Micron Document
<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>Generalised logistic function</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Generalised_logistic_function"> <link href="./mw/ext.cite.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Generalised_logistic_function rootpage-Generalised_logistic_function skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Generalised logistic function</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">







<p>The <b>generalized logistic function</b> or <b>curve</b> is an extension of the <a href="Logistic_function" title="Logistic function">logistic</a> or <a href="Sigmoid_function" title="Sigmoid function">sigmoid</a> functions. Originally developed for growth modelling, it allows for more flexible S-shaped curves. The function is sometimes named <b>Richards's curve</b> after <a href="Francis_John_Richards" title="Francis John Richards">F.<span class="nowrap">&nbsp;</span>J.<span class="nowrap">&nbsp;</span>Richards</a>, who proposed the general form for the family of models in 1959.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>Richards's curve has the following form:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(t)=A+{K-A \over (C+Qe^{-Bt})^{1/\nu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>K</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>C</mi>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mi>t</mi>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y(t)=A+{K-A \over (C+Qe^{-Bt})^{1/\nu }}}</annotation>
</semantics>
</math></span><img src="./25076dc2b208b66357ea02b249c2da195d352fcc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:28.377ex; height:6.343ex;" alt="{\displaystyle Y(t)=A+{K-A \over (C+Qe^{-Bt})^{1/\nu }}}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> = weight, height, size etc., and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> = time. It has six parameters:
</p>
<ul><li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A}</annotation>
</semantics>
</math></span><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"></span>: the left horizontal asymptote;</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
</semantics>
</math></span><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span>: the right horizontal asymptote when <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C=1}</annotation>
</semantics>
</math></span><img src="./605a443c472808db9a502a801b8a2dfa5ee15d08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C=1}" loading="lazy"></span>. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A=0}</annotation>
</semantics>
</math></span><img src="./75c34024483e6fb7c89e45aff3882ebf11d95a00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.004ex; height:2.176ex;" alt="{\displaystyle A=0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C=1}</annotation>
</semantics>
</math></span><img src="./605a443c472808db9a502a801b8a2dfa5ee15d08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C=1}" loading="lazy"></span> then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K}</annotation>
</semantics>
</math></span><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"></span> is called the <a href="Carrying_capacity" title="Carrying capacity">carrying capacity</a>;</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>B</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle B}</annotation>
</semantics>
</math></span><img src="./47136aad860d145f75f3eed3022df827cee94d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}" loading="lazy"></span>: the growth rate;</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ν<!-- ν --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nu &gt;0}</annotation>
</semantics>
</math></span><img src="./bb9a5b220856a9047b20d89d6c17f7724232f230.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu >0}" loading="lazy"></span>&nbsp;: affects near which asymptote maximum growth occurs.</li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span>: is related to the value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(0)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo stretchy="false">(</mo>
<mn>0</mn>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y(0)}</annotation>
</semantics>
</math></span><img src="./fbafe23a79d3d97c9d84e9e25fb6c25c252692ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.745ex; height:2.843ex;" alt="{\displaystyle Y(0)}" loading="lazy"></span></li>
<li><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span>: typically takes a value of 1. Otherwise, the upper asymptote is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A+{K-A \over C^{\,1/\nu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>A</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>K</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
</mrow>
<msup>
<mi>C</mi>
<mrow class="MJX-TeXAtom-ORD">
<mspace width="thinmathspace"></mspace>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle A+{K-A \over C^{\,1/\nu }}}</annotation>
</semantics>
</math></span><img src="./fb0f8a4766956e80e68b85545f45a34040791f09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.069ex; height:5.843ex;" alt="{\displaystyle A+{K-A \over C^{\,1/\nu }}}" loading="lazy"></span></li></ul>
<p>The equation can also be written:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(t)=A+{K-A \over (C+e^{-B(t-M)})^{1/\nu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>K</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>C</mi>
<mo>+</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y(t)=A+{K-A \over (C+e^{-B(t-M)})^{1/\nu }}}</annotation>
</semantics>
</math></span><img src="./16bacf0222a8f94ebb774326f1744bd80bc87f2d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:30.824ex; height:6.343ex;" alt="{\displaystyle Y(t)=A+{K-A \over (C+e^{-B(t-M)})^{1/\nu }}}" loading="lazy"></span></dd></dl>
<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> can be thought of as a starting time, at which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(M)=A+{K-A \over (C+1)^{1/\nu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo stretchy="false">(</mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>K</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>C</mi>
<mo>+</mo>
<mn>1</mn>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y(M)=A+{K-A \over (C+1)^{1/\nu }}}</annotation>
</semantics>
</math></span><img src="./ee05ff5090a2d165be3a653086b2681580c77663.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:24.869ex; height:6.343ex;" alt="{\displaystyle Y(M)=A+{K-A \over (C+1)^{1/\nu }}}" loading="lazy"></span>. Including both <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q}</annotation>
</semantics>
</math></span><img src="./8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span> can be convenient:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(t)=A+{K-A \over (C+Qe^{-B(t-M)})^{1/\nu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>A</mi>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>K</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>C</mi>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y(t)=A+{K-A \over (C+Qe^{-B(t-M)})^{1/\nu }}}</annotation>
</semantics>
</math></span><img src="./41a044f0cf1511af56e8c829f8cf539d316eb68b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:32.662ex; height:6.343ex;" alt="{\displaystyle Y(t)=A+{K-A \over (C+Qe^{-B(t-M)})^{1/\nu }}}" loading="lazy"></span></dd></dl>
<p>this representation simplifies the setting of both a starting time and the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>
</semantics>
</math></span><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"></span> at that time.
</p><p>The <a href="Logistic_function" title="Logistic function">logistic function</a>, with maximum growth rate at time <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>M</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle M}</annotation>
</semantics>
</math></span><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"></span>, is the case where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\nu =1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo>=</mo>
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q=\nu =1}</annotation>
</semantics>
</math></span><img src="./ef64fb6474ebbd10d126eee9a19bc2d2f8122be2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.43ex; height:2.509ex;" alt="{\displaystyle Q=\nu =1}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Generalised_logistic_differential_equation">Generalised logistic differential equation</h2></div>
<p>A particular case of the generalised logistic function is:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(t)={K \over (1+Qe^{-\alpha \nu (t-t_{0})})^{1/\nu }}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>K</mi>
<mrow>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>α<!-- α --></mi>
<mi>ν<!-- ν --></mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y(t)={K \over (1+Qe^{-\alpha \nu (t-t_{0})})^{1/\nu }}}</annotation>
</semantics>
</math></span><img src="./3439980d5eadfa8e868b0df4633a17643b6b859d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:27.849ex; height:6.176ex;" alt="{\displaystyle Y(t)={K \over (1+Qe^{-\alpha \nu (t-t_{0})})^{1/\nu }}}" loading="lazy"></span></dd></dl>
<p>which is the solution of the Richards's differential equation (RDE):
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y^{\prime }(t)=\alpha \left(1-\left({\frac {Y}{K}}\right)^{\nu }\right)Y}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>α<!-- α --></mi>
<mrow>
<mo>(</mo>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>Y</mi>
<mi>K</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mrow>
<mo>)</mo>
</mrow>
<mi>Y</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y^{\prime }(t)=\alpha \left(1-\left({\frac {Y}{K}}\right)^{\nu }\right)Y}</annotation>
</semantics>
</math></span><img src="./470edcbb5fc3377eee070a5c2c52b68fc22161a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:27.218ex; height:6.176ex;" alt="{\displaystyle Y^{\prime }(t)=\alpha \left(1-\left({\frac {Y}{K}}\right)^{\nu }\right)Y}" loading="lazy"></span></dd></dl>
<p>with initial condition
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y(t_{0})=Y_{0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Y</mi>
<mo stretchy="false">(</mo>
<msub>
<mi>t</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
<mo stretchy="false">)</mo>
<mo>=</mo>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y(t_{0})=Y_{0}}</annotation>
</semantics>
</math></span><img src="./65008feb81c6ec722bfb3500823717265aeb55f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.98ex; height:2.843ex;" alt="{\displaystyle Y(t_{0})=Y_{0}}" loading="lazy"></span></dd></dl>
<p>where
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=-1+\left({\frac {K}{Y_{0}}}\right)^{\nu }}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Q</mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>+</mo>
<msup>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>K</mi>
<msub>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>0</mn>
</mrow>
</msub>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Q=-1+\left({\frac {K}{Y_{0}}}\right)^{\nu }}</annotation>
</semantics>
</math></span><img src="./520a094fd1739caa55b06a80b0e3a0dc559d1b4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.513ex; height:6.176ex;" alt="{\displaystyle Q=-1+\left({\frac {K}{Y_{0}}}\right)^{\nu }}" loading="lazy"></span></dd></dl>
<p>provided that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ν<!-- ν --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nu &gt;0}</annotation>
</semantics>
</math></span><img src="./bb9a5b220856a9047b20d89d6c17f7724232f230.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu >0}" loading="lazy"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha >0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>&gt;</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha &gt;0}</annotation>
</semantics>
</math></span><img src="./edd4f784b6e8bb68fa774213ceacbab2d97825dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}" loading="lazy"></span>
</p><p>The classical logistic differential equation is a particular case of the above equation, with <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu =1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ν<!-- ν --></mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nu =1}</annotation>
</semantics>
</math></span><img src="./c2ca7fd51cd02b2c4311446c6d6b0f53b2c529cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.493ex; height:2.176ex;" alt="{\displaystyle \nu =1}" loading="lazy"></span>, whereas the <a href="Gompertz_curve" class="mw-redirect" title="Gompertz curve">Gompertz curve</a> can be recovered in the limit <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu \rightarrow 0^{+}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ν<!-- ν --></mi>
<mo stretchy="false">→<!-- → --></mo>
<msup>
<mn>0</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>+</mo>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nu \rightarrow 0^{+}}</annotation>
</semantics>
</math></span><img src="./7e4f2fe26d1f87523547fb47ef6da1390e2f8b41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.519ex; height:2.509ex;" alt="{\displaystyle \nu \rightarrow 0^{+}}" loading="lazy"></span> provided that:
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha =O\left({\frac {1}{\nu }}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>α<!-- α --></mi>
<mo>=</mo>
<mi>O</mi>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>ν<!-- ν --></mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \alpha =O\left({\frac {1}{\nu }}\right)}</annotation>
</semantics>
</math></span><img src="./964ec905fbb336870daf03128c7f8b9ce0124707.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.236ex; height:6.176ex;" alt="{\displaystyle \alpha =O\left({\frac {1}{\nu }}\right)}" loading="lazy"></span></dd></dl>
<p>In fact, for small <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ν<!-- ν --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \nu }</annotation>
</semantics>
</math></span><img src="./c15bbbb971240cf328aba572178f091684585468.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.232ex; height:1.676ex;" alt="{\displaystyle \nu }" loading="lazy"></span> it is
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y^{\prime }(t)=Yr{\frac {1-\exp \left(\nu \ln \left({\frac {Y}{K}}\right)\right)}{\nu }}\approx rY\ln \left({\frac {Y}{K}}\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>Y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi class="MJX-variant" mathvariant="normal">′<!-- ′ --></mi>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi>Y</mi>
<mi>r</mi>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mi>exp</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow>
<mi>ν<!-- ν --></mi>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>Y</mi>
<mi>K</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mi>ν<!-- ν --></mi>
</mfrac>
</mrow>
<mo>≈<!-- ≈ --></mo>
<mi>r</mi>
<mi>Y</mi>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>Y</mi>
<mi>K</mi>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle Y^{\prime }(t)=Yr{\frac {1-\exp \left(\nu \ln \left({\frac {Y}{K}}\right)\right)}{\nu }}\approx rY\ln \left({\frac {Y}{K}}\right)}</annotation>
</semantics>
</math></span><img src="./cafdfaa7db25665481ad599cce358dc22e6ec1fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.522ex; height:8.176ex;" alt="{\displaystyle Y^{\prime }(t)=Yr{\frac {1-\exp \left(\nu \ln \left({\frac {Y}{K}}\right)\right)}{\nu }}\approx rY\ln \left({\frac {Y}{K}}\right)}" loading="lazy"></span></dd></dl>
<p>The RDE models many growth phenomena, arising in fields such as oncology and epidemiology.
</p>
<div class="mw-heading mw-heading2"><h2 id="Gradient_of_generalized_logistic_function">Gradient of generalized logistic function</h2></div>
<p>When estimating parameters from data, it is often necessary to compute the partial derivatives of the logistic function with respect to parameters at a given data point <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>t</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle t}</annotation>
</semantics>
</math></span><img src="./65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"></span> (see<sup id="cite_ref-fekedulegn1999parameter_1-0" class="reference"><a href="#cite_note-fekedulegn1999parameter-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>). For the case where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C=1}</annotation>
</semantics>
</math></span><img src="./605a443c472808db9a502a801b8a2dfa5ee15d08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.027ex; height:2.176ex;" alt="{\displaystyle C=1}" loading="lazy"></span>,
</p>
<dl><dd><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\\{\frac {\partial Y}{\partial A}}&amp;=1-(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial K}}&amp;=(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial B}}&amp;={\frac {(K-A)(t-M)Qe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial \nu }}&amp;={\frac {(K-A)\ln(1+Qe^{-B(t-M)})}{\nu ^{2}(1+Qe^{-B(t-M)})^{\frac {1}{\nu }}}}\\\\{\frac {\partial Y}{\partial Q}}&amp;=-{\frac {(K-A)e^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial M}}&amp;=-{\frac {(K-A)QBe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd></mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Y</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>A</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mn>1</mn>
<mo>−<!-- − --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Y</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>K</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mo>/</mo>
</mrow>
<mi>ν<!-- ν --></mi>
</mrow>
</msup>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Y</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>B</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>K</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow>
<mi>ν<!-- ν --></mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>ν<!-- ν --></mi>
</mfrac>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Y</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>ν<!-- ν --></mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>K</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mi>ln</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<mo stretchy="false">)</mo>
</mrow>
<mrow>
<msup>
<mi>ν<!-- ν --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mn>2</mn>
</mrow>
</msup>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>ν<!-- ν --></mi>
</mfrac>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Y</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Q</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>K</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow>
<mi>ν<!-- ν --></mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>ν<!-- ν --></mi>
</mfrac>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
</mtr>
<mtr>
<mtd>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>Y</mi>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>M</mi>
</mrow>
</mfrac>
</mrow>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>K</mi>
<mo>−<!-- − --></mo>
<mi>A</mi>
<mo stretchy="false">)</mo>
<mi>Q</mi>
<mi>B</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
</mrow>
<mrow>
<mi>ν<!-- ν --></mi>
<mo stretchy="false">(</mo>
<mn>1</mn>
<mo>+</mo>
<mi>Q</mi>
<msup>
<mi>e</mi>
<mrow class="MJX-TeXAtom-ORD">
<mo>−<!-- − --></mo>
<mi>B</mi>
<mo stretchy="false">(</mo>
<mi>t</mi>
<mo>−<!-- − --></mo>
<mi>M</mi>
<mo stretchy="false">)</mo>
</mrow>
</msup>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mi>ν<!-- ν --></mi>
</mfrac>
</mrow>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
</mrow>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\\{\frac {\partial Y}{\partial A}}&amp;=1-(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial K}}&amp;=(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial B}}&amp;={\frac {(K-A)(t-M)Qe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial \nu }}&amp;={\frac {(K-A)\ln(1+Qe^{-B(t-M)})}{\nu ^{2}(1+Qe^{-B(t-M)})^{\frac {1}{\nu }}}}\\\\{\frac {\partial Y}{\partial Q}}&amp;=-{\frac {(K-A)e^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial M}}&amp;=-{\frac {(K-A)QBe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\end{aligned}}}</annotation>
</semantics>
</math></span><img src="./0f335ca231f791095a64730f765b345037b0f8a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -29.481ex; margin-bottom: -0.19ex; width:36.438ex; height:60.509ex;" alt="{\displaystyle {\begin{aligned}\\{\frac {\partial Y}{\partial A}}&amp;=1-(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial K}}&amp;=(1+Qe^{-B(t-M)})^{-1/\nu }\\\\{\frac {\partial Y}{\partial B}}&amp;={\frac {(K-A)(t-M)Qe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial \nu }}&amp;={\frac {(K-A)\ln(1+Qe^{-B(t-M)})}{\nu ^{2}(1+Qe^{-B(t-M)})^{\frac {1}{\nu }}}}\\\\{\frac {\partial Y}{\partial Q}}&amp;=-{\frac {(K-A)e^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\\{\frac {\partial Y}{\partial M}}&amp;=-{\frac {(K-A)QBe^{-B(t-M)}}{\nu (1+Qe^{-B(t-M)})^{{\frac {1}{\nu }}+1}}}\\\end{aligned}}}" loading="lazy"></span></dd></dl>
<p><br>
</p>
<div class="mw-heading mw-heading2"><h2 id="Special_cases">Special cases</h2></div>
<p>The following functions are specific cases of Richards's curves:
</p>
<ul><li><a href="Logistic_function" title="Logistic function">Logistic function</a></li>
<li><a href="Gompertz_curve" class="mw-redirect" title="Gompertz curve">Gompertz curve</a></li>
<li><a href="Von_Bertalanffy_function" title="Von Bertalanffy function">Von Bertalanffy function</a></li>
<li>Monomolecular curve</li></ul>
<div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
/* start https://en.wikipedia.org/ */


.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}


/* end https://en.wikipedia.org/ */
</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-fekedulegn1999parameter-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-fekedulegn1999parameter_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */


.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}


/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFFekedulegnMairitin_P._Mac_SiurtainJim_J._Colbert1999" class="citation journal cs1">Fekedulegn, Desta; Mairitin P. Mac Siurtain; Jim J. Colbert (1999). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110929005929/http://www.metla.fi/silvafennica/full/sf33/sf334327.pdf">"Parameter Estimation of Nonlinear Growth Models in Forestry"</a> <span class="cs1-format">(PDF)</span>. <i>Silva Fennica</i>. <b>33</b> (4): <span class="nowrap">327–</span>336. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.14214%2Fsf.653">10.14214/sf.653</a>. Archived from <a rel="nofollow" class="external text" href="http://www.metla.fi/silvafennica/full/sf33/sf334327.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2011-09-29<span class="reference-accessdate">. Retrieved <span class="nowrap">2011-05-31</span></span>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li><cite id="CITEREFRichards1959" class="citation journal cs1">Richards, F. J. (1959). "A Flexible Growth Function for Empirical Use". <i><a href="Journal_of_Experimental_Botany" title="Journal of Experimental Botany">Journal of Experimental Botany</a></i>. <b>10</b> (2): <span class="nowrap">290–</span>300. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fjxb%2F10.2.290">10.1093/jxb/10.2.290</a>.</cite></li>
<li><cite id="CITEREFPellaTomlinson1969" class="citation journal cs1">Pella, J. S.; Tomlinson, P. K. (1969). "A Generalised Stock-Production Model". <i>Bull. Inter-Am. Trop. Tuna Comm</i>. <b>13</b>: <span class="nowrap">421–</span>496.</cite></li>
<li><cite id="CITEREFLeiZhang2004" class="citation journal cs1">Lei, Y. C.; Zhang, S. Y. (2004). <a rel="nofollow" class="external text" href="https://doi.org/10.15388%2FNA.2004.9.1.15171">"Features and Partial Derivatives of Bertalanffy–Richards Growth Model in Forestry"</a>. <i>Nonlinear Analysis: Modelling and Control</i>. <b>9</b> (1): <span class="nowrap">65–</span>73. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.15388%2FNA.2004.9.1.15171">10.15388/NA.2004.9.1.15171</a></span>.</cite></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2025-06-09" href="https://en.wikipedia.org/wiki/?title=Generalised_logistic_function&amp;oldid=1294792726">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>

</body></html>