In ecology: modeling population growth A typical application of the logistic equation is a common model of
population growth (see also
population dynamics), originally due to
Pierre-François Verhulst in 1838, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal. The Verhulst equation was published after Verhulst had read
Thomas Malthus'
An Essay on the Principle of Population, which describes the
Malthusian growth model of simple (unconstrained) exponential growth. Verhulst derived his logistic equation to describe the self-limiting growth of a
biological population. The equation was rediscovered in 1911 by
A. G. McKendrick for the growth of bacteria in broth and experimentally tested using a technique for nonlinear parameter estimation. The equation is also sometimes called the
Verhulst-Pearl equation following its rediscovery in 1920 by
Raymond Pearl (1879–1940) and
Lowell Reed (1888–1966) of the
Johns Hopkins University. Another scientist,
Alfred J. Lotka derived the equation again in 1925, calling it the
law of population growth. Letting P represent population size (N is often used in ecology instead) and t represent time, this model is formalized by the
differential equation: \frac{dP}{dt}=r P \left(1 - \frac{P}{K}\right), where the constant r defines the
growth rate and K is the
carrying capacity. In the equation, the early, unimpeded growth rate is modeled by the first term +rP. The value of the rate r represents the proportional increase of the population P in one unit of time. Later, as the population grows, the modulus of the second term (which multiplied out is -r P^2 / K) becomes almost as large as the first, as some members of the population P interfere with each other by competing for some critical resource, such as food or living space. This antagonistic effect is called the
bottleneck, and is modeled by the value of the parameter K. The competition diminishes the combined growth rate, until the value of P ceases to grow (this is called
maturity of the population). The solution to the equation (with P_0 being the initial population) is P(t) = \frac{K P_0 e^{rt}}{K + P_0 \left( e^{rt} - 1\right)} = \frac{K}{1+\left(\frac{K-P_0}{P_0}\right)e^{-rt}}, where \lim_{t\to\infty} P(t) = K, where K is the limiting value of P, 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 P(0) > 0, and also in the case that P(0) > K. In ecology,
species are sometimes referred to as
r-strategist or K-strategist depending upon the
selective processes that have shaped their
life history strategies.
Choosing the variable dimensions so that n measures the population in units of carrying capacity, and \tau measures time in units of 1/r, gives the dimensionless differential equation \frac{dn}{d\tau} = n (1-n).
Integral The
antiderivative of the ecological form of the logistic function can be computed by the
substitution u = K + P_0 \left( e^{rt} - 1\right), since du = r P_0 e^{rt} dt \int \frac{K P_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
Time-varying carrying capacity Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying, with K(t) > 0, leading to the following mathematical model: \frac{dP}{dt} = rP \cdot \left(1 - \frac{P}{K(t)}\right). A particularly important case is that of carrying capacity that varies periodically with period T: K(t + T) = K(t). It can be shown that in such a case, independently from the initial value P(0) > 0, P(t) will tend to a unique periodic solution P_*(t), whose period is T. A typical value of T is one year: In such case K(t) may reflect periodical variations of weather conditions. Another interesting generalization is to consider that the carrying capacity K(t) 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, punctuated unlimited growth (i.e., multiple S-shapes), punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death.
In statistics and machine learning Logistic functions are used in several roles in statistics. For example, they are the
cumulative distribution function of the
logistic family of distributions, and they are, a bit simplified, used to model the chance a chess player has to beat their opponent in the
Elo rating system. More specific examples now follow.
Logistic regression Logistic functions are used in
logistic regression to model how the probability p of an event may be affected by one or more
explanatory variables: an example would be to have the model p = f(a + bx), where x is the explanatory variable, a and b are model parameters to be fitted, and f is the standard logistic function. Logistic regression and other
log-linear models are also commonly used in
machine learning. A generalisation of the logistic function to multiple inputs is the
softmax activation function, used in
multinomial logistic regression. Another application of the logistic function is in the
Rasch model, used in
item response theory. In particular, the Rasch model forms a basis for
maximum likelihood estimation of the locations of objects or persons on a
continuum, based on collections of
categorical data, for example the abilities of persons on a continuum based on responses that have been categorized as correct and incorrect.
Neural networks Logistic functions are often used in
artificial neural networks to introduce
nonlinearity in the model or to clamp signals to within a specified
interval. A popular
neural net element computes a
linear combination of its input signals, and applies a bounded logistic function as the
activation function to the result; this model can be seen as a "smoothed" variant of the classical
threshold neuron.y' = y(1-y). The right hand side is a low-degree polynomial. Furthermore, the polynomial has factors y and 1 − y, both of which are simple to compute. Given y = sig(t) at a particular t, the derivative of the logistic function at that t can be obtained by multiplying the two factors together. --> A common choice for the activation or "squashing" functions, used to clip large magnitudes to keep the response of the neural network bounded, is g(h) = \frac{1}{1 + e^{-2 \beta h}}, which is a logistic function. These relationships result in simplified implementations of
artificial neural networks with
artificial neurons. Practitioners caution that sigmoidal functions which are
antisymmetric about the origin (e.g. the
hyperbolic tangent) lead to faster convergence when training networks with
backpropagation. The logistic function is itself the derivative of another proposed activation function, the
softplus.
In medicine: modeling of growth of tumors Another application of logistic curve is in medicine, where the logistic differential equation can be used to model the growth of
tumors. This application can be considered an extension of the above-mentioned use in the framework of ecology (see also the
Generalized logistic curve, allowing for more parameters). Denoting with X(t) the size of the tumor at time t, its dynamics are governed by X' = r\left(1 - \frac X K \right)X, which is of the type X' = F(X)X, \quad F'(X) \le 0, where F(X) is the proliferation rate of the tumor. If a course of
chemotherapy is started with a log-kill effect, the equation may be revised to be X' = r\left(1 - \frac X K \right)X - c(t) X, where c(t) is the therapy-induced death rate. In the idealized case of very long therapy, c(t) can be modeled as a
periodic function (of period T) or (in case of continuous infusion therapy) as a
constant function, and one has that \frac 1 T \int_0^T c(t)\, dt > r \to \lim_{t \to +\infty} x(t) = 0, 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. For example, it does not take into account the evolution of clonal resistance, or the side-effects of the therapy on the patient. These factors can result in the eventual failure of chemotherapy, or its discontinuation.
In medicine: modeling of a pandemic 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
COVID-19 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
herd immunity) 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
Pierre-François Verhulst, as noted above) and then reaching a maximal limit. A logistic function, or related functions (e.g. the
Gompertz function) 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.
Modeling early COVID-19 cases (Richards growth curve) in epidemiological modeling A
generalized logistic function, also called the Richards growth curve, has been applied to model the early phase of the
COVID-19 outbreak. The authors fit the generalized logistic function to the cumulative number of infected cases, here referred to as
infection trajectory. There are different parameterizations of the
generalized logistic function in the literature. One frequently used forms is f(t ; \theta_1,\theta_2,\theta_3, \xi) = \frac{\theta_1}{{\left[1 + \xi \exp \left(-\theta_2 \cdot (t - \theta_3) \right) \right]}^{1/\xi}} where \theta_1,\theta_2,\theta_3 are real numbers, and \xi is a positive real number. The flexibility of the curve f is due to the parameter \xi : (i) if \xi = 1 then the curve reduces to the logistic function, and (ii) as \xi approaches zero, the curve converges to the
Gompertz function. In epidemiological modeling, \theta_1, \theta_2, and \theta_3 represent the final epidemic size, infection rate, and lag phase, respectively. See the right panel for an example infection trajectory when (\theta_1,\theta_2,\theta_3) is set to (10000,0.2,40). One of the benefits of using a growth function such as the
generalized logistic function in epidemiological modeling is its relatively easy application to the
multilevel model framework, where information from different geographic regions can be pooled together.
In chemistry: reaction models The concentration of reactants and products in
autocatalytic reactions follow the logistic function. The degradation of
Platinum group metal-free (PGM-free) oxygen reduction reaction (ORR) catalyst in fuel cell cathodes follows the logistic decay function, suggesting an autocatalytic degradation mechanism.
In physics: Fermi–Dirac distribution The logistic function determines the statistical distribution of fermions over the energy states of a system in
thermal equilibrium. In particular, it is the distribution of the probabilities that each possible energy level is occupied by a fermion, according to
Fermi–Dirac statistics.
In optics: mirage The logistic function also finds applications in optics, particularly in modelling phenomena such as
mirages. 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. 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 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. an innovation that is at first marginal begins to spread more quickly with time, and then more slowly as it becomes more universally adopted.
In agriculture: modeling crop response The logistic S-curve can be used for modeling the crop response to changes in growth factors. There are two types of response functions:
positive and
negative growth curves. For example, the crop yield may
increase with increasing value of the growth factor up to a certain level (positive function), or it may
decrease with increasing growth factor values (negative function owing to a negative growth factor), which situation requires an
inverted S-curve.
In economics and sociology: diffusion of innovations The logistic function can be used to illustrate the progress of the
diffusion of an innovation through its life cycle. In
The Laws of Imitation (1890),
Gabriel Tarde describes the rise and spread of new ideas through imitative chains. In particular, Tarde identifies three main stages through which innovations spread: the first one corresponds to the difficult beginnings, during which the idea has to struggle within a hostile environment full of opposing habits and beliefs; the second one corresponds to the properly exponential take-off of the idea, with f(x)=2^x; finally, the third stage is logarithmic, with f(x)=\log(x), and corresponds to the time when the impulse of the idea gradually slows down while, simultaneously new opponent ideas appear. The ensuing situation halts or stabilizes the progress of the innovation, which approaches an asymptote. In a
sovereign state, 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
scarcity constraints, especially the resources the banks can lend (due to their
equity or
Basel limits). These restrictions, which represent a saturation level, along with an exponential rush in an
economic competition for money, create a
public finance diffusion of credit pleas and the aggregate national response is a
sigmoid curve. Historically, when new products are introduced there is an intense amount of
research and development which leads to dramatic improvements in quality and reductions in cost. This leads to a period of rapid industry growth. Some of the more famous examples are: railroads, incandescent light bulbs,
electrification, 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. Logistic analysis was used in papers by several researchers at the International Institute of Applied Systems Analysis (
IIASA). These papers deal with the diffusion of various innovations, infrastructures and energy source substitutions and the role of work in the economy as well as with the long economic cycle. Long economic cycles were investigated by Robert Ayres (1989). Cesare Marchetti published on
long economic cycles 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. Carlota Perez used a logistic curve to illustrate the long (
Kondratiev)
business cycle with the following labels: beginning of a technological era as
irruption, the ascent as
frenzy, the rapid build out as
synergy and the completion as
maturity.
Inflection Point Determination in Logistic Growth Regression Logistic growth regressions carry significant uncertainty when data is available only up to around the inflection point of the growth process. Under these conditions, estimating the height at which the inflection point will occur may have uncertainties comparable to the carrying capacity (K) of the system. A method to mitigate this uncertainty involves using the carrying capacity from a surrogate logistic growth process as a reference point. By incorporating this constraint, even if K is only an estimate within a factor of two, the regression is stabilized, which improves accuracy and reduces uncertainty in the prediction parameters. This approach can be applied in fields such as economics and biology, where analogous surrogate systems or populations are available to inform the analysis.
Sequential analysis Link created an extension of
Wald's theory of sequential analysis to a distribution-free accumulation of random variables until either a positive or negative bound is first equaled or exceeded. Link derives the probability of first equaling or exceeding the positive boundary as 1/(1+e^{-\theta A}), 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. ==See also==