Von Bertalanffy function

The von Bertalanffy growth function (VBGF), or von Bertalanffy curve, is a type of growth curve for a time series and is named after Ludwig von Bertalanffy. It is a special case of the generalised logistic function. The growth curve is used to model mean length from age in animals.^[1] The function is commonly applied in ecology to model fish growth^[2] and in paleontology to model sclerochronological parameters of shell growth.^[3]

The model can be written as the following:

${\displaystyle L(a)=L_{\infty }(1-\exp(-k(a-t_{0})))}$

where ${\displaystyle a}$ is age, ${\displaystyle k}$ is the growth coefficient, ${\displaystyle t_{0}}$ is the theoretical age when size is zero, and ${\displaystyle L_{\infty }}$ is asymptotic size.^[4] It is the solution of the following linear differential equation:

${\displaystyle {\frac {dL}{da}}=k(L_{\infty }-L)}$

In 1920, August Pütter proposed that growth was the result of a balance between anabolism and catabolism.^[5] von Bertalanffy, citing Pütter, borrowed this concept and published its equation first in 1941,^[6] and elaborated on it later on.^[7] The original equation was under the following form: $${\displaystyle {\frac {dW}{dt}}=\eta W^{m}-\kappa W^{n}}$$with ${\textstyle W}$ the weight, ${\textstyle \eta }$ and ${\textstyle \kappa }$ constants of anabolism and catabolism respectively, and ${\textstyle m}$, ${\textstyle n}$ constant exponants. Von Bertalanffy gave himself the resulting equation for ${\textstyle W}$ as a function of ${\textstyle t}$, assuming that ${\textstyle n=1}$ and ${\textstyle m\leq 1}$ :^[7]

${\displaystyle W={\Biggl (}{\frac {\eta }{\kappa }}-{\Bigl (}{\frac {\eta }{\kappa }}-W_{0}^{1-m}{\Bigr )}e^{-(1-m)\kappa t}{\Biggr )}^{\frac {1}{1-m}}}$

Prior to von Bertalanffy, in 1921, J. A. Murray wrote a similar differential equation,^[8] with ${\textstyle m={\frac {2}{3}}}$, according to the then-called "surface law", and ${\textstyle n=1}$, but Murray's article does not appear in van Bertalanffy sources.

The seasonally-adjusted von Bertalanffy is an extension of this function that accounts for organism growth that occurs seasonally. It was created by I. F. Somers in 1988.^[9]

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