Jacobson et al.: A biomass-based assessment model for Engraulis mordax 



715 



Population dynamics 



Fishing seasons were used as annual time steps, and 

 ages to 4+ were included (age group 4+ includes 

 northern anchovy age 4 and older). Fish were aged 

 in the model at the beginning of each fishing season 

 on 1 July when recruitment of age-0 northern an- 

 chovy was assumed to occur (Methot, 1989). In real- 

 ity, some recruitment of northern anchovy occurs 

 throughout the year (MacCall and Prager, 1988). 

 Therefore, our estimates of recruitment should be 

 regarded as estimates of "effective" recruitment, i.e. 

 biomass of age-0 fish that would have been neces- 

 sary on 1 July to account for the biomass of the co- 

 hort in later years. 



Numbers of northern anchovy were not included 

 in SMPAR; abundance was measured solely in units 

 of biomass because weight at age for northern an- 

 chovy changes rapidly throughout the year, and de- 

 pends on where samples are taken (Parrish et al., 

 1985). In addition, weight-at-age data from commer- 

 cial fisheries for northern anchovy were not avail- 

 able for recent fishing seasons. 



Biomass dynamics were modeled as 



B. 



H - fi o., 



(5) 



where B av is the biomass of northern anchovy age a 

 (a>0, i.e. excluding new recruits) at the beginning of 

 fishing season y and n is the net instantaneous rate 

 of change for northern anchovy in fishing season y. 

 Random process errors (e.g. variation in growth and 

 natural mortality, Hilborn and Walters, 1992) were 

 captured in the model by recruitment estimates. 



For modeling purposes, recruitment of northern 

 anchovy in each year was assumed independent of 

 spawning stock size: 



B n 



By 



(6) 



where B Q is recruitment ( biomass age-0 fish ) in fish- 

 ing seasony, B Q is mean recruitment during the study 

 period, and 8 y is a log-normally distributed error term 

 for fishing season y with mean zero and standard 

 deviation o. Recruitments in each fishing season 

 (S 0y ) were treated as parameters and estimated by 

 the model. 



The net instantaneous rate of change for northern 

 anchovy biomass in each fishing season (r; in Eqn. 

 5) is the sum of rates for fishing mortality growth, 

 and natural mortality: 



n v =F v + M-G, 



(7) 



where F y is the fishing mortality rate in fishing sea- 

 son y, M is the natural mortality rate, and G is the 



growth rate. All rates are defined as positive values. 

 The fishing mortality rate for each fishing season 

 (F y ) was assumed constant over ages but variable 

 over time, whereas rates for natural mortality (M) 

 and growth (G) were assumed constant over ages and 

 time. Fishing mortality rates were calculated by us- 

 ing the "forward solution" algorithm in Sims (1982) 

 and actual catch data (Table 1; Fig. 1). 



The rate of natural mortality (M) for northern an- 

 chovy was assumed to be 0.8 yr _1 , which is reason- 

 able for a fish that seldom exceeds seven years in 

 age (Hoenig, 1983). Methot ( 1989) found that differ- 

 ent levels of natural mortality had only modest ef- 

 fects on biomass estimates for northern anchovy be- 

 cause the estimates were anchored by DEP spawn- 

 ing biomass measurements. 



Modeling growth as an instantaneous rate (G) is 

 appropriate for northern anchovy because fish grow 

 rapidly throughout the fishing season (Zhang and 

 Sullivan, 1988). By treating growth as an instanta- 

 neous rate, northern anchovy are, in effect, allowed 

 to continue growing in the model until they are 

 caught. 



The rate for growth used in the SMPAR model for 

 northern anchovy (G=0. 198 yr" 1 , SE=0.0166) was es- 

 timated by fitting an exponential growth model to 

 mean weight at age data from three sources (Methot, 

 1989). The exponential growth model was logarith- 

 mically transformed to give 



\n(W da ) = \n(W d0 ) + aG, 



(8) 



where W da is the mean weight of northern anchovy 

 age a in data set d, and W d is the estimated weight 

 at age 0. The approach assumes that northern an- 

 chovy may differ in initial weight as measured by 

 the W (l0 parameters but experience the same rate of 

 exponential growth (G). Parameter estimates for 

 Equation 8 were obtained by linear regression and 

 standard general linear model techniques (Weisberg, 

 1980). Residuals were dome-shaped because of the 

 linear approximation to the asymptotic growth pat- 

 tern but the linear regression model explained most 

 of the variation in log-scale size at age (R 2 =93%). 



Abundance data 



Abundance data (EPI, HEP, SONAR, DEP, and 

 SPOTTER abundance indices) were assumed to be 

 measured with log-normally distributed random er- 

 rors. Predicted values for abundance data during 

 each fishing season were calculated in the model as: 



l,y=Q t ^P t , a B ay e-"', 



(9) 



