Verhulst's logistic curve
History and Overview
2007-06-22 v1 Classical Analysis and ODEs
Abstract
We observe that the elementary logistic differential equation dP/dt=(1-P/M)kP may be solved by first changing the variable to R=(M-P)/P. This reduces the logistic differential equation to the simple linear differential equation dR/dt=-kR, which can be solved without using the customary but slightly more elaborate methods applied to the original logistic DE. The resulting solution in terms of R can be converted by simple algebra to the familiar sigmoid expression involving P. A biological argument is given for introducing logistic growth via the simpler DE for R. It is also shown that the sigmoid P may be written in terms of the hyperbolic tangent by a simple translation that is also motivated by a biological argument.
Cite
@article{arxiv.0706.3163,
title = {Verhulst's logistic curve},
author = {David M. Bradley},
journal= {arXiv preprint arXiv:0706.3163},
year = {2007}
}