Logistic function



A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.

As shown below, the untrammeled growth can be modelled as a rate term +rKP (a percentage of P). But then, as the population grows, some members of P (modelled as &minus;rP2) interfere with each other in competition for some critical resource (which can be called the bottleneck, modelled by K). This competition diminishes the growth rate, until the set P ceases to grow (this is called maturity).

A logistic function is defined by the mathematical formula:


 * $$P(t;a,m,n,\tau) = a\frac{1 + m e^{-t/\tau}}{1 + n e^{-t/\tau}} \!$$

for real parameters a, m, n, and $$\tau$$. These functions find applications in a range of fields, including biology and economics.

Concentration of reactants and products in autocatalytical reactions follow the logistic function.

An important 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.

The Verhulst equation
A typical application of the logistic equation is a common model of population growth, which states that:
 * the rate of reproduction is proportional to the existing population, all else being equal
 * the rate of reproduction is proportional to the amount of available resources, all else being equal. Thus the second term models the competition for available resources, which tends to limit the 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:


 * $$\frac{dP}{dt}=rP\left(1 - \frac{P}{K}\right)$$

where the constant $$r$$ defines the growth rate and $$K$$ is the carrying capacity. 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. The solution to the equation (with $$P_0$$ being the initial population) is


 * $$P(t) = \frac{K P_0 e^{rt}}{K + P_0 \left( e^{rt} - 1\right)} $$

where


 * $$\lim_{t\to\infty} P(t) = K.\,$$

Sigmoid function
The special case of the logistic function with a = 1, m = 0, n = 1, &tau; = 1, namely


 * $$P(t) = \frac{1}{1 + e^{-t}}\!$$

is called sigmoid function or sigmoid curve. The name is due to the sigmoid shape of its graph. This function is also called the standard logistic function and is often encountered in many technical domains, especially in artificial neural networks as a transfer function, probability, statistics, biomathematics, mathematical psychology and economics.

Properties of the sigmoid function
The (standard) sigmoid function is the solution of the first-order non-linear differential equation


 * $$\frac{dP}{dt}=P(1-P), \quad\mbox{(2)}\!$$

with boundary condition P(0) = 1/2. Equation (2) is the continuous version of the logistic map.

The sigmoid curve shows early exponential growth for negative t, which slows to linear growth of slope 1/4 near t = 0, then approaches y = 1 with an exponentially decaying gap.

The logistic function is the inverse of the natural logit function and so can be used to convert the logarithm of odds into a probability; the conversion from the log-likelihood ratio of two alternatives also takes the form of a sigmoid curve.

History
The Verhulst equation, (1), was first published by Pierre F. Verhulst in 1838 after he had read Thomas Malthus' An Essay on the Principle of Population.

Verhulst derived his logistic equation to describe the self-limiting growth of a biological population. The equation is also sometimes called the Verhulst-Pearl equation following its rediscovery in 1920. Alfred J. Lotka derived the equation again in 1925, calling it the law of population growth.