Porch et al : A catch-free assessment model with application to Epinephelus ita/ara 



93 



log P(c\0} = 



'Iog,,(c,,,)-log,, 



II 



2a- 



+ logCT,., 



(19) 



Anecdotal observations may be treated in similar 

 fashion. For example the perceptions of constituents on 

 the abundance of the resource relative to virgin levels 

 {/}) can be modeled as 



<F,';r,*M„)A 



V 1 AT -''■»'•. <.+'"a 



n,- 



r„,y 



a 



7„ ^. ~ Nor?7ial{0,(j„). 



-IM 



(20) 



Here A^ is the relative contribution of each age class in 

 forming the perception of total abundance (e.g., fisher- 

 men may never encounter very young fish), A is the time 

 of the year most reflective of the period upon which the 

 perceptions were based (e.g., the peak of the fishing 

 season), and a„ is the standard deviation of the fluctua- 

 tions in log. n^,. 



It is not generally possible to obtain consistent esti- 

 mates for all of the elements of the covariance matrix 

 V (i.e., pp, cfip, p,., cfi^, cfl. , and o'~^). In the case of survey 

 data, the variances associated with sampling variability 

 are often estimated extraneous to the population model 

 (e.g., during the standardization procedure). However, 

 there may be additional variance owing to fluctuations 

 in the distribution of the stock in relation to the survey 

 area (IWC, 1994). To accommodate such possibilities, 

 the survey variance parameters are modeled as 



The model outlined above was implemented by using 

 the nonlinear optimization package AD Model Builder 

 (version 4.5, Otter Research Ltd., Sidney, Canada), which 

 provides facilities for estimating the mode and shape of 

 the posterior distribution. Confidence intervals for the 

 probability of recovery were generated directly from the 

 posteriors approximated by the likelihood profile method 

 (the accuracy of which was checked by replicating the 

 prior distributions without data and by comparing the 

 modes of the posterior with the HPD estimates). For 

 some quantities confidence limits were computed by us- 

 ing normal approximations centered at the HPD esti- 

 mate with variances obtained by inverting the Hessian 

 matrix. This approach reduced computing time consid- 

 erably, but the approximations were poor for confidence 

 intervals broader than 80 percent owing to the thick 

 tails and skewed nature of the posterior distributions. 



9 



■■x„.,+P„<^ • 



(21) 



where the x^c.,.y and /^ , , 



the annual observation vari- 

 ances (estimated outside 

 the model); 



a- reflects some overall process 

 variance (estimated within 

 the model); and 



/j = constant multipliers (usu- 

 ally fixed by the analyst 

 based on a careful con- 

 sideration of the inherent 

 variability of the underly- 

 ing processes). 



The recruitment variance and correlation coefficient are 

 generally inestimable without a good index of recruit- 

 ment and may have to be fixed to some moderate values 

 (say 0^=0.4 and p=0.5). 



Application to goliath grouper 



Goliath grouper are large, long-lived predators found 

 predominantly in the tropical western Atlantic and 

 Caribbean Sea. They are among the least wary of reef 

 fishes, easily approached by spearfishers and readily 

 caught in traps or by hook and line gear. Not surpris- 

 ingly, they have declined considerably throughout much 

 of their range (Sadovy and Eklund, 1999). Although 

 there are few data on the historic abundance of these 

 animals in southern Florida, anecdotal reports suggest 

 that they were much more abundant during the 1950s 

 and 60s than they are now (Table 1). Concerns of over- 

 fishing prompted regulators in the U.S. to impose a 

 moratorium on the harvest of goliath grouper that has 

 remained in effect since 1990. To date, the duration of 

 the moratorium has not been specified owing to the pau- 

 city of information on their potential recovery rates. 



Spawner-recruit relationship There does not appear 

 to be any reliable information on the nature of the 



