394 



Fishery Bulletin 103(2) 



size was assumed a function of age (a) and followed the 

 von Bertalanffy model, 



Ha) 



LJl-e-^-V], 



(1) 



where /(a)=the length-at-age of an individual; 

 L r = the theoretical maximum length; 

 K =the growth rate, and 



t =the theoretical age when size would have 

 been zero. 



In our study, each individual's age and size were updated 

 at each monthly time step. 



Survival was computed differently for the two popula- 

 tions. In the unfished population, individuals survived 

 with a probability depending only on the natural mor- 

 tality rate (M/yr). In the fished population, individuals 

 survived with a probability depending on both the natu- 

 ral mortality rate and the size-specific fishing mortality 

 rate. Size selectivity [s(/)] by the fishery increased with 

 length according to the logistic equation 



s(l)- 



1 + e 



■P s u-l,) 



(2) 



where /3 S = the slope of the selectivity curve; and 

 L s = the length at 50% selectivity. 



The function s(l) describes the proportion of the fully- 

 selected fishing mortality rate IF) experienced by indi- 

 viduals of length /. The size-specific fishing mortality 

 rate, therefore, is s(l)F per year. Fishing was applied 

 over a fishing season of duration D F . 



The probability of reproduction was assumed equal to 

 the probability of maturity [m(a)\. In the model, matu- 

 rity increases with age and is independent of length. Al- 

 though maturity likely relates to length through bioen- 

 ergetics, the relationship was not modeled here because 

 it is, in general, poorly understood. Like selectivity (Eq. 

 2), m(a) was modeled by a logistic equation, but with a 

 slope parameter, )3 m , and age at 50% maturity, A m . 



In nature, values of life-history parameters K and A m 

 are related to a stock's natural mortality rate. A higher 

 natural mortality rate reduces the expected lifespan 

 and consequently tends to be associated with a higher 

 growth rate (K) and a younger age at maturity (A m ). 

 In the simulation, K and A m were related to natural 

 mortality by life-history invariants (detailed later). 

 Life-history invariants have a strong theoretical and 

 empirical basis (Roff, 1984; Beverton, 1992; Charnov, 

 1993) and have been valuable in other fishery applica- 

 tions (Mangel, 1996; Charnov and Skuladottir, 2000; 

 Frisk et al., 2001; Williams and Shertzer, 2003). 



Simulation 



To initialize the simulation, individuals were assigned 

 at random to a cohort. The number of cohorts was deter- 

 mined as the age at which approximately 1% of the 

 population would be expected to remain under natural 



mortality [-ln(0.01)/M, rounded to the nearest integer]. 

 Probabilities of cohort membership decayed exponen- 

 tially with age according to M; the probability of the 

 oldest cohort was adjusted to include the remaining 

 fraction offish (i.e., a plus group). The probabilities were 

 scaled to sum to one, and a uniform random number was 

 drawn to determine an individual's cohort. 



Next, individuals were assigned parameter values for 

 von Bertalanffy growth. The value of t was fixed at 0.5 

 yr. Values of L x and K were chosen uniquely for each in- 

 dividual. Following Xiao (1994), L x and if were assumed 

 to follow a bivariate normal distribution with standard 

 deviations a L and a A -, respectively, and correlation p. 



Finally, individuals were assigned a time step (month) 

 within the year to attempt spawning. The time step 

 was chosen from months distributed uniformly over a 

 spawning season of duration, Z) s . 



Once assigned parameter values, each individual was 

 duplicated. One copy entered the unfished population, 

 the other the fished population. The populations were 

 simulated in parallel over a single model year. 



The simulation iterated each individual through 

 monthly time steps. At each step, the simulation com- 

 puted growth and checked for survival and reproduc- 

 tion. In the unfished population, the monthly probability 

 of survival was exp(-M/12). In the fished population, 

 the monthly probability of survival during the fishing 

 season depended on natural mortality and on the size- 

 specific fishing mortality. For simplicity, we assumed 

 size within a month was fixed so that that the prob- 

 ability of survival was exp[(-M/12-s(/ )F)/D F ], where l 

 was an individual's size at the beginning of the month. 

 Outside the fishing season, only natural mortality ap- 

 plied. To check for survival, a uniform random number 

 was drawn and compared to the survival probability 

 appropriate for the population. 



Each individual surviving to its assigned spawning 

 time had the opportunity to reproduce. In that case, a 

 uniform random number was drawn and compared to 

 the probability of reproduction. If reproduction was suc- 

 cessful, the individual's growth parameters went into a 

 pool of parents used to compute selection differentials. 



Growth parameters L x and K jointly determine size- 

 at-age, and it is on these parameters that we describe 

 selection differentials. At the end of the simulation year, 

 we computed a selection differential on each growth 

 parameter as the percent difference between mean trait 

 values (L r or K) of the unfished and fished parents. 

 Based on the differences in L x and K, we also computed 

 upper and lower bounds of selection differentials on 

 size-at-age. The bounds occur where age approaches t 

 or oc. Because each population consisted of the same set 

 of individuals at the beginning of the year, any differ- 

 ence in growth traits between parents at the end of the 

 year could be attributed solely to fishing. 



Base model and variations 



We began with a base model built on parameter values 

 chosen or computed to represent common life-history and 



