logistic function

logistic function, an equation describing growth rates of quantities, such as populations, over time and graphed as an S-shaped, or sigmoid, curve. The equation is applied in many fields, such as biology, sociology, agriculture, and economics.

The logistic function is writtenP(t) = P₀/(1 + P₀[e^rt − 1]/K),where the function P(t) is the quantity that changes with time t; r is the quantity’s growth (or decay) rate, represented by the slope of the S-shaped graph; e is the base of the exponential function; P₀ is the initial value of the quantity; and K is the maximum or saturation value of P(t). In population studies, K is referred to as the carrying capacity of an environment.

Belgian mathematician Pierre-François Verhulst developed the logistic function after reading An Essay on the Principle of Population (1798), by English economist and demographer Thomas Malthus. Malthus postulated that the unrestricted exponential growth of populations would always outrun the merely linear growth of subsistence production.

Verhulst disagreed with Malthus’s ideas about population growth. He started with the assumption that the rate of reproduction is proportional to both the existing population size and the amount of resources available. This better reflected real-world circumstances, where conditions often limit the extent of growth until a point of stability, or no growth, is reached. This type of growth is called logistic growth, represented by the S-shaped curve.

Verhulst published his ideas of constrained or self-limiting growth in three papers between 1838 and 1847. In 1845 he named his equation the logistic function. (The term logistic should not be confused with the term logistics, as used in military and business operations.)

The logistic function is commonly used to graph the growth of specific quantities, such as product sales, bacteria populations, crop yields, and so on. The function is often reduced to what is called the standard logistic, or sigmoid, functionf(t) = 1/(1 + e^−t).

The logistic function curve is marked by three distinct regions of change: (1) an initial slow-to-rapid growth stage, represented by a rising exponential curve, (2) a stage marked by a midpoint, where the positive curvature, or increasing exponential growth, inflects to a negative curvature, or decreasing exponential growth, and (3) a stage in which growth no longer occurs as rapidly and the function levels off toward K.