The Critical Point of a Sigmoidal Curve: the Generalized Logistic Equation Example

TL;DR

This paper investigates the critical point of sigmoidal curves, proving conditions under which this point is at zero and analyzing the behavior of generalized logistic functions through Fourier transforms.

Contribution

It provides new theoretical results on the location of the critical point for sigmoidal curves, including generalized logistic functions, based on properties of their derivatives and Fourier transforms.

Findings

The critical point is at zero when the derivative's magnitude is monotone and the function is even.

For non-even functions, the critical point remains at zero under certain Fourier phase conditions.

Fourier transforms of generalized logistic functions are computed and used to illustrate the theory.

Abstract

Let $y(t)$ be a smooth sigmoidal curve, $y^{(n)}(t)$ be its $n$th derivative, $\{t_{m,i}\}$ and $\{t_{a,i}\}$, $i=1,2,\dots$ be the set of points where respectively the derivatives of odd and even order reach their extreme values. The "critical point of the sigmoidal curve" is defined to be the common limit of the sequences $\{t_{m,i}\}$ and $\{t_{a,i}\}$, provided that the limit exists. We prove that if $f(t)=\frac{dy}{dt}$ is an even function such that the magnitude of the analytic representation $|f_A(t)|=|f(t)+if_h(t)|$, where $f_h(t)$ is the Hilbert transform of $f(t)$, is monotone on $(0,\infty)$, then the point $t=0$ is the critical point in the sense above. For the general case, where $f(t)=\frac{dy}{dx}$ is not even, we prove that if $|f_A(t)|$ monotone on $(0,\infty)$ and if the phase of its Fourier transform $F(\omega)$ has a limit as $\omega\to \pm\infty$, then $t=0$ is still the critical point but as opposed to the previous case, the maximum of $f(t)$ is located away from $t=0$. We compute the Fourier transform of the generalized logistic growth functions and illustrate the notions above on these examples.