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.{{cite book|author1=Daniel Pauly|author2=G. R. Morgan|title=Length-based Methods in Fisheries Research|url=https://books.google.com/books?id=R4DC-ALyducC&pg=PA299|year=1987|publisher=WorldFish|isbn=978-971-10-2228-0|pages=299}} The function is commonly applied in ecology to model fish growth{{cite book|author=Food and Agriculture Organization of the United Nations|title=Management Techniques for Elasmobranch Fisheries|url=https://books.google.com/books?id=KT0jXz2AyIsC&pg=PA93|year=2005|publisher=Food & Agriculture Org.|isbn=978-92-5-105403-1|pages=93}} and in paleontology to model sclerochronological parameters of shell growth.{{cite journal | last1 = Moss | first1 = D.K. | last2 = Ivany | first2 = L.C. | last3 = Jones | first3 = D.S. | title = Fossil bivalves and the sclerochronological reawakening | date = 2021 | journal = Paleobiology | volume = 47 | issue = 4 | pages = 551–573 | doi = 10.1017/pab.2021.16| s2cid = 234844791 | doi-access = free }}

The model can be written as the following:

: $L(a)= L_\infty(1-\exp(-k(a-t_0)))$

where $a$ is age, $k$ is the growth coefficient, $t_0$ is the theoretical age when size is zero, and $L_\infty$ is asymptotic size.{{cite book|author1=John K. Carlson|author2=Kenneth J. Goldman|title=Special Issue: Age and Growth of Chondrichthyan Fishes: New Methods, Techniques and Analysis|url=https://books.google.com/books?id=ESUyc2dMrnEC&pg=PA301|date=5 April 2007|publisher=Springer Science & Business Media|isbn=978-1-4020-5570-6}} It is the solution of the following linear differential equation:

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

History

In 1920, August Pütter proposed that growth was the result of a balance between anabolism and catabolism.{{Cite journal |last=Pütter |first=August |date=1920 |title=Studien über physiologische Ähnlichkeit VI. Wachstumsähnlichkeiten |journal=Pflüger's Archiv für die Gesamte Physiologie des Menschen und der Tiere |volume=180 |issue=1 |pages=298-340}} von Bertalanffy, citing Pütter, borrowed this concept and published its equation first in 1941,{{Cite journal |last=von Bertalanffy |first=Ludwig |date=1941 |title=Untersuchungen uber die Gesetzlichkeit des Wachstums. VII. Stoffwechseltypen und Wachstumstypen |journal=Biologisches Zentralblatt |volume=61 |pages=510-532}} and elaborated on it later on.{{Cite journal |last=von Bertalanffy |first=Ludwig |date=1957 |title=Quantitative laws in metabolism and growth |url=http://www.jstor.org/stable/2815257 |journal=The Quarterly Review of Biology |volume=32 |issue=3 |pages=217-231}} The original equation was under the following form: $\frac{dW}{dt} = \eta W^m - \kappa W^n$ with $W$ the weight, $\eta$ and $\kappa$ constants of anabolism and catabolism respectively, and $m$ , $n$ constant exponants. Von Bertalanffy gave himself the resulting equation for $W$ as a function of $t$ , assuming that $n=1$ and $m \leq 1$ :

$W = \Biggl(\frac{\eta}{\kappa}-\Bigl(\frac{\eta}{\kappa}-W_0^{1-m}\Bigr)e^{-(1-m)\kappa t}\Biggr)^{\frac{1}{1-m}}$

Prior to von Bertalanffy, in 1921, J. A. Murray wrote a similar differential equation,{{Cite journal |last=Murray |first=J Alan |date=1921 |title=Normal growth in animals |journal=The Journal of Agricultural Science |volume=11 |issue=3 |pages=258-274 |via=Cambridge University Press}} with $m = \frac{2}{3}$ , according to the then-called "surface law", and $n = 1$ , but Murray's article does not appear in van Bertalanffy 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.{{cite journal | last = Somers | first = I.F. | date = 1988 | title = On a seasonally oscillating growth function | journal = Fishbyte | volume = 6 | issue = 1 | pages = 8–11 | url = https://econpapers.repec.org/RePEc:wfi:wfbyte:39518}}

References

Category:Growth curves

Category:Mathematical modeling

Von Bertalanffy function

History

Seasonally-adjusted von Bertalanffy

See also

References