View source for Logistic function

{{For|the recurrence relation|Logistic map}}
{{short description|S-shaped curve}}
[[File:Logistic-curve.svg|thumb|320px|right|Standard logistic sigmoid function i.e. <math>L=1,k=1,x_0=0</math>]]

A '''logistic function''' or '''logistic curve''' is a common "S" shape ([[sigmoid function|sigmoid curve]]), with equation:

:<math>f(x) = \frac{L}{1 + e^{-k(x-x_0)}} </math>

where
*<math>e</math> = the [[natural logarithm]] base (also known as [[e (mathematical constant)|Euler's number]]),
*<math>x_0</math> = the <math>x</math>-value of the sigmoid's midpoint,
*<math>L</math> = the curve's maximum value, and
*<math>k</math> = the logistic growth rate or steepness of the curve.<ref>{{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/?id=8GsEAAAAYAAJ
| format = PDF| accessdate = 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>.

The logistic function finds applications in a range of fields, including [[artificial neural network]]s, [[biology]] (especially [[ecology]]), [[biomathematics]], [[chemistry]], [[demography]], [[economics]], [[geoscience]], [[mathematical psychology]], [[probability]], [[sociology]], [[political science]], [[linguistics]], and [[statistics]].

== History ==
[[File:Courbe_logistique,_Verhulst,_1845.png|thumb|upright|300px|Original image of a logistic curve, contrasted with a logarithmic curve]]

The logistic function was introduced in a series of three papers by [[Pierre François Verhulst]] between 1838 and 1847, who devised it as a model of [[population growth]] by adjusting the [[exponential growth]] model, under the guidance of [[Adolphe Quetelet]].{{sfn|Cramer|2002|pp=3–5}} Verhulst first devised the function in the mid 1830s, publishing a brief note in 1838,{{sfn|Verhulst|1838}} then presented an expanded analysis and named the function in 1844 (published 1845);{{efn|1=The paper was presented in 1844, and published in 1845: "(Lu à la séance du 30 novembre 1844)." "(Read at the session of 30 November 1844).", p. [https://gdz.sub.uni-goettingen.de/id/PPN129323640_0018?tify={%22pages%22:%5B14%5D,%22view%22:%22info%22} 1].}}<ref>{{cite journal|first= Pierre-François |last=Verhulst |year= 1845| title = Recherches mathématiques sur la loi d'accroissement de la population | journal = Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles |volume = 18 | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN129323640_0018&DMDID=dmdlog7| accessdate = 2013-02-18|trans-title= Mathematical Researches into the Law of Population Growth Increase |page=[https://gdz.sub.uni-goettingen.de/id/PPN129323640_0018?tify={%22pages%22:%5B21%5D,%22view%22:%22info%22} 8] |quote=Nous donnerons le nom de ''logistique'' à la courbe [We will give the name ''logistic'' to the curve]}}</ref> the third paper adjusted the correction term in his model of Belgian population growth.<ref>{{cite journal|first= Pierre-François |last=Verhulst |year= 1847| title = Deuxième mémoire sur la loi d'accroissement de la population | journal = Mémoires de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique |volume = 20| pages = 1–32 | url = http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN129323659_0020&DMDID=dmdlog29| accessdate = 2013-02-18}}</ref>

The initial stage of growth is approximately exponential (geometric); then, as saturation begins, the growth slows to linear (arithmetic), and at maturity, growth stops. Verhulst did not explain the choice of the term "logistic" ({{lang-fr|logistique}}), but it is presumably in contrast to the ''logarithmic'' curve,<ref>{{cite journal |title=Math-alive! using original sources to teach mathematics in social context |journal=[[PRIMUS (journal)|PRIMUS]] |volume=8 |first=Bonnie |last=Shulman |pages=1–14 |issue=March |year=1998 |doi=10.1080/10511979808965879 |url=https://www.researchgate.net/publication/233238354 |quote=The diagram clinched it for me: there two curves labeled "Logistique" and "Logarithmique" are drawn on the same axes, and one can see that there is a region where they match almost exactly, and then diverge.<br>I concluded that Verhulst's intention in naming the curve was indeed to suggest this comparison, and that "logistic" was meant to convey the curve's "log-like" quality.}}</ref>{{efn|1=Verhulst first refers to arithmetic ''progression'' and geometric ''progression'', and refers to the geometric growth curve as a ''logarithmic'' curve (confusingly, the modern term is instead ''exponential'' curve, which is the inverse). He then calls his curve ''logistic'', in contrast to ''logarithmic'', and compares the logarithmic curve and logistic curve in the [https://gdz.sub.uni-goettingen.de/id/PPN129323640_0018?tify={%22pages%22:%5B54%5D,%22panX%22:0.487,%22panY%22:0.616,%22view%22:%22info%22,%22zoom%22:0.592} figure] of his paper.}} and by analogy with arithmetic and geometric. His growth model is preceded by a discussion of [[arithmetic growth]] and [[geometric growth]] (whose curve he calls a [[logarithmic curve]], instead of the modern term [[exponential curve]]), and thus "logistic growth" is presumably named by analogy, ''logistic'' being from {{lang-grc|λογῐστῐκός|logistikós}}, a traditional division of [[Greek mathematics]].{{efn|1=In Ancient Greece, {{lang|grc|λογῐστῐκός}} referred to practical computation and accounting, in contrast to {{lang|grc|ἀριθμητική}} (''{{lang|grc-Latn|arithmētikḗ}}''), the theoretical or philosophical study of numbers. Confusingly, in English, ''[[arithmetic]]'' refers to practical computation, even though it derives from {{lang|grc|ἀριθμητική}}, not {{lang|grc|λογῐστῐκός}}. See for example {{w|Louis Charles Karpinski}}, ''[https://babel.hathitrust.org/cgi/pt?id=mdp.39015005675411 Nicomachus of Gerasa: Introduction to Arithmetic]'' (1926) p. 3: "Arithmetic is fundamentally associated by modern readers, particularly by scientists and mathematicians, with the art of computation. For the ancient Greeks after [[Pythagoras]], however, arithmetic was primarily a philosophical study, having no necessary connection with practical affairs. Indeed the Greeks gave a separate name to the arithmetic of business, ''λογιστική'' [accounting or practical logistic] ... In general the philosophers and mathematicians of Greece undoubtedly considered it beneath their dignity to treat of this branch, which probably formed a part of the elementary instruction of children."}} The term is unrelated to the military and management term ''[[logistics]]'', which is instead from {{lang-fr|{{wikt-lang|fr|logis}}}} "lodgings", though some believe the Greek term also influenced ''logistics''; see {{slink|Logistics|Origin}} for details.

== Mathematical properties ==
<!--{{move section portions from|Pierre François Verhulst|date=February 2014}} Please see next section-->
The '''{{visible anchor|standard logistic function}}''' is the logistic function with parameters (<math>k = 1</math>, <math>x_0 = 0</math>, <math>L = 1</math>) which yields

: <math>\begin{align}
f(x) &= \frac{1}{1 + e^{-x}} = \frac{e^x}{e^x + 1} = \tfrac12 + \tfrac12 \tanh(\tfrac{x}{2}) \\
\end{align}</math>

In practice, due to the nature of the [[exponential function]] <math>e^{-x}</math>, it is often sufficient to compute the standard logistic function for <math>x</math> over a small range of real numbers such as a range contained in [−6,&nbsp;+6] as it quickly converges very close to its saturation values of 0 and 1.

The logistic function has the symmetry property that:

: <math>1-f(x) = f(-x).</math>

Thus, <math>x \mapsto f(x) - 1/2</math> is an [[odd function]].

The logistic function is an offset and scaled [[hyperbolic tangent]] function
:<math>f(x) = \tfrac12 + \tfrac12 \tanh(\tfrac{x}{2})</math>
or
:<math>\tanh(x) = 2 \, f(2x) - 1</math>.

This follows from
:<math>
\begin{align}
\tanh(x) & = \frac{e^x - e^{-x}}{e^x + e^{-x}} = \frac{e^x \cdot \left(1 - e^{-2x}\right)}{e^x \cdot \left(1 + e^{-2x}\right)} \\[6pt]
& = f(2x) - \frac{e^{-2x}}{1+e^{-2x}} = f(2x) - \frac{e^{-2x} + 1 - 1}{1+e^{-2x}} = 2f(2x)-1.
\end{align}
</math>

=== Derivative ===
The standard logistic function has an easily calculated [[derivative]]. The derivative is known as the [[logistic distribution]] (not to be confused with the [[normal distribution]]).

<math>f(x) = \frac 1 {1+\mathrm{e}^{-x}}= \frac{\mathrm{e}^{x}} {1+\mathrm{e}^{x}},</math>

<math>\frac{\operatorname{d}}{\operatorname{d}x}f(x) = \frac{\mathrm{e}^{x} \cdot (1+\mathrm{e}^{x})-\mathrm{e}^{x} \cdot \mathrm{e}^{x}}{(1 + \mathrm{e}^{x})^2} =  \frac{\mathrm{e}^{x}}{(1 + \mathrm{e}^{x})^2} = f(x)(1-f(x)).</math>

The derivative of the logistic function is an [[even function]], that is,

<math>f'(-x)=f'(x).</math>

=== Integral ===
Conversely, its [[antiderivative]] can be computed by the [[Integration by substitution|substitution]] <math>u = 1 + e^x</math>, since <math>f(x) = \frac {e^{x}} {1+e^{x}} = \frac {u'} {u}</math>, so (dropping the [[constant of integration]]):

:<math>\int \frac {e^{x}} {1+e^{x}}\,dx = \int \frac 1 {u}\,du = \log u = \log (1 + e^x)</math>

In [[artificial neural network]]s, this is known as the ''[[softplus]]'' function, and (with scaling) is a smooth approximation of the [[ramp function]], just as the logistic function (with scaling) is a smooth approximation of the [[Heaviside step function]].

=== Logistic differential equation ===
The standard logistic function is the solution of the simple first-order non-linear [[ordinary differential equation]]

:<math>\frac{d}{dx}f(x) = f(x)(1-f(x)) </math>

with [[boundary condition]] <math>f(0) = 1/2</math>.  This equation is the continuous version of the [[logistic map]].

The qualitative behavior is easily understood in terms of the [[Phase line (mathematics)|phase line]]: the derivative is 0 when the function is 1; and the derivative is positive for <math>f</math> between 0 and 1, and negative for <math>f</math> above 1 or less than 0 (though negative populations do not generally accord with a physical model). This yields an unstable equilibrium at 0, and a stable equilibrium at 1, and thus for any function value greater than 0 and less than 1, it grows to 1.

<!-- The above equation can be rewritten in the following steps:

:<math>\frac{d}{dx}f(x) = f(x)(1-f(x)) </math>

:<math>\frac{dy}{dx} = y(1-y) </math>

:<math>\frac{dy}{dx} = y - y^2 </math>

:<math>\frac{dy}{dx} - y = -y^2 </math> trivial algebraic manipulation-->
The logistic equation is a special case of the [[Bernoulli differential equation]] and has the following solution:

:<math>f(x)=\frac{e^{x}}{e^{x}+C}</math>

Choosing the constant of integration <math>C=1</math> gives the other well-known form of the definition of the logistic curve

:<math>f(x) = \frac{e^x}{e^x + 1} = \frac{1}{1 + e^{-x}}</math>

More quantitatively, as can be seen from the analytical solution, the logistic curve shows early [[exponential growth]] for negative argument, which slows to linear growth of slope 1/4 for an argument near 0, then approaches 1 with an exponentially decaying gap.

The logistic function is the inverse of the natural [[logit]] function and so can be used to convert the logarithm of [[odds]] into a [[probability]]. In mathematical notation the logistic function is sometimes written as ''expit''<ref>[http://www.inside-r.org/packages/cran/clusterPower/docs/expit expit documentation for R's clusterPower package]</ref> in the same form as ''logit''. The conversion from the [[log-likelihood ratio]] of two alternatives also takes the form of a logistic curve.

The hyperbolic tangent relationship leads to another form for the logistic function's derivative:

:<math>\frac d {dx} f(x) = \frac 1 4 \operatorname{sech}^2\left(\frac{x}{2} \right), </math>

which ties the logistic function into the [[logistic distribution]].

===Rotational symmetry about (0, &frac12;)===
The sum of the logistic function and its reflection about the vertical axis, <math>f(-x)</math>, is

:<math>\frac {1}{1+e^{-x}} + \frac 1 {1+e^{-(-x)}} = \frac{e^x}{e^x+1} + \frac{1}{e^x+1} = 1 .</math>

The logistic function is thus rotationally symmetrical about the point (0,&thinsp;{{sfrac|1|2}}).<ref>{{cite book|title=Neural Networks - A Systematic Introduction|author=Raul Rojas|url=http://page.mi.fu-berlin.de/rojas/neural/chapter/K11.pdf|accessdate=15 October 2016}}</ref>

== Applications ==

Link<ref name="A sequential theory of psychological discrimination">S. W. Link, Psychometrika, 1975, 40, 1, 77-105</ref> created an extension of Wald's theory of sequential analysis to a distribution-free accumulation of random variables until either a positive or negative bound is first equaled or exceeded. Link<ref name="The Relative Judgment Theory of the Psychometric Function">S. W. Link, Attention and Performance VII, 1978, 619-630</ref> derives the probability of first equaling or exceeding the positive boundary as <math>(1+e^{-\theta A})</math>, the Logistic function. This is the first proof that the Logistic function may have a stochastic process as its basis. Link<ref name="The wave theory of difference and similarity">S. W. Link, The wave theory of difference and similarity (book), Taylor and Francis, 1992</ref> provides a century of examples of “Logistic” experimental results and a newly derived relation between this probability and the time of absorption at the boundaries.

=== In ecology: modeling population growth ===
[[File:Pierre Francois Verhulst.jpg|right|thumb|150px|Pierre-François Verhulst (1804–1849)]]
A typical application of the logistic equation is a common model of [[population growth]] (see also [[population dynamics]]), originally due to [[Pierre François Verhulst|Pierre-François Verhulst]] in 1838, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal. The Verhulst equation was published after Verhulst had read [[Thomas Malthus]]' ''[[An Essay on the Principle of Population]]''. Verhulst derived his logistic equation to describe the self-limiting growth of a [[biology|biological]] population. The equation was rediscovered in 1911 by [[Anderson Gray McKendrick|A. G. McKendrick]] for the growth of bacteria in broth and experimentally tested using a technique for nonlinear parameter estimation.<ref name="McKendric Logistic">{{Cite journal | doi = 10.1017/S0370164600025426|journal=Proceedings of the Royal Society of Edinburgh|volume= 31 |date= January 1912|pages= 649–653 |title=XLV.—The Rate of Multiplication of Micro-organisms: A Mathematical Study|author= A. G. McKendricka; M. Kesava Paia1|url=https://zenodo.org/record/1543653}}</ref> The equation is also sometimes called the ''Verhulst-Pearl equation'' following its rediscovery in 1920 by [[Raymond Pearl]] (1879–1940) and [[Lowell Reed]] (1888–1966) of the [[Johns Hopkins University]].<ref>{{cite news|author=[[Raymond Pearl]] and [[Lowell Reed]]|title=On the Rate of Growth of the Population of the United States|url=http://math.bu.edu/people/mak/MA565/Pearl_Reed_PNAS_1920.pdf|date=June 1920|journal=[[Proceedings of the National Academy of Sciences of the United States of America]]|pages=275|number=6|volume=6|month=}}</ref> Another scientist, [[Alfred J. Lotka]] derived the equation again in 1925, calling it the ''law of population growth''.

Letting <math>P</math> represent population size (<math>N</math> is often used in ecology instead) and <math>t</math> represent time, this model is formalized by the [[differential equation]]:

: <math>\frac{dP}{dt}=r P \cdot \left(1 - \frac{P}{K}\right),</math>

where the constant <math>r</math> defines the [[population growth rate|growth rate]] and <math>K</math> is the [[carrying capacity]].

In the equation, the early, unimpeded growth rate is modeled by the first term <math>+rP</math>. The value of the rate <math>r</math> represents the proportional increase of the population <math>P</math> in one unit of time. Later, as the population grows, the modulus of the second term (which multiplied out is <math>-r P^2 / K</math>) becomes almost as large as the first, as some members of the population <math>P</math> interfere with each other by competing for some critical resource, such as food or living space. This antagonistic effect is called the ''bottleneck'', and is modeled by the value of the parameter <math>K</math>. The competition diminishes the combined growth rate, until the value of <math>P</math> ceases to grow (this is called ''maturity'' of the population).
The solution to the equation (with <math>P_0</math> being the initial population) is

:<math>P(t) = \frac{K P_0 e^{rt}}{K + P_0 \left( e^{rt} - 1\right)} = \frac{K}{1+\left(\frac{K-P_0}{P_0}\right)e^{-rt}}, </math>

where

:<math>\lim_{t\to\infty} P(t) = K.</math>

Which is to say that <math>K</math> is the limiting value of <math>P</math>: the highest value that the population can reach given infinite time (or come close to reaching in finite time). It is important to stress that the carrying capacity is asymptotically reached independently of the initial value <math>P(0) > 0</math>, and also in the case that <math>P(0) > K</math>.

In [[ecology]], [[species]] are sometimes referred to as [[r/K selection theory|<math>r</math>-strategist or <math>K</math>-strategist]] depending upon the [[Natural selection|selective]] processes that have shaped their [[Biological life cycle|life history]] strategies.
[[Dimensional analysis|Choosing the variable dimensions]] so that <math>n</math> measures the population in units of carrying capacity, and <math>\tau</math> measures time in units of <math>1/r</math>, gives the dimensionless
differential equation

: <math>\frac{dn}{d\tau} =  n (1-n).</math>

==== Time-varying carrying capacity ====
Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying: <math>K(t) > 0</math>, leading to the following mathematical model:

: <math>\frac{dP}{dt}=rP \cdot \left(1 - \frac{P}{K(t)}\right)</math>

A particularly important case is that of carrying capacity that varies periodically with period <math>T</math>:

: <math>K(t+T) = K(t).</math>

It can be shown that in such a case, independently from the initial value <math>P(0) > 0</math>, <math>P(t)</math> will tend to a unique periodic solution <math>P_*(t)</math>, whose period is <math>T</math>.

A typical value of <math>T</math> is one year: In such case <math>K(t)</math> may reflect periodical variations of weather conditions.

Another interesting generalization is to consider that the carrying capacity <math>K(t)</math> is a function of the population at an earlier time, capturing a delay in the way population modifies its environment. This leads to a logistic delay equation,<ref name="delay carrying">{{Cite journal | last1 = Yukalov | first1 = V. I. | last2 = Yukalova | first2 = E. P. | last3 = Sornette | first3 = D. | doi = 10.1016/j.physd.2009.05.011 | title = Punctuated evolution due to delayed carrying capacity | journal = Physica D: Nonlinear Phenomena | volume = 238 | issue = 17 | pages = 1752–1767 | year = 2009 | pmid =  | pmc = | arxiv = 0901.4714 | bibcode = 2009PhyD..238.1752Y }}</ref> which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth (i.e., multiple S-shapes), punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death.

=== In statistics and machine learning ===
Logistic functions are used in several roles in [[statistics]]. For example, they are the [[cumulative distribution function]] of the [[Logistic distribution|logistic family of distributions]], and they are, a bit simplified, used to model the chance a chess player has to beat his opponent in the [[Elo rating system]]. More specific examples now follow.

==== Logistic regression ====
{{Main|Logistic regression}}

Logistic functions are used in [[logistic regression]] to model how the probability <math>p</math> of an event may be affected by one or more [[explanatory variables]]: an example would be to have the model

: <math>p=f(a + bx)</math>

where <math>x</math> is the explanatory variable and <math>a</math> and <math>b</math> are model parameters to be fitted and <math>f</math> is the standard logistic function.

Logistic regression and other [[log-linear model]]s are also commonly used in [[machine learning]]. A generalisation of the logistic function to multiple inputs is the [[softmax activation function]], used in [[multinomial logistic regression]].

Another application of the logistic function is in the [[Rasch model]], used in [[item response theory]]. In particular, the Rasch model forms a basis for [[maximum likelihood]] estimation of the locations of objects or persons on a [[Continuum (theory)|continuum]], based on collections of categorical data, for example the abilities of persons on a continuum based on responses that have been categorized as correct and incorrect.

==== Neural networks ====
Logistic functions are often used in [[neural network]]s to introduce [[nonlinearity]] in the model or to clamp signals to within a specified [[Range (mathematics)|range]].  A popular [[artificial neuron|neural net element]] computes a [[linear combination]] of its input signals, and applies a bounded logistic function to the result; this model can be seen as a "smoothed" variant of the classical [[perceptron|threshold neuron]].

<!-- A reason for its popularity in neural networks is because the logistic function satisfies the differential equation

:<math>y' = y(1-y).</math>

The right hand side is a low-degree polynomial. Furthermore, the polynomial has factors <math>y</math> and <math>1 − y</math>, both of which are simple to compute. Given <math>y = sig(t)</math> at a particular <math>t</math>, the derivative of the logistic function at that <math>t</math> can be obtained by multiplying the two factors together. -->
A common choice for the activation or "squashing" functions, used to clip for large magnitudes to keep the response of the neural network bounded<ref name="Gershenfeld-1999">Gershenfeld 1999, p.150</ref> is

:<math>g(h) = \frac{1}{1 + e^{-2 \beta h}}</math>

which is a logistic function.
These relationships result in simplified implementations of [[artificial neural network]]s with [[artificial neuron]]s. Practitioners caution that sigmoidal functions which are [[Odd functions|antisymmetric]] about the origin (e.g. the [[hyperbolic tangent]]) lead to faster convergence when training networks with [[backpropagation]].<ref name="LeCun-1998">{{cite book | author1 = LeCun, Y.  | author2 = Bottou, L.  | author3 = Orr, G. | author4 = Muller, K. | editor = Orr, G. | editor2 = Muller, K. | year = 1998 | title = Efficient BackProp  | work = Neural Networks: Tricks of the trade  | isbn = 3-540-65311-2 | publisher = Springer | url = http://yann.lecun.com/exdb/publis/pdf/lecun-98b.pdf }}</ref>

The logistic function is itself the derivative of another proposed activation function, the [[softplus]].

=== In medicine: modeling of growth of tumors ===
{{See also|Gompertz curve#Growth of tumors}}
Another application of logistic curve is in medicine, where the logistic differential equation is used to model the growth of tumors. This application can be considered an extension of the above-mentioned use in the framework of ecology (see also the [[Generalized logistic curve]], allowing for more parameters). Denoting with <math>X(t)</math> the size of the tumor at time <math>t</math>, its dynamics are governed by:

: <math>X'=r\left(1 - \frac X K \right)X</math>

which is of the type:

:<math>X'=F(X)X, \qquad F'(X) \le 0 </math>

where <math>F(X)</math> is the proliferation rate of the tumor.

If a chemotherapy is started with a log-kill effect, the equation may be revised to be

:<math>X'=r\left(1 - \frac X K \right)X - c(t)X,</math>

where <math>c(t)</math> is the therapy-induced death rate. In the idealized case of very long therapy, <math>c(t)</math> can be modeled as a periodic function (of period <math>T</math>) or (in case of continuous infusion therapy) as a constant function, and one has that

:<math> \frac 1 T \int_0^T c(t)\, dt > r \to \lim_{t \to +\infty} x(t) = 0 </math>

i.e. if the average therapy-induced death rate is greater than the baseline proliferation rate then there is the eradication of the disease. Of course, this is an oversimplified model of both the growth and the therapy (e.g. it does not take into account the phenomenon of clonal resistance).

=== In chemistry: reaction models ===
The concentration of reactants and products in [[autocatalysis|autocatalytic reactions]] follow the logistic function.
The degradation of [[Platinum group|platinum group metal]]-free (PGM-free) oxygen reduction reaction (ORR) catalyst in fuel cell cathodes follows the logistic decay function,<ref>{{cite journal |last1=Yin |first1=Xi |last2=Zelenay |first2=Piotr |title=Kinetic Models for the Degradation Mechanisms of PGM-Free ORR Catalysts |journal=ECS Transactions |date=13 July 2018 |volume=85 |issue=13 |pages=1239–1250 |doi=10.1149/08513.1239ecst}}</ref> suggesting an autocatalytic degradation mechanism.

=== In physics: Fermi distribution ===
The logistic function determines the statistical distribution of fermions over the energy states of a system in thermal equilibrium. In particular, it is the distribution of the probabilities that each possible energy level is occupied by a fermion, according to [[Fermi function|Fermi–Dirac statistics]].

=== In material science: Phase diagrams ===
[[Diffusion bonding]].

=== In linguistics: language change ===
In linguistics, the logistic function can be used to model [[language change]]:<ref name="probabilistic linguistics">Bod, Hay, Jennedy (eds.) 2003, pp. 147–156</ref> an innovation that is at first marginal begins to spread more quickly with time, and then more slowly as it becomes more universally adopted.

=== In agriculture: modeling crop response ===
The logistic S-curve can be used for modeling the crop response to changes in growth factors. There are two types of response functions:  ''positive''  and ''negative'' growth curves. For example, the crop yield may ''increase'' with increasing value of the growth factor up to a certain level (positive function), or it may ''decrease'' with increasing growth factor values (negative function owing to a negative growth factor), which situation requires an ''inverted'' S-curve.
{| class="wikitable"
|The logistic S-curve can be used to model the relation between crop yield and depth of the '''[[water table]]''' in the soil.<ref name="calc">Calculator for crop response to changes in growth factors using [[segmented regression]], S-curves and [[parabola]]s. On line: [https://www.waterlog.info/segreg.htm].</ref> <br>The inverted logistic S-curve can be used to model the relation between crop yield and '''[[soil salinity]]'''.<ref>[https://www.waterlog.info/sigmoid.htm Software for fitting S-curves to data sets]</ref> 
[[File:Sugarcane S-curve.png|thumb|300px|left|S-curve model for yield versus depth of watertable.<ref>Collection of data on crop production and depth of the [[water table]] in the soil of various authors. On line: [https://www.waterlog.info/cropwat.htm]</ref> ]]
 [[File:Barley S-curve.png|thumb|300px|right|Inverted S-curve model for yield versus soil salinity.<ref>Collection of data on crop production and [[soil salinity]] of various authors. On line: [https://www.waterlog.info/croptol.htm]</ref>]] 
|}

=== In economics and sociology: diffusion of innovations ===
The logistic function can be used to illustrate the progress of the [[Diffusion of innovations|diffusion of an innovation]] through its life cycle.

In ''The Laws of Imitation'' (1890), [[Gabriel Tarde]] describes the rise and spread of new ideas through imitative chains. In particular, Tarde identifies three main stages through which innovations spread: the first one corresponds to the difficult beginnings, during which the idea has to struggle within a hostile environment full of opposing habits and beliefs; the second one corresponds to the properly exponential take-off of the idea, with <math>f(x)=2^x</math>; finally, the third stage is logarithmic, with <math>f(x)=\log(x)</math>, and corresponds to the time when the impulse of the idea gradually slows down while, simultaneously new opponent ideas appear. The ensuing situation halts or stabilizes the progress of the innovation, which approaches an asymptote.

In the history of economy, when new products are introduced there is an intense amount of research and development which leads to dramatic improvements in quality and reductions in cost.  This leads to a period of rapid industry growth. Some of the more famous examples are: railroads, incandescent light bulbs, [[electrification]], [[car]]s and air travel. Eventually, dramatic improvement and cost reduction opportunities are exhausted, the product or process are in widespread use with few remaining potential new customers, and markets become saturated.

Logistic analysis was used in papers by several researchers at the International Institute of Applied Systems Analysis ([[IIASA]]). These papers deal with the diffusion of various innovations, infrastructures and energy source substitutions and the role of work in the economy as well as with the long economic cycle. Long economic cycles were investigated by Robert Ayres (1989).<ref>{{cite journal
 | last1 = Ayres
 | first1 = Robert
 | author1-link = Robert Ayres (scientist)
 | title =Technological Transformations and Long Waves
 | year = 1989
 | url = http://www.iiasa.ac.at/Admin/PUB/Documents/RR-89-001.pdf
 }}</ref> Cesare Marchetti published on [[Kondratiev wave|long economic cycles]] and on diffusion of innovations.<ref>{{cite journal
 |last1       = Marchetti
 |first1      = Cesare
 |title       = Pervasive Long Waves: Is Society Cyclotymic
 |year        = 1996
 |url         = http://www.agci.org/dB/PDFs/03S2_CMarchetti_Cyclotymic.pdf
 |url-status     = dead
 |archiveurl  = https://web.archive.org/web/20120305095553/http://www.agci.org/dB/PDFs/03S2_CMarchetti_Cyclotymic.pdf
 |archivedate = 2012-03-05
}}</ref><ref>{{cite journal
 | last1 = Marchetti
 | first1 = Cesare
 | title = Kondratiev Revisited-After One Cycle
 | year = 1988
 | url = http://www.cesaremarchetti.org/archive/scan/MARCHETTI-037.pdf
 }}</ref> Arnulf Grübler's book (1990) gives a detailed account of the diffusion of infrastructures including canals, railroads, highways and airlines, showing that their diffusion followed logistic shaped curves.<ref name="Grubler1990">{{cite book
 | last1 = Grübler
 | first1 = Arnulf
 | author1-link =
 | title = The Rise and Fall of Infrastructures: Dynamics of Evolution and Technological Change in Transport
 | year = 1990
|publisher=Physica-Verlag
|location= Heidelberg and New York
|isbn=
|pages=
 | url = http://www.iiasa.ac.at/Admin/PUB/Documents/XB-90-704.pdf
 }}</ref>

Carlota Perez used a logistic curve to illustrate the long ([[Kondratiev wave|Kondratiev]]) business cycle with the following labels: beginning of a technological era as ''irruption'', the ascent as ''frenzy'', the rapid build out as ''synergy'' and the completion as ''maturity''.<ref name="Perez2002">{{cite book |title= Technological Revolutions and Financial Capital: The Dynamics of Bubbles and Golden Ages
|last1=Perez
|first1= Carlota
|year=2002 |publisher= Edward Elgar Publishing Limited |location=UK  |isbn=1-84376-331-1  |pages=  |url=https://archive.org/details/technologicalrev00carl|url-access= registration
}}</ref>

== See also ==
{{div col|colwidth=30em}}
* [[Exponential growth]]
* [[Hyperbolic growth]]
* [[Diffusion of innovations]]
* [[Generalised logistic function]]
* [[Gompertz curve]]
* [[Heaviside step function]]
* [[Hubbert curve]]
* [[Logistic distribution]]
* [[Logistic map]]
* [[Logistic regression]]
* [[Star model|Logistic smooth-transmission model]]
* [[Logit]]
* [[Log-likelihood ratio]]
* [[Malthusian growth model]]
* [[Population dynamics]]
* [[r/K selection theory]]
* [[Shifted Gompertz distribution]]
* [[Tipping point (sociology)]]
* [[Rectifier (neural networks)]]
* [[Cross fluid]]
* [[Hill equation (biochemistry)]]
* [[Michaelis Menten equation]]
{{div col end}}

== Notes ==
{{notelist}}

== References ==
{{Reflist|30em}}
{{ref begin}}
* {{cite techreport
|last=Cramer|first=J. S.
|title=The origins of logistic regression
|institution=Tinbergen Institute
|date=2002|volume=119|issue=4|pages=167–178
|doi=10.2139/ssrn.360300
|url=https://papers.tinbergen.nl/02119.pdf
|ref=harv
}}
** Published as:{{cite journal
|title=The early origins of the logit model
|last=Cramer|first=J. S.
|journal=Studies in History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences
|volume=35
|number=4
|year=2004
|pages=613–626
|doi=10.1016/j.shpsc.2004.09.003
}}
* {{cite book
 |author1=Jannedy, Stefanie |author2=Bod, Rens |author3=Hay, Jennifer | year = 2003
 | title =Probabilistic Linguistics
 | isbn = 0-262-52338-8
 | publisher = MIT Press
 | location = Cambridge, Massachusetts
 | url = https://books.google.com/books?id=Ac5bgYVk8kwC&printsec=frontcover&dq=isbn:0262523388&hl=en&sa=X&ved=0ahUKEwjrgJrJnsHjAhWCVs0KHe4PDOAQ6AEIKjAA#v=onepage&q=logistic&f=false
}}
* {{cite book
 | author = Gershenfeld, Neil A.
 | year = 1999
 | title =The Nature of Mathematical Modeling
 | isbn = 978-0-521-57095-4
 | publisher = Cambridge University Press
 | location = Cambridge, UK
 | oclc =
 | url=https://books.google.com/books?id=zYAcGbp17nYC&printsec=frontcover&dq=isbn:9780521570954&hl=en&sa=X&ved=0ahUKEwiRqdXnnsHjAhVEGM0KHZlMBZEQ6AEIKjAA#v=onepage&q=logistic&f=false
}}
* {{cite book
| author = Kingsland, Sharon E.
| title = Modeling nature: episodes in the history of population ecology
| publisher = University of Chicago Press
| location = Chicago
| year = 1995
| pages =
| isbn = 0-226-43728-0
| oclc =
| doi =
| accessdate =
}}
* {{MathWorld |title=Logistic Equation |urlname=LogisticEquation}}
{{ref end}}

== External links ==
{{commons|Logistic functions}}
* L.J. Linacre, [http://rasch.org/rmt/rmt64k.htm Why logistic ogive and not autocatalytic curve?], accessed 2009-09-12.
* https://web.archive.org/web/20060914155939/http://luna.cas.usf.edu/~mbrannic/files/regression/Logistic.html
* {{MathWorld |title=Sigmoid Function |urlname= SigmoidFunction}}
* [http://jsxgraph.uni-bayreuth.de/wiki/index.php/Logistic_process Online experiments with JSXGraph]
* [https://www.sciencedaily.com/releases/2011/07/110720151541.htm Esses are everywhere.]
* [http://www.sciencecodex.com/seeing_the_scurve_in_everything Seeing the s-curve is everything.]
* [http://jeffreyfreeman.me/restricted-logarithmic-growth-with-injection/ Restricted Logarithmic Growth with Injection]

[[Category:Special functions]]
[[Category:Differential equations]]
[[Category:Population]]
[[Category:Population ecology]]
[[Category:Logistic regression]]
[[Category:Growth curves]]