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>]] | ||
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| + | A '''logistic function''' or '''logistic curve''' is a common S-shaped curve ([[sigmoid function|sigmoid curve]]) with equation | ||
| + | |||
| + | : <math>f(x) = \frac{L}{1 + e^{-k(x-x_0)}},</math> | ||
| + | |||
| + | where | ||
| + | : <math>x_0</math>, the <math>x</math> value of the sigmoid's midpoint; | ||
| + | : <math>L</math>, the curve's maximum value; | ||
| + | : <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> | ||
| + | |||
| + | 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>. | ||
| + | |||
| + | ==Resources== | ||
* [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. | ||
Revision as of 09:39, 10 October 2021
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.[1]
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 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 growth and decay models. Written notes created by Professor Esparza, UTSA.
- Logistic growth and decay models Continued. Written notes created by Professor Esparza, UTSA.
