Specific Growth Models

Many attempts have been made to develop functions which are able to describe patterns of individual growth. These "specific" growth models are widely used in invertebrate and vertebrate growth analysis. They have between 2 and 4 parameters:

Parameter
 Symbol
Significance
Infinite Size
Soo
 Size (mass) reached after an infinite time of growth
Growth constant
K
 Defines "speed" of growth
Age t-zero
to
 Age at which size would be zero
Age t-star
t*
 Age of growth inflexion (is often denoted to, too)
Shape parameter
D
 Determines shape of the curve (more or less sigmoid)

These are the most commonly used specific growth models:

Function Name
Function
Parameters
specialised
von Bertalanffy
 St = Soo * (1 - e -K * (t - to)) Soo, K, to,
generalised
von Bertalanffy
 St = Soo * (1 - e -K * (t - to))D Soo, K, to, D
Gompertz St = Soo * e -e^[-K * (t - t*)] Soo, K, t*,
Richards St = Soo * (1 + 1/D * e -K * (t - t*))-D Soo, K, t*, D
Single Logistic St = Soo / (1 + e -K * (t - t*)) Soo, K, t*,

The corresponding inverse growth model is computed by solving the growth model for t, e.g.:

Function Name
Function
Parameters
specialised
von Bertalanffy
 t = ln(1 - (St / Soo) / -K + to Soo, K, to,
generalised
von Bertalanffy
 t = ln(1 - (St / Soo)D) / -K + to Soo, K, to, D
 
 
Obviously all these models are closely related. Nevertheless, fitting different models to the same set of data may result in quite different parameter values. Hence, parameter values derived by different models are not easily comparable, see example.
  
The Tanaka function (Tanaka 1982, 88), see Ebert (1999) for a very good description, is quite unique as it models a sigmoid growth pattern but without an asymptode (infinite size). It has been found to provide superior fit in rather long lived species with apparent continuous growth such as some sea urchins (see Ebert 1999 and references therein).

St = 1/(f0.5) * ln(2 * f * (t - c) + 2 * [f2 * (t - c)2 + f * a ]0.5) + d

(Tanaka 1988) explains the biological meaning of the four parameters as follows:

a related to maximum growth rate (approx. 1/a0.5)
c age at which growth rate is maximum
d shifts body size at which growth is maximum
f measure of rate of change of growth rate

 
  Application: Besides some linearisation approaches for von Bertalanffy and Gompertz (see Analysis), the models are fitted to the data by a nonlinear fitting algorithm.
  See also transformations required to fit models to size-increment data.
  Download computation spread sheet for nonlinear iterative growth curve fit.
 

Schnute's general growth model
"Specific" growth models
Seasonally oscill. growth models
Growth performance