Logistic function

A logistic function or logistic curve models the S-curve of growth of some set P. The initial stage of growth is approximately exponential; then, as competition arises, the growth slows, and at maturity, growth stops.

As shown below, the untrammeled growth can be modelled as a rate term +rKP (a percentage of P). But then, as the population grows, some members of P (modelled as $-rP^2$ ) interfere with each other in competition for some critical resource (which can be called the bottleneck, modelled by K). This competition diminishes the growth rate, until the set P ceases to grow (this is called maturity).

A logistic function is defined by the mathematical formula:

P(t) = a\frac{1 + m e^{-t/\tau}}{1 + n e^{-t/\tau}} \!

for real parameters a, m, n, and $\tau$ . These functions find applications in a range of fields, from biology to economics.

For example, in the development of an embryo, a fertilized ovum splits, and the cell count grows: 1, 2, 4, 8, 16, 32, 64, etc. This is exponential growth. But the fetus can grow only as large as the uterus can hold; thus other factors start slowing down the increase in the cell count, and the rate of growth slows (but the baby is still growing, of course). After a suitable time, the child is born and keeps growing. Ultimately, the cell count is stable; the person's height is constant; the growth has stopped, at maturity.

Concentration of reactants and products in autocatalytical reactions follow the logistic function.

In such examples, relationships between variables are modelled. In addition, an important logistic function is the Rasch model, which is a general stochastic measurement model. This model is used as a basis for measurement rather than for modelling relationships between variables for which measurements have already been obtained (as in the preceding example). In particular, the Rasch model forms a basis for maximum likelihood estimation of the locations of objects to be measured on a continuum, based on collections of categorical data. For example, the model can be applied in order to estimate the abilities of persons on a continuum based on assessment responses that have been categorized as correct and incorrect.

The Verhulst equation

A typical application of the logistic equation is a common model of population growth, which 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 (N is often used in ecology instead) and t represent time, this model is formalized by the differential equation:

\frac{dP}{dt}=rP\left(1 - \frac{P}{K}\right) \qquad \mbox{(1)}, \!

where the constant $r$ defines the growth rate and $K$ is the carrying capacity. The general solution to this equation is a logistic function. 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 $P_0$ being the initial population) is

P(t) = \frac{K P_0 e^{rt}}{K + P_0 \left( e^{rt} - 1\right)}

where

\lim_{t\to\infty} P(t) = K.\,

Sigmoid function

The special case of the logistic function with $a=1, m=0, n=1, \tau=1$ , namely

P(t) = \frac{1}{1 + e^{-t}}\!

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 in artificial neural networks as a transfer function, probability, statistics, biomathematics, and economics.

Properties of the sigmoid function

The (standard) sigmoid function is the solution of the first-order non-linear differential equation

\frac{dP}{dt}=P(1-P),  \quad\mbox{(2)}\!

with boundary condition $P(0)=1/2$ . Equation (2) 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' An 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.

Critics

Despite its persistent popularity as a model for population growth in the field of population dynamics, this use of the logistic function has been heavily criticised. One critic, demographer and Professor of Population, Joel E. Cohen (How Many People Can The Earth Support, 1995) explains that Verhulst attempted to fit a logistic curve based on the logistic function to 3 separate censuses of the population of the United States of America in order to predict future growth. All 3 sets of predictions failed.

In 1924, Professor Ray Pearl and Lowell J. Reed used Verhulst's model to predict an upper limit of 2 billion for the world population. This was passed in 1930. A later attempt by Pearl and an associate Sophia Gould in 1936 then estimated an upper limit of 2.6 billion. This was passed in 1955.

These criticisms are echoed by Professor Peter Turchin (Complex Population Dynamics, 2003), who nonetheless concludes that it provides a useful framework for single-species dynamics and contributes to models for multispecies interactions.

Nevertheless, the logistics curve has been a staple of models both mathematical and sociological, for instance the transformation theory of George Land, which uses the concept of the S-curve to prescribe appropriate business behaviour in various stages of a technology's growth.

References

Kingsland, S. E. (1995) Modeling nature ISBN 0226437280

External links

http://www.dartmouth.edu/~math3f98/csc98/chap5/CSC.USAPop5.html
http://www.ento.vt.edu/~sharov/PopEcol/lec5/logist.html
http://www.nlreg.com/aids.htm
http://luna.cas.usf.edu/~mbrannic/files/regression/Logistic.html
MathWorld: Sigmoid Function
Professor Joel E. Cohen's web page
Professor Peter Turchin's web page

Special functions | Differential equations | Population | Curves

Logistische Funktion | 로지스틱 방정식 | Logistische functie | ロジスティック式

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