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For the recurrence relation, see logistic map.
A logistic function or logistic curve is a common sigmoid curve, given its name in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. It can model the "Sshaped" curve (abbreviated Scurve) of growth of some population P. The initial stage of growth is approximately exponential; then, as saturation begins, the growth slows, and at maturity, growth stops.
A simple logistic function may be defined by the formula
where the variable P might be considered to denote a population and the variable t might be thought of as time^{[1]}. For values of t in the range of real numbers from −∞ to +∞, the Scurve shown is obtained. In practice, due to the nature of the exponential function e^{−t}, it is sufficient to compute t over a small range of real numbers such as [−6, +6].
The logistic function finds applications in a range of fields, including artificial neural networks, biology, biomathematics, demography, economics, chemistry, mathematical psychology, probability, sociology, political science, and statistics. It has an easily calculated derivative:
It also has the property that
In other words, the function P  1/2 is odd.
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