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

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.