Abstract. —Dynamic pool models 

 were originally developed to evalu- 

 ate the effect of exploitation on 

 gonochoristic species of fishes. Con- 

 sequently, application of these mod- 

 els to hermaphroditic species may 

 lead to erroneous conclusions. The 

 objective of our analysis was to de- 

 velop an alternative method for es- 

 timation of yield-per-recruit and 

 spawning stock biomass-per-recruit 

 that incorporates the effect of sex 

 transformation. Life-history charac- 

 teristics such as the rate of sex 

 change, growth, and mortality are 

 size-specific; therefore, a length- 

 based model was produced using a 

 series of time-invariant distributed 

 delays. The model incorporates a se- 

 quence of length-stages through 

 which the cohort moves, at rates de- 

 termined from empirical length data. 

 At each length-interval, fish may die, 

 move to the next interval, or switch 

 sex according to a probability distri- 

 bution of transformation. The model 

 tracks the cohort through time and 

 length categories as an aggregation 

 of individuals rather than a repre- 

 sentative average. The estimate of 

 yield-per-recruit for a protogynous 

 hermaphrodite, black sea bass 

 Centropristis striata, differed little 

 from traditional models. The distrib- 

 uted delay model with a sex-trans- 

 formation phase produced consider- 

 ably lower estimates of spawning 

 stock biomass-per-recruit than the 

 model with traditional assumptions. 

 Potentially erroneous estimates of 

 biological reference points using 

 standard methods for hermaphro- 

 ditic species may be overcome with 

 use of a distributed delay model, 

 which can incorporate size-specific 

 life-history dynamics. 



Length-based analyses of yield 

 and spawning biomass per recruit 

 for black sea bass Centropristis 

 striata, a protogynous 

 hermaphrodite 



Gary R. Shepherd 

 Josef S. Idoine 



Woods Hole Laboratory, Northeast Fisheries Science Center 

 National Marine Fisheries Service. NOAA 

 Woods Hole, Massachusetts 02543 



Manuscript accepted 28 January 1993. 

 Fishery Bulletin, U.S. 91:328-337 (1993). 



Analytical models describing the po- 

 tential yield of a fisheries resource 

 consist of three basic dynamic ele- 

 ments: population growth, recruit- 

 ment, and mortality (Russell 1931). 

 Dynamic pool models (e.g., Thomp- 

 son & Bell 1934, Ricker 1954, 

 Beverton & Holt 1957) are tradition- 

 ally used to evaluate potential yield- 

 per-recruit (Y/R) and spawning stock 

 biomass-per-recruit (SSB/R) relative 

 to fishing mortality and size-at- 

 recruitment. Potential yield and 

 spawning biomass of a cohort are 

 typically calculated as a summation 

 across discrete age-increments, un- 

 der specified conditions of growth and 

 natural mortality. Within each age- 

 interval, growth and fishing mortal- 

 ity rates are assumed to apply uni- 

 formly to all members of the cohort 

 (Gulland 1977). Not all individuals 

 follow the same growth trajectory, 

 however, and the susceptibility of in- 

 dividual fish to mortality is often a 

 size — rather than age-dependent phe- 

 nomena (Kirkpatrick 1984). Conse- 

 quently, length-based models can pro- 

 vide a more precise and accurate 

 depiction of cohort dynamics, in ad- 

 dition to using directly measurable 

 variables that are often linked to 

 management measures. 



Traditional age-based models of Y/ 

 R and SSB/R may be particularly in- 

 appropriate for species with unusual 

 and/or complex life histories. This 



may be true for hermaphroditic spe- 

 cies that have unique reproductive 

 stages that change as a function of 

 fish size. For example, significant 

 growth and mortality changes that 

 occur following sexual transformation 

 would not be accounted for in con- 

 ventional Y/R models. 



Hermaphroditism will also influ- 

 ence estimation of sex-specific mor- 

 talities. For example, as transforma- 

 tions occur in a protogynous 

 hermaphrodite, females are lost from 

 the cohort, creating the same effect 

 as increased female mortality. The 

 transformations increase the number 

 of males in the cohort, reducing the 

 apparent mortality of males. Conse- 

 quently, the mortality estimates for 

 each sex are not independent, and a 

 potential bias may exist in estimat- 

 ing Y/R and SSB/R when sex-specific 

 characteristics are not considered. 



Black sea bass Centropristis striata 

 is a protogynous hermaphrodite com- 

 mon along the continental shelf of 

 the northwest Atlantic between Mas- 

 sachusetts and Florida. Growth in 

 black sea bass is sexually dimorphic, 

 with faster growth but lower maxi- 

 mum age in females (Lavenda 1949, 

 Mercer 1978, Alexander 1981). Fe- 

 males commonly reach sizes of 38 cm 

 and age 8yr (Lavenda 1949); how- 

 ever, at sizes of 8-35 cm females may 

 transform into males and enter the 

 male growth regime. Males may grow 



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