A Model for the Mass-Growth of Wild-Caught Fish

Katharina Renner-Martin,Klaus Scheicher,Manfred Kühleitner,Norbert Brunner,Werner-Georg Nowak

doi:10.4236/ojmsi.2019.71002

Abstract

The paper searched for raw data about wild-caught fish, where a sigmoidal growth function described the mass growth significantly better than non-sigmoidal functions. Specifically, von Bertalanffy’s sigmoidal growth function (metabolic exponent-pair a = 2/3, b = 1) was compared with unbounded linear growth and with bounded exponential growth using the Akaike information criterion. Thereby the maximum likelihood fits were compared, assuming a lognormal distribution of mass (i.e. a higher variance for heavier animals). Starting from 70+ size-at-age data, the paper focused on 15 data coming from large datasets. Of them, six data with 400 - 20,000 data-points were suitable for sigmoidal growth modeling. For these, a custom-made optimization tool identified the best fitting growth function from the general von Bertalanffy-Putter class of models. This class generalizes the well-known models of Verhulst (logistic growth), Gompertz and von Bertalanffy. Whereas the best-fitting models varied widely, their exponent-pairs displayed a remarkable pattern, as their difference was close to 1/3 (example: von Bertalanffy exponent-pair). This defined a new class of models, for which the paper provided a biological motivation that relates growth to food consumption.

Highlights

The von Bertalanffy growth function for length (VBGF) fits to the present framework, too: It is a special case of Equation (1) using the exponent pair a = 0, b = 1
The traditional explanation of differential Equation (1) proposed that the rate of growth would be proportional to the difference between anabolism and catabolism, both of which would be proportional to a power of mass
Specific values of the exponents were derived from biophysical arguments; e.g.: b = 1, as catabolism would be proportional to mass and a = 2/3, as anabolism would be proportional to the gills’ surface and to the 2/3th power of mass [1]

Summary

Introduction

The von Bertalanffy [1] and Pütter [2] differential Equation (1) provides a comprehensive framework for the most common models for mass-growth: dm (t ) = p ⋅ m (t )a − q ⋅ m (t )b (1) dt It describes body mass m(t) > 0 as a function of age t, using the following five model parameters: The exponent-pair a < b (“metabolic scaling exponents”) and the constants p and q are assumed to be non-negative; m0 > 0 is an initial value, i.e. m(0) = m0. The von Bertalanffy growth function for length (VBGF) fits to the present framework, too: It is a special case of Equation (1) using the exponent pair a = 0, b = 1 (bounded exponential growth).

Methods

Results

Discussion

Conclusion