Verhulst's logistic curve

Details Verhulst's logistic curve Verhulst's logistic curve

2007-06-21

arxiv.org/abs/0706.3163v1

The College Mathematics Journal, Vol. 32, No. 2, March 2001, pp. 94--98. [MR 1833354] (2002c:26002)

Mathematics

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5 pages AMSLaTeX, 2 figures

Scientific paper

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.

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