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.
  
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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>.
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For values of <math>x</math> in the domain of real numbers 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>.
  
 
==Resources==
 
==Resources==
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* [https://openstax.org/books/calculus-volume-2/pages/4-4-the-logistic-equation The Logistic Equation], OpenStax Calculus Volume 2
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Exponential%20growth%20and%20decay%20models/Esparza%201093%20Notes%207.6.pdf Logistic growth and decay models]. Written notes created by Professor Esparza, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Exponential%20growth%20and%20decay%20models/Esparza%201093%20Notes%207.6.pdf Logistic growth and decay models]. Written notes created by Professor Esparza, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Exponential%20growth%20and%20decay%20models/Esparza%201093%20Notes%207.6B.pdf Logistic growth and decay models Continued]. Written notes created by Professor Esparza, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Exponential%20growth%20and%20decay%20models/Esparza%201093%20Notes%207.6B.pdf Logistic growth and decay models Continued]. Written notes created by Professor Esparza, UTSA.
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== Licensing ==
 +
Content obtained and/or adapted from:
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* [https://en.wikipedia.org/wiki/Logistic_function Logistic function, Wikipedia] under a CC BY-SA license

Latest revision as of 17:24, 28 October 2021

Standard logistic sigmoid function where

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 to , the S-curve shown on the right is obtained, with the graph of approaching as approaches and approaching zero as approaches .

Resources

Licensing

Content obtained and/or adapted from:

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