On the 3/4-exponent von Bertalanffy equation for ontogenetic growth

Abstract

Bertalanffy proposed the differential equation m´(t) = p × m (t) a-q × m (t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 2/3 (von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, as for certain species other models would provide a better fit. This paper reconsiders this question. Using the Akaike information criterion it proposes a testable definition of 'weak universality' for a taxonomic group of species. (It roughly means that a model has an acceptable fit to most data sets of that group.) This definition was applied to 60 data sets from literature (37 about fish and 23 about non-fish species) and for each dataset an optimal metabolic scaling exponent 0 ≤ a opt < 1 was identified, where the model function m(t) achieved the best fit to the data. Although in general this optimal exponent differed widely from a = 2/3 of the VBGF, the VBGF was weakly universal for fish, but not for non-fish. This observation supported the conjecture that the pattern of growth for fish may be distinct. The paper discusses this conjecture. PeerJ Preprints | https://doi.org/10.7287/peerj.preprints.3303v1 | CC BY 4.0 Open Access | rec Abstract. Bertalanffy proposed the differential equation m´(t) = pm(t) a-qm(t) for the description of the mass growth 10 of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 2/3 11 (von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, 12 as for certain species other models would provide a better fit. This paper reconsiders this question. Using the Akaike 13 information criterion it proposes a testable definition of 'weak universality' for a taxonomic group of species. (It 14 roughly means that a model has an acceptable fit to most data sets of that group.) This definition was applied to 60 15 data sets from literature (37 about fish and 23 about non-fish species) and for each dataset an optimal metabolic scaling 16 exponent 0 ≤ a opt < 1 was identified, where the model function m(t) achieved the best fit to the data. Although in 17 general this optimal exponent differed widely from a = 2/3 of the VBGF, the VBGF was weakly universal for fish, 18 but not for non-fish. This observation supported the conjecture that the pattern of growth for fish may be distinct. The 19 paper discusses this conjecture. 20 Keywords: Akaike's information criteria (AIC), multi-model inference, von Bertalanffy growth function (VBGF), 21 metabolic scaling exponent, weak universality 22 1. Introduction 23 Growth models: Size at age is a key metric of productivity for any animal population (MacNeil 24 et al., 2017) and since Verhulst' (1838) seminal work about the logistic function a wide range of 25 growth models to describe the size of animals as a function of time has been developed. Amongst 26 applications are improved otolith analysis fo...

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