90 



Fishery Bulletin 104(1) 



The purpose of this article is to introduce a model- 

 based framework for estimating reference points, stock 

 status, and recovery times in situations where catch 

 data and other measures of absolute abundance are 

 unreliable. The specific estimator developed in this 

 study is essentially an age-structured production model 

 recast in terms relative to pre-exploitation levels. A 

 Bayesian estimation scheme is adopted to allow the 

 incorporation of pertinent auxiliary information such as 

 might be obtained from meta-analyses of similar stocks 

 or anecdotal observations. The approach is applied to 

 the population of goliath grouper off southern Florida, 

 which is believed to have been severely depleted during 

 the 1980s and has been protected from all harvest since 

 1990 (GMFMC-). 



Materials and methods 



v^ = the relative vulnerability of the remaining 

 age classes (which implicitly includes factors 

 such as gear selectivity, size limit regula- 

 tions, and the fraction of the stock exposed 

 to the fishery). 



Relative recruitment (r) is modeled as a first-order log- 

 normal autoregressive process. 



 P,-f v-l + Ir.y 



(3) 



where f/^ 



Pr 



n 



the median expectation 

 the correlation coefficient; and 

 normal-distributed random variates having 

 mean and standard deviation a, (ostensi- 

 bly representing the effect on recruitment 

 of fluctuations in the environment). 



Population dynamics model 



The study period begins when the stock is believed to 

 be near virgin levels, such that the relative abundance 

 N of each age class a at the beginning of the first year 

 is given by 



Na.l 



1 



N„ 



Na-u^ ■" '/(l-e"'"'). 



-Af J . 



a = a^ 



a^<a<A (1) 



where a^ = the youngest age class in the analysis; 



A = a "plus-group" representing age classes A 



and older; and 

 M = the natural mortality rate. 



The relative abundance at the beginning of subsequent 

 years (y) is modeled by the recursion 



N„ 



A^„-i,v-ie 



-n-l'a-l-A^a-l 



•'',i-l,v-l^ ^ 



N 



A,y-1"= 



-fy-ifA-'^A 



c,. <a<A 

 a = A 



(2) 



where r^ = the annual recruitment to age class a,, rela- 

 tive to virgin levels; 

 F = the fishing mortality rate on the most vul- 

 nerable age class; and 



2 GMFMC (Gulf of Mexico Fishery Management Council). 

 1990. Amendment number 2 to the fishery manage- 

 ment Plan for the reef fish fishery of the Gulf of Mexico, 

 31 p. Gulf of Mexico Fishery Management Council. 3018 

 North US Highway 301, Suite 1000, Tampa, FL 33619. 



The median is modeled by the Ricker or Beverton-Holt 

 spawner-recruit functions recast in terms of the maxi- 

 mum lifetime reproductive rate a and relative spawning 

 biomass s: 



^'r 



(4) 



'.v=ISa-""^'""^''""'^«,.v/I^.-"^"'^A^«-.- 



where E = an index of the per-capita number of eggs 

 produced by each age class {E)\ and 

 t^ = the fraction of the year elapsed at the time 

 of spawning. 



The shapes of these two curves are essentially the same 

 as the conventional relationships (Fig. 1); however their 

 domain is implicitly limited to the interval Osssl (see 

 Appendix 1 for a derivation). 



The fishing mortality rate on the most vulnerable 

 age class F is also modeled as a first-order lognormal 

 autoregressive process, 





(5) 



where f)p = the median level; 



pp. = the correlation coefficient; and 

 11 = normal-distributed random variates having 

 mean and standard deviation Op.. 



The median annual is generally assumed to be propor- 

 tional to an index of fishing effort /: 



/'f = <Pf, 



(6) 



