{{Short description|Growth curve model}} 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.<ref name="PaulyMorgan1987">{{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}}</ref> The function is commonly applied in ecology to model fish growth<ref name="Nations2005">{{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}}</ref> and in paleontology to model sclerochronological parameters of shell growth.<ref name="moss2021">{{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 }}</ref>
The model can be written as the following:
: <math>L(a)= L_\infty(1-\exp(-k(a-t_0)))</math>
where <math>a</math> is age, <math>k</math> is the growth coefficient, <math>t_0</math> is the theoretical age when size is zero, and <math>L_\infty</math> is asymptotic size.<ref name="CarlsonGoldman2007">{{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}}</ref> It is the solution of the following linear differential equation:
: <math> \frac{dL}{da} = k (L_{\infty} - L ) </math>
== History == In 1920, August Pütter proposed that growth was the result of a balance between anabolism and catabolism.<ref>{{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}}</ref> von Bertalanffy, citing Pütter, borrowed this concept and published its equation first in 1941,<ref>{{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}}</ref> and elaborated on it later on.<ref name=":0">{{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}}</ref> The original equation was under the following form: <math display="block">\frac{dW}{dt} = \eta W^m - \kappa W^n</math>with <math display="inline">W</math> the weight, <math display="inline">\eta</math> and <math display="inline">\kappa</math> constants of anabolism and catabolism respectively, and <math display="inline">m</math>, <math display="inline">n</math> constant exponants. Von Bertalanffy gave himself the resulting equation for <math display="inline">W</math> as a function of <math display="inline">t</math>, assuming that <math display="inline">n=1</math> and <math display="inline">m \leq 1</math> :<ref name=":0" />
<math>W = \Biggl(\frac{\eta}{\kappa}-\Bigl(\frac{\eta}{\kappa}-W_0^{1-m}\Bigr)e^{-(1-m)\kappa t}\Biggr)^{\frac{1}{1-m}}</math>
Prior to von Bertalanffy, in 1921, J. A. Murray wrote a similar differential equation,<ref>{{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}}</ref> with <math display="inline">m = \frac{2}{3}</math>, according to the then-called "surface law", and <math display="inline">n = 1</math>, 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.<ref>{{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}}</ref>
==See also== {{Commons category|Von Bertalanffy curve}} * Gompertz function * Monod equation * Michaelis–Menten kinetics
==References== {{reflist}}
Category:Growth curves Category:Mathematical modeling Category:Sigmoid functions