247.]. In all the cases above, the adopted approach includes fitting the trend curve to the data by means of a well-known estimation procedure, such as least squares. We suggest a somewhat different approach, which consists of expressing the model through its differential equation and searching for a functional specification for the variable representing growth rate. Two series have been chosen from the recent literature in order to illustrate the methodology presented.