logistic growth

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. It can model the "S-shaped" curve (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. If the initial size is above the supported ecologic capacity, the size of the population decreases down toward this capacity.

A simple logistic function may be defined by the formula

[[Math:c|P(t) = \frac{1}{1 \plus e^{-t}}]]

where the variable P might be considered to denote a population and the variable t might 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:

[[Math:c|\frac{d}{dt}P(t) = P(t)\cdot(1-P(t))]]

The function P has the intuitively appealing quality that

[[Math:c|1-P(t) = P(-t)]]

Logistic differential equation

The logistic function is the solution of the simple first-order non-linear differential equation

[[Math:c|\frac{d}{dt}P(t) = P(t)(1-P(t))]]

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. One may readily find the (symbolic) solution to be


Choosing the constant of integration ec = 1 gives the other well-known form of the definition of the logistic curve

[[Math:c|P(t) = \frac{e^{t}}{e^{t} \plus 1} = \frac{1}{1 \plus e^{-t}}]]

The logistic 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 logistical 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 logistic curve.

The logistic sigmoid function is related to the hyperbolic tangent, A.p. by

[[Math:c|P(t) = 1 \plus tanh \frac{t}{2} ]]

In ecology: modeling population growth

Logistic growth. P0 defines the population at the zero time. Values under negative time is an estimation about the probable size in the past. The model also works with initial sizes above ecological capacity, predicting decay
A typical application of the logistic equation is a common model of population growth, originally due to Pierre-François Verhulst in 1838, where the rate of reproduction is proportional to both the existing population and the amount of available resources, all else being equal. The Verhulst equation was published after Verhulst had read Thomas Malthus' An Essay on the Principle of Population. Verhulst derived his 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.

Letting P represent population size (N is often used in ecology instead) and t represent time, this model is formalized by the differential equation:

[[Math:c|\frac{dP}{dt}=rP(1 - \frac{P}{K})]]

where the constant r defines the growth rate and K is the carrying capacity.

In the equation, the early, unimpeded growth rate is modeled by the first term +rP. The value of the rate r represents the proportional increase of the population P in one unit of time. Later, as the population grows, the second term, which multiplied out is −rP2/K, becomes larger than the first as some members of the population P interfere with each other by competing for some critical resource, such as food or living space. This antagonistic effect is called the bottleneck, and is modeled by the value of the parameter K. The competition diminishes the combined growth rate, until the value of P ceases to grow (this is called maturity of the population).

Dividing both sides of the equation by K gives

[[Math:c|\frac{d}{dt}\frac{P}{K}=r\frac{P}{K}1 - \frac{P}{K}]]

Now setting x = P / K gives the differential equation

[[Math:c|\frac{dx}{dt} = r x (1-x)]]

For r = 1 we have the particular case with which we started.

In ecology, species are sometimes referred to as r-strategist or K-strategist depending upon the selective processes that have shaped their life history strategies. The solution to the equation (with P0 being the initial population) is

[[Math:c|P(t) = \frac{K P_0 e^{rt}}{K \plus P_0 ( e^{rt} - 1)} ]]


[[Math:c|lim_{t\to\infty} P(t) = K]]

Which is to say that K is the limiting value of P: the highest value that the population can reach given infinite time (or come close to reaching in finite time). It is important to stress that the carrying capacity is asymptotically reached independently of the initial value P(0) > 0, also in case that P(0) > K.

Acknowledgements and notes

This article reuses material from Wikipedia page http://en.wikipedia.org/wiki/Logistic_growth under the rights of CC-BY-SA license. It is too similar and requires further work to integrate more sources.