Logistic growth model

For the recurrence relation, see Logistic map.


A logistic function or logistic curve is a common sigmoid function, given its name (in reference to its S-shape) in 1844 or 1845 by Pierre François Verhulst who studied it in relation to population growth. A generalized logistic curve can model the "S-shaped" behaviour (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.

The logistic function is the sigmoid curve with equation:

f(x) = \frac{1}{1 + \mathrm e^{-x}}

where e is Euler's number.[1] For values of x in the range of real numbers from −∞ to +∞, the S-curve shown is obtained. In practice, due to the nature of the exponential function ex, it is often sufficient to compute x 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:

\frac{d}{dx}f(x) = f(x)\cdot(1-f(x)).\,

It also has the property that

1-f(x) = f(-x).\,

Thus, the function x \mapsto f(x) - 1/2 is odd.

Logistic differential equation

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

\frac{d}{dx}f(x) = f(x)(1-f(x))

with boundary condition f(0) = 1/2. This equation is the continuous version of the logistic map.

The qualitative behavior is easily understood in terms of the phase line: the derivative is null when function is unit and the derivative is positive for f between 0 and 1, and negative for f above 1 or less than 0 (though negative populations do not generally accord with a physical model). This yields an unstable equilibrium at 0, and a stable equilibrium at 1, and thus for any function value greater than zero and less than unit, it grows to unit.

One may readily find the (symbolic) solution to be

f(x)=\frac{e^{x}}{e^{x}+e^{x_0}}

Choosing the constant of integration ex0 = 1 gives the other well-known form of the definition of the logistic curve

f(x) = \frac{e^x}{e^x + 1} \! = \frac{1}{1 + e^{-x}} \!

More quantitatively, as can be seen from the analytical solution, the logistic curve shows early exponential growth for negative argument, which slows to linear growth of slope 1/4 for an argument near zero, then approaches one 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 logistic curve.

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

2 \, f(x) = 1 + \tanh \left( \frac{x}{2} \right).

In ecology: modeling population growth

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:

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

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

\frac{d}{dt}\frac{P}{K}=r\frac{P}{K}\left(1 - \frac{P}{K}\right)

Now setting x=P/K gives the differential equation

\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 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.\,

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.

Time-varying carrying capacity

Since the environmental conditions influence the carrying capacity, as a consequence it can be time-varying: K(t) > 0, leading to the following mathematical model:

\frac{dP}{dt}=rP\left(1 - \frac{P}{K(t)}\right)

A particularly important case is that of carrying capacity that varies periodically with period T:

K(t+T) = K(t).\,

It can be shown that in such a case, independently from the initial value P(0) > 0, P(t) will tend to a unique periodic solution P*(t), whose period is T.

A typical value of T is one year: in such case K(t) reflects periodical variations of weather conditions.

Another interesting generalization is to consider that the carrying capacity K(t) is a function of the population at an earlier time, capturing a delay in the way population modifies its environment. This leads to a logistic delay equation,[2] which has a very rich behavior, with bistability in some parameter range, as well as a monotonic decay to zero, smooth exponential growth, punctuated unlimited growth (i.e., multiple S-shapes), punctuated growth or alternation to a stationary level, oscillatory approach to a stationary level, sustainable oscillations, finite-time singularities as well as finite-time death.

In statistics and machine learning

Logistic regression

Logistic functions are used in several roles in statistics. Firstly, they are the cumulative distribution function of the logistic family of distributions. Secondly they are used in logistic regression to model how the probability p of an event may be affected by one or more explanatory variables: an example would be to have the model

p=P(a + bx)\,

where x is the explanatory variable and a and b are model parameters to be fitted.

An important application of the logistic function is in the Rasch model, used in item response theory. In particular, the Rasch model forms a basis for maximum likelihood estimation of the locations of objects or persons on a continuum, based on collections of categorical data, for example the abilities of persons on a continuum based on responses that have been categorized as correct and incorrect.

Logistic regression and other log-linear models are also commonly used in machine learning.

Neural networks

Logistic functions are often used in neural networks to introduce nonlinearity in the model and/or to clamp signals to within a specified range. A popular neural net element computes a linear combination of its input signals, and applies a bounded logistic function to the result; this model can be seen as a "smoothed" variant of the classical threshold neuron.

A common choice for the activation or "squashing" functions, used to clip for large magnitudes to keep the response of the neural network bounded[3] is

g(h) = \frac{1}{1 + e^{-2 \beta h}} \!

which is a logistic function. These relationships result in simplified implementations of artificial neural networks with artificial neurons. Practitioners caution that sigmoidal functions which are antisymmetric about the origin (e.g. the hyperbolic tangent) lead to faster convergence when training networks with backpropagation.[4]

A generalisation and extension of the logistic function to multiple inputs is the softmax activation function.

In medicine: modeling of growth of tumors

Another application of logistic curve is in medicine, where the logistic differential equation is used to model the growth of tumors. This application can be considered an extension of the above mentioned use in the framework of ecology (see also the Generalized logistic curve, allowing for more parameters). Denoting with X(t) the size of the tumor at time t, its dynamics are governed by:

X^{\prime}=r\left(1 - \frac{X}{K}\right)X

which is of the type:

X^{\prime}=F\left(X\right)X, F^{\prime}(X) \le 0

where F(X) is the proliferation rate of the tumor.

If a chemotherapy is started with a log-kill effect, the equation may be revised to be

X^{\prime}=r\left(1 - \frac{X}{K}\right)X - c(t)X,

where c(t) is the therapy-induced death rate. In the idealized case of very long therapy, c(t) can be modeled as a periodic function (of period T) or (in case of continuous infusion therapy) as a constant function, and one has that

\frac{1}{T}\int_{0}^{T}{c(t)\, dt} > r \rightarrow \lim_{t \rightarrow +\infty}x(t)=0

i.e. if the average therapy-induced death rate is greater than the baseline proliferation rate then there is the eradication of the disease. Of course, this is an oversimplified model of both the growth and the therapy (e.g. it does not take into account the phenomenon of clonal resistance).

In chemistry: reaction models

The concentration of reactants and products in autocatalytic reactions follow the logistic function.

In physics: Fermi distribution

The logistic function determines the statistical distribution of fermions over the energy states of a system in thermal equilibrium. In particular, it is the distribution of the probabilities that each possible energy level is occupied by a fermion, according to Fermi–Dirac statistics.

In linguistics: language change

In linguistics, the logistic function can be used to model language change:[5] an innovation that is at first marginal begins to spread more quickly with time, and then more slowly as it becomes more universally adopted.

In economics: diffusion of innovations

The logistic function can be used to illustrate the progress of the diffusion of an innovation through its life cycle. Historically, when new products are introduced there is an intense amount of research and development which leads to dramatic improvements in quality and reductions in cost. This leads to a period of rapid industry growth. Some of the more famous examples are: railroads, incandescent light bulbs, electrification, the Ford Model T, air travel and computers. Eventually, dramatic improvement and cost reduction opportunities are exhausted, the product or process are in widespread use with few remaining potential new customers, and markets become saturated.

Logistic analysis was used in papers by several researchers at the International Institute of Applied Systems Analysis (IIASA). These papers deal with the diffusion of various innovations, infrastructures and energy source substitutions and the role of work in the economy as well as with the long economic cycle. Long economic cycles were investigated by Robert Ayres (1989).[6] Cesare Marchetti published on long economic cycles and on diffusion of innovations.[7][8] Arnulf Grübler’s book (1990) gives a detailed account of the diffusion of infrastructures including canals, railroads, highways and airlines, showing that their diffusion followed logistic shaped curves.[9]

Carlota Perez used a logistic curve to illustrate the long (Kondratiev) business cycle with the following labels: beginning of a technological era as irruption, the ascent as frenzy, the rapid build out as synergy and the completion as maturity.[10]

In design in nature: spreading and collecting flows

The logistic S-shaped history of areas and volumes filled by spreading or collecting flows was predicted based on the Constructal law of design and evolution in nature. [11][12] The natural phenomenon of S-curves everywhere is the universal tendency of natural flow systems (animate or inanimate) to morph and to generate designs that flow more easily and offer greater access over time.

Double logistic function


The double logistic is a function similar to the logistic function with numerous applications . Its general formula is:

f(x) = \mathrm{sgn}(x-d) \, \Bigg(1-\exp\bigg(-\bigg(\frac{x-d}{s}\bigg)^2\bigg)\Bigg),

where d is its centre and s is the steepness factor. Here "sgn" represents the sign function.

It is based on the Gaussian curve and graphically it is similar to two identical logistic sigmoids bonded together at the point x = d.

One of its applications is non-linear normalization of a sample, as it has the property of eliminating outliers.

See also

Notes

References

  1. MathWorld.

External links

  • L.J. Linacre, Why logistic ogive and not autocatalytic curve?, accessed 2009-09-12.
  • http://luna.cas.usf.edu/~mbrannic/files/regression/Logistic.html
  • Modeling Market Adoption in Excel with a simplified s-curve
  • MathWorld.
  • Online experiments with JSXGraph
  • Esses are everywhere.
  • Seeing the s-curve is everything.
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