Restrepo et al : Monte Carlo simulation applied to Xiphias gladius and Cadus morhua 



741 



implies that annual adjustments in the quota will be 

 proscribed even when no changes are in fact necessary. 

 Instead of letting the quota "float" from year to year, 

 one can stabilize the quota and let the risks float from 

 year to year. Thus, as long as the risks remain within 

 certain limits, there is no need to adjust the quota. 

 (Here, the risks can include potential stock collapse as 

 well as foregone potential yield.) 



The sequential population 

 analysis model: ADAPT 



The examples presented below use data from two very 

 different fisheries that are assessed with the same SPA 

 approach, known as ADAPT (Gavaris 1988). ADAPT 

 is widely used in the eastern United States and Canada. 

 Here we describe the basic method briefly and direct 

 the interested reader to more details in Parrack (1986), 

 Gavaris (1988), Conser and Powers (1990), and Powers 

 and Restrepo (1992). 



The objective in ADAPT is to minimize deviations 

 between observed (age-specific) indices of abundance 

 and those predicted by what is commonly referred to 

 as virtual population analysis (VPA). Let the subscripts 

 t, a, and i denote time, age, and abundance-index 

 sequence number, respectively. The basic equations 

 governing the model are 



Nat = Na,i,t,, ez«, (1) 



Cat = Fat Na^i.t.i (eZa,-i)/Z,t, and (2) 



I.t = q, Na, (l-e-Z3,)/Zat, (3) 



where N = stock size in numbers of fish, C = catch 

 in numbers, I = index of relative stock abundance (each 

 index is associated with one or more ages which must 

 be specified), F = instantaneous fishing mortality rate, 

 Z = total instantaneous mortality rate (Z = F -i- M), and 

 q = coefficient of proportionality between relative 

 abundance and absolute abundance. Inputs to the 

 model are the catch, natural mortality, and relative 

 abundance indices. Given that there are T years of data, 

 A ages, and Y indices, a search algorithm, e.g., Mar- 

 quardt-Levenberg (Seber and Wild 1989), is used to 

 estimate the parameters q^ (i = 1 . . . Y) and Na, t + 1 (a = 

 2 ... A) that minimize the weighted residual sum of 

 squares: 



RSS = min2:.I.A,(I,-i;)2, (4) 



where the weights, Aj, may be input or estimated via 



iteratively-reweighted least squares. 



ADAPT, like other VPA calibration procedures, re- 

 quires model constraints in order to reduce the number 

 of parameters. Hence, the stock sizes for the last age 

 each year are not normally estimated but are instead 

 derived from a specified relationship between Fai and 

 Fa-i, f Additional constraints may be required when 

 the amount of relative-abundance data does not sup- 

 port the estimation of a large number of parameters. 

 Often, as in the swordfish example below, this involves 

 estimating the relative selectivities of the various age- 

 groups in year T in some fashion external to the calibra- 

 tion process. This leads us to add to our explanation 

 the notion that ADAPT is generally thought of as a 

 framework rather than a rigid model. Thus the reader 

 is likely to encounter applications that deviate from the 

 model in equations (1) through (3). For example, for 

 swordfish, A is a "plus" group consisting of ages A and 

 older. For cod in Atlantic Canada, the objective func- 

 tion (4) is modified to allow for lognormal errors. A 

 detailed presentation of some of the most commonly 

 used options in ADAPT can be found in Powers and 

 Restrepo (1992). 



Assessment uncertainty: Application 

 to North Atlantic swordfish 



Swordfish in the North Atlantic Ocean are assessed by 

 the International Commission for the Conservation of 

 Atlantic Tunas (ICCAT). Interest is centered on the 

 level of fishing mortality relative to reference values 

 (e.g., Fmax)i and on trends in mortality and stock 

 abundance. Potential management options involve 

 restrictions aimed at controlling fishing mortality. The 

 assessment procedure is continually changing as ex- 

 perience is gained. The procedure below was used for 

 the 1989 assessment (ICCAT 1990). 



Assessment procedure 



Nine age-groups were recognized in the commercial 

 catch, ages 1 to 9-1- . There were 11 years of catch-at- 

 age data from 1978 to 1988. Fleets from the United 

 States, Japan, and Spain accounted for most of the 

 catch. Eleven abundance indices were available based 

 on fleet-specific catch rates from the longline fisheries 

 (ICCAT 1990). 



Details of this assessment of the stock are presented 

 in ICCAT (1990). Briefly, the procedure used was as 

 follows. (1) A separable virtual population analysis, 

 SVPA (Pope and Shepherd 1982), was computed in 

 order to obtain estimates of the age-effects or partial 

 recruitment in the last year for which data were avail- 

 able. Data from 1983 to 1988 were used for this under 



