Chen and Montgomery: Modeling the dynamics of Jasus verreauxi 



27 



important in terms of the reliability of estimated 

 parameters as the specification of the algebraic form 

 of the underlying population dynamic model (Punt, 

 1988, 1993; Polacheck et al., 1993). Several ap- 

 proaches have been proposed to estimate parameters 

 in production models when only indices of abundance 

 and catch are available (Hilbom and Walters, 1992). 

 The four most commonly used approaches are equilib- 

 rium estimators (Gulland, 1961), effort-averaging es- 

 timators (Fox, 1975), process-error estimators (Walters 

 and Hilbom, 1976; Schnute, 1977), and observation- 

 error estimators (Butterworth and Andrew, 1984; 

 Ludwig and Walters, 1985). These approaches differ in 

 how observation and process errors are introduced into 

 the models that describe the dynamics of populations. 

 Recently, it has been suggested in some studies that 

 observation-error estimators tend to perform better 

 than others in parameter estimation (Punt, 1988, 

 1993; Hilborn and Walters, 1992; Polacheck et al., 

 1993). These estimators are constructed by assum- 

 ing that the population dynamic equations are de- 

 terministic (thus there is no process error) and that 

 all of the error occurs in the relationship between 

 stock biomass and relative abundance index. This 

 assumption can be written as 



log(/,) = \og{qB^ + e,. 



With the assumption that the e, are independent, 

 normally distributed variates, the estimates of the 

 model parameters (S^^^^^^^,, q, r, and K) are obtained 

 by maximizing the appropriate likelihood function 

 (Polacheck et al., 1993) or by minimizing the sum of 

 squared e^ (Hilborn and Walters, 1992). The time 

 series of stock biomass are estimated by projecting 

 the biomass at the start of the catch series forward 

 by using the historical annual catches. Because the 

 estimation methods for observation-error estimators 

 are least-squares types, they are sensitive to the as- 

 sumption about the error structure of the model. 

 Thus, parameter estimates tend to be unreliable if 

 the specification of error structure (i.e. no process 

 error, no error in observed catch, and log-normal er- 

 ror in observed abundance index) is not correct. How- 

 ever, in practice, it is almost impossible to know the 

 true error structure. It is therefore desirable to use 

 an estimation approach that is robust to the assump- 

 tion about the model error structure for observation- 

 error estimators. 



An observation-error estimator, which minimizes 

 the median of squared differences between observed 

 and predicted log CPUEs, has been found to be ro- 

 bust with respect to incorrect specification of error 

 structure (Chen and Andrew, 1998). This estimator 

 can be written as 



Minimize median log(/,)-log(/,) 

 / = l,...,iV 



It is an extension of the linear robust regression 

 method used by Chen and Paloheimo (1994) and 

 Chen et al.(1994) for nonlinear parameter estima- 

 tion. It should be noted that the algorithm developed 

 for the linear parameter estimation (Rousseeuw, 

 1984) can not be used for the above estimator. The 

 simplex method of Nelder and Mead ( 1965 ) was used 

 to conduct the nonlinear parameter estimation for 

 the above estimator (Press et al., 1992; Chen and 

 Andrew, 1998). 



Estimation of stock parameters for eastern 

 rock lobster 



Catch and CPUE data were available for two time 

 periods. The first period (hereafter referred to as 

 period I) was from 1903 to 1936, and the second (pe- 

 riod II) from 1969-70 to 1993-94. The fishery was 

 confined to grounds close to shore in period I, whereas 

 from the beginning of period II, the fishery expanded 

 to the continental slope. Therefore, it is highly likely 

 that large differences in catchability existed between 

 these two periods. Both the size and structure of the 

 rock lobster stock on the NSW coast may have 

 changed greatly over the two periods (Montgomery, 

 1995), and it is likely that the growth rate of the NSW 

 rock lobster stock differed between these two peri- 

 ods of time. Parameters r and q were thus assumed 

 to be different in these two time periods. Parameter 

 K was assumed to be the same for these two time 

 periods. This assumption was considered to be rea- 

 sonable because the harvesting on the expanded fish- 

 ing grounds at the beginning of period II was not 

 from an unexploited portion of the stock. It is thought 

 that eastern rock lobsters along the NSW coast dis- 

 play a movement that is typical of several other spe- 

 cies of rock lobster, moving between inshore and off- 

 shore grounds and along the coast (see Herrnkind et 

 al., 1994). Hence, lobsters on the grounds on the con- 

 tinental slope likely had been exposed to fishing on 

 more traditional shallower grounds at other times. 



The LMSE method was applied to fit the model to 

 data observed in period I and estimate parameters 

 ^1903' '"i' 9i' ^"^^ ^i' where subscript I refers to pe- 

 riod I. Because the year 1903 was early in the devel- 

 opment of the fishery, it is reasonable to assume that 

 BjgQ3 is the same as K^ in parameter estimation 

 (Hilborn and Walters, 1992). Thus, there are only 

 three parameters to be estimated with data observed 

 in the first time period. 



An algorithm that incorporates a bootstrap ap- 

 proach into the LMSE method was developed to es- 



