Difference between revisions of "Logistic growth and decay models"

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[[File:Logistic-curve.svg|thumb|320px|right|Standard logistic sigmoid function where <math>L=1,k=1,x_0=0</math>]]
 
[[File:Logistic-curve.svg|thumb|320px|right|Standard logistic sigmoid function where <math>L=1,k=1,x_0=0</math>]]
  
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A '''logistic function''' or '''logistic curve''' is a common S-shaped curve ([[sigmoid function|sigmoid curve]]) with equation
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A '''logistic function''' or '''logistic curve''' is a common S-shaped curve sigmoid curve with equation
  
 
: <math>f(x) = \frac{L}{1 + e^{-k(x-x_0)}},</math>
 
: <math>f(x) = \frac{L}{1 + e^{-k(x-x_0)}},</math>
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: <math>x_0</math>, the <math>x</math> value of the sigmoid's midpoint;
 
: <math>x_0</math>, the <math>x</math> value of the sigmoid's midpoint;
 
: <math>L</math>, the curve's maximum value;
 
: <math>L</math>, the curve's maximum value;
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: <math>k</math>, the logistic growth rate or steepness of the curve.<ref name=verhulst1838>{{cite journal |first= Pierre-François |last=Verhulst |year= 1838 |title = Notice sur la loi que la population poursuit dans son accroissement |journal = Correspondance Mathématique et Physique |volume = 10 |pages = 113–121 |url = https://books.google.com/books?id=8GsEAAAAYAAJ |format = PDF |access-date = 3 December 2014}}</ref>
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: <math>k</math>, the logistic growth rate or steepness of the curve.
  
 
For values of <math>x</math> in the domain of [[real number]]s from <math>-\infty</math> to <math>+\infty</math>, the S-curve shown on the right is obtained, with the graph of <math>f</math> approaching <math>L</math> as <math>x</math> approaches <math>+\infty</math> and approaching zero as <math>x</math> approaches <math>-\infty</math>.
 
For values of <math>x</math> in the domain of [[real number]]s from <math>-\infty</math> to <math>+\infty</math>, the S-curve shown on the right is obtained, with the graph of <math>f</math> approaching <math>L</math> as <math>x</math> approaches <math>+\infty</math> and approaching zero as <math>x</math> approaches <math>-\infty</math>.

Revision as of 09:40, 10 October 2021

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

where

, the value of the sigmoid's midpoint;
, the curve's maximum value;
, the logistic growth rate or steepness of the curve.

For values of in the domain of real numbers from 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} to 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} , the S-curve shown on the right is obtained, with the graph of 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} approaching 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} 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

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