On the exponent in the von Bertalanffy growth model

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 nonfish. This observation supported the conjecture that the pattern of growth for fish may be distinct. The paper discusses this conjecture. E-mail corresponding author: katharina.renner-martin@boku.ac.at 8 9 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. PeerJ Preprints20 Keywords: Akaike's information criteria (AIC), multi-model inference, von Bertalanffy growth function (VBGF), 21 metabolic scaling exponent, weak universality 22 Introduction23 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 for age estimat...