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Fishery Bulletin 100(2) 



computer program,^ by using the two fishery-independent 

 abundance indices (hook-and-line and extended chevron 

 trap). Most calibrated VPA runs were made with both 

 indices, but a few were made with individual indices to 

 assess sensitivity. The hook-and-line index included esti- 

 mates of zero in 1992 and 1996, to which a small value 

 (0.0001) was added before log transformation, a technique 

 often used in statistical AN OVA models (e.g. Snedecor and 

 Cochran, 1980). Effects of that approach were explored by 

 assuming missing values in place of zeroes in an addition- 

 al calibrated VPA run. 



Several additional techniques were used to examine 

 sensitivity to assumptions. Sensitivity of estimated F and 

 recruitment to age 1 to uncertainty in M was investigat- 

 ed by conducting the separable and calibrated VPAs with 

 alternate values of M (0.20/yr and 0..35/vr). Retrospective 

 analyses of the calibrated VPA were conducted to inves- 

 tigate the possibility of systematic deviations ("retrospec- 

 tive patterns") in estimates of F and recruitment in most 

 recent years, in comparison to estimates obtained for the 

 same years from later analyses. In the retrospective anal- 

 yses, we varied the final year of data used from 1992 to 

 1997; the initial year was 1972 throughout. 



For presentation of results, we averaged estimated 

 quantities within three time periods: 1972-78, repre- 

 senting a lightly fished stock; 1982-86. representing the 

 stock after increasing exploitation during the early 1980s; 

 and 1992-96, representing conditions since the last ma- 

 jor change in management (amendment 4|SAFMC'^|), but 

 omitting the terminal year (1997) to reduce retrospective 

 effects (discussed below). We refer to these as early, mid- 

 dle, and recent periods. Recruitment was defined as the 

 number offish at age 1 on January 1. 



Yield per recruit 



Equilibrium yield per recruit (YPR) was estimated by the 

 method of Ricker (1975), which divides the exploited life 

 span into phases of constant mortality and gi'owth rates; 

 total YPR is obtained by summing yield across phases. 

 Parameter estimates from YPR analysis were for both 

 sexes and across the three time periods described above. It 

 has long been recognized that fishing at F^^^.^^ (F that maxi- 

 mizes YPR) can cause recruitment failure (Gulland and 

 Boerema, 1973; Clarke, 1991; Caddy and Mahon, 1995), 

 and the reference point F^ , was introduced as a more con- 

 servative alternative (Gulland and Boerema, 1973). Using 

 most recent growth and selectivity, we computed both F,,,.,^ 

 and Fq j. 



Spawning potential ratio 



Static spawning potential ratio (static SPR) — also known 

 as equilibrium SPR or maximum spawning potential 

 (Gabriel et al., 1989) — is a measure of fishing mortality 

 rate scaled to a species' biology. Static SPR for a given 



fishing mortality rate F* is computed as the ratio SPR = 

 SS(F*) / SS(0), where SS(F*) is the spawning-stock size 

 (lifetime cohort spawning contribution) expected under 

 F  , and SS(0) is the corresponding spawning-stock size 

 expected under F = 0. Other life history parameters are 

 assumed constant at most recent values. Increases in F 

 reduce static SPR. 



In species that do not change sex, spawning-stock bio- 

 mass is usually computed as total mature female biomass 

 or total egg production (Prager et al., 1987), quantities 

 that are highly correlated. Because red porgy change sex, 

 we used four representations of spawning-stock biomass: 

 female mature biomass, male mature biomass, total (male 

 + female) mature biomass, and total egg production, each 

 assuming the sex ratios at age given above ( Roumillat and 

 Waltz-). 



To compute total egg production, we used the relation- 

 ship between fecundity i£, number of eggs) and total 

 length (TL, mm) of Manooch (1976): 



In £ = -14.1325 + 4.3598 (In TL). 



(2) 



Sex ratio in the mature population can be changed by 

 fishing. We estimated reduction in the proportion of males 

 among mature fish from the proportion expected with no 

 fishing and assuming that the rate of sex transformation 

 is not affected by changes in population structure. 



Spawner-recruit relationships 



The relationship between spawning-stock biomass and 

 resulting recruitment to age 1 was modeled by using the 

 Beverton-Holt spawner-recruit function (Ricker, 1975): 



/? = SSB/(6„ 



SSB-^6,), 



(3) 



'■ FADAPT by V. R. Restrepo (International Commission for the 

 Conservation of Atlantic Tunas, Calle Corazon de Maria, 8, 

 Sixth Floor. 28002 Madrid, Spain). 



where R = recruitment; 



SSB = spawning-stock biomass, and 

 6,i, fe, = fitted parameters. 



To reduce the influence of retrospective patterns, SSB and 

 R series (estimated from VPA) used in recruitment mod- 

 eling were terminated at 1992. We used total mature bio- 

 mass, rather than the more usual female mature biomass, 

 to represent SSB in recruitment modeling. 



With the estimated stock-recruitment relationship, 500- 

 year simulations of the stock were made to estimate equilibri- 

 um yield and spawning-stock biomass as functions of fishing 

 mortality rate. The simulations used mean population num- 

 bers, 1992-96, as starting values. Age-specific selectivity for 

 the same period and the most recent gi-owth patterns were 

 used throughout. We thus estimated MSY, B^^^, and F^^a,-^ 

 for each catch matrix. 



Surplus-production model 



A surplus-production model was fitted to obtain additional 

 estimates of management benchmarks and stock status. 

 We used the Prager (1994, 1995) formulation and imple- 

 mentation of the continuous-time Graham-Schaefer (logis- 



