A von Bertalanffy growth model for Division 58.5.2 fitted to length-at-age data and a comparison of predicted with observed growth from mark–recapture data

Length-at-age data for the Patagonian toothfish (Dissostichus eleginoides) caught by trawl in Heard Island fishery were fitted using a von Bertalanffy (VB) growth model taking into account variable probability (VP) sampling of fish that were aged. Sub-sampling of random length frequency (LF) data used to obtain the samples of fish for ageing used length-bin sampling involving a fixed sample size per bin. Estimation of the VB parameters used a definition of the maximum likelihood which took into account VP sampling due to length-dependent selectivity of trawl fishing and also accounted for the additional affect of length-bin sampling on sampling probabilities. Estimation of the trawl selectivity curve is described elsewhere and assumed known for this study with its most important feature being a sharp decline in selectivity from 100% at 1000 mm down to 1% by 1600 mm. The VB curve fitted to the length-at-age data assuming normal errors with constant coefficient of variation using iteratively weighted least squares (IWLS) substantially under-predicted mean length-at-age for older ages compared to VB curve fitted using the same error model and VP maximum likelihood (MLP) with length-bin relative probabilities defined using fishing selectivity alone. When length bin sampling frequencies for aged fish relative to those for the LF sample were also included in defining relative probabilities, the VP maximum likelihood (MLPLB) and IWLS estimated curves were more similar. Predicted and observed values of annual growth rate (AGR) for lengths measured at release and first recapture in mark-recapture studies were compared where predictions used the VB parameter estimates obtained from the length-at-age data and the Fabens form of the VB growth model. Formulae for adjusting predictions for bias imparted by the use of the Fabens model were developed based on assumptions about the distribution of age at first capture and showed that the bias is relatively small for the range of release lengths in the data. Predictions of AGR using the MLPLB estimated VB parameters were closer to the mean trend in observed values with release length than those obtained using IWLS estimated parameters. A young-age adjustment (less than 5 years old) to the VB model is also given in order to give more realistic predictions of mean length-at-age for young fish.