Fitting the logistic function
The logistic function is widely used in many disciplines to study the asymptotic growth behaviour of various populations. The multiscience interest in the logistic function is due to the fact that with it one can describe growth following or approximating a cumulative normal distribution.
Despite its positive qualities, the logistic function has proved difficult to apply to empirical data in the case with three unknown parameters, one parameter being the asymptotic maximum size of the population. In geographical research the maximum size of the population is usually known, leaving only two parameters to be estimated. This can easily be done either by the linear regression formula ‑ a method generally recommended in the geograpical literature ‑ or by the method presented by Croxton & al.
In this paper it is shown that both of these simple methods for fitting the logistic function produce inaccurate estimates of the parameters. The use of these simple methods should be abandoned and in the future the logistic function should be applied to empirical data with some iterative optimization algorithm, like the Gauss‑Newton algorithm applied in this paper. This would ensure the compatibility and reliability of the estimation results.