Editing Generalised logistic function (section)

==Generalised logistic differential equation==
A particular case of the generalised logistic function is:

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

which is the solution of the Richards's differential equation (RDE):

:<math>Y^{\prime}(t) = \alpha  \left(1 - \left(\frac{Y}{K} \right)^{\nu} \right)Y </math>

with initial condition

:<math>Y(t_0) = Y_0 </math>

where

:<math>Q = -1 + \left(\frac {K}{Y_0} \right)^{\nu}</math>

provided that <math>\nu > 0</math> and <math>\alpha > 0</math>

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

:<math>\alpha = O\left(\frac{1}{\nu}\right)</math>

In fact, for small <math>\nu</math> it is

:<math>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) </math>

The RDE models many growth phenomena, arising in fields such as oncology and epidemiology.