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. The function is commonly applied in ecology to model fish growth and in paleontology to model sclerochronological parameters of shell growth.
The model can be written as the following:
L ( a ) = L ∞ ( 1 − exp ( − k ( a − t 0 ) ) ) {\displaystyle L(a)=L_{\infty }(1-\exp(-k(a-t_{0})))}where a {\displaystyle a} is age, k {\displaystyle k} is the growth coefficient, t 0 {\displaystyle t_{0}} is the theoretical age when size is zero, and L ∞ {\displaystyle L_{\infty }} is asymptotic size. It is the solution of the following linear differential equation:
d L d a = k ( L ∞ − L ) {\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: d W d t = η W m − κ W n {\displaystyle {\frac {dW}{dt}}=\eta W^{m}-\kappa W^{n}} with W {\textstyle W} the weight, η {\textstyle \eta } and κ {\textstyle \kappa } constants of anabolism and catabolism respectively, and m {\textstyle m} , n {\textstyle n} constant exponants. Von Bertalanffy gave himself the resulting equation for W {\textstyle W} as a function of t {\textstyle t} , assuming that n = 1 {\textstyle n=1} and m ≤ 1 {\textstyle m\leq 1} :8
W = ( η κ − ( η κ − W 0 1 − m ) e − ( 1 − m ) κ t ) 1 1 − m {\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,9 with m = 2 3 {\textstyle m={\frac {2}{3}}} , according to the then-called "surface law", and n = 1 {\textstyle 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 1988.10
See also
Wikimedia Commons has media related to Von Bertalanffy curve.References
Daniel Pauly; G. R. Morgan (1987). Length-based Methods in Fisheries Research. WorldFish. p. 299. ISBN 978-971-10-2228-0. 978-971-10-2228-0 ↩
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. 978-92-5-105403-1 ↩
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. https://doi.org/10.1017%2Fpab.2021.16 ↩
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. 978-1-4020-5570-6 ↩
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. ↩
von Bertalanffy, Ludwig (1941). "Untersuchungen uber die Gesetzlichkeit des Wachstums. VII. Stoffwechseltypen und Wachstumstypen". Biologisches Zentralblatt. 61: 510–532. ↩
von Bertalanffy, Ludwig (1957). "Quantitative laws in metabolism and growth". The Quarterly Review of Biology. 32 (3): 217–231. http://www.jstor.org/stable/2815257 ↩
von Bertalanffy, Ludwig (1957). "Quantitative laws in metabolism and growth". The Quarterly Review of Biology. 32 (3): 217–231. http://www.jstor.org/stable/2815257 ↩
Murray, J Alan (1921). "Normal growth in animals". The Journal of Agricultural Science. 11 (3): 258–274 – via Cambridge University Press. ↩
Somers, I.F. (1988). "On a seasonally oscillating growth function". Fishbyte. 6 (1): 8–11. https://econpapers.repec.org/RePEc:wfi:wfbyte:39518 ↩