334 



Fishery Bulletin 91(2), 1993 



700 



10 11 



Figure 8 



Spawning stock biomass-at-age for black sea 

 bass Centropristis striata estimated with a dis- 

 tributed delay model with transitions (dashed) 

 and without transitions (solid) for length-at- 

 recruitment of 25cm. M = 0.3 (1-10 cm) and 0.2 

 ( 11-max. length). 



transformation in the 16-23 cm range had the greatest 

 effect, decreasing SSB/R by 39.1% with F=0 (Table 3). 

 Changes in transformation rates in the tails of the trans- 

 formation probability distribution had little impact. In 

 all cases, increases in F decreased the relative impact of 

 transformation loss and, consequently, the sensitivity to 

 changes in transformation rates decreased. 



Discussion 



Ignoring the unique population dynamics that may oc- 

 cur in hemaphroditic fishes will increase the risk of 

 incorrectly estimating optimal exploitation levels from 

 yield-per-recruit models (Bannerot et al. 1987). The 

 problems can be compounded by disregarding the in- 

 creased size variability within the system resulting 

 from sexual transformations. Distributed delay mod- 

 els provide a mechanism for handling length-specific 

 life-history changes, and results can be summarized 

 as an aggregate of individuals and the associated vari- 

 ability rather than a simple mean estimate without 

 error. The consequence is greater flexibility in dealing 

 with changes between length categories without ignor- 

 ing the size variability inherent within the system. 



In the black sea bass example presented, the incor- 

 poration of a transitional term in the delay model had 

 little effect on the estimate of yield-per-recruit. Since 

 the sexes were combined to produce total yield, the 

 growth differences between sexes were not large enough 

 to produce significant changes in overall yield. Also, 

 the inclusion of fishing mortality quickly removed any 

 added yield contributed by transformed females. It is 

 likely that the result would diverge further from a 

 traditional model estimate as the differences in lon- 

 gevity or growth rate between the sexes increased. 



The effect of modeling length- and sex-specific 

 characteristics with a delay model was more appar- 

 ent in spawning stock biomass-per-recruit estimates. 

 The removal of female biomass via transformations 

 was similar to increasing natural mortality at each 

 length step, and consequently the spawning poten- 

 tial was reduced more quickly than under typical 

 effects of fishing mortality and a constant M of 0.2. 

 The system was more sensitive to changes in size- 

 at-recruitment to the fishery than reduction in F at 

 a constant size-at-recruitment. 



The differences in spawning biomass-per-recruit es- 

 timates resulting from the distributed delay model have 

 important management implications. If management 



