Verhulst's logistic curve

TL;DR

This paper presents a straightforward method to solve Verhulst's logistic differential equation by variable substitution, simplifying the derivation of the logistic growth model and its sigmoid solution.

Contribution

It introduces a simple variable change that reduces the logistic differential equation to a linear form, providing clearer biological motivation and an easier derivation of the sigmoid solution.

Findings

The logistic differential equation can be solved via substitution to a linear differential equation.

The sigmoid solution can be expressed using hyperbolic tangent functions.

Biological arguments support the simplified derivation of logistic growth models.

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.