308 



Fishery Bulletin 89(2), 1991 



simulate a population of opakapaka Pristipomoides 

 filamentosus, a Hawaiian deepwater snapper. Mean 

 length-at-age (L t ) is described by a von Bertalanffy 

 growth equation: 



L t = LJl-e-KC-W); 



(1) 



where L^, K, and t are parameters. Estimates for 

 these parameters (L M = 66cm, K = 0.24/year, and t 

 = - 0.78 year) are from Ralston and Miyamoto (1983). 

 As a computational convenience, however, growth is 

 considered an incremental process: 



t-i 



Lt = U + Z DL i 



(2) 



i = l 



where the annual growth increment (DL t ) is ex- 

 pressed as a time-differenced form of Equation (1): 



DL t = L t+1 - L t = L TC e-K(t-t ) (l - e -K). (3) 



Variance in length-at-age (V t ) is described by an 

 asymptotic function of age: 



V t = C(l-e- Dt ), 



(4) 



where C and D are parameters. Estimates of these 

 parameters can be obtained by fitting this function to 

 size and age data obtained from Ralston and Miyamoto 

 (1983), but the predicted variance is unrealistically 

 large. Consequently, the parameters C = 10 and D = 

 0.1 are chosen to best fit the predicted-to-observed 

 length-frequency distributions. Variance in annual 

 growth increment is described as a time-differenced 

 form of Equation (4): 



DV t = V t+1 - V t - C(l-e- D )e- 



iM 



(5) 



Recruitment is assumed to occur instantaneously at 

 age-1 and, unless experimentally manipulated, to be 

 identical each year. The length distribution of recruits 

 is assumed to be normal with a mean and variance 

 defined by Equations (1) and (4), respectively. The 

 length distribution of the surviving members of this 

 age-class in subsequent years (y) is estimated with the 

 recursive relationship: 



N U+ i,y + i = I N Sjt , y e-(M+QF)p(l_ s |t); (6) 



s = l 



where M and F are the instantaneous rates of natural 

 and fishing mortality, Q is a size-dependent selectivity 

 coefficient, 1 is the smallest individual in each age 



group and P(l - s|t) is the conditional probability of 

 growing an amount (1-s) at t years of age. The 

 growth-increment probability function is assumed to 

 be normal with a mean and variance computed from 

 Equations (3) and (5). An estimate of natural mortal- 

 ity (M = 0.3) is obtained from Ralston (1987); estimates 

 of fishing mortality (F = 0.3 and 0.6) are chosen 

 somewhat arbitrarily to bracket the probable true value 

 for opakapaka. The selectivity coefficient Q is repre- 

 sented by a reparameterized logistic function of length: 



Q = 



l + e-EC-iso) 



(7) 



where E is a parameter controlling the steepness of 

 the function, and 1 50 is a parameter controlling the 

 length of 50% selectivity. The parameter values of 

 E = 0.5 and l 5fl = 45cm are chosen arbitrarily. The 

 population length-distribution is calculated by summing 

 across all age-classes, i.e., 



N l.-.y = X N l,t+l,y> 



t 



(8) 



and the catch length-frequency distribution is calcu- 

 lated as 



C|,..y = N!,., y 



QF 



M + QF 



(1 _ e -(M + QF))_ 



(9) 



The parameters Z/K and L^, are estimated from the 

 simulated catch length-frequency distribution as 

 follows. First, the length at full vulnerability (L c ) is 

 estimated as one length interval (1cm) larger than the 

 rightmost mode in the catch length-frequency distribu- 

 tion (Polovina 1989). Second, Tj, the mean length of 

 all fish >\ CI , is calculated for each l ci >L c . Third, lj is 

 regressed on l ci by using weighted linear regression 

 with weights equal to the sample size (i.e., ZQ for 1> 

 l c ). Fourth, Z/K and L^ are then computed from the 

 previously specified functions of the regression coef- 

 ficients. 



The disequilibrium experiments are conducted as 

 follows. For the experiments examining the fishing-up 

 phase of a fishery, the initial population length- 

 frequency distribution is set at the equilibrium distribu- 

 tion in the absence of fishing. This length distribution 

 is generated by running the model with F = 0.0 until 

 the population is in equilibrium (i.e., until the total 

 population size is identical on two successive iterations). 

 Fishing mortality at each of the two specified levels 

 is then applied instantaneously, and the simulation is 

 run until the population is again at equilibrium. For the 

 experiments examining the effect of perturbation dur- 



