The von Bertalanffy growth equation

von Bertalanffy derived this equation in 1938 from simple physiological arguments. It is the most widely used growth curve and is especially important in fisheries studies.

Assume that the rate of growth of an organism declines with size so that the rate of change in length, l,  may be described by:

where t is time,

l is length (or some other measure of size),

K is the growth rate and

L∞, termed 'L infinity' in fisheries science, is the asymptotic length at which growth is zero.

Integrating this becomes:

The parameter t0 is included to adjust the equation for the initial size of the organism and is defined as age at which the organisms would have had zero size. Thus to fit this equation you need to fit 3 parameters (L∞, K and t0 ) by nonlinear regression.         

The following graph shows an example plot of this equation.

To fit this curve we must therefore estimate 3 parameters, L∞, K and t0. While this was once done graphically, it is now accomplished using the Levenberg-Marquardt Method for non-linear regression. Growth II offers both the graphical and the more accurate nonlinear numerical method.