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Logistic Equation -- from Wolfram MathWorld

  • ️Weisstein, Eric W.
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The logistic equation (sometimes called the Verhulst model or logistic growth curve) is a model of population growth first published by Pierre Verhulst (1845, 1847). The model is continuous in time, but a modification of the continuous equation to a discrete quadratic recurrence equation known as the logistic map is also widely used.

LogisticEquationContinuous

The continuous version of the logistic model is described by the differential equation

(dN)/(dt)=(rN(K-N))/K,

(1)

where r is the Malthusian parameter (rate of maximum population growth) and K is the so-called carrying capacity (i.e., the maximum sustainable population). Dividing both sides by K and defining x=N/K then gives the differential equation

(dx)/(dt)=rx(1-x),

(2)

which is known as the logistic equation and has solution

x(t)=1/(1+(1/(x_0)-1)e^(-rt)).

(3)

The function x(t) is sometimes known as the sigmoid function.

While r is usually constrained to be positive, plots of the above solution are shown for various positive and negative values of r and initial conditions x_0=x(t=0) ranging from 0.00 to 1.00 in steps of 0.05.

The discrete version of the logistic equation (3) is known as the logistic map.

The curve

x=a/(1+bq^t)

(4)

obtained from (3) is sometimes known as the logistic curve. Similarly, a normalized form of equation (3) is commonly used as a statistical distribution known as the logistic distribution.

See also

Gompertz Curve, Law of Growth, Life Expectancy, Logistic Distribution, Logistic Map, Makeham Curve, Malthusian Parameter, Population Growth, Sigmoid Function

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References

Verhulst, P.-F. "Recherches mathématiques sur la loi d'accroissement de la population." Nouv. mém. de l'Academie Royale des Sci. et Belles-Lettres de Bruxelles 18, 1-41, 1845.Verhulst, P.-F. "Deuxième mémoire sur la loi d'accroissement de la population." Mém. de l'Academie Royale des Sci., des Lettres et des Beaux-Arts de Belgique 20, 1-32, 1847.Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 918, 2002.

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Logistic Equation

Cite this as:

Weisstein, Eric W. "Logistic Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LogisticEquation.html

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