Estimating fishing gear selectivity for trawlers using length-frequency data from concurrent commercial trawl and longline fishing for Patagonian toothfish in Division 58.5.2 and the ratio of their hazard functions

The upper-arm of the length-dependent fishing selectivity function for trawlers fishing for Patagonian toothfish around Heard Island (Division 58.5.2) was modelled as the exponential of a quadratic function in length. This function was calibrated using random length frequency (LF) data collected from commercial operations of each of trawlers and longliners fishing concurrently for 3 seasons assuming that, for the upper range of lengths, fish are fully selected by longliners but not by trawlers. To estimate the trawl gear selectivity while simultaneously adjusting for relative availability of length classes as a function of fishing depth, a log-linear hazard function for each gear type was fitted by a technique used to model grouped survival data. The hazard function relates to the event of a fish falling in a particular length bin, analagous to a death in a given time interval in survival analysis, and the ratio of the hazard function for trawlers to that of longliners for the length range where this ratio is less than 1 defines the upper-arm of the trawl gear selectivity function. This model of selectivity was estimated using the LF data and a fitted binomial generalized linear mixed model (GLMM). The GLMM was fitted to the number of fish in a bin using a complementary log-log link function and binomial sample size defined as the number of fish in that bin or a larger length bin. The resultant conditional binomial probability of the event is linked to an approximate integral of the hazard function to give the GLMM formulation. Random effects of haul identifier and ‘season by length bin’ were included in the GLMM which was fitted using an approximation to the marginal quasi-likelihood (MQL). The fitted trawl gear selectivity function showed a decline from 100% to 1% selection for a corresponding length range of 1030 to 1610 mm and relative availability of fish was also predicted to decrease across a similar length range for the median trawl depth of 580 m relative to the median depth of longline sets of 1200 m.