Generalized Two-Parameter Equation of Growth

We considered the general growth equation y′ = ay^α + by^β that encompasses the well-known growth equations by Bertalanffy and Verhulst-Pearl (logistic) as special cases. The solutions of this equation for some values of α and/or β can be found in the literature. Here we present the general solution for any real α and β and discuss its behavior. We found that the generalized Gompertz growth equation y′ = ay^α + by^α ln y and its solution could be considered a limiting case of the general growth equation and its solution, respectively. Thus, all three classical growth models (Bertalanffy, logistic, and Gompertz) are nested within the general model. This nesting relationship makes it possible to compare statistically the applicability of models to data.