26 



Fishery Bulletin 97(1), 1999 



year (i.e. from July 1 1994 to June 30 1995), this fish- 

 ery has been managed under an output-control scheme 

 with an annual total allowable catch (TAG) of 106 t. 

 No attempt has been made previously to quantify 

 the dynamics of this fishery. 



Abundance indices, catch per unit of effort (CPUE), 

 have been developed by Montgomery ( 1995) from data 

 collected in the commercial fishery for the period of 

 1903-36 and that fi-om 1969-70 (i.e. 1 July 1969 to 30 

 June 1970) to 1993-94. Because data are mainly col- 

 lected and derived fi-om the NSW commercial fishery, 

 large errors are likely to exist in the catch and CPUE 

 data, and there is a concern that the quality of the data 

 is perhaps not good for the purpose of modeling. 



Production models are fitted to catch and CPUE 

 data by using an observation-error estimator that 

 minimizes the sum of squared differences between 

 log-observed and predicted CPUEs (Hilborn and 

 Walters, 1992). This estimator assumes that there 

 is only error in the observed abundance index and 

 that there are no observation errors in catch or pro- 

 cess errors in the dynamics of the stock biomass. 

 Because the least-squares method is sensitive to the 

 assumption on the error structure in the model 

 (Rousseeuw and Leroy, 1987), the unrealistic error 

 assumption associated with the observation-error 

 estimator tends to result in large errors in estimated 

 parameters when models are fitted to data (Schnute, 

 1989; Chen and Andrew, 1998). 



A more realistic error structure should include 

 process error in the dynamics of the stock biomass 

 and observation errors in both CPUE and catch. In 

 our case, if the distribution of all error terms can be 

 fully defined, we can apply the Kalman filter to gen- 

 erate a likelihood function and then maximize this 

 likelihood function to yield parameter estimates 

 (Sullivan, 1992), or we can define an appropriate 

 variance-covariance matrix based on the defined er- 

 ror structure and then apply a generalized least- 

 squares method to estimate parameters in the model 

 (Paloheimo and Chen, 1996). However, the former 

 approach is rather complicated because the dynamic 

 model is nonlinear and there are two observation 

 models (i.e. one for CPUE and the other for catch; 

 Reed and Simons, 1996). The latter approach needs 

 information on process and observation errors 

 (Paloheimo and Chen, 1996). Such information is 

 probably nonexistent in most fisheries. Moreover, the 

 parametric assumption on error distribution (e.g. 

 normal, log-normal, etc) may not be true. 



Because of all these difficulties with the error struc- 

 ture for production models, it is desirable to have an 

 estimation method that is robust with respect to as- 

 sumptions concerning model error structure. Least 

 median of squared errors (LMSE; Rousseeuw, 1984), 



which minimizes the median of squared differences 

 between predicted and observed log CPUEs, is such 

 an estimator (Chen and Andrew, 1998). A bootstrap 

 procedure (Efron, 1979) was incorporated into the 

 LMSE estimator to estimate the parameters and 

 their uncertainties in this study. The probability of 

 short-term overexploitation, defined as a fishing 

 mortality rate higher than the selected biological 

 reference points, was estimated for the next fishing 

 season with different levels of TAC. 



Production models 



Production models are the simplest stock assessment 

 models that are commonly used in fisheries (Hilborn 

 and Walters, 1992). The input data for these models 

 are the time series of catch and associated abundance 

 index. Several variants of production models have 

 been proposed (e.g. Pella and Tomlinson, 1969; 

 Walters and Hilborn, 1976; Schnute, 1977, 1989; 

 Punt, 1993). Without considering the structure of 

 observation and process errors, the deterministic 

 production model that is most commonly used can 

 be written as 



B 



i^\ 



B.+g.-C, 



(1) 



where B 



c, 



s, 



= the stock biomass; 

 - the catch; and 



= the growth of population in biomass, all 

 in year i. 



The g^ is often referred to as surplus production and 

 often described by the logistic or Schaefer function 

 written as giB^) - rB^il - B/K), where r is a para- 

 meter describing the intrinsic rate of population 

 growth in biomass and AT is a parameter correspond- 

 ing to the unfished equilibrium stock size (often re- 

 ferred to as the carrying capacity or virgin biomass). 

 The stock biomass in year ;' is often assumed to be 

 directly related to a relative abundance index that 

 can be observed in fisheries. This assumption can be 

 written as 



where q - the catchability coefficient; and 



I^ = the abundance index in year i (Hilborn 

 and Waters, 1992). 



Methods for the parameter estimation 



The use of an appropriate method to fit a production 

 model to the observed data has been shown to be as 



