## 2 Materials and Methods  
### Logistic Function and Its Derivatives  
For the convenience of the reader, we now briefly describe the logistic function and some of its properties, which have been proven mainly in our paper [18]. Equation (1) is a particular case of [PERSON]'s equation with constant coefficients  
\[u^{\prime}(t)=r(u-u_{1})(u-u_{2}). \tag{5}\]  
The constants \(r\
eq 0,\ u_{1},\ u_{2}\) can be real or complex numbers. If \(u(t)\) is a solution of (5) then the \(n\)th derivative \(u^{(n)}(t)\) (\(n=2,3,4,\ldots\)) of \(u(t)\) can be expressed as a polynomial of the function \(u(t)\) itself:  
\[u^{(n)}(t)=r^{n}\sum_{k=0}^{n-1}\genfrac{\langle}{\rangle}{0.0 pt}{}{n}{k}(u-u _{1})^{k+1}(u-u_{2})^{n-k} \tag{6}\]  
where \(n=2,3,\ldots\) and \(\genfrac{\langle}{\rangle}{0.0 pt}{}{n}{k}\) denotes the Eulerian number (number of permutations of the set \(\{1,2,\ldots,n\}\) having \(k\), \((k=0,1,2,\ldots,n-1)\) permutation ascents, see [PERSON] et al. [19]). The first few Eulerian numbers are given in the Table 1.  
Formula (6) was discussed during the Conference ICNAAM 2006 and it appeared, with an inductive proof, in paper [20] (see also [21]). Independently the formula has been considered and proved, with the proof based on generating functions, by [PERSON] [22]. The polynomial of \(u\), of order \((n+1)\), appearing on the right-hand side of (1) is known in the literature as the so-called derivative polynomial. In [23], we proved that all \((n+1)\) zeros of the polynomial are simple and lie in the interval \([u_{1},u_{2}]\). The derivative polynomials have been recently intensively studied. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\mathbf{n}\) & \(\genfrac{\langle}{\rangle}{0.0 pt}{}{n}{0}\) & \(\genfrac{\langle}{\rangle}{0.0 pt}{}{n}{1}\) & \(\genfrac{\langle}{\rangle}{0.0 pt}{}{n}{2}\) & \(\genfrac{\langle}{\rangle}{0.0 pt}{}{3}\) & \(\genfrac{\langle}{\rangle}{0.0 pt}{}{n}{4}\) & \(\genfrac{\langle}{\rangle}{0.0 pt}{}{5}\) & \(\genfrac{\langle}{\rangle}{0.0 pt}{}{n}{6}\) & \(\genfrac{\langle}{\rangle}{0.0 pt}{}{7}\) \\ \hline
2 & 1 & 1 & 0 & & & & \\
3 & 1 & 4 & 1 & 0 & & & \\
4 & 1 & 11 & 11 & 1 & 0 & & \\
5 & 1 & 26 & 66 & 26 & 1 & 0 & \\
6 & 1 & 57 & 302 & 302 & 57 & 1 & 0 & \\
7 & 1 & 120 & 1191 & 2416 & 1191 & 120 & 1 & 0 \\ \hline \end{tabular}
\end{table} 
Table 1: Eulerian numbers. Formula (6) applied to the particular case of the logistic Equation (1) is as follows:  
\[u^{(n)}(t)=\Big{(}-\frac{s}{u_{max}}\Big{)}^{n}\sum_{k=0}^{n-1}\Big{\langle} \begin{matrix}n\\ k\end{matrix} \Big{\rangle}u^{k+1}(u-u_{max})^{n-k}. \tag{7}\]  
It is easily seen that if \(u_{0}\) is a zero of the polynomial on the right-hand side of (7) then \(u_{0}/u_{max}\) is the zero of the polynomial  
\[P_{n+1}(u)=(-1)^{n}\sum_{k=0}^{n-1}\Big{\langle} \begin{matrix}n\\ k\end{matrix} \Big{\rangle}u^{k+1}(u-1)^{n-k}. \tag{8}\]  
The first few derivatives (7) (for \(u_{max}=1\), \(s=1\)) are as follows  
\[u^{\prime}(t)= u(1-u)=-u(u-1)=P_{2}(u),\] \[u^{\prime\prime}(t)= u(u-1)^{2}+u^{2}(u-1)=P_{3}(u),\] \[u^{\prime\prime\prime}(t)= -u(u-1)^{3}-4u^{2}(u-1)^{2}-u^{3}(u-1)=P_{4}(u),\] \[u^{(4)}(t)= u(u-1)^{4}+11u^{2}(u-1)^{3}+11u^{3}(u-1)^{2}+u^{4}(u-1)=P_{5}(u),\] \[u^{(5)}(t)= -u(u-1)^{5}-26u^{2}(u-1)^{4}-66u^{3}(u-1)^{3}-26u^{4}(u-1)^{2}-u^ {5}(u-1)=P_{6}(u).\]  
By direct calculation, one can show that the smallest positive zeros of the polynomials \(P_{4}(u)\), \(P_{5}(u)\), \(P_{6}(u)\) are, respectively, \(0.211\), \(0.0917\), \(0.0413\). From this it follows that if, e.g., \(t_{1}\) is the smallest root of the equation \(u^{\prime\prime\prime}(t)=0\), where \(u(t)\) is given by (2), then \(u(t_{1})=0.211u_{max}\). Simultaneously, at \(t=t_{1}\), \(u^{\prime\prime}(t)\) attains its maximum. Similarly, if \(t_{2}\) and \(t_{3}\) are the smallest roots of the equations \(u^{(4)}(t)=0\) and \(u^{(5)}(t)=0\), respectively, then \(u(t_{2})=0.0917u_{max}\) and \(u(t_{3})=0.0413u_{max}\). One can also calculate distances of points \(t_{1}\), \(t_{2}\), \(t_{3}\) from the inflection point \(t_{0}\).