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The logistic function or logistic curve is defined by the mathematical formula:
- <math>P(t) = a\frac{1 + m e^{-t/\tau}}{1 + n e^{-t/\tau}}<math>
for real parameters a, m, n, and <math>\tau<math>. These functions are found in a range of fields, from biology to economics.
The Verhulst equation
A typical application of the logistic equation is a common model of population growth 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 and t represent time, this model is formalized by the differential equation:
- <math>\frac{dP}{dt}=kP(C-P) \qquad \mbox{(1)},<math>
where the constant k defines the growth rate and C is the carrying capacity. The general solution to this equation is a logistic function.
Sigmoid function
The special case of the logistic function with <math> a=1, m=0, n=1, \tau=1 <math>, namely
- <math>P(t) = \frac{1}{1 + e^{-t}}<math>
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 probability, statistics, biomathematics, and economics.
Beware that other functions with "S"-shaped graphs may also be called by this name: see sigmoid function.
Properties of the sigmoid function
The (standard) sigmoid function is the solution of the first-order non-linear differential equation
- <math>\frac{dP}{dt}=P(1-P), \quad\mbox{(1)}<math>
with boundary condition <math>P(0)=1/2<math>. Equation (1) 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' Essay on the Principle of Population. Verhulst derived his equation logistique (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.
See also
References
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