330 



Fishery Bulletin 9 1(2). 1993 



t 



Figure 2 



Erlang probability density function for various values of k 

 and a mean t = 10. 



to state variables (e.g., numbers of individuals dead 

 due to fishing mortality at a given size) where neces- 

 sary for analyses. Individuals in the cohort passed 

 through each stage at a rate determined by an Erlang 

 probability density function. The shape, or the order, 

 of the density function was determined by the value of 

 k (Manetsch 1976) (Fig. 2). Transit time of an indi- 

 vidual through stage i was a function of the mean 

 delay period (D) and associated variance. The greater 

 the variance relative to the mean, the greater the value 

 of k. This continuous function was approximated, us- 

 ing difference equations, by a series of substages. The 

 number of substages within a stage was equal to the 

 integer value of k, since fractional substages cannot be 

 used in the discrete approximation necessary for the 

 model. 



A cohort entered length stage i at time t and moved 

 through the substages at stage-specific rates. At time 

 t+1, fish either moved into length stage i+1, remained 

 in length stage i (at some substage), or were removed 

 from the system at a defined rate of mortality due to 

 fishing and/or natural mortality. At time t+2, some 

 fish were still in stage i, some in i+1, and possibly 

 some in i+2, etc. The biomass of the cohort and mean 

 size changed as the distribution moved through the 

 series of length stages until all individuals reached a 

 maximum size or the cohort was eliminated through 

 mortality. Any individuals moving to a size beyond the 

 maximum were assumed to undergo senescence and 

 were removed from the system at that maximum size. 

 Yield was calculated from the number of individuals 

 removed at each length stage > the size of recruitment 

 to the fishery x the ratio of fishing mortality/total 



mortality (F/Z). The biomass offish removed via fishing 

 mortality was calculated using the appropriate sex- 

 specific length-weight equations. Spawning stock was 

 calculated as the sum of the number of females per 

 length remaining in the system at the time of spawn- 

 ing (1 June) x the probability of female maturity- 

 at-length. The number of mature females-at-length was 

 converted to biomass using a length-weight equation. 



With this model, hermaphroditism was included by 

 splitting the cohort into three growth regimes and us- 

 ing a size-specific probability of sex transformation. At 

 the initial stage, the cohort was divided into an appro- 

 priate number of males and females, then passed 

 through length stages at sex-specific rates. At a desig- 

 nated length-interval, females began transformation 

 by passing through an intermediate transitional stage 

 prior to entering the male growth sequence (Fig. 1). 



Input parameters 



Input parameters required in the model were mean 

 transit time (D) and its associated variance (S 2 ) per 

 1 cm length-category in days and days 2 respectively, 

 by sex; maximum potential length for each sex in the 

 population; probability of sex transformation-at-length 

 for females; a range of lengths-at-recruitment (L c ) to 

 the fishery; instantaneous natural mortality rate (spe- 

 cific to length groups and sex if appropriate); a range 

 of instantaneous fishing mortality rates; and the per- 

 centage of females mature-at-length. In addition, 

 length-weight equations by sex were necessary for con- 

 version of numbers-at-length to biomass. 



Estimates of mean transit times and their associ- 

 ated variances for black sea bass were determined from 

 back-calculation of scale data. Sea bass scales were 

 collected in coastal Long Island during 1979-80, aged, 

 and length-at-age back-calculated (Mark Alexander, 

 Conn. Dep. Environ. Prot, Old Lyme, CT, pers. commun.). 

 Daily time-increments of growth were chosen to pro- 

 vide estimates of k. Transit time (D) by cm-intervals 

 was calculated as 



D = 365/ [(S I ,-S„. 1 )*SL1 

 S 



where S n = scale annulus n, 

 S n .! = scale annulus n-1, 

 S = total scale size, 

 SL = standard length offish, 



with the assumption of linear growth between annuli 

 (i.e., D equal between cm-intervals (S^/S^SL and 

 (S„/S)*SL). Thus, each individual scale provided esti- 

 mates of transit times (D) by cm, up to the maximum 

 length of the fish (SL). The means of D and S 2 were 



