Generalised logistic function: Information from Answers.com

A=0, K=1, B=1.5, Q=ν=0.5, M=0.5

The generalized logistic curve or function, also known as Richards' curve is a widely-used and flexible sigmoid function for growth modelling, extending the well-known logistic curve.

$Y(t) = A + { K-A \over (1 + Q e^{-B(t - M)}) ^ {1 / \nu} }$

where Y = weight, height, size etc., and t = time.

It has six parameters:

A: the lower asymptote;
K: the upper asymptote. If A=0 then K is called the carrying capacity;
B: the growth rate;
ν>0 : affects near which asymptote maximum growth occurs.
Q: depends on the value Y(0)
M: the time of maximum growth if Q=ν

1 Generalized logistic differential equation
2 See also
3 References
4 See also

Generalized logistic differential equation

A particular case of Richard's function is:

$Y(t) = { K \over (1 + Q e^{- \alpha \nu (t - t_0)}) ^ {1 / \nu} }$

which is the solution of the so called Richard's differential equation (RDE):

$Y^{\prime}(t) = \alpha \left(1 - \left(\frac{Y}{K} \right)^{\nu} \right)Y$

with initial condition

Y (t 0) = Y 0

where

$Q = -1 + \left(\frac {K}{Y_0} \right)^{\nu}$

provided that ν > 0 and α > 0.

The classical logistic differential equation is a particular case of the above equation, with ν =1, whereas the Gompertz curve can be recovered in the limit $\nu \rightarrow 0^+$ provided that:

$\alpha = O\left(\frac{1}{\nu}\right)$

In fact, for small ν it is

$Y^{\prime}(t) = Y r \frac{1-\exp\left(\nu \ln\left(\frac{Y}{K}\right) \right)}{\nu} \approx r Y \ln\left(\frac{Y}{K}\right)$

The RDE suits to model many growth phenomena, including the growth of tumors. Concerning its applications in oncology, its main biological features are similar to those of Logistic curve model.

References

Richards, F.J. 1959 A flexible growth function for empirical use. J. Exp. Bot. 10: 290--300.
Pella JS and PK Tomlinson. 1969. A generalised stock-production model. Bull. IATTC 13: 421-496.
Lei, Y.C. and Zhang, S.Y. 2004. Features and Partial Derivatives of Bertalanffy-Richards Growth Model in Forestry. Nonlinear Analysis: Modelling and Control, Vol 9, No. 1:65-73

	Sigmoid function
	List of mathematics articles (G)
	List of mathematics articles (G-I)

	At which incident facility are primary logistics functions coordinated and administered? Read answer...
	At what incident facility are primary logistics and administrative functions coordinated and administered? Read answer...
	At which incident facility are primary logistics and administrative functions coordinated and administered? Read answer...

	What is the domain of a logistic function?
	What r the function of logistics managment?
	Where is a logistic function used?

Generalised logistic function

Contents

Generalized logistic differential equation

See also

References

See also