Logistic growth and decay models

Standard logistic sigmoid function where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=1,k=1,x_0=0}

A logistic function or logistic curve is a common S-shaped curve sigmoid curve with equation

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \frac{L}{1 + e^{-k(x-x_0)}},}

where

$x_{0}$, the $x$ value of the sigmoid's midpoint;

$L$, the curve's maximum value;

$k$, the logistic growth rate or steepness of the curve.

For values of $x$ in the domain of real numbers from $-\infty$ to $+\infty$, the S-curve shown on the right is obtained, with the graph of $f$ approaching $L$ as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle +\infty} and approaching zero as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\infty} .

Resources

• Logistic Function, Wikipedia
• The Logistic Equation, OpenStax Calculus Volume 2
• Logistic growth and decay models. Written notes created by Professor Esparza, UTSA.
• Logistic growth and decay models Continued. Written notes created by Professor Esparza, UTSA.