Shepherd and Idoine Yield- and spawning biomass-per-recruit for Centropnstis striata 



329 



o s 



sub- 

 tfage 



sut> 

 stags 



to 60 cm and age 20 yr (Lavenda 

 1949) with greater variability in 

 size-at-age than females. Conse- 

 quently, a black sea bass has three 

 possible growth rates: female, 

 male, and transitional. These dif- 

 ferences in growth rates and vari- 

 ability in size-at-age between the 

 sexes will influence the potential 

 yield from a cohort. 



An alternative approach for 

 modeling changes in a cohort 

 over time is use of delay models. 

 Delay models were first devel- 

 oped in the field of industrial dy- 

 namics as a technique to moni- 

 tor movement of individuals 

 through a system of substages 

 (Forrester 1961), and have since 

 been modified to incorporate ef- 

 fects of attrition (mortality) and 

 variability in timing of move- 

 ments (Manetsch 1976, Vansickle 1977). The subclass 

 of distributed delay models can be a useful tool to 

 describe the movement of any item through a process 

 or, in a biological context, through developmental stages 

 (Manetsch 1976). Extending the model to include a 

 series of consecutive processes, it has been used to 

 simulate growth dynamics in marine crustaceans 

 (Idoine & Finn 1984), insects (Ravlin et al. 1978, 

 Schaub & Baumgartner 1989), and agricultural crops 

 (Gutierrez et al. 1984, 1988). The major advantage of 

 this model type is its focus on the aggregated behavior 

 of individuals rather than a representative mean. 



To estimate Y/R and SSB7R in black sea bass, we 

 modeled the growth and mortality of a cohort as a se- 

 ries of distributed delays with an associated mortality, 

 using length categories as individual developmental 

 stages. This approach allowed us to simulate the de- 

 cline of the cohort while retaining information about 

 variation in size composition and size-specific mortality 

 in the cohort. In addition, we were able to evaluate the 

 influence of additional mortality on the population dy- 

 namics of a species with an hermaphroditic life history. 



Methods 



The model structure 



Time-invariant distributed delay models with a mor- 

 tality term (Vansickle 1977) can be characterized as a 

 sequence of stages, with flow through each stage i rep- 

 resented by a series of differential equations (Fig. 1): 



TRANSITION 



Cf 



<? 



MORTALITIES 



1 



Figure 1 



(A) Schematic of intra-stage flow of the distributed delay model: D = delay period 

 defined as mean number of days necessary to move through a stage; S 2 = variance of D; 

 A = instantaneous mortality rate; i = index of stage; INi = number of individuals 

 entering stage i; k = integer value of D-7 (S 2 ), which defines the number of substages 

 within a stage; r, = flowrate between substages; OUTi = number of individuals moving 

 into next stage; mortalities = number of individuals removed from stage i through 

 attrition. (B) Schematic of flow between cm stages in the model. 



dr u /dt = k/D[(x(t)-r u (t))-(A(t)*D/k)] 

 dr l2 /dt = k/D[(r u (t)-r,, 2 (t))-(A(t)*D/k)] 



dr, k /dt=k/D[(r 1 , k . 1 (t)-r lk (t))-(A(t)*D/k)] 



where x(t) = input rate at time t, 



k = integer value of D 2 /variance of D (S 2 ) 



which defines the number of substages 



within a stage, 

 i = index of stage, 

 j = index of substage from 1 to k, and 

 r, = flowrate between substages j and j-1, 



stage i; 



or r v = (k,/D,)*Q 1J 



where Q,j = the number of individuals in substage j, 

 stage i, 

 D = delay period defined as the mean num- 

 ber of days necessary to move through a 

 stage (the value of D at stage i varies by 

 sex), and 

 A(t)= mortality rate at time t. 



All rates in the model are instantaneous. The model 

 was simulated as a series of difference equations with 

 a time step equal to 1 day. The delay model can be 

 expressed either in terms of flows (r u ) or state vari- 

 ables (Q.j) (Vansickle 1977). For simulation purposes, 

 this model utilizes flows within, between, and out of 

 stages representing 1cm length categories through 

 which a cohort will pass. These flows were converted 



