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

Abstract

The von Bertalanffy equation has been widely used to describe the ontogenetic growth of animals. Recently a new ontogenetic growth model (OGM) was proposed based on allometric scaling and has gained very good attention from the readers. Mathematically speaking, this model is a special case of von Bertalanffy equation with a scaling exponent (α) being 3/4. This new OGM has been criticized on several grounds, such as contradicting the law of energy conservation. Its generality has also been questioned as it fits poorly to the growth of many crop species. Moreover, fish growth fits the von Bertalanffy equation better when α=2/3 rather than when α=3/4. Here, we fit the von Bertalanffy equation with α=2/3 or 3/4 or unknown, and also the logistic equation, to the body length data of three freshwater fish species. This allows us to test: (i) how the choice of the scaling exponent α in the von Bertalanffy equation affects its performance, and (ii) whether the logistic equation provides a better fit than the von Bertalanffy equation for fish growth, as already demonstrated for crop growth. Results showed that the OGM (α=3/4) fitted better than unknown α or α=2/3 in the von Bertalanffy equation but worse than the logistic equation. When choosing α values between 1/2 and 1, we found that increasing the value of α could improve the goodness-of-fit but potentially lead to overfitting and unreliable estimates of model parameters. This suggests that there is no universal value of α for different species or taxa. Use of 2/3 or 3/4 value for the scaling exponent α in the von Bertalanffy equation should be carefully decided based on observed relationships between metabolic rate and body weight, and not merely based on the goodness-of-fit. As a rule of thumb, the logistic equation is still the best model for describing the ontogenetic growth of animals and plants. Future research should be aimed at clarifying the potential linkages between the logistic equation and the metabolic theories.