604:
3638:
3077:
9039:) are usually used in a descriptive or phenomenological manner because they fit well not only to the early exponential rise, but to the eventual levelling off of the pandemic as the population develops a herd immunity. This is in contrast to actual models of pandemics which attempt to formulate a description based on the dynamics of the pandemic (e.g. contact rates, incubation times, social distancing, etc.). Some simple models have been developed, however, which yield a logistic solution.
6286:
3196:
9717:
9596:
329:
9733:
9048:
2037:
3633:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} x}}f(x)&={\frac {e^{x}\cdot (1+e^{x})-e^{x}\cdot e^{x}}{(1+e^{x})^{2}}}\\&={\frac {e^{x}}{(1+e^{x})^{2}}}\\&=\left({\frac {e^{x}}{1+e^{x}}}\right)\left({\frac {1}{1+e^{x}}}\right)\\&=\left({\frac {e^{x}}{1+e^{x}}}\right)\left(1-{\frac {e^{x}}{1+e^{x}}}\right)\\&=f(x)\left(1-f(x)\right)\end{aligned}}}
1656:
7686:
9769:
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
8104:
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, which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth,
9003:
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).
9655:
A mirage, resulting from a temperature gradient that modifies the refractive index related to the density/concentration of the material over distance, can be modelled using a fluid with a refractive index gradient due to the concentration gradient. This mechanism can be equated to a limiting
10002:, the logistic function. This is the first proof that the logistic function may have a stochastic process as its basis. Link 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.
10661:
9908:). 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). Cesare Marchetti published on
6939:
7448:
2032:{\displaystyle {\begin{aligned}\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)}}\\&=f(2x)-{\frac {e^{-2x}}{1+e^{-2x}}}=f(2x)-{\frac {e^{-2x}+1-1}{1+e^{-2x}}}=2f(2x)-1.\end{aligned}}}
9635:
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
5725:
627:. Verhulst first devised the function in the mid 1830s, publishing a brief note in 1838, then presented an expanded analysis and named the function in 1844 (published 1845); the third paper adjusted the correction term in his model of Belgian population growth.
630:
The initial stage of growth is approximately exponential (geometric); then, as saturation begins, the growth slows to linear (arithmetic), and at maturity, growth approaches the limit with an exponentially decaying gap, like the initial stage in reverse.
1391:
4626:
8496:
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
4235:
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.
3066:
981:
2132:
9912:
and on diffusion of innovations. 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.
9247:
8105:
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.
9948:
of sequential analysis to a distribution-free accumulation of random variables until either a positive or negative bound is first equaled or exceeded. Link derives the probability of first equaling or exceeding the positive boundary as
4006:
9859:, 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.
9028:) or reduction in the accessibility of potential hosts through physical distancing measures, may result in exponential-looking epidemic curves first linearizing (replicating the "logarithmic" to "logistic" transition first noted by
8999:
2990:
1481:
1590:
9901:, cars 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.
9023:
exhibited exponential growth early in the course of infection in several countries in early 2020. Factors including a lack of susceptible hosts (through the continued spread of infection until it passes the threshold for
6756:
7819:
9018:
A novel infectious pathogen to which a population has no immunity will generally spread exponentially in the early stages, while the supply of susceptible individuals is plentiful. The SARS-CoV-2 virus that causes
10469:
4438:
3190:
3870:
11110:
6305:
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
6494:
10874:
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.
10190:... 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."
8807:
4130:
2680:
2774:
8613:
9897:
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,
7041:, the highest value that the population can reach given infinite time (or come close to reaching in finite time). The carrying capacity is asymptotically reached independently of the initial value
1242:
9552:
8690:
4481:
9302:
3201:
1661:
134:
9656:
population growth model, where the concentrated region attempts to diffuse into the lower concentration region, while seeking equilibrium with gravity, thus yielding a logistic function curve.
8448:
5895:
5807:
5276:
819:
7384:
5502:
5108:
4310:
2046:
9866:, the subnational units (constituent states or cities) may use loans to finance their projects. However, this funding source is usually subject to strict legal rules as well as to economy
7681:{\displaystyle \int {\frac {KP_{0}e^{rt}}{K+P_{0}\left(e^{rt}-1\right)}}\,dt=\int {\frac {K}{r}}{\frac {1}{u}}\,du={\frac {K}{r}}\ln u+C={\frac {K}{r}}\ln \left(K+P_{0}(e^{rt}-1)\right)+C}
7302:
6997:
515:
10186:, 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,
4681:
10937:
Tableau
Analytique des principales combinaisons De La Guerre, Et De Leurs Rapports Avec La Politique Des États: Pour Servir D'Introduction Au Traité Des Grandes Opérations Militaires
9079:
3883:
10000:
10182:(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
1649:
10460:
7442:
6171:
1196:
9652:. Under certain conditions, such as the presence of a temperature or concentration gradient due to diffusion and balancing with gravity, logistic curve behaviours can emerge.
4862:
8894:
5497:
2995:
452:
387:
9627:
metal-free (PGM-free) oxygen reduction reaction (ORR) catalyst in fuel cell cathodes follows the logistic decay function, suggesting an autocatalytic degradation mechanism.
9590:
2901:
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1102:
303:
274:
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7896:
2247:
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6046:
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1513:
9810:
8214:
2816:
2402:
10876:
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.
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1016:
10255:
7211:
6749:
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6006:
5401:
5370:
5339:
5308:
4944:
4906:
4833:
4801:
4755:
3703:
2525:
2332:
1235:
218:
9408:
9362:
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8528:
8102:
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2851:
7239:
6573:
4721:
4338:
3672:
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2358:
812:
753:
6268:
In statistics, where the logistic function is interpreted as the probability of one of two alternatives, the generalization to three or more alternatives is the
6235:
5141:
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10219:
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8887:
8548:
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6613:
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6393:
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2489:
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1124:
1036:
323:
238:
189:
159:
11039:
Kyurkchiev, Nikolay, and
Svetoslav Markov. "Sigmoid functions: some approximation and modelling aspects". LAP LAMBERT Academic Publishing, Saarbrucken (2015).
6328:
for the growth of bacteria in broth and experimentally tested using a technique for nonlinear parameter estimation. The equation is also sometimes called the
2934:
1413:
10892:
4345:
3097:
6424:
6087:, so the other logits are interpreted as odds versus this reference. This is generally done with the first alternative, hence the choice of numbering:
4044:
3790:
10094:
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. 1.
8379:
A common choice for the activation or "squashing" functions, used to clip large magnitudes to keep the response of the neural network bounded, is
11080:
4446:
for negative argument, which reaches to linear growth of slope 1/4 for an argument near 0, then approaches 1 with an exponentially decaying gap.
10656:{\displaystyle \operatorname {LSE_{0}} ^{+}(x_{1},\dots ,x_{n}):=\operatorname {LSE} (0,x_{1},\dots ,x_{n})=\ln(1+e^{x_{1}}+\cdots +e^{x_{n}}).}
9680:: an innovation that is at first marginal begins to spread more quickly with time, and then more slowly as it becomes more universally adopted.
8731:
12077:
9495:
represent the final epidemic size, infection rate, and lag phase, respectively. See the right panel for an example infection trajectory when
11792:
10784:
1038:
over a small range of real numbers, such as a range contained in , as it quickly converges very close to its saturation values of 0 and 1.
9688:
The logistic S-curve can be used for modeling the crop response to changes in growth factors. There are two types of response functions:
8555:
11763:
8384:
6676:
interfere with each other by competing for some critical resource, such as food or living space. This antagonistic effect is called the
11302:"Estimation of COVID-19 dynamics "on a back-of-envelope": Does the simplest SIR model provide quantitative parameters and predictions?"
4449:
The differential equation derived above is a special case of a general differential equation that only models the sigmoid function for
11726:
6934:{\displaystyle P(t)={\frac {KP_{0}e^{rt}}{K+P_{0}\left(e^{rt}-1\right)}}={\frac {K}{1+\left({\frac {K-P_{0}}{P_{0}}}\right)e^{-rt}}},}
10719:
607:
Original image of a logistic curve, contrasted with what
Verhulst called a "logarithmic curve" (in modern terms, "exponential curve")
4248:
2575:
689:
As a word derived from ancient Greek mathematical terms, the name of this function is unrelated to the military and management term
7246:
9599:
Extrapolated infection trajectories of 40 countries severely affected by COVID-19 and grand (population) average through May 14th
8622:
6948:
6312:
2685:
6320:
of simple (unconstrained) exponential growth. Verhulst derived his logistic equation to describe the self-limiting growth of a
8340:, for example the abilities of persons on a continuum based on responses that have been categorized as correct and incorrect.
12015:
11962:
Studies in
History and Philosophy of Science Part C: Studies in History and Philosophy of Biological and Biomedical Sciences
9066:
outbreak. The authors fit the generalized logistic function to the cumulative number of infected cases, here referred to as
9498:
9013:
10813:
10772:
9254:
57:
9904:
Logistic analysis was used in papers by several researchers at the
International Institute of Applied Systems Analysis (
9704:
with increasing growth factor values (negative function owing to a negative growth factor), which situation requires an
2992:, while the parametrization of the unit hyperbola (for the hyperbolic tangent) corresponds to the linear transformation
1595:
5720:{\displaystyle e^{x_{1}-x_{0}}/(e^{x_{1}-x_{0}}+1)=1/\left(1+e^{-(x_{1}-x_{0})}\right)=e^{x_{1}}/(e^{x_{0}}+e^{x_{1}})}
11279:
Villalobos-Arias, Mario (2020). "Using generalized logistics regression to forecast population infected by Covid-19".
12034:
11994:
11913:
11861:
11252:
5730:
5182:
7321:
5823:
5017:
11357:
Saito, Takesi (June 2020). "A Logistic Curve in the SIR Model and Its
Application to Deaths by COVID-19 in Japan".
8114:
6254:
6083:
not being log-odds on their own. Often one of the options is used as a reference ("pivot"), and its value fixed as
1051:
11518:
Tanalikhit, Pattarapon; Worakitthamrong, Thanabodi; Chaidet, Nattanon; Kanchanapusakit, Wittaya (24–25 May 2021).
10981:
Kocian, Alexander; Carmassi, Giulia; Cela, Fatjon; Incrocci, Luca; Milazzo, Paolo; Chessa, Stefano (7 June 2020).
7848:
7695:
Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying, with
12136:
8314:
4240:
2823:
17:
10915:
10745:
8172:
460:
11174:
Yukalov, V. I.; Yukalova, E. P.; Sornette, D. (2009). "Punctuated evolution due to delayed carrying capacity".
10037:
4037:
1386:{\displaystyle {\frac {1}{1+e^{-x}}}+{\frac {1}{1+e^{-(-x)}}}={\frac {e^{x}}{e^{x}+1}}+{\frac {1}{e^{x}+1}}=1.}
10940:
4631:
10069:
10047:
9604:
9071:
9059:
9051:
6615:
in one unit of time. Later, as the population grows, the modulus of the second term (which multiplied out is
6258:
4621:{\displaystyle {\frac {df(x)}{dx}}={\frac {k}{a}}f(x){\big (}a-f(x){\big )},\quad f(0)={\frac {a}{1+e^{kr}}}}
672:
11622:
Collection of data on crop production and depth of the water table in the soil of various authors. On line:
8121:, and they are, a bit simplified, used to model the chance a chess player has to beat their opponent in the
12151:
12141:
12078:
https://web.archive.org/web/20060914155939/http://luna.cas.usf.edu/~mbrannic/files/regression/Logistic.html
10027:
12146:
12116:
10064:
9952:
8310:
7316:
3744:
9029:
6302:
3061:{\displaystyle {\tfrac {1}{2}}{\bigl (}{\begin{smallmatrix}1&1\\-1&1\end{smallmatrix}}{\bigr )}}
976:{\displaystyle f(x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{e^{x}+1}}={\frac {e^{x/2}}{e^{x/2}+e^{-x/2}}}.}
612:
12131:
10416:
8498:
7389:
6246:
6123:
2127:{\displaystyle {\frac {d}{dx}}f(x)={\frac {1}{4}}\operatorname {sech} ^{2}\left({\frac {x}{2}}\right),}
1152:
9878:
limits). These restrictions, which represent a saturation level, along with an exponential rush in an
4838:
10269:
10074:
9648:
The logistic function also finds applications in optics, particularly in modelling phenomena such as
8456:
8349:
5455:
4011:
3643:
2362:) of tanh. Analogously, the logistic function can be viewed as the hyperbolic angle on the hyperbola
642:
curve, and by analogy with arithmetic and geometric. His growth model is preceded by a discussion of
588:
400:
335:
9557:
2857:
11722:
10313:
10275:
9894:
9755:
6341:
6325:
4192:
2153:
279:
250:
9815:
2199:
2041:
The hyperbolic-tangent relationship leads to another form for the logistic function's derivative:
12106:
11831:
The Rise and Fall of
Infrastructures: Dynamics of Evolution and Technological Change in Transport
10175:
10022:
9879:
9471:
9444:
9417:
6317:
6262:
6011:
5907:
5411:
4442:
More quantitatively, as can be seen from the analytical solution, the logistic curve shows early
4025:
3875:
564:
11803:
11568:
11363:
10732:
9773:
9700:
with increasing value of the growth factor up to a certain level (positive function), or it may
8491:
7823:
A particularly important case is that of carrying capacity that varies periodically with period
5343:, which then generalizes naturally to more than two alternatives. Given two real number inputs,
4875:
for the first alternative (relative to the other alternative), measured in "logistic units" (or
2779:
2365:
12156:
11102:
10381:
9770:
and beliefs; the second one corresponds to the properly exponential take-off of the idea, with
8357:
7903:
7698:
7079:
7044:
6618:
6519:
4768:
3749:
2819:
663:
12111:
9242:{\displaystyle f(t;\theta _{1},\theta _{2},\theta _{3},\xi )={\frac {\theta _{1}}{^{1/\xi }}}}
7967:
4140:
2407:
12101:
11774:
11074:
8164:
8118:
7166:
6416:
6184:
6090:
5951:
4977:
4001:{\displaystyle \int {\frac {e^{x}}{1+e^{x}}}\,dx=\int {\frac {1}{u}}\,du=\ln u=\ln(1+e^{x}).}
3089:
2137:
758:
10466:. Softplus thus generalizes as (note the 0 and the corresponding 1 for the reference class)
9367:
8478:
The logistic function is itself the derivative of another proposed activation function, the
5152:
4452:
4036:
The unique standard logistic function is the solution of the simple first-order non-linear
991:
11737:
11668:
11647:"Is the public sector of your country a diffusion borrower? Empirical evidence from Brazil"
11580:
11531:
11422:
11358:
11313:
11193:
11138:
10994:
10955:
10932:
10228:
10052:
8337:
8325:
7196:
7170:
7119:
6727:
6059:
5984:
5379:
5348:
5317:
5286:
4917:
4884:
4806:
4774:
4728:
3681:
2498:
2305:
1495:
1208:
986:
196:
11544:
11519:
9393:
9347:
9307:
9062:, also called the Richards growth curve, has been applied to model the early phase of the
8843:
8814:
8697:
8504:
8078:
8046:
7938:
4771:); the two alternatives are complementary, so the probability of the other alternative is
2827:
8:
11928:
S. W. Link, The wave theory of difference and similarity (book), Taylor and
Francis, 1992
11897:
11877:
Link, S. W.; Heath, R. A. (1975). "A sequential theory of psychological discrimination".
11829:
8369:
8140:
8134:
7216:
6552:
6298:
4700:
4692:
4317:
3651:
2906:
2534:
2341:
791:
732:
11852:
Technological Revolutions and Financial Capital: The Dynamics of Bubbles and Golden Ages
11672:
11584:
11535:
11513:
11511:
11426:
11401:"Estimation of COVID-19 spread curves integrating global data and borrowing information"
11317:
11197:
11142:
10998:
8994:{\displaystyle {\frac {1}{T}}\int _{0}^{T}c(t)\,dt>r\to \lim _{t\to +\infty }x(t)=0,}
6217:
5123:
2985:{\displaystyle {\bigl (}{\begin{smallmatrix}1&1\\0&1\end{smallmatrix}}{\bigr )}}
2254:
11699:
11658:
11646:
11596:
11549:
11497:
11445:
11412:
11400:
11379:
11334:
11301:
11280:
11209:
11183:
11017:
10982:
10907:
10204:
9945:
9611:
framework, where information from different geographic regions can be pooled together.
9327:
8889:) or (in case of continuous infusion therapy) as a constant function, and one has that
8872:
8726:
If a chemotherapy is started with a log-kill effect, the equation may be revised to be
8533:
8468:
8365:
8333:
8329:
8281:
8261:
8241:
8221:
8146:
8026:
8003:
7826:
7176:
7142:
7122:
7024:
7004:
6703:
6683:
6659:
6598:
6578:
6525:
6501:
6398:
6378:
6358:
5981:. Notably, the softmax function is invariant under adding a constant to all the logits
4951:
4443:
4218:
4198:
4135:
3712:
2474:
2450:
2281:
1507:
1129:
1109:
1021:
620:
308:
223:
174:
144:
12048:
11237:
10983:"Bayesian Sigmoid-Type Time Series Forecasting with Missing Data for Greenhouse Crops"
8113:
Logistic functions are used in several roles in statistics. For example, they are the
1476:{\displaystyle \operatorname {logit} p=\log {\frac {p}{1-p}}{\text{ for }}0<p<1}
1395:
The logistic function is thus rotationally symmetrical about the point (0, 1/2).
12087:
12084:
12045:
12030:
12011:
11990:
11909:
11857:
11850:
11704:
11686:
11633:
Collection of data on crop production and soil salinity of various authors. On line:
11592:
11553:
11508:
11501:
11489:
11450:
11339:
11248:
11156:
11022:
10851:
10818:
Mémoires de l'Académie Royale des Sciences, des Lettres et des Beaux-Arts de Belgique
10358:
10017:
9665:
9411:
9036:
8460:
8361:
8122:
7162:
6543:
6294:
6250:
4835:. The two alternatives are coded as 1 and 0, corresponding to the limiting values as
4767:
can be interpreted as the probability of one of two alternatives (the parameter of a
2565:
683:
655:
651:
643:
616:
576:
162:
11600:
11213:
10911:
1585:{\displaystyle f(x)={\frac {1}{2}}+{\frac {1}{2}}\tanh \left({\frac {x}{2}}\right),}
603:
11969:
11947:
11886:
11870:
11694:
11676:
11588:
11539:
11481:
11440:
11430:
11329:
11321:
11201:
11146:
11062:
11012:
11002:
10865:
10856:
10825:
10127:, and compares the logarithmic curve and logistic curve in the figure of his paper.
9871:
9608:
8306:
8302:
6269:
5813:
2144:
647:
624:
48:
10777:
Nouveaux Mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles
12005:
11984:
11904:
Link, S. W. (1978). "The Relative Judgment Theory of the Psychometric Function".
11681:
11435:
10686:
9917:
9909:
9898:
9863:
9677:
8472:
6345:
4195:: the derivative is 0 when the function is 1; and the derivative is positive for
4184:. Note that the reciprocal logistic function is solution to a simple first-order
2569:
694:
11973:
11325:
11205:
10365:, and in this sense it is the "natural parametrization" of a binary probability.
9920:) business cycle with the following labels: beginning of a technological era as
4914:
for the first event (relative to the second), and, recalling that given odds of
4024:, just as the logistic function (with scaling) is a smooth approximation of the
3076:
3020:
2947:
2903:: the parametrization of the hyperbola for the logistic function corresponds to
11645:
Rocha, Leno S.; Rocha, Frederico S. A.; Souza, Thársis T. P. (5 October 2017).
11517:
11111:
Proceedings of the National Academy of Sciences of the United States of America
10362:
10156:), the theoretical or philosophical study of numbers. Confusingly, in English,
9883:
9637:
9624:
8840:
is the therapy-induced death rate. In the idealized case of very long therapy,
8372:
to the result; this model can be seen as a "smoothed" variant of the classical
7312:
6549:
In the equation, the early, unimpeded growth rate is modeled by the first term
6307:
3740:
2148:
544:
11151:
11126:
11066:
10869:
10775:[Mathematical Researches into the Law of Population Growth Increase].
12125:
11690:
11267:
11160:
11094:
10032:
9893:
Historically, when new products are introduced there is an intense amount of
9887:
9766:
9739:
9620:
9025:
8464:
8075:
Another interesting generalization is to consider that the carrying capacity
6333:
4021:
1205:
The sum of the logistic function and its reflection about the vertical axis,
700:
11922:
710:
11708:
11485:
11454:
11378:
Reiser, Paul A. (2020). "Modified SIR Model Yielding a Logistic Solution".
11343:
11026:
10852:"Math-alive! using original sources to teach mathematics in social context"
10829:
8353:
6285:
6272:, which is vector-valued, as it gives the probability of each alternative.
4181:
1199:
31:
11946:(Technical report). Vol. 119. Tinbergen Institute. pp. 167–178.
11051:"XLV.—The Rate of Multiplication of Micro-organisms: A Mathematical Study"
7900:
It can be shown that in such a case, independently from the initial value
7814:{\displaystyle {\frac {dP}{dt}}=rP\cdot \left(1-{\frac {P}{K(t)}}\right).}
6700:. The competition diminishes the combined growth rate, until the value of
6656:) becomes almost as large as the first, as some members of the population
11470:"Kinetic Models for the Degradation Mechanisms of PGM-Free ORR Catalysts"
11098:
10012:
9723:
8321:
6337:
4340:
gives the other well known form of the definition of the logistic curve:
1491:
580:
568:
535:
The logistic function finds applications in a range of fields, including
244:
11951:
11938:
11634:
11623:
10378:
function (the integral of the logistic function) is a smooth version of
6237:
term in many expressions for the logistic function and generalizations.
11982:
11890:
10183:
10158:
10042:
9870:
constraints, especially the resources the banks can lend (due to their
9749:
8373:
7315:
of the ecological form of the logistic function can be computed by the
3085:
1018:, it is often sufficient to compute the standard logistic function for
584:
560:
552:
11493:
11469:
11007:
10773:"Recherches mathématiques sur la loi d'accroissement de la population"
9607:
in epidemiological modeling is its relatively easy application to the
8309:. A generalisation of the logistic function to multiple inputs is the
1126:
is small is symmetric with the decay of the gap to the limit (1) when
12092:
12053:
10687:"Notice sur la loi que la population poursuit dans son accroissement"
10463:
9875:
8368:
of its input signals, and applies a bounded logistic function as the
572:
556:
548:
11235:
11050:
9754:
The logistic function can be used to illustrate the progress of the
9716:
9595:
6293:
A typical application of the logistic equation is a common model of
4433:{\displaystyle f(x)={\frac {e^{x}}{e^{x}+1}}={\frac {1}{1+e^{-x}}}.}
3185:{\displaystyle f(x)={\frac {1}{1+e^{-x}}}={\frac {e^{x}}{1+e^{x}}},}
634:
Verhulst did not explain the choice of the term "logistic" (French:
611:
The logistic function was introduced in a series of three papers by
11663:
11417:
11384:
11285:
10375:
9886:
diffusion of credit pleas and the aggregate national response is a
9867:
9063:
9020:
8479:
4872:
4016:
166:
12071:
11188:
6245:
In growth modeling, numerous generalizations exist, including the
5900:
More subtly, the symmetric form emphasizes interpreting the input
2445:, and thus has asymptotes the lines through the origin with slope
2249:, and thus has asymptotes the lines through the origin with slope
10142:
referred to practical computation and accounting, in contrast to
9732:
7115:
6489:{\displaystyle {\frac {dP}{dt}}=rP\left(1-{\frac {P}{K}}\right),}
6321:
4628:
can be desirable. Its solution is the shifted and scaled sigmoid
540:
536:
328:
11048:
4974:), the probability is the ratio of for over (for plus against),
9649:
9047:
3646:, which corresponds to parameters of the logistic function. If
705:"lodgings", though some believe the Greek term also influenced
12027:
Modeling nature: episodes in the history of population ecology
11960:
Cramer, J. S. (2004). "The early origins of the logit model".
11103:"On the Rate of Growth of the Population of the United States"
10814:"Deuxième mémoire sur la loi d'accroissement de la population"
5812:
This form immediately generalizes to more alternatives as the
5281:
This can be framed more symmetrically in terms of two inputs,
4191:
The qualitative behavior is easily understood in terms of the
4125:{\displaystyle {\frac {d}{dx}}f(x)=f(x){\big (}1-f(x){\big )}}
3865:{\displaystyle f(x)={\frac {e^{x}}{1+e^{x}}}={\frac {u'}{u}},}
2675:{\displaystyle \left((e^{t}+e^{-t})/2,(e^{t}-e^{-t})/2\right)}
658:), and thus "logistic growth" is presumably named by analogy,
10357:
In fact, the logistic function is the inverse mapping to the
10169:
10163:
10162:
refers to practical computation, even though it derives from
10150:
10143:
10137:
9905:
9390:
then the curve reduces to the logistic function, and (ii) as
4876:
4020:
function and (with scaling) is a smooth approximation of the
1498:
of two alternatives also takes the form of a logistic curve.
1404:
529:
9916:
Carlota Perez used a logistic curve to illustrate the long (
8485:
8455:
These relationships result in simplified implementations of
191:
is the logistic growth rate, the steepness of the curve; and
9676:
In linguistics, the logistic function can be used to model
9603:
One of the benefits of using a growth function such as the
8463:. Practitioners caution that sigmoidal functions which are
7193:
measures the population in units of carrying capacity, and
4911:
2818:, with quotient the logistic function. These correspond to
1487:
677:
12043:
11566:
10980:
2769:{\displaystyle {\bigl (}e^{t/2}+e^{-t/2},e^{t/2}{\bigr )}}
12082:
11607:
9683:
8802:{\displaystyle X'=r\left(1-{\frac {X}{K}}\right)X-c(t)X,}
8471:) lead to faster convergence when training networks with
8072:
may reflect periodical variations of weather conditions.
6280:
5110:
is the probability of the first alternative. Conversely,
11093:
9324:
is a positive real number. The flexibility of the curve
3084:
The standard logistic function has an easily calculated
11567:
López-Arias, T; Calzà, G; Gratton, L M; Oss, S (2009).
11520:"Measuring refractive index gradient of sugar solution"
11173:
10893:"Ancient logistics – historical timeline and etymology"
9630:
8356:
in the model or to clamp signals to within a specified
8332:
estimation of the locations of objects or persons on a
8320:
Another application of the logistic function is in the
6724:
of the population). The solution to the equation (with
6595:
represents the proportional increase of the population
10119:
curve, which is the inverse). He then calls his curve
9659:
9007:
5826:
3000:
2143:
Geometrically, the hyperbolic tangent function is the
10472:
10419:
10384:
10316:
10278:
10231:
10207:
9955:
9818:
9776:
9560:
9547:{\displaystyle (\theta _{1},\theta _{2},\theta _{3})}
9501:
9474:
9447:
9420:
9396:
9370:
9350:
9330:
9310:
9257:
9082:
9032:, as noted above) and then reaching a maximal limit.
8897:
8875:
8846:
8817:
8734:
8700:
8625:
8558:
8536:
8507:
8387:
8284:
8264:
8244:
8224:
8175:
8149:
8108:
8081:
8049:
8029:
8006:
7970:
7941:
7906:
7851:
7829:
7738:
7701:
7451:
7392:
7324:
7249:
7219:
7199:
7179:
7145:
7125:
7082:
7047:
7027:
7007:
6951:
6759:
6730:
6706:
6686:
6662:
6621:
6601:
6581:
6555:
6528:
6504:
6427:
6401:
6381:
6361:
6324:
population. The equation was rediscovered in 1911 by
6220:
6187:
6126:
6093:
6062:
6014:
5987:
5954:
5910:
5733:
5505:
5458:
5414:
5382:
5351:
5320:
5289:
5185:
5155:
5126:
5020:
4980:
4954:
4920:
4887:
4841:
4809:
4777:
4731:
4723:, the value of the logistic function is in the range
4703:
4634:
4484:
4455:
4348:
4320:
4251:
4221:
4201:
4143:
4047:
3886:
3793:
3752:
3715:
3684:
3654:
3199:
3100:
2998:
2937:
2909:
2860:
2830:
2782:
2688:
2578:
2537:
2501:
2477:
2453:
2410:
2368:
2344:
2308:
2284:
2257:
2202:
2156:
2049:
1659:
1598:
1516:
1416:
1245:
1211:
1155:
1132:
1112:
1054:
1046:
The logistic function has the symmetry property that
1024:
994:
822:
794:
761:
735:
463:
403:
338:
311:
282:
253:
226:
199:
177:
147:
60:
12024:
12003:
11983:
Jannedy, Stefanie; Bod, Rens; Hay, Jennifer (2003).
11734:
International Institute for Applied Systems Analysis
10752:
9750:
In economics and sociology: diffusion of innovations
9035:
A logistic function, or related functions (e.g. the
2682:, with quotient the hyperbolic tangent. Similarly,
1403:
The logistic function is the inverse of the natural
615:
between 1838 and 1847, who devised it as a model of
11236:LeCun, Y.; Bottou, L.; Orr, G.; Muller, K. (1998).
11049:A. G. McKendricka; M. Kesava Paia1 (January 1912).
10890:
9297:{\displaystyle \theta _{1},\theta _{2},\theta _{3}}
9054:(Richards growth curve) in epidemiological modeling
8608:{\displaystyle X'=r\left(1-{\frac {X}{K}}\right)X,}
8328:. In particular, the Rasch model forms a basis for
6348:derived the equation again in 1925, calling it the
129:{\displaystyle f(x)={\frac {L}{1+e^{-k(x-x_{0})}}}}
11849:
11124:
11042:
10655:
10454:
10405:
10340:
10302:
10249:
10213:
9994:
9851:
9804:
9671:
9584:
9546:
9487:
9460:
9433:
9402:
9382:
9356:
9336:
9316:
9296:
9241:
8993:
8881:
8861:
8832:
8801:
8715:
8684:
8607:
8542:
8522:
8443:{\displaystyle g(h)={\frac {1}{1+e^{-2\beta h}}},}
8442:
8290:
8270:
8250:
8230:
8208:
8155:
8096:
8064:
8035:
8012:
7992:
7956:
7927:
7890:
7835:
7813:
7722:
7680:
7436:
7378:
7296:
7233:
7205:
7185:
7151:
7131:
7103:
7068:
7033:
7013:
6991:
6933:
6743:
6712:
6692:
6668:
6648:
6607:
6587:
6567:
6534:
6510:
6488:
6407:
6387:
6367:
6229:
6206:
6165:
6112:
6075:
6040:
6000:
5973:
5936:
5889:
5801:
5719:
5491:
5440:
5395:
5364:
5333:
5302:
5270:
5171:
5135:
5102:
5006:
4960:
4938:
4900:
4856:
4827:
4795:
4749:
4715:
4675:
4620:
4467:
4432:
4332:
4304:
4227:
4207:
4172:
4124:
4000:
3864:
3777:
3721:
3697:
3666:
3632:
3184:
3060:
2984:
2923:
2895:
2845:
2810:
2768:
2674:
2551:
2519:
2483:
2459:
2437:
2396:
2352:
2326:
2290:
2266:
2241:
2188:
2126:
2031:
1643:
1584:
1475:
1385:
1229:
1190:
1138:
1118:
1096:
1030:
1010:
975:
806:
780:
747:
509:
446:
381:
317:
297:
268:
232:
212:
183:
153:
128:
9623:follow the logistic function. The degradation of
8869:can be modeled as a periodic function (of period
7690:
4180:. This equation is the continuous version of the
4031:
3080:The logistic function and its first 3 derivatives
12123:
11399:Lee, Se Yoon; Lei, Bowen; Mallick, Bani (2020).
11278:
11220:
10891:Tepic, J.; Tanackov, I.; Stojić, Gordan (2011).
10733:expit documentation for R's clusterPower package
10420:
10385:
10180:Nicomachus of Gerasa: Introduction to Arithmetic
10111:, and refers to the geometric growth curve as a
9812:; finally, the third stage is logarithmic, with
9614:
9074:in the literature. One frequently used forms is
9042:
8952:
7241:, gives the dimensionless differential equation
6953:
6415:represent time, this model is formalized by the
3088:. The derivative is known as the density of the
12072:Why logistic ogive and not autocatalytic curve?
11644:
10931:
10115:curve (confusingly, the modern term is instead
9696:growth curves. For example, the crop yield may
9619:The concentration of reactants and products in
9070:. There are different parameterizations of the
8501:, allowing for more parameters). Denoting with
7730:, leading to the following mathematical model:
6680:, and is modeled by the value of the parameter
5890:{\textstyle e^{x_{i}}/\sum _{i=0}^{n}e^{x_{i}}}
5802:{\displaystyle e^{x_{0}}/(e^{x_{0}}+e^{x_{1}})}
5271:{\displaystyle e^{-x}/(e^{-x}+1)=1/(1+e^{x})=q}
4686:
4239:The logistic equation is a special case of the
11727:"Technological Transformations and Long Waves"
11167:
11125:Griffiths, Graham; Schiesser, William (2009).
10680:
10678:
10413:, while the relative form is a smooth form of
7379:{\displaystyle u=K+P_{0}\left(e^{rt}-1\right)}
5727:is the probability of option 1, and similarly
5278:is the probability of the second alternative.
5103:{\displaystyle e^{x}/(e^{x}+1)=1/(1+e^{-x})=p}
4305:{\displaystyle f(x)={\frac {e^{x}}{e^{x}+C}}.}
1506:The logistic function is an offset and scaled
12010:. Cambridge, UK: Cambridge University Press.
11764:"Pervasive Long Waves: Is Society Cyclotymic"
11398:
11055:Proceedings of the Royal Society of Edinburgh
9738:Inverted S-curve model for crop yield versus
9722:S-curve model for crop yield versus depth of
4668:
4643:
4566:
4541:
4117:
4092:
3053:
3013:
2977:
2940:
2761:
2691:
12117:Restricted Logarithmic Growth with Injection
9410:approaches zero, the curve converges to the
7297:{\displaystyle {\frac {dn}{d\tau }}=n(1-n).}
5179:is the odds for the second alternative, and
11821:
11079:: CS1 maint: numeric names: authors list (
10957:Neural Networks – A Systematic Introduction
10746:"Scipy.special.expit — SciPy v1.7.1 Manual"
10675:
8685:{\displaystyle X'=F(X)X,\quad F'(X)\leq 0,}
8163:of an event may be affected by one or more
6992:{\displaystyle \lim _{t\to \infty }P(t)=K,}
5448:is the log-odds for option 1 (the log-odds
2529:, corresponding to the range and midpoint (
638:), but it is presumably in contrast to the
11908:. Taylor & Francis. pp. 619–630.
11837:. Heidelberg and New York: Physica-Verlag.
11613:Bod, Hay, Jennedy (eds.) 2003, pp. 147–156
11467:
11371:
11268:Worldometer: COVID-19 CORONAVIRUS PANDEMIC
11229:
10361:of the Bernoulli distribution, namely the
5816:, which is a vector-valued function whose
4475:. In many modeling applications, the more
2136:which ties the logistic function into the
1106:This reflects that the growth from 0 when
716:
510:{\displaystyle f(x)={\frac {1}{1+e^{-x}}}}
11876:
11841:
11790:
11761:
11698:
11680:
11662:
11543:
11444:
11434:
11416:
11383:
11362:
11333:
11299:
11284:
11187:
11150:
11016:
11006:
8935:
8486:In medicine: modeling of growth of tumors
7567:
7534:
3945:
3922:
2336:corresponding to the range and midpoint (
729:is the logistic function with parameters
12029:. Chicago: University of Chicago Press.
11468:Yin, Xi; Zelenay, Piotr (13 July 2018).
10811:
10770:
10720:"Sigmoid — PyTorch 1.10.1 documentation"
10684:
9594:
9046:
8723:is the proliferation rate of the tumor.
8167:: an example would be to have the model
7964:will tend to a unique periodic solution
6284:
4759:and can be interpreted as a probability
4676:{\displaystyle aS{\big (}k(x-r){\big )}}
3075:
2572:give coordinates on the unit hyperbola:
602:
327:
11989:. Cambridge, Massachusetts: MIT Press.
11856:. UK: Edward Elgar Publishing Limited.
11827:
11114:. Vol. 6, no. 6. p. 275.
10849:
10694:Correspondance Mathématique et Physique
8278:are model parameters to be fitted, and
6313:An Essay on the Principle of Population
5948:to some reference point, implicitly to
2822:(and rescaling the parametrization) of
14:
12124:
11959:
11936:
11793:"Kondratiev Revisited-After One Cycle"
11377:
10953:
10886:
10884:
10758:
9939:
9684:In agriculture: modeling crop response
8492:Gompertz curve § Growth of tumors
8128:
6395:is often used in ecology instead) and
6281:In ecology: modeling population growth
6008:, which corresponds to the difference
985:In practice, due to the nature of the
12083:
12044:
11847:
11721:
11524:Journal of Physics: Conference Series
11356:
8348:Logistic functions are often used in
8125:. More specific examples now follow.
6332:following its rediscovery in 1920 by
4314:Choosing the constant of integration
1501:
243:The logistic function has domain the
11903:
11245:Neural Networks: Tricks of the trade
10103:Verhulst first refers to arithmetic
9643:
9631:In physics: Fermi–Dirac distribution
9014:Compartmental models in epidemiology
6289:Pierre-François Verhulst (1804–1849)
592:
528:, being the inverse function of the
12007:The Nature of Mathematical Modeling
10881:
9995:{\displaystyle 1/(1+e^{-\theta A})}
9660:In material science: Phase diagrams
9008:In medicine: modeling of a pandemic
8298:is the standard logistic function.
1398:
24:
11940:The origins of logistic regression
11300:Postnikov, Eugene B. (June 2020).
10812:Verhulst, Pierre-François (1847).
10771:Verhulst, Pierre-François (1845).
10685:Verhulst, Pierre-François (1838).
10482:
10478:
10475:
10326:
10288:
8965:
8343:
8109:In statistics and machine learning
6963:
6240:
4851:
4215:between 0 and 1, and negative for
3213:
3207:
1644:{\displaystyle \tanh(x)=2f(2x)-1.}
524:. It is also sometimes called the
292:
263:
25:
12168:
12112:Seeing the s-curve in everything.
12064:
11243:. In Orr, G.; Muller, K. (eds.).
10455:{\displaystyle \max(x_{0},x_{1})}
7437:{\displaystyle du=rP_{0}e^{rt}dt}
7165:processes that have shaped their
6751:being the initial population) is
6166:{\displaystyle x_{i}=x_{i}-x_{0}}
4867:In this interpretation the input
3019:
2946:
1486:and so converts the logarithm of
1191:{\displaystyle x\mapsto f(x)-1/2}
332:Standard logistic function where
240:value of the function's midpoint.
30:For the recurrence relation, see
12102:Online experiments with JSXGraph
9731:
9715:
8119:logistic family of distributions
8115:cumulative distribution function
7171:Choosing the variable dimensions
6263:hyperbolastic function of type I
6255:cumulative distribution function
5809:is the probability of option 0.
4857:{\displaystyle x\to \pm \infty }
4243:and has the following solution:
4188:ordinary differential equation.
11784:
11755:
11715:
11638:
11627:
11616:
11560:
11545:10.1088/1742-6596/2145/1/012072
11461:
11392:
11350:
11293:
11272:
11261:
11118:
11087:
11033:
10974:
10947:
10925:
10368:
10351:
10262:
10193:
10130:
10097:
9672:In linguistics: language change
9414:. In epidemiological modeling,
8655:
8550:, its dynamics are governed by
8315:multinomial logistic regression
8139:Logistic functions are used in
6720:ceases to grow (this is called
6275:
5492:{\displaystyle e^{x_{1}-x_{0}}}
5405:, interpreted as logits, their
4574:
4241:Bernoulli differential equation
3642:The logistic distribution is a
520:and is sometimes simply called
447:{\displaystyle L=1,k=1,x_{0}=0}
382:{\displaystyle L=1,k=1,x_{0}=0}
11306:Chaos, Solitons & Fractals
11226:Gershenfeld 1999, p. 150.
11176:Physica D: Nonlinear Phenomena
10843:
10805:
10764:
10738:
10726:
10712:
10647:
10595:
10583:
10545:
10533:
10501:
10449:
10423:
10400:
10388:
10329:
10320:
10291:
10282:
10244:
10232:
10088:
10038:List of mathematical functions
9989:
9964:
9846:
9840:
9828:
9822:
9786:
9780:
9585:{\displaystyle (10000,0.2,40)}
9579:
9561:
9541:
9502:
9219:
9215:
9212:
9193:
9174:
9156:
9137:
9086:
8979:
8973:
8959:
8948:
8932:
8926:
8856:
8850:
8827:
8821:
8790:
8784:
8710:
8704:
8670:
8664:
8646:
8640:
8530:the size of the tumor at time
8517:
8511:
8452:which is a logistic function.
8397:
8391:
8301:Logistic regression and other
8200:
8185:
8091:
8085:
8059:
8053:
7987:
7981:
7951:
7945:
7916:
7910:
7882:
7876:
7867:
7855:
7797:
7791:
7711:
7705:
7691:Time-varying carrying capacity
7664:
7642:
7288:
7276:
7092:
7086:
7057:
7051:
6977:
6971:
6960:
6769:
6763:
6048:being the log-odds for option
5796:
5756:
5714:
5674:
5639:
5613:
5580:
5541:
5259:
5240:
5226:
5204:
5091:
5069:
5055:
5036:
5001:
4989:
4845:
4744:
4732:
4663:
4651:
4584:
4578:
4561:
4555:
4536:
4530:
4500:
4494:
4358:
4352:
4261:
4255:
4153:
4147:
4112:
4106:
4087:
4081:
4072:
4066:
4038:ordinary differential equation
4032:Logistic differential equation
3992:
3973:
3803:
3797:
3707:is the location and the slope
3618:
3612:
3595:
3589:
3383:
3363:
3328:
3308:
3277:
3258:
3232:
3226:
3110:
3104:
2931:and the linear transformation
2896:{\displaystyle (e^{-t},e^{t})}
2890:
2861:
2656:
2627:
2613:
2584:
2514:
2502:
2426:
2414:
2321:
2309:
2230:
2218:
2215:
2203:
2074:
2068:
2016:
2007:
1934:
1925:
1869:
1860:
1676:
1670:
1632:
1623:
1611:
1605:
1526:
1520:
1306:
1297:
1224:
1215:
1171:
1165:
1159:
1088:
1079:
1070:
1064:
832:
826:
473:
467:
286:
257:
169:of the values of the function;
118:
99:
70:
64:
13:
1:
12025:Kingsland, Sharon E. (1995).
12004:Gershenfeld, Neil A. (1999).
11906:Attention and Performance VII
11771:Aspen Global Change INstitute
11364:10.1101/2020.06.25.20139865v2
10669:
10341:{\displaystyle f(+\infty )=1}
10303:{\displaystyle f(-\infty )=0}
10070:Shifted Gompertz distribution
9944:Link created an extension of
9615:In chemistry: reaction models
9605:generalized logistic function
9072:generalized logistic function
9060:generalized logistic function
9052:Generalized logistic function
9043:Modeling early COVID-19 cases
8238:is the explanatory variable,
8143:to model how the probability
7306:
7118:are sometimes referred to as
6259:shifted Gompertz distribution
3071:
2189:{\displaystyle x^{2}-y^{2}=1}
1097:{\displaystyle 1-f(x)=f(-x).}
1041:
654:, instead of the modern term
623:model, under the guidance of
298:{\displaystyle x\to +\infty }
269:{\displaystyle x\to -\infty }
11682:10.1371/journal.pone.0185257
11436:10.1371/journal.pone.0236860
11127:"Linear and nonlinear waves"
10348:, matching the limit values.
10268:This can be extended to the
10151:
10028:Hill equation (biochemistry)
9852:{\displaystyle f(x)=\log(x)}
7891:{\displaystyle K(t+T)=K(t).}
7076:, and also in the case that
6056:, but the individual logits
4687:Probabilistic interpretation
3676:is fixed, then the midpoint
2561:) of the logistic function.
2242:{\displaystyle (x+y)(x-y)=1}
682:, a traditional division of
678:
47:is a common S-shaped curve (
7:
11974:10.1016/j.shpsc.2004.09.003
11326:10.1016/j.chaos.2020.109841
11206:10.1016/j.physd.2009.05.011
10065:Rectifier (neural networks)
10005:
9488:{\displaystyle \theta _{3}}
9461:{\displaystyle \theta _{2}}
9434:{\displaystyle \theta _{1}}
8467:about the origin (e.g. the
8311:softmax activation function
6375:represent population size (
6173:is the log-odds for option
6041:{\displaystyle x_{j}-x_{i}}
5937:{\displaystyle x_{1}-x_{0}}
5441:{\displaystyle x_{1}-x_{0}}
3734:
2776:parametrizes the hyperbola
1494:. The conversion from the
397:, depicted at right, where
10:
12173:
11791:Marchetti, Cesare (1988).
11762:Marchetti, Cesare (1996).
11593:10.1088/0031-9120/44/6/002
10170:
10164:
10144:
10138:
9805:{\displaystyle f(x)=2^{x}}
9756:diffusion of an innovation
9011:
8499:Generalized logistic curve
8489:
8457:artificial neural networks
8350:artificial neural networks
8336:, based on collections of
8305:are also commonly used in
8209:{\displaystyle p=f(a+bx),}
8132:
8043:is one year: In such case
7213:measures time in units of
6247:generalized logistic curve
4690:
4012:artificial neural networks
2811:{\displaystyle xy-y^{2}=1}
2397:{\displaystyle xy-y^{2}=1}
725:standard logistic function
667:
598:
595:, depending on the field.
589:artificial neural networks
395:standard logistic function
29:
11986:Probabilistic Linguistics
11152:10.4249/scholarpedia.4308
11067:10.1017/S0370164600025426
10870:10.1080/10511979808965879
10796:Nous donnerons le nom de
10406:{\displaystyle \max(0,x)}
10270:Extended real number line
10075:Tipping point (sociology)
10048:Michaelis–Menten kinetics
9928:, the rapid build out as
7928:{\displaystyle P(0)>0}
7723:{\displaystyle K(t)>0}
7104:{\displaystyle P(0)>K}
7069:{\displaystyle P(0)>0}
7021:is the limiting value of
6649:{\displaystyle -rP^{2}/K}
3778:{\displaystyle u=1+e^{x}}
11828:Grübler, Arnulf (1990).
10850:Shulman, Bonnie (1998).
10081:
9895:research and development
9758:through its life cycle.
9344:is due to the parameter
9030:Pierre-François Verhulst
7993:{\displaystyle P_{*}(t)}
6575:. The value of the rate
6350:law of population growth
6342:Johns Hopkins University
6303:Pierre-François Verhulst
5149:the second alternative,
5118:the second alternative,
4173:{\displaystyle f(0)=1/2}
2438:{\displaystyle y(x-y)=1}
693:, which is instead from
650:(whose curve he calls a
613:Pierre François Verhulst
11848:Perez, Carlota (2002).
10176:Louis Charles Karpinski
10023:Heaviside step function
9621:autocatalytic reactions
6330:Verhulst-Pearl equation
6318:Malthusian growth model
6207:{\displaystyle e^{0}=1}
6113:{\displaystyle x_{0}=0}
5974:{\displaystyle x_{0}=0}
5007:{\displaystyle O/(O+1)}
4026:Heaviside step function
4014:, this is known as the
3876:constant of integration
3743:can be computed by the
2854:, with parametrization
781:{\displaystyle x_{0}=0}
717:Mathematical properties
711:Logistics § Origin
565:mathematical psychology
276:is 0, and the limit as
12137:Differential equations
12074:, accessed 2009-09-12.
11937:Cramer, J. S. (2002).
11486:10.1149/08513.1239ecst
10830:10.3406/marb.1847.3457
10657:
10456:
10407:
10342:
10304:
10251:
10223:for the parameter and
10215:
9996:
9932:and the completion as
9853:
9806:
9638:Fermi–Dirac statistics
9600:
9586:
9548:
9489:
9462:
9435:
9404:
9384:
9383:{\displaystyle \xi =1}
9358:
9338:
9318:
9304:are real numbers, and
9298:
9243:
9055:
8995:
8883:
8863:
8834:
8803:
8717:
8686:
8609:
8544:
8524:
8444:
8292:
8272:
8252:
8232:
8210:
8157:
8098:
8066:
8037:
8014:
7994:
7958:
7929:
7892:
7837:
7815:
7724:
7682:
7438:
7380:
7298:
7235:
7207:
7187:
7153:
7133:
7105:
7070:
7035:
7015:
6993:
6935:
6745:
6714:
6694:
6670:
6650:
6609:
6589:
6569:
6536:
6512:
6490:
6409:
6389:
6369:
6316:, which describes the
6290:
6231:
6208:
6167:
6114:
6077:
6042:
6002:
5975:
5938:
5891:
5869:
5803:
5721:
5493:
5442:
5397:
5366:
5335:
5304:
5272:
5173:
5172:{\displaystyle e^{-x}}
5137:
5104:
5008:
4962:
4940:
4902:
4858:
4829:
4797:
4769:Bernoulli distribution
4751:
4717:
4677:
4622:
4469:
4468:{\displaystyle x>0}
4434:
4334:
4306:
4229:
4209:
4174:
4126:
4002:
3866:
3779:
3723:
3699:
3668:
3634:
3186:
3081:
3062:
2986:
2925:
2897:
2847:
2820:linear transformations
2812:
2770:
2676:
2553:
2521:
2485:
2461:
2439:
2398:
2354:
2328:
2292:
2268:
2243:
2190:
2128:
2033:
1645:
1586:
1477:
1387:
1231:
1192:
1140:
1120:
1098:
1032:
1012:
1011:{\displaystyle e^{-x}}
977:
808:
782:
749:
699:
698:
635:
608:
511:
448:
390:
383:
319:
299:
270:
234:
214:
185:
155:
130:
12107:Esses are everywhere.
11569:"Mirages in a bottle"
10658:
10457:
10408:
10343:
10305:
10252:
10250:{\displaystyle (x,y)}
10216:
9997:
9854:
9807:
9763:The Laws of Imitation
9598:
9587:
9549:
9490:
9463:
9436:
9405:
9385:
9359:
9339:
9319:
9299:
9244:
9050:
8996:
8884:
8864:
8835:
8804:
8718:
8687:
8617:which is of the type
8610:
8545:
8525:
8445:
8293:
8273:
8253:
8233:
8211:
8165:explanatory variables
8158:
8099:
8067:
8038:
8015:
7995:
7959:
7930:
7893:
7838:
7816:
7725:
7683:
7439:
7381:
7299:
7236:
7208:
7206:{\displaystyle \tau }
7188:
7154:
7134:
7106:
7071:
7036:
7016:
6994:
6936:
6746:
6744:{\displaystyle P_{0}}
6715:
6695:
6671:
6651:
6610:
6590:
6570:
6537:
6513:
6491:
6417:differential equation
6410:
6390:
6370:
6344:. Another scientist,
6301:), originally due to
6288:
6232:
6209:
6168:
6115:
6078:
6076:{\displaystyle x_{i}}
6043:
6003:
6001:{\displaystyle x_{i}}
5976:
5939:
5892:
5849:
5804:
5722:
5494:
5443:
5398:
5396:{\displaystyle x_{1}}
5367:
5365:{\displaystyle x_{0}}
5336:
5334:{\displaystyle x_{1}}
5305:
5303:{\displaystyle x_{0}}
5273:
5174:
5138:
5105:
5009:
4963:
4941:
4939:{\displaystyle O=O:1}
4903:
4901:{\displaystyle e^{x}}
4859:
4830:
4828:{\displaystyle p+q=1}
4798:
4796:{\displaystyle q=1-p}
4752:
4750:{\displaystyle (0,1)}
4718:
4691:Further information:
4678:
4623:
4470:
4435:
4335:
4307:
4230:
4210:
4175:
4127:
4003:
3867:
3780:
3724:
3700:
3698:{\displaystyle x_{0}}
3669:
3644:location–scale family
3635:
3187:
3090:logistic distribution
3079:
3063:
2987:
2926:
2898:
2848:
2813:
2771:
2677:
2554:
2522:
2520:{\displaystyle (2,1)}
2486:
2462:
2440:
2399:
2355:
2329:
2327:{\displaystyle (1,0)}
2293:
2269:
2244:
2191:
2138:logistic distribution
2129:
2034:
1646:
1587:
1478:
1388:
1232:
1230:{\displaystyle f(-x)}
1193:
1141:
1121:
1099:
1033:
1013:
978:
809:
783:
750:
606:
512:
449:
384:
331:
320:
300:
271:
235:
215:
213:{\displaystyle x_{0}}
186:
156:
131:
11238:"Efficient BackProp"
10470:
10417:
10382:
10314:
10276:
10259:for the coordinates.
10229:
10205:
9953:
9910:long economic cycles
9882:for money, create a
9880:economic competition
9816:
9774:
9558:
9499:
9472:
9445:
9418:
9403:{\displaystyle \xi }
9394:
9368:
9357:{\displaystyle \xi }
9348:
9328:
9317:{\displaystyle \xi }
9308:
9255:
9080:
9068:infection trajectory
8895:
8873:
8862:{\displaystyle c(t)}
8844:
8833:{\displaystyle c(t)}
8815:
8732:
8716:{\displaystyle F(X)}
8698:
8623:
8556:
8534:
8523:{\displaystyle X(t)}
8505:
8385:
8326:item response theory
8282:
8262:
8242:
8222:
8173:
8147:
8097:{\displaystyle K(t)}
8079:
8065:{\displaystyle K(t)}
8047:
8027:
8004:
7968:
7957:{\displaystyle P(t)}
7939:
7904:
7849:
7827:
7736:
7699:
7449:
7390:
7322:
7247:
7217:
7197:
7177:
7143:
7123:
7080:
7045:
7025:
7005:
6949:
6757:
6728:
6704:
6684:
6660:
6619:
6599:
6579:
6553:
6526:
6502:
6425:
6399:
6379:
6359:
6218:
6185:
6124:
6091:
6060:
6012:
5985:
5952:
5908:
5824:
5731:
5503:
5456:
5412:
5380:
5349:
5318:
5287:
5183:
5153:
5124:
5018:
4978:
4952:
4918:
4885:
4839:
4807:
4775:
4729:
4701:
4632:
4482:
4453:
4346:
4318:
4249:
4219:
4199:
4141:
4045:
3884:
3791:
3750:
3713:
3682:
3652:
3197:
3098:
2996:
2935:
2907:
2858:
2846:{\displaystyle xy=1}
2828:
2780:
2686:
2576:
2535:
2499:
2475:
2451:
2408:
2366:
2342:
2306:
2282:
2255:
2200:
2154:
2047:
1657:
1596:
1514:
1496:log-likelihood ratio
1414:
1243:
1209:
1153:
1130:
1110:
1052:
1022:
992:
987:exponential function
820:
792:
759:
733:
591:. There are various
461:
401:
336:
309:
280:
251:
224:
197:
175:
145:
58:
51:) with the equation
12152:Logistic regression
12142:Population dynamics
12049:"Logistic Equation"
11952:10.2139/ssrn.360300
11673:2017PLoSO..1285257R
11585:2009PhyEd..44..582L
11536:2021JPhCS2145a2072T
11427:2020PLoSO..1536860L
11318:2020CSF...13509841P
11198:2009PhyD..238.1752Y
11143:2009SchpJ...4.4308G
10999:2020Senso..20.3246K
10136:In Ancient Greece,
9940:Sequential analysis
8922:
8370:activation function
8141:logistic regression
8135:Logistic regression
8129:Logistic regression
8023:A typical value of
7234:{\displaystyle 1/r}
7161:depending upon the
6568:{\displaystyle +rP}
6498:where the constant
6340:(1888–1966) of the
6299:population dynamics
4716:{\displaystyle L=1}
4693:Logistic regression
4333:{\displaystyle C=1}
3667:{\displaystyle L=1}
2924:{\displaystyle t/2}
2552:{\displaystyle 1/2}
2404:, which factors as
2353:{\displaystyle {1}}
2196:, which factors as
807:{\displaystyle L=1}
748:{\displaystyle k=1}
454:, has the equation
12147:Population ecology
12088:"Sigmoid Function"
12085:Weisstein, Eric W.
12046:Weisstein, Eric W.
11891:10.1007/BF02291481
10653:
10452:
10403:
10338:
10300:
10247:
10211:
10174:. See for example
9992:
9849:
9802:
9601:
9582:
9544:
9485:
9458:
9431:
9400:
9380:
9354:
9334:
9314:
9294:
9239:
9056:
8991:
8969:
8908:
8879:
8859:
8830:
8799:
8713:
8682:
8605:
8540:
8520:
8469:hyperbolic tangent
8461:artificial neurons
8440:
8366:linear combination
8362:neural net element
8330:maximum likelihood
8288:
8268:
8248:
8228:
8206:
8153:
8094:
8062:
8033:
8010:
8000:, whose period is
7990:
7954:
7925:
7888:
7833:
7811:
7720:
7678:
7434:
7376:
7294:
7231:
7203:
7183:
7149:
7129:
7101:
7066:
7031:
7011:
6989:
6967:
6931:
6741:
6710:
6690:
6666:
6646:
6605:
6585:
6565:
6532:
6508:
6486:
6405:
6385:
6365:
6291:
6230:{\displaystyle +1}
6227:
6214:, this yields the
6204:
6163:
6110:
6073:
6038:
5998:
5971:
5934:
5887:
5820:-th coordinate is
5799:
5717:
5489:
5438:
5393:
5362:
5331:
5300:
5268:
5169:
5136:{\displaystyle -x}
5133:
5100:
5004:
4958:
4936:
4898:
4854:
4825:
4793:
4763:. In more detail,
4747:
4713:
4697:When the capacity
4673:
4618:
4465:
4444:exponential growth
4430:
4330:
4302:
4225:
4205:
4170:
4136:boundary condition
4122:
3998:
3862:
3775:
3719:
3695:
3664:
3630:
3628:
3182:
3082:
3058:
3049:
3048:
3009:
2982:
2973:
2972:
2921:
2893:
2843:
2808:
2766:
2672:
2549:
2517:
2481:
2457:
2435:
2394:
2350:
2324:
2288:
2267:{\displaystyle -1}
2264:
2239:
2186:
2124:
2029:
2027:
1653:This follows from
1641:
1582:
1508:hyperbolic tangent
1502:Hyperbolic tangent
1473:
1383:
1227:
1188:
1136:
1116:
1094:
1028:
1008:
973:
804:
778:
745:
621:exponential growth
609:
507:
444:
391:
379:
315:
295:
266:
230:
210:
181:
151:
126:
12132:Special functions
12017:978-0-521-57095-4
11725:(February 1989).
11573:Physics Education
11480:(13): 1239–1250.
11182:(17): 1752–1767.
11008:10.3390/s20113246
10900:Technical Gazette
10374:For example, the
10359:natural parameter
10214:{\displaystyle t}
10123:, in contrast to
10018:Hyperbolic growth
9666:Diffusion bonding
9644:In optics: mirage
9412:Gompertz function
9337:{\displaystyle f}
9237:
9037:Gompertz function
8951:
8906:
8882:{\displaystyle T}
8768:
8592:
8543:{\displaystyle t}
8435:
8303:log-linear models
8291:{\displaystyle f}
8271:{\displaystyle b}
8251:{\displaystyle a}
8231:{\displaystyle x}
8156:{\displaystyle p}
8123:Elo rating system
8036:{\displaystyle T}
8013:{\displaystyle T}
7836:{\displaystyle T}
7801:
7757:
7613:
7585:
7565:
7555:
7532:
7268:
7186:{\displaystyle n}
7152:{\displaystyle K}
7132:{\displaystyle r}
7034:{\displaystyle P}
7014:{\displaystyle K}
6952:
6926:
6903:
6852:
6713:{\displaystyle P}
6693:{\displaystyle K}
6669:{\displaystyle P}
6608:{\displaystyle P}
6588:{\displaystyle r}
6544:carrying capacity
6535:{\displaystyle K}
6511:{\displaystyle r}
6476:
6446:
6408:{\displaystyle t}
6388:{\displaystyle N}
6368:{\displaystyle P}
6295:population growth
6251:Gompertz function
4961:{\displaystyle O}
4616:
4525:
4512:
4425:
4394:
4297:
4228:{\displaystyle f}
4208:{\displaystyle f}
4061:
3943:
3920:
3874:so (dropping the
3857:
3839:
3722:{\displaystyle k}
3569:
3522:
3472:
3439:
3393:
3338:
3221:
3177:
3142:
3008:
2566:hyperbolic cosine
2484:{\displaystyle 1}
2460:{\displaystyle 0}
2291:{\displaystyle 1}
2115:
2088:
2063:
1996:
1917:
1845:
1744:
1573:
1553:
1540:
1456:
1451:
1375:
1347:
1312:
1272:
1139:{\displaystyle x}
1119:{\displaystyle x}
1031:{\displaystyle x}
968:
899:
864:
684:Greek mathematics
676:
656:exponential curve
652:logarithmic curve
644:arithmetic growth
619:by adjusting the
617:population growth
577:political science
505:
318:{\displaystyle L}
233:{\displaystyle x}
184:{\displaystyle k}
163:carrying capacity
154:{\displaystyle L}
124:
41:logistic function
16:(Redirected from
12164:
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12021:
12000:
11977:
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11855:
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11825:
11819:
11818:
11816:
11814:
11808:
11802:. Archived from
11800:Cesare Marchetti
11797:
11788:
11782:
11781:
11780:on 5 March 2012.
11779:
11773:. Archived from
11768:
11759:
11753:
11752:
11750:
11748:
11742:
11736:. Archived from
11731:
11719:
11713:
11712:
11702:
11684:
11666:
11657:(10): e0185257.
11642:
11636:
11631:
11625:
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11614:
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11605:
11604:
11564:
11558:
11557:
11547:
11515:
11506:
11505:
11474:ECS Transactions
11465:
11459:
11458:
11448:
11438:
11420:
11396:
11390:
11389:
11387:
11375:
11369:
11368:
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10978:
10972:
10971:
10969:
10967:
10962:
10951:
10945:
10944:
10929:
10923:
10922:
10921:on 9 March 2019.
10920:
10914:. Archived from
10897:
10888:
10879:
10878:
10847:
10841:
10840:
10838:
10836:
10809:
10803:
10802:
10793:
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10497:
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10435:
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10412:
10410:
10409:
10404:
10372:
10366:
10355:
10349:
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10309:
10307:
10306:
10301:
10266:
10260:
10258:
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10254:
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10248:
10222:
10220:
10218:
10217:
10212:
10197:
10191:
10173:
10172:
10167:
10166:
10154:
10147:
10146:
10141:
10140:
10134:
10128:
10101:
10095:
10092:
10060:selection theory
10001:
9999:
9998:
9993:
9988:
9987:
9963:
9924:, the ascent as
9858:
9856:
9855:
9850:
9811:
9809:
9808:
9803:
9801:
9800:
9735:
9719:
9609:multilevel model
9591:
9589:
9588:
9583:
9553:
9551:
9550:
9545:
9540:
9539:
9527:
9526:
9514:
9513:
9494:
9492:
9491:
9486:
9484:
9483:
9467:
9465:
9464:
9459:
9457:
9456:
9440:
9438:
9437:
9432:
9430:
9429:
9409:
9407:
9406:
9401:
9389:
9387:
9386:
9381:
9363:
9361:
9360:
9355:
9343:
9341:
9340:
9335:
9323:
9321:
9320:
9315:
9303:
9301:
9300:
9295:
9293:
9292:
9280:
9279:
9267:
9266:
9248:
9246:
9245:
9240:
9238:
9236:
9235:
9234:
9230:
9211:
9210:
9189:
9188:
9154:
9153:
9144:
9130:
9129:
9117:
9116:
9104:
9103:
9000:
8998:
8997:
8992:
8968:
8921:
8916:
8907:
8899:
8888:
8886:
8885:
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8868:
8866:
8865:
8860:
8839:
8837:
8836:
8831:
8808:
8806:
8805:
8800:
8774:
8770:
8769:
8761:
8742:
8722:
8720:
8719:
8714:
8691:
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8663:
8633:
8614:
8612:
8611:
8606:
8598:
8594:
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8585:
8566:
8549:
8547:
8546:
8541:
8529:
8527:
8526:
8521:
8449:
8447:
8446:
8441:
8436:
8434:
8433:
8432:
8404:
8374:threshold neuron
8338:categorical data
8307:machine learning
8297:
8295:
8294:
8289:
8277:
8275:
8274:
8269:
8257:
8255:
8254:
8249:
8237:
8235:
8234:
8229:
8215:
8213:
8212:
8207:
8162:
8160:
8159:
8154:
8103:
8101:
8100:
8095:
8071:
8069:
8068:
8063:
8042:
8040:
8039:
8034:
8019:
8017:
8016:
8011:
7999:
7997:
7996:
7991:
7980:
7979:
7963:
7961:
7960:
7955:
7934:
7932:
7931:
7926:
7897:
7895:
7894:
7889:
7842:
7840:
7839:
7834:
7820:
7818:
7817:
7812:
7807:
7803:
7802:
7800:
7783:
7758:
7756:
7748:
7740:
7729:
7727:
7726:
7721:
7687:
7685:
7684:
7679:
7671:
7667:
7657:
7656:
7641:
7640:
7614:
7606:
7586:
7578:
7566:
7558:
7556:
7548:
7533:
7531:
7530:
7526:
7519:
7518:
7501:
7500:
7484:
7483:
7482:
7470:
7469:
7456:
7443:
7441:
7440:
7435:
7427:
7426:
7414:
7413:
7385:
7383:
7382:
7377:
7375:
7371:
7364:
7363:
7346:
7345:
7303:
7301:
7300:
7295:
7269:
7267:
7259:
7251:
7240:
7238:
7237:
7232:
7227:
7212:
7210:
7209:
7204:
7192:
7190:
7189:
7184:
7158:
7156:
7155:
7150:
7138:
7136:
7135:
7130:
7110:
7108:
7107:
7102:
7075:
7073:
7072:
7067:
7040:
7038:
7037:
7032:
7020:
7018:
7017:
7012:
6998:
6996:
6995:
6990:
6966:
6940:
6938:
6937:
6932:
6927:
6925:
6924:
6923:
6908:
6904:
6902:
6901:
6892:
6891:
6890:
6874:
6858:
6853:
6851:
6850:
6846:
6839:
6838:
6821:
6820:
6804:
6803:
6802:
6790:
6789:
6776:
6750:
6748:
6747:
6742:
6740:
6739:
6719:
6717:
6716:
6711:
6699:
6697:
6696:
6691:
6675:
6673:
6672:
6667:
6655:
6653:
6652:
6647:
6642:
6637:
6636:
6614:
6612:
6611:
6606:
6594:
6592:
6591:
6586:
6574:
6572:
6571:
6566:
6541:
6539:
6538:
6533:
6517:
6515:
6514:
6509:
6495:
6493:
6492:
6487:
6482:
6478:
6477:
6469:
6447:
6445:
6437:
6429:
6414:
6412:
6411:
6406:
6394:
6392:
6391:
6386:
6374:
6372:
6371:
6366:
6336:(1879–1940) and
6326:A. G. McKendrick
6270:softmax function
6236:
6234:
6233:
6228:
6213:
6211:
6210:
6205:
6197:
6196:
6180:
6176:
6172:
6170:
6169:
6164:
6162:
6161:
6149:
6148:
6136:
6135:
6119:
6117:
6116:
6111:
6103:
6102:
6086:
6082:
6080:
6079:
6074:
6072:
6071:
6055:
6051:
6047:
6045:
6044:
6039:
6037:
6036:
6024:
6023:
6007:
6005:
6004:
5999:
5997:
5996:
5980:
5978:
5977:
5972:
5964:
5963:
5943:
5941:
5940:
5935:
5933:
5932:
5920:
5919:
5903:
5896:
5894:
5893:
5888:
5886:
5885:
5884:
5883:
5868:
5863:
5848:
5843:
5842:
5841:
5840:
5819:
5814:softmax function
5808:
5806:
5805:
5800:
5795:
5794:
5793:
5792:
5775:
5774:
5773:
5772:
5755:
5750:
5749:
5748:
5747:
5726:
5724:
5723:
5718:
5713:
5712:
5711:
5710:
5693:
5692:
5691:
5690:
5673:
5668:
5667:
5666:
5665:
5648:
5644:
5643:
5642:
5638:
5637:
5625:
5624:
5593:
5573:
5572:
5571:
5570:
5558:
5557:
5540:
5535:
5534:
5533:
5532:
5520:
5519:
5498:
5496:
5495:
5490:
5488:
5487:
5486:
5485:
5473:
5472:
5447:
5445:
5444:
5439:
5437:
5436:
5424:
5423:
5404:
5402:
5400:
5399:
5394:
5392:
5391:
5373:
5371:
5369:
5368:
5363:
5361:
5360:
5342:
5340:
5338:
5337:
5332:
5330:
5329:
5311:
5309:
5307:
5306:
5301:
5299:
5298:
5277:
5275:
5274:
5269:
5258:
5257:
5239:
5219:
5218:
5203:
5198:
5197:
5178:
5176:
5175:
5170:
5168:
5167:
5145:is the log-odds
5144:
5142:
5140:
5139:
5134:
5114:is the log-odds
5113:
5109:
5107:
5106:
5101:
5090:
5089:
5068:
5048:
5047:
5035:
5030:
5029:
5013:
5011:
5010:
5005:
4988:
4973:
4969:
4967:
4965:
4964:
4959:
4945:
4943:
4942:
4937:
4909:
4907:
4905:
4904:
4899:
4897:
4896:
4870:
4863:
4861:
4860:
4855:
4834:
4832:
4831:
4826:
4802:
4800:
4799:
4794:
4766:
4762:
4758:
4756:
4754:
4753:
4748:
4722:
4720:
4719:
4714:
4682:
4680:
4679:
4674:
4672:
4671:
4647:
4646:
4627:
4625:
4624:
4619:
4617:
4615:
4614:
4613:
4591:
4570:
4569:
4545:
4544:
4526:
4518:
4513:
4511:
4503:
4486:
4474:
4472:
4471:
4466:
4439:
4437:
4436:
4431:
4426:
4424:
4423:
4422:
4400:
4395:
4393:
4386:
4385:
4375:
4374:
4365:
4339:
4337:
4336:
4331:
4311:
4309:
4308:
4303:
4298:
4296:
4289:
4288:
4278:
4277:
4268:
4234:
4232:
4231:
4226:
4214:
4212:
4211:
4206:
4179:
4177:
4176:
4171:
4166:
4131:
4129:
4128:
4123:
4121:
4120:
4096:
4095:
4062:
4060:
4049:
4007:
4005:
4004:
3999:
3991:
3990:
3944:
3936:
3921:
3919:
3918:
3917:
3901:
3900:
3891:
3871:
3869:
3868:
3863:
3858:
3853:
3845:
3840:
3838:
3837:
3836:
3820:
3819:
3810:
3784:
3782:
3781:
3776:
3774:
3773:
3739:Conversely, its
3730:
3728:
3726:
3725:
3720:
3706:
3704:
3702:
3701:
3696:
3694:
3693:
3675:
3673:
3671:
3670:
3665:
3639:
3637:
3636:
3631:
3629:
3625:
3621:
3579:
3575:
3571:
3570:
3568:
3567:
3566:
3550:
3549:
3540:
3527:
3523:
3521:
3520:
3519:
3503:
3502:
3493:
3481:
3477:
3473:
3471:
3470:
3469:
3450:
3444:
3440:
3438:
3437:
3436:
3420:
3419:
3410:
3398:
3394:
3392:
3391:
3390:
3381:
3380:
3361:
3360:
3351:
3343:
3339:
3337:
3336:
3335:
3326:
3325:
3306:
3305:
3304:
3292:
3291:
3276:
3275:
3254:
3253:
3243:
3222:
3220:
3216:
3210:
3205:
3191:
3189:
3188:
3183:
3178:
3176:
3175:
3174:
3158:
3157:
3148:
3143:
3141:
3140:
3139:
3117:
3067:
3065:
3064:
3059:
3057:
3056:
3050:
3017:
3016:
3010:
3001:
2991:
2989:
2988:
2983:
2981:
2980:
2974:
2944:
2943:
2930:
2928:
2927:
2922:
2917:
2902:
2900:
2899:
2894:
2889:
2888:
2876:
2875:
2852:
2850:
2849:
2844:
2817:
2815:
2814:
2809:
2801:
2800:
2775:
2773:
2772:
2767:
2765:
2764:
2758:
2757:
2753:
2737:
2736:
2732:
2713:
2712:
2708:
2695:
2694:
2681:
2679:
2678:
2673:
2671:
2667:
2663:
2655:
2654:
2639:
2638:
2620:
2612:
2611:
2596:
2595:
2564:Parametrically,
2560:
2558:
2556:
2555:
2550:
2545:
2528:
2526:
2524:
2523:
2518:
2493:, and vertex at
2492:
2490:
2488:
2487:
2482:
2468:
2466:
2464:
2463:
2458:
2444:
2442:
2441:
2436:
2403:
2401:
2400:
2395:
2387:
2386:
2361:
2359:
2357:
2356:
2351:
2349:
2335:
2333:
2331:
2330:
2325:
2300:, and vertex at
2299:
2297:
2295:
2294:
2289:
2275:
2273:
2271:
2270:
2265:
2248:
2246:
2245:
2240:
2195:
2193:
2192:
2187:
2179:
2178:
2166:
2165:
2145:hyperbolic angle
2133:
2131:
2130:
2125:
2120:
2116:
2108:
2099:
2098:
2089:
2081:
2064:
2062:
2051:
2038:
2036:
2035:
2030:
2028:
1997:
1995:
1994:
1993:
1971:
1958:
1957:
1941:
1918:
1916:
1915:
1914:
1892:
1891:
1876:
1850:
1846:
1844:
1843:
1839:
1838:
1837:
1808:
1807:
1797:
1796:
1792:
1791:
1790:
1761:
1760:
1750:
1745:
1743:
1742:
1741:
1726:
1725:
1715:
1714:
1713:
1698:
1697:
1687:
1650:
1648:
1647:
1642:
1591:
1589:
1588:
1583:
1578:
1574:
1566:
1554:
1546:
1541:
1533:
1482:
1480:
1479:
1474:
1457:
1454:
1452:
1450:
1436:
1399:Inverse function
1392:
1390:
1389:
1384:
1376:
1374:
1367:
1366:
1353:
1348:
1346:
1339:
1338:
1328:
1327:
1318:
1313:
1311:
1310:
1309:
1278:
1273:
1271:
1270:
1269:
1247:
1236:
1234:
1233:
1228:
1197:
1195:
1194:
1189:
1184:
1145:
1143:
1142:
1137:
1125:
1123:
1122:
1117:
1103:
1101:
1100:
1095:
1037:
1035:
1034:
1029:
1017:
1015:
1014:
1009:
1007:
1006:
982:
980:
979:
974:
969:
967:
966:
965:
961:
942:
941:
937:
923:
922:
918:
905:
900:
898:
891:
890:
880:
879:
870:
865:
863:
862:
861:
839:
813:
811:
810:
805:
787:
785:
784:
779:
771:
770:
754:
752:
751:
746:
727:
726:
681:
671:
669:
648:geometric growth
625:Adolphe Quetelet
516:
514:
513:
508:
506:
504:
503:
502:
480:
453:
451:
450:
445:
437:
436:
388:
386:
385:
380:
372:
371:
324:
322:
321:
316:
304:
302:
301:
296:
275:
273:
272:
267:
239:
237:
236:
231:
219:
217:
216:
211:
209:
208:
190:
188:
187:
182:
160:
158:
157:
152:
135:
133:
132:
127:
125:
123:
122:
121:
117:
116:
77:
21:
12172:
12171:
12167:
12166:
12165:
12163:
12162:
12161:
12122:
12121:
12067:
12062:
12037:
12018:
11997:
11943:
11932:
11927:
11923:
11916:
11902:
11898:
11875:
11871:
11864:
11846:
11842:
11834:
11826:
11822:
11812:
11810:
11809:on 9 March 2012
11806:
11795:
11789:
11785:
11777:
11766:
11760:
11756:
11746:
11744:
11743:on 1 March 2012
11740:
11729:
11720:
11716:
11643:
11639:
11632:
11628:
11621:
11617:
11612:
11608:
11565:
11561:
11516:
11509:
11466:
11462:
11411:(7): e0236860.
11397:
11393:
11376:
11372:
11355:
11351:
11298:
11294:
11277:
11273:
11266:
11262:
11255:
11240:
11234:
11230:
11225:
11221:
11172:
11168:
11123:
11119:
11105:
11092:
11088:
11072:
11071:
11047:
11043:
11038:
11034:
10979:
10975:
10965:
10963:
10960:
10952:
10948:
10933:Baron de Jomini
10930:
10926:
10918:
10895:
10889:
10882:
10875:
10864:(March): 1–14.
10848:
10844:
10834:
10832:
10810:
10806:
10789:
10787:
10769:
10765:
10761:, pp. 3–5.
10757:
10753:
10744:
10743:
10739:
10731:
10727:
10718:
10717:
10713:
10703:
10701:
10689:
10683:
10676:
10672:
10667:
10666:
10639:
10635:
10634:
10630:
10613:
10609:
10608:
10604:
10577:
10573:
10558:
10554:
10527:
10523:
10508:
10504:
10492:
10485:
10481:
10474:
10473:
10471:
10468:
10467:
10462:, specifically
10443:
10439:
10430:
10426:
10418:
10415:
10414:
10383:
10380:
10379:
10373:
10369:
10356:
10352:
10315:
10312:
10311:
10277:
10274:
10273:
10267:
10263:
10230:
10227:
10226:
10224:
10206:
10203:
10202:
10200:
10198:
10194:
10135:
10131:
10102:
10098:
10093:
10089:
10084:
10079:
10008:
9977:
9973:
9959:
9954:
9951:
9950:
9942:
9899:electrification
9864:sovereign state
9817:
9814:
9813:
9796:
9792:
9775:
9772:
9771:
9752:
9747:
9746:
9745:
9744:
9743:
9736:
9728:
9727:
9720:
9686:
9678:language change
9674:
9662:
9646:
9633:
9617:
9559:
9556:
9555:
9535:
9531:
9522:
9518:
9509:
9505:
9500:
9497:
9496:
9479:
9475:
9473:
9470:
9469:
9452:
9448:
9446:
9443:
9442:
9425:
9421:
9419:
9416:
9415:
9395:
9392:
9391:
9369:
9366:
9365:
9349:
9346:
9345:
9329:
9326:
9325:
9309:
9306:
9305:
9288:
9284:
9275:
9271:
9262:
9258:
9256:
9253:
9252:
9226:
9222:
9218:
9206:
9202:
9184:
9180:
9155:
9149:
9145:
9143:
9125:
9121:
9112:
9108:
9099:
9095:
9081:
9078:
9077:
9045:
9016:
9010:
8955:
8917:
8912:
8898:
8896:
8893:
8892:
8874:
8871:
8870:
8845:
8842:
8841:
8816:
8813:
8812:
8760:
8753:
8749:
8735:
8733:
8730:
8729:
8699:
8696:
8695:
8656:
8626:
8624:
8621:
8620:
8584:
8577:
8573:
8559:
8557:
8554:
8553:
8535:
8532:
8531:
8506:
8503:
8502:
8494:
8488:
8473:backpropagation
8419:
8415:
8408:
8403:
8386:
8383:
8382:
8346:
8344:Neural networks
8283:
8280:
8279:
8263:
8260:
8259:
8243:
8240:
8239:
8223:
8220:
8219:
8174:
8171:
8170:
8148:
8145:
8144:
8137:
8131:
8111:
8080:
8077:
8076:
8048:
8045:
8044:
8028:
8025:
8024:
8005:
8002:
8001:
7975:
7971:
7969:
7966:
7965:
7940:
7937:
7936:
7905:
7902:
7901:
7850:
7847:
7846:
7828:
7825:
7824:
7787:
7782:
7775:
7771:
7749:
7741:
7739:
7737:
7734:
7733:
7700:
7697:
7696:
7693:
7649:
7645:
7636:
7632:
7625:
7621:
7605:
7577:
7557:
7547:
7511:
7507:
7506:
7502:
7496:
7492:
7485:
7475:
7471:
7465:
7461:
7457:
7455:
7450:
7447:
7446:
7419:
7415:
7409:
7405:
7391:
7388:
7387:
7356:
7352:
7351:
7347:
7341:
7337:
7323:
7320:
7319:
7309:
7260:
7252:
7250:
7248:
7245:
7244:
7223:
7218:
7215:
7214:
7198:
7195:
7194:
7178:
7175:
7174:
7144:
7141:
7140:
7139:-strategist or
7124:
7121:
7120:
7081:
7078:
7077:
7046:
7043:
7042:
7026:
7023:
7022:
7006:
7003:
7002:
6956:
6950:
6947:
6946:
6913:
6909:
6897:
6893:
6886:
6882:
6875:
6873:
6869:
6862:
6857:
6831:
6827:
6826:
6822:
6816:
6812:
6805:
6795:
6791:
6785:
6781:
6777:
6775:
6758:
6755:
6754:
6735:
6731:
6729:
6726:
6725:
6705:
6702:
6701:
6685:
6682:
6681:
6661:
6658:
6657:
6638:
6632:
6628:
6620:
6617:
6616:
6600:
6597:
6596:
6580:
6577:
6576:
6554:
6551:
6550:
6527:
6524:
6523:
6503:
6500:
6499:
6468:
6461:
6457:
6438:
6430:
6428:
6426:
6423:
6422:
6400:
6397:
6396:
6380:
6377:
6376:
6360:
6357:
6356:
6346:Alfred J. Lotka
6283:
6278:
6243:
6241:Generalizations
6219:
6216:
6215:
6192:
6188:
6186:
6183:
6182:
6178:
6177:against option
6174:
6157:
6153:
6144:
6140:
6131:
6127:
6125:
6122:
6121:
6098:
6094:
6092:
6089:
6088:
6084:
6067:
6063:
6061:
6058:
6057:
6053:
6052:against option
6049:
6032:
6028:
6019:
6015:
6013:
6010:
6009:
5992:
5988:
5986:
5983:
5982:
5959:
5955:
5953:
5950:
5949:
5928:
5924:
5915:
5911:
5909:
5906:
5905:
5901:
5879:
5875:
5874:
5870:
5864:
5853:
5844:
5836:
5832:
5831:
5827:
5825:
5822:
5821:
5817:
5788:
5784:
5783:
5779:
5768:
5764:
5763:
5759:
5751:
5743:
5739:
5738:
5734:
5732:
5729:
5728:
5706:
5702:
5701:
5697:
5686:
5682:
5681:
5677:
5669:
5661:
5657:
5656:
5652:
5633:
5629:
5620:
5616:
5609:
5605:
5598:
5594:
5589:
5566:
5562:
5553:
5549:
5548:
5544:
5536:
5528:
5524:
5515:
5511:
5510:
5506:
5504:
5501:
5500:
5481:
5477:
5468:
5464:
5463:
5459:
5457:
5454:
5453:
5432:
5428:
5419:
5415:
5413:
5410:
5409:
5387:
5383:
5381:
5378:
5377:
5375:
5356:
5352:
5350:
5347:
5346:
5344:
5325:
5321:
5319:
5316:
5315:
5313:
5294:
5290:
5288:
5285:
5284:
5282:
5253:
5249:
5235:
5211:
5207:
5199:
5190:
5186:
5184:
5181:
5180:
5160:
5156:
5154:
5151:
5150:
5125:
5122:
5121:
5119:
5111:
5082:
5078:
5064:
5043:
5039:
5031:
5025:
5021:
5019:
5016:
5015:
4984:
4979:
4976:
4975:
4971:
4953:
4950:
4949:
4947:
4919:
4916:
4915:
4892:
4888:
4886:
4883:
4882:
4880:
4868:
4840:
4837:
4836:
4808:
4805:
4804:
4776:
4773:
4772:
4764:
4760:
4730:
4727:
4726:
4724:
4702:
4699:
4698:
4695:
4689:
4667:
4666:
4642:
4641:
4633:
4630:
4629:
4606:
4602:
4595:
4590:
4565:
4564:
4540:
4539:
4517:
4504:
4487:
4485:
4483:
4480:
4479:
4454:
4451:
4450:
4415:
4411:
4404:
4399:
4381:
4377:
4376:
4370:
4366:
4364:
4347:
4344:
4343:
4319:
4316:
4315:
4284:
4280:
4279:
4273:
4269:
4267:
4250:
4247:
4246:
4220:
4217:
4216:
4200:
4197:
4196:
4162:
4142:
4139:
4138:
4116:
4115:
4091:
4090:
4053:
4048:
4046:
4043:
4042:
4034:
3986:
3982:
3935:
3913:
3909:
3902:
3896:
3892:
3890:
3885:
3882:
3881:
3846:
3844:
3832:
3828:
3821:
3815:
3811:
3809:
3792:
3789:
3788:
3769:
3765:
3751:
3748:
3747:
3737:
3714:
3711:
3710:
3708:
3689:
3685:
3683:
3680:
3679:
3677:
3653:
3650:
3649:
3647:
3627:
3626:
3602:
3598:
3577:
3576:
3562:
3558:
3551:
3545:
3541:
3539:
3532:
3528:
3515:
3511:
3504:
3498:
3494:
3492:
3488:
3479:
3478:
3465:
3461:
3454:
3449:
3445:
3432:
3428:
3421:
3415:
3411:
3409:
3405:
3396:
3395:
3386:
3382:
3376:
3372:
3362:
3356:
3352:
3350:
3341:
3340:
3331:
3327:
3321:
3317:
3307:
3300:
3296:
3287:
3283:
3271:
3267:
3249:
3245:
3244:
3242:
3235:
3212:
3211:
3206:
3204:
3200:
3198:
3195:
3194:
3170:
3166:
3159:
3153:
3149:
3147:
3132:
3128:
3121:
3116:
3099:
3096:
3095:
3074:
3052:
3051:
3047:
3046:
3041:
3032:
3031:
3026:
3018:
3012:
3011:
2999:
2997:
2994:
2993:
2976:
2975:
2971:
2970:
2965:
2959:
2958:
2953:
2945:
2939:
2938:
2936:
2933:
2932:
2913:
2908:
2905:
2904:
2884:
2880:
2868:
2864:
2859:
2856:
2855:
2829:
2826:
2825:
2796:
2792:
2781:
2778:
2777:
2760:
2759:
2749:
2745:
2741:
2728:
2721:
2717:
2704:
2700:
2696:
2690:
2689:
2687:
2684:
2683:
2659:
2647:
2643:
2634:
2630:
2616:
2604:
2600:
2591:
2587:
2583:
2579:
2577:
2574:
2573:
2570:hyperbolic sine
2541:
2536:
2533:
2532:
2530:
2500:
2497:
2496:
2494:
2476:
2473:
2472:
2470:
2469:and with slope
2452:
2449:
2448:
2446:
2409:
2406:
2405:
2382:
2378:
2367:
2364:
2363:
2345:
2343:
2340:
2339:
2337:
2307:
2304:
2303:
2301:
2283:
2280:
2279:
2277:
2276:and with slope
2256:
2253:
2252:
2250:
2201:
2198:
2197:
2174:
2170:
2161:
2157:
2155:
2152:
2151:
2107:
2103:
2094:
2090:
2080:
2055:
2050:
2048:
2045:
2044:
2026:
2025:
1983:
1979:
1972:
1947:
1943:
1942:
1940:
1904:
1900:
1893:
1881:
1877:
1875:
1848:
1847:
1827:
1823:
1816:
1812:
1803:
1799:
1798:
1780:
1776:
1769:
1765:
1756:
1752:
1751:
1749:
1734:
1730:
1721:
1717:
1716:
1706:
1702:
1693:
1689:
1688:
1686:
1679:
1660:
1658:
1655:
1654:
1597:
1594:
1593:
1565:
1561:
1545:
1532:
1515:
1512:
1511:
1504:
1455: for
1453:
1440:
1435:
1415:
1412:
1411:
1401:
1362:
1358:
1357:
1352:
1334:
1330:
1329:
1323:
1319:
1317:
1293:
1289:
1282:
1277:
1262:
1258:
1251:
1246:
1244:
1241:
1240:
1210:
1207:
1206:
1180:
1154:
1151:
1150:
1131:
1128:
1127:
1111:
1108:
1107:
1053:
1050:
1049:
1044:
1023:
1020:
1019:
999:
995:
993:
990:
989:
957:
950:
946:
933:
929:
925:
924:
914:
910:
906:
904:
886:
882:
881:
875:
871:
869:
854:
850:
843:
838:
821:
818:
817:
814:, which yields
793:
790:
789:
766:
762:
760:
757:
756:
734:
731:
730:
724:
723:
719:
601:
593:generalizations
495:
491:
484:
479:
462:
459:
458:
432:
428:
402:
399:
398:
367:
363:
337:
334:
333:
310:
307:
306:
281:
278:
277:
252:
249:
248:
247:, the limit as
241:
225:
222:
221:
204:
200:
198:
195:
194:
192:
176:
173:
172:
170:
146:
143:
142:
112:
108:
92:
88:
81:
76:
59:
56:
55:
35:
28:
23:
22:
18:Logistic growth
15:
12:
11:
5:
12170:
12160:
12159:
12154:
12149:
12144:
12139:
12134:
12120:
12119:
12114:
12109:
12104:
12099:
12080:
12075:
12070:L.J. Linacre,
12066:
12065:External links
12063:
12061:
12060:
12041:
12035:
12022:
12016:
12001:
11995:
11980:
11979:
11978:
11968:(4): 613–626.
11933:
11931:
11930:
11921:
11914:
11896:
11869:
11862:
11840:
11820:
11783:
11754:
11714:
11637:
11626:
11615:
11606:
11559:
11507:
11460:
11391:
11370:
11349:
11292:
11271:
11260:
11253:
11228:
11219:
11166:
11117:
11086:
11041:
11032:
10973:
10946:
10924:
10880:
10842:
10804:
10763:
10751:
10737:
10725:
10711:
10673:
10671:
10668:
10665:
10664:
10652:
10649:
10642:
10638:
10633:
10629:
10626:
10623:
10616:
10612:
10607:
10603:
10600:
10597:
10594:
10591:
10588:
10585:
10580:
10576:
10572:
10569:
10566:
10561:
10557:
10553:
10550:
10547:
10544:
10541:
10538:
10535:
10530:
10526:
10522:
10519:
10516:
10511:
10507:
10503:
10500:
10495:
10488:
10484:
10480:
10477:
10451:
10446:
10442:
10438:
10433:
10429:
10425:
10422:
10402:
10399:
10396:
10393:
10390:
10387:
10367:
10363:logit function
10350:
10337:
10334:
10331:
10328:
10325:
10322:
10319:
10299:
10296:
10293:
10290:
10287:
10284:
10281:
10261:
10246:
10243:
10240:
10237:
10234:
10210:
10192:
10129:
10107:and geometric
10096:
10086:
10085:
10083:
10080:
10078:
10077:
10072:
10067:
10062:
10050:
10045:
10040:
10035:
10030:
10025:
10020:
10015:
10009:
10007:
10004:
9991:
9986:
9983:
9980:
9976:
9972:
9969:
9966:
9962:
9958:
9941:
9938:
9884:public finance
9848:
9845:
9842:
9839:
9836:
9833:
9830:
9827:
9824:
9821:
9799:
9795:
9791:
9788:
9785:
9782:
9779:
9751:
9748:
9737:
9730:
9729:
9721:
9714:
9713:
9712:
9711:
9710:
9685:
9682:
9673:
9670:
9661:
9658:
9645:
9642:
9632:
9629:
9625:Platinum group
9616:
9613:
9581:
9578:
9575:
9572:
9569:
9566:
9563:
9543:
9538:
9534:
9530:
9525:
9521:
9517:
9512:
9508:
9504:
9482:
9478:
9455:
9451:
9428:
9424:
9399:
9379:
9376:
9373:
9353:
9333:
9313:
9291:
9287:
9283:
9278:
9274:
9270:
9265:
9261:
9233:
9229:
9225:
9221:
9217:
9214:
9209:
9205:
9201:
9198:
9195:
9192:
9187:
9183:
9179:
9176:
9173:
9170:
9167:
9164:
9161:
9158:
9152:
9148:
9142:
9139:
9136:
9133:
9128:
9124:
9120:
9115:
9111:
9107:
9102:
9098:
9094:
9091:
9088:
9085:
9044:
9041:
9012:Main article:
9009:
9006:
8990:
8987:
8984:
8981:
8978:
8975:
8972:
8967:
8964:
8961:
8958:
8954:
8950:
8947:
8944:
8941:
8938:
8934:
8931:
8928:
8925:
8920:
8915:
8911:
8905:
8902:
8878:
8858:
8855:
8852:
8849:
8829:
8826:
8823:
8820:
8798:
8795:
8792:
8789:
8786:
8783:
8780:
8777:
8773:
8767:
8764:
8759:
8756:
8752:
8748:
8745:
8741:
8738:
8712:
8709:
8706:
8703:
8681:
8678:
8675:
8672:
8669:
8666:
8662:
8659:
8654:
8651:
8648:
8645:
8642:
8639:
8636:
8632:
8629:
8604:
8601:
8597:
8591:
8588:
8583:
8580:
8576:
8572:
8569:
8565:
8562:
8539:
8519:
8516:
8513:
8510:
8487:
8484:
8439:
8431:
8428:
8425:
8422:
8418:
8414:
8411:
8407:
8402:
8399:
8396:
8393:
8390:
8345:
8342:
8287:
8267:
8247:
8227:
8205:
8202:
8199:
8196:
8193:
8190:
8187:
8184:
8181:
8178:
8152:
8133:Main article:
8130:
8127:
8110:
8107:
8093:
8090:
8087:
8084:
8061:
8058:
8055:
8052:
8032:
8009:
7989:
7986:
7983:
7978:
7974:
7953:
7950:
7947:
7944:
7924:
7921:
7918:
7915:
7912:
7909:
7887:
7884:
7881:
7878:
7875:
7872:
7869:
7866:
7863:
7860:
7857:
7854:
7832:
7810:
7806:
7799:
7796:
7793:
7790:
7786:
7781:
7778:
7774:
7770:
7767:
7764:
7761:
7755:
7752:
7747:
7744:
7719:
7716:
7713:
7710:
7707:
7704:
7692:
7689:
7677:
7674:
7670:
7666:
7663:
7660:
7655:
7652:
7648:
7644:
7639:
7635:
7631:
7628:
7624:
7620:
7617:
7612:
7609:
7604:
7601:
7598:
7595:
7592:
7589:
7584:
7581:
7576:
7573:
7570:
7564:
7561:
7554:
7551:
7546:
7543:
7540:
7537:
7529:
7525:
7522:
7517:
7514:
7510:
7505:
7499:
7495:
7491:
7488:
7481:
7478:
7474:
7468:
7464:
7460:
7454:
7433:
7430:
7425:
7422:
7418:
7412:
7408:
7404:
7401:
7398:
7395:
7374:
7370:
7367:
7362:
7359:
7355:
7350:
7344:
7340:
7336:
7333:
7330:
7327:
7313:antiderivative
7308:
7305:
7293:
7290:
7287:
7284:
7281:
7278:
7275:
7272:
7266:
7263:
7258:
7255:
7230:
7226:
7222:
7202:
7182:
7148:
7128:
7100:
7097:
7094:
7091:
7088:
7085:
7065:
7062:
7059:
7056:
7053:
7050:
7030:
7010:
6988:
6985:
6982:
6979:
6976:
6973:
6970:
6965:
6962:
6959:
6955:
6930:
6922:
6919:
6916:
6912:
6907:
6900:
6896:
6889:
6885:
6881:
6878:
6872:
6868:
6865:
6861:
6856:
6849:
6845:
6842:
6837:
6834:
6830:
6825:
6819:
6815:
6811:
6808:
6801:
6798:
6794:
6788:
6784:
6780:
6774:
6771:
6768:
6765:
6762:
6738:
6734:
6709:
6689:
6665:
6645:
6641:
6635:
6631:
6627:
6624:
6604:
6584:
6564:
6561:
6558:
6531:
6507:
6485:
6481:
6475:
6472:
6467:
6464:
6460:
6456:
6453:
6450:
6444:
6441:
6436:
6433:
6404:
6384:
6364:
6308:Thomas Malthus
6282:
6279:
6277:
6274:
6242:
6239:
6226:
6223:
6203:
6200:
6195:
6191:
6160:
6156:
6152:
6147:
6143:
6139:
6134:
6130:
6109:
6106:
6101:
6097:
6070:
6066:
6035:
6031:
6027:
6022:
6018:
5995:
5991:
5970:
5967:
5962:
5958:
5931:
5927:
5923:
5918:
5914:
5882:
5878:
5873:
5867:
5862:
5859:
5856:
5852:
5847:
5839:
5835:
5830:
5798:
5791:
5787:
5782:
5778:
5771:
5767:
5762:
5758:
5754:
5746:
5742:
5737:
5716:
5709:
5705:
5700:
5696:
5689:
5685:
5680:
5676:
5672:
5664:
5660:
5655:
5651:
5647:
5641:
5636:
5632:
5628:
5623:
5619:
5615:
5612:
5608:
5604:
5601:
5597:
5592:
5588:
5585:
5582:
5579:
5576:
5569:
5565:
5561:
5556:
5552:
5547:
5543:
5539:
5531:
5527:
5523:
5518:
5514:
5509:
5484:
5480:
5476:
5471:
5467:
5462:
5435:
5431:
5427:
5422:
5418:
5390:
5386:
5359:
5355:
5328:
5324:
5297:
5293:
5267:
5264:
5261:
5256:
5252:
5248:
5245:
5242:
5238:
5234:
5231:
5228:
5225:
5222:
5217:
5214:
5210:
5206:
5202:
5196:
5193:
5189:
5166:
5163:
5159:
5132:
5129:
5099:
5096:
5093:
5088:
5085:
5081:
5077:
5074:
5071:
5067:
5063:
5060:
5057:
5054:
5051:
5046:
5042:
5038:
5034:
5028:
5024:
5014:, we see that
5003:
5000:
4997:
4994:
4991:
4987:
4983:
4957:
4935:
4932:
4929:
4926:
4923:
4895:
4891:
4853:
4850:
4847:
4844:
4824:
4821:
4818:
4815:
4812:
4792:
4789:
4786:
4783:
4780:
4746:
4743:
4740:
4737:
4734:
4712:
4709:
4706:
4688:
4685:
4670:
4665:
4662:
4659:
4656:
4653:
4650:
4645:
4640:
4637:
4612:
4609:
4605:
4601:
4598:
4594:
4589:
4586:
4583:
4580:
4577:
4573:
4568:
4563:
4560:
4557:
4554:
4551:
4548:
4543:
4538:
4535:
4532:
4529:
4524:
4521:
4516:
4510:
4507:
4502:
4499:
4496:
4493:
4490:
4464:
4461:
4458:
4429:
4421:
4418:
4414:
4410:
4407:
4403:
4398:
4392:
4389:
4384:
4380:
4373:
4369:
4363:
4360:
4357:
4354:
4351:
4329:
4326:
4323:
4301:
4295:
4292:
4287:
4283:
4276:
4272:
4266:
4263:
4260:
4257:
4254:
4224:
4204:
4169:
4165:
4161:
4158:
4155:
4152:
4149:
4146:
4119:
4114:
4111:
4108:
4105:
4102:
4099:
4094:
4089:
4086:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4059:
4056:
4052:
4033:
4030:
3997:
3994:
3989:
3985:
3981:
3978:
3975:
3972:
3969:
3966:
3963:
3960:
3957:
3954:
3951:
3948:
3942:
3939:
3934:
3931:
3928:
3925:
3916:
3912:
3908:
3905:
3899:
3895:
3889:
3861:
3856:
3852:
3849:
3843:
3835:
3831:
3827:
3824:
3818:
3814:
3808:
3805:
3802:
3799:
3796:
3772:
3768:
3764:
3761:
3758:
3755:
3741:antiderivative
3736:
3733:
3731:is the scale.
3718:
3692:
3688:
3663:
3660:
3657:
3624:
3620:
3617:
3614:
3611:
3608:
3605:
3601:
3597:
3594:
3591:
3588:
3585:
3582:
3580:
3578:
3574:
3565:
3561:
3557:
3554:
3548:
3544:
3538:
3535:
3531:
3526:
3518:
3514:
3510:
3507:
3501:
3497:
3491:
3487:
3484:
3482:
3480:
3476:
3468:
3464:
3460:
3457:
3453:
3448:
3443:
3435:
3431:
3427:
3424:
3418:
3414:
3408:
3404:
3401:
3399:
3397:
3389:
3385:
3379:
3375:
3371:
3368:
3365:
3359:
3355:
3349:
3346:
3344:
3342:
3334:
3330:
3324:
3320:
3316:
3313:
3310:
3303:
3299:
3295:
3290:
3286:
3282:
3279:
3274:
3270:
3266:
3263:
3260:
3257:
3252:
3248:
3241:
3238:
3236:
3234:
3231:
3228:
3225:
3219:
3215:
3209:
3203:
3202:
3181:
3173:
3169:
3165:
3162:
3156:
3152:
3146:
3138:
3135:
3131:
3127:
3124:
3120:
3115:
3112:
3109:
3106:
3103:
3073:
3070:
3055:
3045:
3042:
3040:
3037:
3034:
3033:
3030:
3027:
3025:
3022:
3021:
3015:
3007:
3004:
2979:
2969:
2966:
2964:
2961:
2960:
2957:
2954:
2952:
2949:
2948:
2942:
2920:
2916:
2912:
2892:
2887:
2883:
2879:
2874:
2871:
2867:
2863:
2842:
2839:
2836:
2833:
2824:the hyperbola
2807:
2804:
2799:
2795:
2791:
2788:
2785:
2763:
2756:
2752:
2748:
2744:
2740:
2735:
2731:
2727:
2724:
2720:
2716:
2711:
2707:
2703:
2699:
2693:
2670:
2666:
2662:
2658:
2653:
2650:
2646:
2642:
2637:
2633:
2629:
2626:
2623:
2619:
2615:
2610:
2607:
2603:
2599:
2594:
2590:
2586:
2582:
2548:
2544:
2540:
2516:
2513:
2510:
2507:
2504:
2480:
2456:
2434:
2431:
2428:
2425:
2422:
2419:
2416:
2413:
2393:
2390:
2385:
2381:
2377:
2374:
2371:
2348:
2323:
2320:
2317:
2314:
2311:
2287:
2263:
2260:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2185:
2182:
2177:
2173:
2169:
2164:
2160:
2149:unit hyperbola
2123:
2119:
2114:
2111:
2106:
2102:
2097:
2093:
2087:
2084:
2079:
2076:
2073:
2070:
2067:
2061:
2058:
2054:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1992:
1989:
1986:
1982:
1978:
1975:
1970:
1967:
1964:
1961:
1956:
1953:
1950:
1946:
1939:
1936:
1933:
1930:
1927:
1924:
1921:
1913:
1910:
1907:
1903:
1899:
1896:
1890:
1887:
1884:
1880:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1851:
1849:
1842:
1836:
1833:
1830:
1826:
1822:
1819:
1815:
1811:
1806:
1802:
1795:
1789:
1786:
1783:
1779:
1775:
1772:
1768:
1764:
1759:
1755:
1748:
1740:
1737:
1733:
1729:
1724:
1720:
1712:
1709:
1705:
1701:
1696:
1692:
1685:
1682:
1680:
1678:
1675:
1672:
1669:
1666:
1663:
1662:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1581:
1577:
1572:
1569:
1564:
1560:
1557:
1552:
1549:
1544:
1539:
1536:
1531:
1528:
1525:
1522:
1519:
1503:
1500:
1484:
1483:
1472:
1469:
1466:
1463:
1460:
1449:
1446:
1443:
1439:
1434:
1431:
1428:
1425:
1422:
1419:
1400:
1397:
1382:
1379:
1373:
1370:
1365:
1361:
1356:
1351:
1345:
1342:
1337:
1333:
1326:
1322:
1316:
1308:
1305:
1302:
1299:
1296:
1292:
1288:
1285:
1281:
1276:
1268:
1265:
1261:
1257:
1254:
1250:
1226:
1223:
1220:
1217:
1214:
1187:
1183:
1179:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1135:
1115:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1043:
1040:
1027:
1005:
1002:
998:
972:
964:
960:
956:
953:
949:
945:
940:
936:
932:
928:
921:
917:
913:
909:
903:
897:
894:
889:
885:
878:
874:
868:
860:
857:
853:
849:
846:
842:
837:
834:
831:
828:
825:
803:
800:
797:
777:
774:
769:
765:
744:
741:
738:
718:
715:
600:
597:
545:biomathematics
518:
517:
501:
498:
494:
490:
487:
483:
478:
475:
472:
469:
466:
443:
440:
435:
431:
427:
424:
421:
418:
415:
412:
409:
406:
378:
375:
370:
366:
362:
359:
356:
353:
350:
347:
344:
341:
314:
294:
291:
288:
285:
265:
262:
259:
256:
229:
207:
203:
193:
180:
171:
150:
141:
137:
136:
120:
115:
111:
107:
104:
101:
98:
95:
91:
87:
84:
80:
75:
72:
69:
66:
63:
45:logistic curve
27:S-shaped curve
26:
9:
6:
4:
3:
2:
12169:
12158:
12157:Growth curves
12155:
12153:
12150:
12148:
12145:
12143:
12140:
12138:
12135:
12133:
12130:
12129:
12127:
12118:
12115:
12113:
12110:
12108:
12105:
12103:
12100:
12095:
12094:
12089:
12086:
12081:
12079:
12076:
12073:
12069:
12068:
12056:
12055:
12050:
12047:
12042:
12038:
12036:0-226-43728-0
12032:
12028:
12023:
12019:
12013:
12009:
12008:
12002:
11998:
11996:0-262-52338-8
11992:
11988:
11987:
11981:
11975:
11971:
11967:
11963:
11958:Published as:
11957:
11956:
11953:
11949:
11942:
11941:
11935:
11934:
11925:
11917:
11915:9781003310228
11911:
11907:
11900:
11892:
11888:
11885:(1): 77–105.
11884:
11880:
11879:Psychometrika
11873:
11865:
11863:1-84376-331-1
11859:
11854:
11853:
11844:
11833:
11832:
11824:
11805:
11801:
11794:
11787:
11776:
11772:
11765:
11758:
11739:
11735:
11728:
11724:
11723:Ayres, Robert
11718:
11710:
11706:
11701:
11696:
11692:
11688:
11683:
11678:
11674:
11670:
11665:
11660:
11656:
11652:
11648:
11641:
11635:
11630:
11624:
11619:
11610:
11602:
11598:
11594:
11590:
11586:
11582:
11578:
11574:
11570:
11563:
11555:
11551:
11546:
11541:
11537:
11533:
11530:(1): 012072.
11529:
11525:
11521:
11514:
11512:
11503:
11499:
11495:
11491:
11487:
11483:
11479:
11475:
11471:
11464:
11456:
11452:
11447:
11442:
11437:
11432:
11428:
11424:
11419:
11414:
11410:
11406:
11402:
11395:
11386:
11381:
11374:
11365:
11360:
11353:
11345:
11341:
11336:
11331:
11327:
11323:
11319:
11315:
11311:
11307:
11303:
11296:
11287:
11282:
11275:
11269:
11264:
11256:
11254:3-540-65311-2
11250:
11246:
11239:
11232:
11223:
11215:
11211:
11207:
11203:
11199:
11195:
11190:
11185:
11181:
11177:
11170:
11162:
11158:
11153:
11148:
11144:
11140:
11136:
11132:
11128:
11121:
11113:
11112:
11104:
11101:(June 1920).
11100:
11096:
11095:Raymond Pearl
11090:
11082:
11076:
11068:
11064:
11060:
11056:
11052:
11045:
11036:
11028:
11024:
11019:
11014:
11009:
11004:
11000:
10996:
10992:
10988:
10984:
10977:
10959:
10958:
10950:
10942:
10938:
10934:
10928:
10917:
10913:
10909:
10905:
10901:
10894:
10887:
10885:
10877:
10871:
10867:
10863:
10859:
10858:
10853:
10846:
10831:
10827:
10823:
10819:
10815:
10808:
10801:
10799:
10786:
10782:
10778:
10774:
10767:
10760:
10755:
10747:
10741:
10734:
10729:
10721:
10715:
10699:
10695:
10688:
10681:
10679:
10674:
10650:
10640:
10636:
10631:
10627:
10624:
10621:
10614:
10610:
10605:
10601:
10598:
10592:
10589:
10586:
10578:
10574:
10570:
10567:
10564:
10559:
10555:
10551:
10548:
10542:
10539:
10536:
10528:
10524:
10520:
10517:
10514:
10509:
10505:
10498:
10493:
10486:
10465:
10444:
10440:
10436:
10431:
10427:
10397:
10394:
10391:
10377:
10371:
10364:
10360:
10354:
10335:
10332:
10323:
10317:
10297:
10294:
10285:
10279:
10271:
10265:
10241:
10238:
10235:
10208:
10196:
10189:
10185:
10181:
10177:
10161:
10160:
10155:
10153:
10133:
10126:
10122:
10118:
10114:
10110:
10106:
10100:
10091:
10087:
10076:
10073:
10071:
10068:
10066:
10063:
10061:
10059:
10055:
10051:
10049:
10046:
10044:
10041:
10039:
10036:
10034:
10033:Hubbert curve
10031:
10029:
10026:
10024:
10021:
10019:
10016:
10014:
10011:
10010:
10003:
9984:
9981:
9978:
9974:
9970:
9967:
9960:
9956:
9947:
9946:Wald's theory
9937:
9935:
9931:
9927:
9923:
9919:
9914:
9911:
9907:
9902:
9900:
9896:
9891:
9889:
9888:sigmoid curve
9885:
9881:
9877:
9873:
9869:
9865:
9860:
9843:
9837:
9834:
9831:
9825:
9819:
9797:
9793:
9789:
9783:
9777:
9768:
9767:Gabriel Tarde
9764:
9759:
9757:
9741:
9740:soil salinity
9734:
9725:
9718:
9709:
9707:
9703:
9699:
9695:
9691:
9681:
9679:
9669:
9667:
9657:
9653:
9651:
9641:
9639:
9628:
9626:
9622:
9612:
9610:
9606:
9597:
9593:
9576:
9573:
9570:
9567:
9564:
9536:
9532:
9528:
9523:
9519:
9515:
9510:
9506:
9480:
9476:
9453:
9449:
9426:
9422:
9413:
9397:
9377:
9374:
9371:
9351:
9331:
9311:
9289:
9285:
9281:
9276:
9272:
9268:
9263:
9259:
9249:
9231:
9227:
9223:
9207:
9203:
9199:
9196:
9190:
9185:
9181:
9177:
9171:
9168:
9165:
9162:
9159:
9150:
9146:
9140:
9134:
9131:
9126:
9122:
9118:
9113:
9109:
9105:
9100:
9096:
9092:
9089:
9083:
9075:
9073:
9069:
9065:
9061:
9053:
9049:
9040:
9038:
9033:
9031:
9027:
9026:herd immunity
9022:
9015:
9005:
9001:
8988:
8985:
8982:
8976:
8970:
8962:
8956:
8945:
8942:
8939:
8936:
8929:
8923:
8918:
8913:
8909:
8903:
8900:
8890:
8876:
8853:
8847:
8824:
8818:
8809:
8796:
8793:
8787:
8781:
8778:
8775:
8771:
8765:
8762:
8757:
8754:
8750:
8746:
8743:
8739:
8736:
8727:
8724:
8707:
8701:
8692:
8679:
8676:
8673:
8667:
8660:
8657:
8652:
8649:
8643:
8637:
8634:
8630:
8627:
8618:
8615:
8602:
8599:
8595:
8589:
8586:
8581:
8578:
8574:
8570:
8567:
8563:
8560:
8551:
8537:
8514:
8508:
8500:
8493:
8483:
8481:
8476:
8474:
8470:
8466:
8465:antisymmetric
8462:
8458:
8453:
8450:
8437:
8429:
8426:
8423:
8420:
8416:
8412:
8409:
8405:
8400:
8394:
8388:
8380:
8377:
8375:
8371:
8367:
8363:
8360:. A popular
8359:
8355:
8352:to introduce
8351:
8341:
8339:
8335:
8331:
8327:
8323:
8318:
8316:
8312:
8308:
8304:
8299:
8285:
8265:
8245:
8225:
8216:
8203:
8197:
8194:
8191:
8188:
8182:
8179:
8176:
8168:
8166:
8150:
8142:
8136:
8126:
8124:
8120:
8116:
8106:
8088:
8082:
8073:
8056:
8050:
8030:
8021:
8007:
7984:
7976:
7972:
7948:
7942:
7922:
7919:
7913:
7907:
7898:
7885:
7879:
7873:
7870:
7864:
7861:
7858:
7852:
7844:
7830:
7821:
7808:
7804:
7794:
7788:
7784:
7779:
7776:
7772:
7768:
7765:
7762:
7759:
7753:
7750:
7745:
7742:
7731:
7717:
7714:
7708:
7702:
7688:
7675:
7672:
7668:
7661:
7658:
7653:
7650:
7646:
7637:
7633:
7629:
7626:
7622:
7618:
7615:
7610:
7607:
7602:
7599:
7596:
7593:
7590:
7587:
7582:
7579:
7574:
7571:
7568:
7562:
7559:
7552:
7549:
7544:
7541:
7538:
7535:
7527:
7523:
7520:
7515:
7512:
7508:
7503:
7497:
7493:
7489:
7486:
7479:
7476:
7472:
7466:
7462:
7458:
7452:
7444:
7431:
7428:
7423:
7420:
7416:
7410:
7406:
7402:
7399:
7396:
7393:
7372:
7368:
7365:
7360:
7357:
7353:
7348:
7342:
7338:
7334:
7331:
7328:
7325:
7318:
7314:
7304:
7291:
7285:
7282:
7279:
7273:
7270:
7264:
7261:
7256:
7253:
7242:
7228:
7224:
7220:
7200:
7180:
7172:
7168:
7164:
7160:
7146:
7126:
7117:
7112:
7098:
7095:
7089:
7083:
7063:
7060:
7054:
7048:
7028:
7008:
6999:
6986:
6983:
6980:
6974:
6968:
6957:
6944:
6941:
6928:
6920:
6917:
6914:
6910:
6905:
6898:
6894:
6887:
6883:
6879:
6876:
6870:
6866:
6863:
6859:
6854:
6847:
6843:
6840:
6835:
6832:
6828:
6823:
6817:
6813:
6809:
6806:
6799:
6796:
6792:
6786:
6782:
6778:
6772:
6766:
6760:
6752:
6736:
6732:
6723:
6707:
6687:
6679:
6663:
6643:
6639:
6633:
6629:
6625:
6622:
6602:
6582:
6562:
6559:
6556:
6547:
6545:
6529:
6521:
6505:
6496:
6483:
6479:
6473:
6470:
6465:
6462:
6458:
6454:
6451:
6448:
6442:
6439:
6434:
6431:
6420:
6418:
6402:
6382:
6362:
6353:
6351:
6347:
6343:
6339:
6335:
6334:Raymond Pearl
6331:
6327:
6323:
6319:
6315:
6314:
6309:
6304:
6300:
6296:
6287:
6273:
6271:
6266:
6264:
6260:
6256:
6252:
6248:
6238:
6224:
6221:
6201:
6198:
6193:
6189:
6158:
6154:
6150:
6145:
6141:
6137:
6132:
6128:
6107:
6104:
6099:
6095:
6068:
6064:
6033:
6029:
6025:
6020:
6016:
5993:
5989:
5968:
5965:
5960:
5956:
5947:
5929:
5925:
5921:
5916:
5912:
5898:
5880:
5876:
5871:
5865:
5860:
5857:
5854:
5850:
5845:
5837:
5833:
5828:
5815:
5810:
5789:
5785:
5780:
5776:
5769:
5765:
5760:
5752:
5744:
5740:
5735:
5707:
5703:
5698:
5694:
5687:
5683:
5678:
5670:
5662:
5658:
5653:
5649:
5645:
5634:
5630:
5626:
5621:
5617:
5610:
5606:
5602:
5599:
5595:
5590:
5586:
5583:
5577:
5574:
5567:
5563:
5559:
5554:
5550:
5545:
5537:
5529:
5525:
5521:
5516:
5512:
5507:
5499:is the odds,
5482:
5478:
5474:
5469:
5465:
5460:
5451:
5433:
5429:
5425:
5420:
5416:
5408:
5388:
5384:
5357:
5353:
5326:
5322:
5295:
5291:
5279:
5265:
5262:
5254:
5250:
5246:
5243:
5236:
5232:
5229:
5223:
5220:
5215:
5212:
5208:
5200:
5194:
5191:
5187:
5164:
5161:
5157:
5148:
5130:
5127:
5117:
5097:
5094:
5086:
5083:
5079:
5075:
5072:
5065:
5061:
5058:
5052:
5049:
5044:
5040:
5032:
5026:
5022:
4998:
4995:
4992:
4985:
4981:
4955:
4933:
4930:
4927:
4924:
4921:
4913:
4893:
4889:
4878:
4874:
4865:
4848:
4842:
4822:
4819:
4816:
4813:
4810:
4790:
4787:
4784:
4781:
4778:
4770:
4741:
4738:
4735:
4710:
4707:
4704:
4694:
4684:
4660:
4657:
4654:
4648:
4638:
4635:
4610:
4607:
4603:
4599:
4596:
4592:
4587:
4581:
4575:
4571:
4558:
4552:
4549:
4546:
4533:
4527:
4522:
4519:
4514:
4508:
4505:
4497:
4491:
4488:
4478:
4462:
4459:
4456:
4447:
4445:
4440:
4427:
4419:
4416:
4412:
4408:
4405:
4401:
4396:
4390:
4387:
4382:
4378:
4371:
4367:
4361:
4355:
4349:
4341:
4327:
4324:
4321:
4312:
4299:
4293:
4290:
4285:
4281:
4274:
4270:
4264:
4258:
4252:
4244:
4242:
4237:
4222:
4202:
4194:
4189:
4187:
4183:
4167:
4163:
4159:
4156:
4150:
4144:
4137:
4132:
4109:
4103:
4100:
4097:
4084:
4078:
4075:
4069:
4063:
4057:
4054:
4050:
4040:
4039:
4029:
4027:
4023:
4022:ramp function
4019:
4018:
4013:
4008:
3995:
3987:
3983:
3979:
3976:
3970:
3967:
3964:
3961:
3958:
3955:
3952:
3949:
3946:
3940:
3937:
3932:
3929:
3926:
3923:
3914:
3910:
3906:
3903:
3897:
3893:
3887:
3879:
3877:
3872:
3859:
3854:
3850:
3847:
3841:
3833:
3829:
3825:
3822:
3816:
3812:
3806:
3800:
3794:
3786:
3770:
3766:
3762:
3759:
3756:
3753:
3746:
3742:
3732:
3716:
3690:
3686:
3661:
3658:
3655:
3645:
3640:
3622:
3615:
3609:
3606:
3603:
3599:
3592:
3586:
3583:
3581:
3572:
3563:
3559:
3555:
3552:
3546:
3542:
3536:
3533:
3529:
3524:
3516:
3512:
3508:
3505:
3499:
3495:
3489:
3485:
3483:
3474:
3466:
3462:
3458:
3455:
3451:
3446:
3441:
3433:
3429:
3425:
3422:
3416:
3412:
3406:
3402:
3400:
3387:
3377:
3373:
3369:
3366:
3357:
3353:
3347:
3345:
3332:
3322:
3318:
3314:
3311:
3301:
3297:
3293:
3288:
3284:
3280:
3272:
3268:
3264:
3261:
3255:
3250:
3246:
3239:
3237:
3229:
3223:
3217:
3192:
3179:
3171:
3167:
3163:
3160:
3154:
3150:
3144:
3136:
3133:
3129:
3125:
3122:
3118:
3113:
3107:
3101:
3093:
3091:
3087:
3078:
3069:
3043:
3038:
3035:
3028:
3023:
3005:
3002:
2967:
2962:
2955:
2950:
2918:
2914:
2910:
2885:
2881:
2877:
2872:
2869:
2865:
2853:
2840:
2837:
2834:
2831:
2821:
2805:
2802:
2797:
2793:
2789:
2786:
2783:
2754:
2750:
2746:
2742:
2738:
2733:
2729:
2725:
2722:
2718:
2714:
2709:
2705:
2701:
2697:
2668:
2664:
2660:
2651:
2648:
2644:
2640:
2635:
2631:
2624:
2621:
2617:
2608:
2605:
2601:
2597:
2592:
2588:
2580:
2571:
2567:
2562:
2546:
2542:
2538:
2511:
2508:
2505:
2478:
2454:
2432:
2429:
2423:
2420:
2417:
2411:
2391:
2388:
2383:
2379:
2375:
2372:
2369:
2346:
2318:
2315:
2312:
2285:
2261:
2258:
2236:
2233:
2227:
2224:
2221:
2212:
2209:
2206:
2183:
2180:
2175:
2171:
2167:
2162:
2158:
2150:
2146:
2141:
2139:
2134:
2121:
2117:
2112:
2109:
2104:
2100:
2095:
2091:
2085:
2082:
2077:
2071:
2065:
2059:
2056:
2052:
2042:
2039:
2022:
2019:
2013:
2010:
2004:
2001:
1998:
1990:
1987:
1984:
1980:
1976:
1973:
1968:
1965:
1962:
1959:
1954:
1951:
1948:
1944:
1937:
1931:
1928:
1922:
1919:
1911:
1908:
1905:
1901:
1897:
1894:
1888:
1885:
1882:
1878:
1872:
1866:
1863:
1857:
1854:
1852:
1840:
1834:
1831:
1828:
1824:
1820:
1817:
1813:
1809:
1804:
1800:
1793:
1787:
1784:
1781:
1777:
1773:
1770:
1766:
1762:
1757:
1753:
1746:
1738:
1735:
1731:
1727:
1722:
1718:
1710:
1707:
1703:
1699:
1694:
1690:
1683:
1681:
1673:
1667:
1664:
1651:
1638:
1635:
1629:
1626:
1620:
1617:
1614:
1608:
1602:
1599:
1579:
1575:
1570:
1567:
1562:
1558:
1555:
1550:
1547:
1542:
1537:
1534:
1529:
1523:
1517:
1509:
1499:
1497:
1493:
1489:
1470:
1467:
1464:
1461:
1458:
1447:
1444:
1441:
1437:
1432:
1429:
1426:
1423:
1420:
1417:
1410:
1409:
1408:
1406:
1396:
1393:
1380:
1377:
1371:
1368:
1363:
1359:
1354:
1349:
1343:
1340:
1335:
1331:
1324:
1320:
1314:
1303:
1300:
1294:
1290:
1286:
1283:
1279:
1274:
1266:
1263:
1259:
1255:
1252:
1248:
1238:
1221:
1218:
1212:
1203:
1201:
1185:
1181:
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713:for details.
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11811:. Retrieved
11804:the original
11799:
11786:
11775:the original
11770:
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11745:. Retrieved
11738:the original
11733:
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11650:
11640:
11629:
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11609:
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11244:
11231:
11222:
11179:
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11169:
11134:
11131:Scholarpedia
11130:
11120:
11109:
11089:
11075:cite journal
11058:
11054:
11044:
11035:
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10990:
10986:
10976:
10964:. Retrieved
10956:
10954:Raul Rojas.
10949:
10936:
10927:
10916:the original
10903:
10899:
10873:
10861:
10855:
10845:
10833:. Retrieved
10821:
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10800:à la courbe
10797:
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10780:
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7167:life history
7114:In ecology,
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6548:
6518:defines the
6497:
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4477:general form
4476:
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4182:logistic map
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245:real numbers
242:
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11137:(7): 4308.
11099:Lowell Reed
11061:: 649–653.
10835:18 February
10790:18 February
10759:Cramer 2002
10272:by setting
10152:arithmētikḗ
10125:logarithmic
10117:exponential
10113:logarithmic
10109:progression
10105:progression
10013:Cross fluid
9724:water table
8364:computes a
8322:Rasch model
7159:-strategist
6520:growth rate
6338:Lowell Reed
6120:, and then
5452:option 0),
1492:probability
662:being from
640:logarithmic
581:linguistics
569:probability
522:the sigmoid
12126:Categories
11813:6 November
11747:6 November
11664:1604.07782
11579:(6): 582.
11418:2005.00662
11385:2006.01550
11312:: 109841.
11286:2004.02406
10966:15 October
10939:. p.
10798:logistique
10704:3 December
10670:References
10184:Pythagoras
10171:λογῐστῐκός
10165:ἀριθμητική
10159:arithmetic
10145:ἀριθμητική
10139:λογῐστῐκός
10043:STAR model
9918:Kondratiev
9554:is set to
8490:See also:
8324:, used in
8313:, used in
6678:bottleneck
6322:biological
6297:(see also
6261:, and the
5407:difference
4193:phase line
3086:derivative
3072:Derivative
1510:function:
1146:is large.
1042:Symmetries
679:logistikós
668:λογῐστῐκός
636:logistique
585:statistics
561:geoscience
553:demography
12093:MathWorld
12054:MathWorld
11691:1932-6203
11554:245811843
11502:103125742
11189:0901.4714
11161:1941-6016
10700:: 113–121
10625:⋯
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10568:…
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10289:∞
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10188:λογιστική
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9364:: (i) if
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9097:θ
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8960:→
8949:→
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8674:≤
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1301:−
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1175:−
1160:↦
1149:Further,
1083:−
1059:−
1001:−
952:−
856:−
707:logistics
691:logistics
673:romanized
573:sociology
557:economics
549:chemistry
497:−
293:∞
287:→
264:∞
261:−
258:→
106:−
94:−
11709:28981532
11651:PLOS ONE
11601:59380632
11455:32726361
11405:PLOS ONE
11344:32501369
11214:14456352
11027:32517314
10935:(1830).
10912:42097070
10824:: 1–32.
10376:softplus
10121:logistic
10006:See also
9934:maturity
9868:scarcity
9765:(1890),
9706:inverted
9702:decrease
9698:increase
9694:negative
9690:positive
9064:COVID-19
9021:COVID-19
8740:′
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7307:Integral
7173:so that
6722:maturity
6355:Letting
6181:. Since
5946:relative
4970:against
4873:log-odds
4017:softplus
3851:′
3785:, since
3735:Integral
660:logistic
167:supremum
11700:5628819
11669:Bibcode
11581:Bibcode
11532:Bibcode
11494:1471365
11446:7390340
11423:Bibcode
11359:medRxiv
11335:7252058
11314:Bibcode
11194:Bibcode
11139:Bibcode
11018:7309099
10995:Bibcode
10987:Sensors
10257:
10225:
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9650:mirages
8117:of the
7116:species
6542:is the
6257:of the
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9468:, and
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6249:, the
4877:logits
4186:linear
1198:is an
709:; see
695:French
587:, and
165:, the
139:where
11944:(PDF)
11835:(PDF)
11807:(PDF)
11796:(PDF)
11778:(PDF)
11767:(PDF)
11741:(PDF)
11730:(PDF)
11659:arXiv
11597:S2CID
11550:S2CID
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11413:arXiv
11380:arXiv
11281:arXiv
11241:(PDF)
11210:S2CID
11184:arXiv
11106:(PDF)
10961:(PDF)
10919:(PDF)
10908:S2CID
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10896:(PDF)
10690:(PDF)
10082:Notes
9906:IIASA
9876:Basel
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9565:10000
8459:with
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4134:with
1418:logit
1405:logit
1237:, is
701:logis
530:logit
526:expit
12031:ISBN
12012:ISBN
11991:ISBN
11910:ISBN
11858:ISBN
11815:2010
11749:2010
11705:PMID
11687:ISSN
11528:2145
11490:OSTI
11451:PMID
11340:PMID
11249:ISBN
11157:ISSN
11081:link
11023:PMID
10968:2016
10837:2013
10792:2013
10706:2014
10310:and
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11310:135
11202:doi
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8512:(
8509:X
8438:,
8430:h
8424:2
8417:e
8413:+
8410:1
8406:1
8401:=
8398:)
8395:h
8392:(
8389:g
8286:f
8266:b
8246:a
8226:x
8204:,
8201:)
8198:x
8195:b
8192:+
8189:a
8186:(
8183:f
8180:=
8177:p
8151:p
8092:)
8089:t
8086:(
8083:K
8060:)
8057:t
8054:(
8051:K
8031:T
8008:T
7988:)
7985:t
7982:(
7973:P
7952:)
7949:t
7946:(
7943:P
7923:0
7917:)
7914:0
7911:(
7908:P
7886:.
7883:)
7880:t
7877:(
7874:K
7871:=
7868:)
7865:T
7862:+
7859:t
7856:(
7853:K
7831:T
7809:.
7805:)
7798:)
7795:t
7792:(
7789:K
7785:P
7777:1
7773:(
7766:P
7763:r
7760:=
7754:t
7751:d
7746:P
7743:d
7718:0
7712:)
7709:t
7706:(
7703:K
7676:C
7673:+
7669:)
7665:)
7662:1
7654:t
7651:r
7647:e
7643:(
7638:0
7634:P
7630:+
7627:K
7623:(
7611:r
7608:K
7603:=
7600:C
7597:+
7594:u
7583:r
7580:K
7575:=
7572:u
7569:d
7563:u
7560:1
7553:r
7550:K
7542:=
7539:t
7536:d
7528:)
7524:1
7516:t
7513:r
7509:e
7504:(
7498:0
7494:P
7490:+
7487:K
7480:t
7477:r
7473:e
7467:0
7463:P
7459:K
7432:t
7429:d
7424:t
7421:r
7417:e
7411:0
7407:P
7403:r
7400:=
7397:u
7394:d
7373:)
7369:1
7361:t
7358:r
7354:e
7349:(
7343:0
7339:P
7335:+
7332:K
7329:=
7326:u
7292:.
7289:)
7286:n
7280:1
7277:(
7274:n
7271:=
7262:d
7257:n
7254:d
7229:r
7225:/
7221:1
7181:n
7147:K
7127:r
7099:K
7093:)
7090:0
7087:(
7084:P
7064:0
7058:)
7055:0
7052:(
7049:P
7029:P
7009:K
6987:,
6984:K
6981:=
6978:)
6975:t
6972:(
6969:P
6958:t
6929:,
6921:t
6918:r
6911:e
6906:)
6899:0
6895:P
6888:0
6884:P
6877:K
6871:(
6867:+
6864:1
6860:K
6855:=
6848:)
6844:1
6836:t
6833:r
6829:e
6824:(
6818:0
6814:P
6810:+
6807:K
6800:t
6797:r
6793:e
6787:0
6783:P
6779:K
6773:=
6770:)
6767:t
6764:(
6761:P
6737:0
6733:P
6708:P
6688:K
6664:P
6644:K
6640:/
6634:2
6630:P
6626:r
6603:P
6583:r
6563:P
6560:r
6557:+
6530:K
6506:r
6484:,
6480:)
6474:K
6471:P
6463:1
6459:(
6455:P
6452:r
6449:=
6443:t
6440:d
6435:P
6432:d
6403:t
6383:N
6363:P
6225:1
6222:+
6202:1
6199:=
6194:0
6190:e
6179:0
6175:i
6159:0
6155:x
6146:i
6142:x
6138:=
6133:i
6129:x
6108:0
6105:=
6100:0
6096:x
6085:0
6069:i
6065:x
6054:i
6050:j
6034:i
6030:x
6021:j
6017:x
5994:i
5990:x
5969:0
5966:=
5961:0
5957:x
5930:0
5926:x
5917:1
5913:x
5902:x
5881:i
5877:x
5872:e
5866:n
5861:0
5858:=
5855:i
5846:/
5838:i
5834:x
5829:e
5818:i
5797:)
5790:1
5786:x
5781:e
5777:+
5770:0
5766:x
5761:e
5757:(
5753:/
5745:0
5741:x
5736:e
5715:)
5708:1
5704:x
5699:e
5695:+
5688:0
5684:x
5679:e
5675:(
5671:/
5663:1
5659:x
5654:e
5650:=
5646:)
5640:)
5635:0
5631:x
5622:1
5618:x
5614:(
5607:e
5603:+
5600:1
5596:(
5591:/
5587:1
5584:=
5581:)
5578:1
5575:+
5568:0
5564:x
5555:1
5551:x
5546:e
5542:(
5538:/
5530:0
5526:x
5517:1
5513:x
5508:e
5483:0
5479:x
5470:1
5466:x
5461:e
5434:0
5430:x
5421:1
5417:x
5389:1
5385:x
5358:0
5354:x
5327:1
5323:x
5296:0
5292:x
5266:q
5263:=
5260:)
5255:x
5251:e
5247:+
5244:1
5241:(
5237:/
5233:1
5230:=
5227:)
5224:1
5221:+
5216:x
5209:e
5205:(
5201:/
5195:x
5188:e
5165:x
5158:e
5131:x
5112:x
5098:p
5095:=
5092:)
5087:x
5080:e
5076:+
5073:1
5070:(
5066:/
5062:1
5059:=
5056:)
5053:1
5050:+
5045:x
5041:e
5037:(
5033:/
5027:x
5023:e
5002:)
4999:1
4996:+
4993:O
4990:(
4986:/
4982:O
4972:1
4956:O
4934:1
4931::
4928:O
4925:=
4922:O
4894:x
4890:e
4869:x
4843:x
4823:1
4820:=
4817:q
4814:+
4811:p
4791:p
4785:1
4782:=
4779:q
4765:p
4761:p
4745:)
4742:1
4739:,
4736:0
4733:(
4711:1
4708:=
4705:L
4669:)
4664:)
4661:r
4655:x
4652:(
4649:k
4644:(
4639:S
4636:a
4611:r
4608:k
4604:e
4600:+
4597:1
4593:a
4588:=
4585:)
4582:0
4579:(
4576:f
4572:,
4567:)
4562:)
4559:x
4556:(
4553:f
4547:a
4542:(
4537:)
4534:x
4531:(
4528:f
4523:a
4520:k
4515:=
4509:x
4506:d
4501:)
4498:x
4495:(
4492:f
4489:d
4463:0
4457:x
4428:.
4420:x
4413:e
4409:+
4406:1
4402:1
4397:=
4391:1
4388:+
4383:x
4379:e
4372:x
4368:e
4362:=
4359:)
4356:x
4353:(
4350:f
4328:1
4325:=
4322:C
4300:.
4294:C
4291:+
4286:x
4282:e
4275:x
4271:e
4265:=
4262:)
4259:x
4256:(
4253:f
4223:f
4203:f
4168:2
4164:/
4160:1
4157:=
4154:)
4151:0
4148:(
4145:f
4118:)
4113:)
4110:x
4107:(
4104:f
4098:1
4093:(
4088:)
4085:x
4082:(
4079:f
4076:=
4073:)
4070:x
4067:(
4064:f
4058:x
4055:d
4051:d
3996:.
3993:)
3988:x
3984:e
3980:+
3977:1
3974:(
3965:=
3962:u
3953:=
3950:u
3947:d
3941:u
3938:1
3930:=
3927:x
3924:d
3915:x
3911:e
3907:+
3904:1
3898:x
3894:e
3860:,
3855:u
3848:u
3842:=
3834:x
3830:e
3826:+
3823:1
3817:x
3813:e
3807:=
3804:)
3801:x
3798:(
3795:f
3771:x
3767:e
3763:+
3760:1
3757:=
3754:u
3717:k
3691:0
3687:x
3662:1
3659:=
3656:L
3623:)
3619:)
3616:x
3613:(
3610:f
3604:1
3600:(
3596:)
3593:x
3590:(
3587:f
3584:=
3573:)
3564:x
3560:e
3556:+
3553:1
3547:x
3543:e
3534:1
3530:(
3525:)
3517:x
3513:e
3509:+
3506:1
3500:x
3496:e
3490:(
3486:=
3475:)
3467:x
3463:e
3459:+
3456:1
3452:1
3447:(
3442:)
3434:x
3430:e
3426:+
3423:1
3417:x
3413:e
3407:(
3403:=
3388:2
3384:)
3378:x
3374:e
3370:+
3367:1
3364:(
3358:x
3354:e
3348:=
3333:2
3329:)
3323:x
3319:e
3315:+
3312:1
3309:(
3302:x
3298:e
3289:x
3285:e
3278:)
3273:x
3269:e
3265:+
3262:1
3259:(
3251:x
3247:e
3240:=
3233:)
3230:x
3227:(
3224:f
3218:x
3214:d
3208:d
3180:,
3172:x
3168:e
3164:+
3161:1
3155:x
3151:e
3145:=
3137:x
3130:e
3126:+
3123:1
3119:1
3114:=
3111:)
3108:x
3105:(
3102:f
3054:)
3044:1
3039:1
3029:1
3024:1
3014:(
3006:2
3003:1
2978:)
2968:1
2963:0
2956:1
2951:1
2941:(
2919:2
2915:/
2911:t
2891:)
2886:t
2882:e
2878:,
2873:t
2866:e
2862:(
2841:1
2838:=
2835:y
2832:x
2806:1
2803:=
2798:2
2794:y
2787:y
2784:x
2762:)
2755:2
2751:/
2747:t
2743:e
2739:,
2734:2
2730:/
2726:t
2719:e
2715:+
2710:2
2706:/
2702:t
2698:e
2692:(
2669:)
2665:2
2661:/
2657:)
2652:t
2645:e
2636:t
2632:e
2628:(
2625:,
2622:2
2618:/
2614:)
2609:t
2602:e
2598:+
2593:t
2589:e
2585:(
2581:(
2547:2
2543:/
2539:1
2515:)
2512:1
2509:,
2506:2
2503:(
2479:1
2455:0
2433:1
2430:=
2427:)
2424:y
2418:x
2415:(
2412:y
2392:1
2389:=
2384:2
2380:y
2373:y
2370:x
2347:1
2322:)
2319:0
2316:,
2313:1
2310:(
2286:1
2262:1
2237:1
2234:=
2231:)
2228:y
2222:x
2219:(
2216:)
2213:y
2210:+
2207:x
2204:(
2184:1
2181:=
2176:2
2172:y
2163:2
2159:x
2122:,
2118:)
2113:2
2110:x
2105:(
2096:2
2086:4
2083:1
2078:=
2075:)
2072:x
2069:(
2066:f
2060:x
2057:d
2053:d
2017:)
2014:x
2011:2
2008:(
2005:f
2002:2
1999:=
1991:x
1988:2
1981:e
1977:+
1974:1
1969:1
1963:1
1960:+
1955:x
1952:2
1945:e
1935:)
1932:x
1929:2
1926:(
1923:f
1920:=
1912:x
1909:2
1902:e
1898:+
1895:1
1889:x
1886:2
1879:e
1870:)
1867:x
1864:2
1861:(
1858:f
1855:=
1841:)
1835:x
1832:2
1825:e
1821:+
1818:1
1814:(
1805:x
1801:e
1794:)
1788:x
1785:2
1778:e
1771:1
1767:(
1758:x
1754:e
1747:=
1739:x
1732:e
1728:+
1723:x
1719:e
1711:x
1704:e
1695:x
1691:e
1684:=
1677:)
1674:x
1671:(
1633:)
1630:x
1627:2
1624:(
1621:f
1618:2
1615:=
1612:)
1609:x
1606:(
1580:,
1576:)
1571:2
1568:x
1563:(
1551:2
1548:1
1543:+
1538:2
1535:1
1530:=
1527:)
1524:x
1521:(
1518:f
1471:1
1465:p
1459:0
1448:p
1442:1
1438:p
1427:=
1424:p
1378:=
1372:1
1369:+
1364:x
1360:e
1355:1
1350:+
1344:1
1341:+
1336:x
1332:e
1325:x
1321:e
1315:=
1307:)
1304:x
1298:(
1291:e
1287:+
1284:1
1280:1
1275:+
1267:x
1260:e
1256:+
1253:1
1249:1
1225:)
1222:x
1216:(
1213:f
1186:2
1182:/
1178:1
1172:)
1169:x
1166:(
1163:f
1157:x
1134:x
1114:x
1092:.
1089:)
1086:x
1080:(
1077:f
1074:=
1071:)
1068:x
1065:(
1062:f
1056:1
1026:x
1004:x
997:e
971:.
963:2
959:/
955:x
948:e
944:+
939:2
935:/
931:x
927:e
920:2
916:/
912:x
908:e
902:=
896:1
893:+
888:x
884:e
877:x
873:e
867:=
859:x
852:e
848:+
845:1
841:1
836:=
833:)
830:x
827:(
824:f
802:1
799:=
796:L
776:0
773:=
768:0
764:x
743:1
740:=
737:k
500:x
493:e
489:+
486:1
482:1
477:=
474:)
471:x
468:(
465:f
442:0
439:=
434:0
430:x
426:,
423:1
420:=
417:k
414:,
411:1
408:=
405:L
389:.
377:0
374:=
369:0
365:x
361:,
358:1
355:=
352:k
349:,
346:1
343:=
340:L
313:L
290:+
284:x
255:x
228:x
206:0
202:x
179:k
149:L
119:)
114:0
110:x
103:x
100:(
97:k
90:e
86:+
83:1
79:L
74:=
71:)
68:x
65:(
62:f
34:.
20:)
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