## Introduction

Growth is one of the most important measurable life-history parameters for individuals and species (Austin *et al*. 2011; Einum, Forseth & Finstad 2012; Paine *et al*. 2012). Recent comparative and analytical work has shown that understanding growth is fundamental to understanding life histories, demography, ecosystem dynamics, and fisheries sustainability (Beddington & Kirkwood 2005; Frisk, Miller & Dulvy 2005). Across species, growth correlates with a number of life-history traits including natural mortality rate (Pauly 1980; Charnov, Gislason & Pope in press), lifespan (Hoenig 1983) and reproductive allocation (Lester, Shuter & Abrams 2004; Charnov 2008); traits that also influence the response of species to exploitation (Jennings, Reynolds & Mills 1998; Frisk, Miller & Dulvy 2005).

A widely used method of describing growth, currently utilised in at least 100 published articles in each of the last 6 years, is the von Bertalanffy growth function, or VBGF (von Bertalanffy 1938, 1957). This model has been used to describe the change in body size over time of fossil and modern species across a wide range of taxa, including mammals (English, Bateman & Clutton-Brock 2012), birds (Tjørve & Tjørve 2010), reptiles (including dinosaurs) (Lehman & Woodward 2008), amphibians (Arntzen 2000), but it is most extensively applied across the most speciose vertebrate taxon – the fishes (Chen, Jackson & Harvey 1992; Frisk, Miller & Fogarty 2001). Most fisheries stock assessment models rely on von Bertalanffy growth models to convert between population numbers and biomass.

Von Bertalanffy hypothesised that net growth, i.e. the change in mass over time resulting from the difference between anabolism and catabolism, is approximately a one-third power function of size describing the net effect of both metabolic processes. By integrating and converting to a length formulation (assuming weight is proportional to the third power of length) von Bertalanffy defined growth in length as:

where *L*(*t*) is length-at-age *t* (age in years, length in cm), *L*_{∞} is the asymptotic size (in cm), *k* is the growth coefficient (in yr^{−1}) and *L*_{0} is the length-at-age zero (in cm) (Fig. 1a). While asymptotic size (*L*_{∞}) is the maximum theoretical size that a species will tend towards, but never actually reach, the growth coefficient (*k*) is the rate at which growth approaches this asymptote such that it takes *ln 2 k*^{−1} units of time to grow halfway towards *L*_{∞} at any given point (Fabens 1965). The third parameter used in the von Bertalanffy growth equation is the size-at-age zero (*L*_{0}) which equates to the y-intercept. Note that two key parameters often lie well beyond the data (the smallest theoretical size *L*_{0} and the largest asymptotic theoretical size *L*_{∞}). Von Bertalanffy growth models are fitted to empirical length-at-age data (in fishes age is usually estimated from tree ring-like growth checks in the otoliths, vertebrae or spines), with age on the *x*-axis and length on the *y*-axis, and models are fit using nonlinear sum-of-squares fitting methods (Appendix I). Some teleost age and growth studies also fix the intercept to zero (McGarvey & Fowler 2002; Taylor, Walters & Martell 2005; Gwinn, Allen & Rogers 2010), but the two-parameter von Bertalanffy growth function is most widely applied to elasmobranchs as they tend to have a large size at hatch or at birth.

Elasmobranchs, like most fishes, grow continuously and asymptotically throughout their lives and their growth is well-described by the von Bertalanffy model (Beverton & Holt 1959; Cailliet *et al*. 2006). In a recent review of elasmobranch age and growth studies, Cailliet *et al*. (2006) recommended the use of the von Bertalanffy growth function based on the *L*_{0} parameter. This formulation then allows fixing *L*_{0} to a known value, the empirical size at birth, and presents the opportunity to save one degree of freedom in the model fitting process. The key assumption is that the *L*_{0} parameter (better described as the theoretical average length when age is zero) is identical to, and can be replaced by, an empirical estimate of size at birth. As a consequence, the two-parameter von Bertalanffy growth function only requires the estimation of the remaining growth parameters *L*_{∞}, and *k* from the available length-at-age data. The use of this two-parameter von Bertalanffy growth function has proliferated in recent years (Neer, Thompson & Carlson 2005; Braccini *et al*. 2007; Pierce & Bennett 2010). While there are specific situations where fixing model parameters may improve growth estimates, such as the case of fledgling growth (Tjørve & Tjørve 2010; Austin *et al*. 2011), the consequences of fixing parameters on model performance in the von Bertalanffy growth function have only rarely been examined. A recent comparison of growth models showed that and even though the two-parameter von Bertalanffy growth model was overall the most parsimonious model (i.e. best ranked using Akaike Information Criteria, or AIC), it appears to perform better, with lower estimating error, only in data-sparse simulations compared to the three-parameter variant which performs best in data-rich settings (Thorson & Simpfendorfer 2009).

In addition to fitting two- and three-parameter von Bertalanffy growth models there is an emerging practice of fitting multiple models (both von Bertalanffy models as well as others, such as Gompertz and logistic), comparing them using (AIC), and reporting parameter estimates of all candidate models or a single set of estimates from multi-model averaging (Katsanevakis 2006; Katsanevakis & Maravelias 2008; Thorson & Simpfendorfer 2009). This approach addresses the question of which model is most parsimonious with the available data, trading off model complexity with goodness-of-fit. Unfortunately, a model can be the most parsimonious while still incorrectly describing the underlying growth trajectory. In this study, we test the performance not by the parsimony approach of AIC, but instead by determining whether the two- or the three-parameter von Bertalanffy growth model provides parameter estimates that are closest to the true (simulated) values.

The specific aims of this study were: (1) to test the equivalence assumption that *L*_{0} is the same or similar to empirically estimated size at birth, (2) to compare, in terms of bias and uncertainty, the estimation of the growth coefficient (*k*) between the two- and three-parameter estimation methods of the von Bertalanffy growth function in data-rich as well as data-sparse scenarios, and (3) to determine whether these differences vary across a range of life histories. We show that *L*_{0} is not equivalent to size at birth, and that assuming so results in severely biased growth estimates, which in turn adversely biases understanding of fisheries stock status. This case study provides a general caution against fixing parameters to save one degree of freedom, especially when the underlying parameters covary.