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:

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

History

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 von Bertalanffy's sources.

Seasonally-adjusted von Bertalanffy

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]

See also

• Gompertz function
• Monod equation
• Michaelis–Menten kinetics

References

1. Daniel Pauly; G. R. Morgan (1987). Length-based Methods in Fisheries Research. WorldFish. p. 299. ISBN   978-971-10-2228-0.
2. Food and Agriculture Organization of the United Nations (2005). Management Techniques for Elasmobranch Fisheries. Food & Agriculture Org. p. 93. ISBN   978-92-5-105403-1.
3. Moss, D.K.; Ivany, L.C.; Jones, D.S. (2021). "Fossil bivalves and the sclerochronological reawakening". Paleobiology. 47 (4): 551–573. doi: 10.1017/pab.2021.16 . S2CID   234844791.
4. John K. Carlson; Kenneth J. Goldman (5 April 2007). Special Issue: Age and Growth of Chondrichthyan Fishes: New Methods, Techniques and Analysis. Springer Science & Business Media. ISBN   978-1-4020-5570-6.
5. Pütter, August (1920). "Studien über physiologische Ähnlichkeit VI. Wachstumsähnlichkeiten". Pflüger's Archiv für die Gesamte Physiologie des Menschen und der Tiere. 180 (1): 298–340.
6. von Bertalanffy, Ludwig (1941). "Untersuchungen uber die Gesetzlichkeit des Wachstums. VII. Stoffwechseltypen und Wachstumstypen". Biologisches Zentralblatt. 61: 510–532.
7. von Bertalanffy, Ludwig (1957). "Quantitative laws in metabolism and growth". The Quarterly Review of Biology. 32 (3): 217–231.
8. Murray, J Alan (1921). "Normal growth in animals". The Journal of Agricultural Science. 11 (3): 258–274 – via Cambridge University Press.
9. Somers, I.F. (1988). "On a seasonally oscillating growth function". Fishbyte. 6 (1): 8–11.