An open-ended logistic-based growth function: analytical solutions and the power-law logistic model

An open-ended form of the logistic equation was recently proposed, using a model comprising two differential equations [Thornley, J.H.M., France, J., 2005. An open-ended logistic-based growth function. Ecol. Model. 184, 257–261]. The equations represent the two processes of growth and development, and are coupled. In this note, an analytical solution is developed for constant parameters. The solution can be expressed as a targetted single-differential-equation model, the θ-logistic or power-law logistic model, which is a well-known empirical growth equation in ecology and elsewhere. The analysis may facilitate mechanistic interpretation and application of the power-law logistic model as well as the original open two-differential-equation model.