**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 −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 τ. These functions find applications in a range of fields, including biology and economics.

Concentration of reactants and products in auto catalytical 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.

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