Logistic Equation

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.

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

(1)

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

(2)

which is known as the logistic equation and has solution

(3)

The function is sometimes known as the sigmoid function.

While is usually constrained to be positive, plots of the above solution are shown for various positive and negative values of and initial conditions 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

(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.