Logistic Function

Logistic function
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For the recurrence relation, see Logistic map.
[pic]
[pic]
Standard logistic sigmoid function
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. A generalized logistic curve can modelthe "S-shaped" behaviour (abbreviated S-curve) 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
[pic]
where the variable P might be considered to denote a population, where e is Euler's number and the variable tmight be thought of as time.[1] For values of t in the range of real numbers from −∞ to +∞, the S-curve 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:
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It also has the property that
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Thus, the function [pic]is odd.
|Contents |
| [hide] |
|1 Logistic differential equation |
|2 In ecology: modeling population growth |
|2.1 Time-varying carrying capacity |
|3 In statistics and machine learning ||3.1 Logistic regression |
|3.2 Neural networks |
|4 In medicine: modeling of growth of tumors |
|5 In chemistry: reaction models |
|6 In physics: Fermidistribution |
|7 In linguistics: language change |
|8 In economics: diffusion of innovations |
|9 Double logistic function |
|10 See also|
|11 Notes |
|12 References |
|13 External links |


[edit] Logistic differential equation
The logistic function is the solution ofthe simple first-order non-linear differential equation
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where P is a variable with respect to time t and with boundary condition P(0) = 1/2. This equation is the continuous version of the logistic map.
The qualitative behavior is easily understood in terms of the phase line: the derivative is 0 at P = 0 or 1 and the derivative is positive for P between 0 and 1, and negative for Pabove 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 value of P greater than 0 and less than 1, P grows to 1.
One may readily find the (symbolic) solution to be
[pic]
Choosing the constant of integration ec = 1 gives the other well-known form of the...