size. The estimates of spawning stock biomass, MSY, and (to a lesser extent) fishing intensity are substantially more biased when the assessment does not consider sexual dimorphism. The ratios of current to unfished spawning stock biomass and to the spawning stock biomass corresponding to MSY were found to be the quantities estimated most robustly of those considered. © 2005 Elsevier B.V. All rights reserved.
Keywords: Sex-specific age-structured assessment method; Simulation; Evaluating model performance; Swordfish; North Pacific; Xiphias gladius
1. Introduction
Swordfish (Xiphias gladius, Linnaeus 1758) is a cos-mopolitan species found in tropical, subtropical, tem-perate, and sometimes cold waters of all oceans and adjacent seas (Nakamura, 1985). In the North Pacific
∗Corresponding author. Tel.: +886 2 23629842;
fax: +886 2 23629842.
E-mail address: chilu@ntu.edu.tw (C.-L. Sun).
Ocean (defined, for the purposes of this paper, to be the area of north of 10◦N and west of 130◦W;Fig. 1), the bulk of the swordfish catch has been taken by Japan, the United States and Taiwan, with very small quan-tities by Korea and China, whose swordfish catch is estimated to be less than 1% of total swordfish catch in the North Pacific (Anon, 2002).
Swordfish were targeted along with albacore in some areas of the North Pacific during 1952–1962 by the Japanese longline fleet when they were fishing at
0165-7836/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.fishres.2005.01.001
Fig. 1. The North Pacific Ocean showing the four regions considered in the analyses and the six “fleets” (four Japanese “fleets”—JPNW, JPSW, JPNE, JPSE and two US “fleets”—HINE, HISE).
night with squid as bait. Since then, this fishery changed to a primarily day operation using a mixture of bait types targeting bluefin tuna, yellowfin tuna and bigeye tuna for the high grade sashimi market. The Japanese fleets began “deep longlining” in 1974 to increase the catch-rate of bigeye tuna (Bartoo and Coan, 1989; Sak-agawa, 1989). The proportion of the total catch taken by Japan decreased from more than 95% in 1969 to about 75% during 1970–1990 because of these changes to fishing practices, the introduction in 1963 of Tai-wanese vessels into the fishery, and increased catches by the US in the late 1960s and early 1970s (Fig. 2). The proportion of the total catch taken by Japan declined
Fig. 2. Annual catch of swordfish from the North Pacific Ocean by Japan, the USA and Taiwan (1952–2001).
even further after 1990 (to about 45–65%) because of the rapid expansion of the Hawaii-based swordfish fisheries (Di Nardo and Kwok, 1998; Holts and Sosa-Nishizaki, 1998; Ito et al., 1998; Skillman, 1998; Anon, 2002).
Although not presently subject to international man-agement, effective management of this commercially important resource will require information on stock status relative to target and limit reference points (FAO, 1995). Common fisheries reference points in-clude maximum sustainable yield, MSY, the biomass at which MSY is achieved, SMSY, and the fishing
mor-tality corresponding to MSY, FMSY. The latter quantity
has been identified as a limit reference point in the U.N. Fish Stocks Agreement (United Nations, 1995).
Most previous assessments of swordfish in the North Pacific Ocean have been based on trends in catch-rates (i.e. catch-per-unit-effort, CPUE) (Yamanaka, 1958; Palko et al., 1981; Bartoo and Coan, 1989; Hanan et al., 1993; Sosa-Nishizaki and Shimizu, 1991; Di Nardo and Kwok, 1998; Kleiber and Bartoo, 1998; Hinton and Deriso, 1998; Nakano, 1998; Uosaki, 1998) and the re-sults of production model assessments (Anon, 1999), althoughKleiber and Yokawa (2002, 2004)attempted to apply MULTIFAN-CL (Fournier et al., 1998) to catch, effort and length–frequency data collected from Japanese and Hawaiian vessels. All of these analyses indicate that the swordfish stock in the North Pacific Ocean is not over-exploited and that it has been rela-tively stable at current levels of exploitation.
the length–frequency of the catches.
Although it is possible to apply ‘length-based syn-thesis’ to the data for swordfish in the North Pacific, it is not clear whether the results will be meaningful given the relatively small sample sizes for length–frequency. Although considerable effort has been directed towards evaluating the ability of production models and age-based methods of fisheries stock assessment to estimate quantities of interest to managers such as MSY, SMSY
and FMSY (e.g.Patterson and Kirkwood, 1995; Punt, 1997; Sampson and Yin, 1998), relatively little atten-tion has been focused on methods which rely primarily on length–frequency data (Quinn et al., 1998; Zheng et al., 1998; Sigler, 1999).
Therefore, prior to applying length-based synthe-sis to the data for North Pacific Ocean swordfish, this study evaluates the likely performance of this method of assessment using Monte Carlo simulation. This is because the performance of this method cannot be in-ferred from the results of previous simulation evalu-ations of estimator performance because the previous studies have not focused on assessment methods that fit to length–frequency data and because most previous studies have focused on highly depleted resources. In contrast, the bulk of the simulations of this paper re-late to a situation in which the resource is not highly depleted.
regular operations (5–9 hooks per basket), and deep operations (more than 10 hooks per basket)]. The data for the Hawaii-based longline fisheries are aggregated by 5◦× 5◦ block, year, month, and are also divided into three categories (less than 3 hooks per basket, 3–5 hooks per basket, and more than 5 hooks per basket).
The nominal catch-rates (defined as the number of fish per 1000 hooks) are standardized using a general linear model (GLM) approach (Gavaris, 1980; Kimura, 1981; Hilborn and Walters, 1992), with the assumption that the errors are log-normally distributed (seeWang (2004)for details).
Length–frequency data are available for the Japanese longline fisheries since 1975 and for the Hawaii-based longline fisheries since 1994. Informa-tion on the sex-structure of the catch by Japan is, how-ever, only available after 1984. The sample sizes for the JPNW, JPSW and JPSE fleets (seeFig. 1) are too small to construct reliable sex-specific length–frequency dis-tributions so the length–frequency data for these three fleets are aggregated over sex when treated as input to population models.
2.2. Evaluating performance using simulation
The steps considered when evaluating the perfor-mance of a stock assessment method using simulation
are:
1. Specify the population dynamics model for the true population (the operating model) and set the val-ues for some of its parameters based on auxiliary information.
2. Fit the operating model to the data for the situation in question (i.e. North Pacific swordfish) to obtain values for the remaining parameters of this model. The results of this model fit provide the ‘true’ values for the quantities of interest (listed below). 3. Generate artificial data sets by adding noise to the
predictions obtained by fitting the operating model to the data (step 2).
(1) Catch data: The simulated catch data (Cft) are generated by adding log-normally distributed observation error (νC; νC∼ N(0, σC2)) to the ‘true’ catch data:
Cft = Cft eνc−σ2c/2 (1)
(2) Catch-rate data: The simulated catch-rate data (Ift) are generated by adding log-normally dis-tributed observation error (νI;νI∼ N(0, σI2)) to the ‘true’ catch-rate data (see Eq.(B.11)):
Ift = ItfeνI−σ2I/2 (2)
(3) Length–frequency data: Following Chen (1996), the simulated catch length–frequency data for each year, sex and fleet (Ps,ft,l ) are gen-erated from the ‘true’ catch length–frequencies (Pt,ls,f) by sampling multinomially from the catches-at-length with a sample size of n. 4. The estimation model (taken here to be the same
as the operating model) is fit to the artificial data sets generated at step 3 to provide estimates of the quantities of interest.
5. Steps 3 and 4 are repeated 100 times—100 rep-etitions was sufficient to enable comparisons to be made among different length–frequency sample sizes and coefficients of variation for the catch and catch-rate data.
The sensitivity of the results to all combinations of three levels of error when quantifying the catch (σC: 0.01, 0.05 and 0.10), four levels of error about the rela-tionship between catch-rate and abundance (σI: 0.01, 0.1, 0.25 and 0.5) and five levels for the extent of sam-pling for length–frequency (sample sizes, n, of 10, 50,
100, 250 and 500 for each year, fleet and sex) is ex-amined. The values forσC,σIand n considered in the simulations are likely to bound the actual values of these parameters for North Pacific swordfish.
2.3. The population dynamics model
The population dynamics model that forms the ba-sis for the operating model and the estimation model is described in Appendix A. This model is sex- and age-structured and considers both sexes from age 0 to age 15 (age 15 being treated as a ‘plus group’). The model assumes that recruitment is related to spawning stock biomass according to a Beverton–Holt stock–recruitment relationship and that the deviations about this relationship are log-normally distributed. Se-lectivity is assumed to be a combination of logistic and dome-shaped components to adequately reflect the length-composition of the JPSE fleet. Different selec-tivity patterns are assumed for the various fleets (com-bination of logistic and dome-shaped for the JPSE fleet and logistic for the other fleets;Fig. 3). Owing to lack of data, the recruitment deviations prior to 1971 and those thereafter are treated differently. For the oper-ating model, the former are generated fromN(0; σv2) while the latter are estimated. For the estimation model, however, the recruitment deviations for the years prior to 1971 are all set to zero because there are no data which could inform year-class strength for these years.
2.4. Parameter estimation
The parameters of the model can be divided into those for which auxiliary information is available (Tables 1 and 2) and those which need to be es-timated from the monitoring data (Table 3). The data available for assessment purposes are: (1) the catches (assumed known without error), (2) the annual length–frequencies by sex and fleet (pooled across sex in some instances), (3) the sex-ratio data by fleet (the ratio of the catch (in numbers) of animals sexed to be females to the total catch of males and females), and (4) the catch-rate-based indices of abundance. The ob-jective function minimized1 to find the estimates of
1 Using AD Model Builder (version 5.0.2) (Otter Research Ltd., 2000).
Fig. 3. “True” age-specific selectivity ogives for the six fleets (solid lines: female; dashed lines: male).
the ‘free’ parameters of the model includes two com-ponents (the data available for assessment purposes and the constraints based on a priori assumptions) (Appendix B).
The values for the parameters of the length–weight relationship, growth and maturity are obtained from Sun et al. (2002) and Wang et al. (2003)
(Tables 1 and 2). However, the values for the param-eters related to natural mortality (M), the steepness of
Table 1
The values assumed for the parameters of the relationships between length and weight, length and age, and maturity and length
Parameter Females Males
Asymptotic size,Ls∞(cm) 300.656 213.052 Growth parameter, ks (year−1) 0.04 0.086 Age-at-zero-length,as0(year) −0.75 −0.626 Shape parameter, ms −0.785 −0.768 Length–weight, As 1.3528 × 10−6 1.3528 × 10−6 Length–weight, Bs 3.4297 3.4297 Length-at-50%-maturity, Lm (cm) 168.16 – Maturity slope, rm −0.1392 –
Maximum age,λ (year) 15 15
the stock–recruitment relationship (h), and the extent of variation in recruitment (σv) cannot be determined from auxiliary information, nor can they be estimated reliably by fitting the model to the data (results not shown) and must therefore be pre-specified. In this study, M is taken to be 0.25 year−1 based on Pauly’s empirical equation (Pauly, 1980), h is assumed to be 0.9 (Anon, 1997; Punt et al., 2001), andσvis assumed to be 0.4 (Punt et al., 2001). The impact of the assess-ment model making erroneous assumptions regarding
Table 2
The values assumed for the standard deviation of the length-at-age
Age Female Male
0 25.00 25.00 1 18.63 15.92 2 10.79 9.59 3 10.27 10.82 4 12.26 10.28 5 13.39 11.46 6 9.79 11.53 7 14.41 13.94 8 12.64 11.47 9 15.45 11.21 ≥10 10.00 10.00
Table 3
The parameters of the population dynamics model not known from auxiliary information (the number of years was 49 and number of fleets was 6)
Parameter No. of parameters
Estimated
Recruitment at unfished equilibrium, R0 1
Process error,vt 1 per year
Selectivity
Length-at-50%-selectivity,Lf50 1 per fleet Length-at-95%-selectivity,Lf95 1 per fleet Length-at-modal-selectivity,Lfmu JPSE fleet only Standard deviation of selectivity,Lfsd JPSE fleet only Weight assigned to dome-shaped ogive,ϕf JPSE fleet only
Pre-specified
Natural mortality, M 1
Steepness, h 1
Variation in recruitment,σv 1
the values for these parameters is examined in the tests of sensitivity.
2.5. Summarizing estimator performance
The operating model includes a large number of out-puts. In this study, focus is placed on the ability to es-timate the following key outputs:
(1) S0, the spawning stock biomass at unfished
equi-librium;
(2) S2000, the spawning stock biomass at the start of
the last year of the assessment period;
(3) F2000, the fleet-aggregated fishing intensity (see
Section 7 of Appendix A.7for the definition of this quantity) during the last year of the assessment period;
(4) MSY, the maximum sustainable yield (see
Shepherd and Pope (2002)for details on how MSY is calculated given the estimates for growth, matu-rity and the stock–recruitment relationship); (5) SMSY, the spawning stock biomass at which MSY
is achieved;
(6) FMSY, the fleet-aggregated fishing intensity at
which MSY is achieved;
(7) S2000/S0, the ratio of the spawning stock biomass
at the start of the last year of the assessment period to S0;
(8) S2000/SMSY, the ratio of the spawning stock
biomass at the start of the last year of the assess-ment period to that at which MSY is achieved;
(9) F2000/FMSY, the ratio of F2000 to the
fleet-aggregated fishing intensity at which MSY is achieved.
Quantities (1) and (2) relate to the ability to esti-mate absolute population size (both at present and in the past), while quantity (3) evaluates the ability to estimate the current exploitation rate. A fleet-aggregated mea-sure of exploitation rate is used in quantity (3) to reduce the volume of results. Quantities (4)–(6) relate to the ability to estimate the yield, spawning stock biomass and exploitation rate at which MSY is achieved. Unlike quantities (1)–(6), quantities (7)–(9) relate to relative measures of spawning stock biomass and exploitation rate. Past evaluations of the performance of stock as-sessment methods (e.g.Punt et al., 2002) suggest that relative measures should be estimated better than ab-solute measures.
Estimation ability is quantified for each Monte Carlo replicate and quantity of interest by the relative error:
Eji = Qˆji − Qj
Qj × 100 (3)
whereEji is the relative error (%) for simulation i and quantity j, ˆQji the value for simulation i and quantity
j provided by the estimation model, and Qj the ‘true’ value for quantity j.
The distributions of median relative errors for each scenario and quantity of interest are summarized by the median relative error (MRE) and inter-quartile range (IQR).
3. Results
3.1. Impacts of individual factors
The first set of results is based on simulations in which only one source of data (catches, catch-rates and length–frequency) is subject to error.
3.1.1. Impact of errors when measuring catch
Fig. 4 shows the distributions for the relative er-rors for the nine quantities of management interest for each of the three levels of observation error for the catch data. The sizes of the relative errors increase with extent of error in catch. As expected, the estimates
Fig. 4. Box plots of the relative errors for the quantities of management interest corresponding to various levels of observation error for the catch data. The centerline represents the median and the box represents the quartiles. The whiskers extend 1.5 times the inter-quartile range.
are almost identical to the true values whenσC= 0.01 (MREs between−1.1 and 3.1% and IQRs between 1.4 and 4.9% for all quantities). Estimation error remains good (MREs between −0.5 and 2.8% and IQRs be-tween 1.4 and 6.0%) for a more realistic extent of error in the catches (σC= 0.05). The ability to determine the quantities of management interest deteriorates some-what whenσC= 0.1. The estimates of relative biomass (i.e. S2000/S0and S2000/SMSY) are close to being
unbi-ased (MREs of 1.6 and 0.3%, respectively). However, noteworthy bias and imprecision is evident for some of the other quantities whenσC= 0.1. For example, FMSY
is positively biased (MRE = 3.2%) while F2000/FMSY
is imprecise (IQR = 9.2%).
The results inFig. 4 are based on the assumption that the estimates of catch are unbiased. Given the lack of management actions for swordfish in the North Pa-cific, there is no obvious reason why the catches by the nations considered in the assessment should be bi-ased. However, catches by nations other than the Japan
and the US may lead to the catches used in the as-sessment being negatively biased estimates of the total removals. The impact of a negative bias of 10% in total catch (over all years) was examined and found to have little impact on the estimates of S2000/S0, S2000/SMSY,
and F2000/FMSY(MREs of 0.3, 0.4 and 1.0%,
respec-tively). In contrast, the estimates of S0, S2000, SMSYand
MSY were negatively biased (MREs∼ −9.0%) when catches were negatively biased and those of F2000and
FMSYwere positively biased (MREs between 5.3 and
6.3%).
3.1.2. Impact of errors in the catch-rate data
Fig. 5shows the distributions for the relative errors for the nine quantities of management interest for each of the four levels of observation error for the catch-rate data. The impact of a low level of observation error in the catch-rate data (σI= 0.01) is minor, but larger than the lowest level of variation in catches (MREs between
Fig. 5. Box plots of the relative errors for the quantities of management interest corresponding to various levels of observation error associated with the catch-rate data. The centerline represents the median and the box represents the quartiles. The whiskers extend 1.5 times the inter-quartile range.
all quantities). The accuracy and precision of the es-timates deteriorates when σI is increased from 0.01 to 0.1 (the MREs now lie between −5.1 and 12.5% and IQRs between 6.7 and 30.6%) and the pattern of greater bias and lower precision with higher values for
σIis clearly evident inFig. 5. As a consequence, the estimates can be very biased (MREs between−17.0 and 29.3%) and highly imprecise (IQRs between 11.4 and 118.5%) for the highest level of observation error considered inFig. 5. It is perhaps of some concern that the relative measures are biased when there is large un-certainty about the relationship between catch-rate and abundance. However, the extent of bias is less for the ratios than for their constituent parts (e.g. F2000/FMSY
is less biased than either F2000 or FMSY) confirming
the expectation that the estimates of ratios of quanti-ties are more reliable than the estimates of absolute values.
3.1.3. Impact of errors in the length–frequency data
The estimates of the management quantities except
S2000/S0 and S2000/SMSY are biased (MREs between −11.9 and 116.4%) when the length–frequency
sam-ple size is quite small (n = 10). However, the MREs are close to zero (−2.0–3.8%) and IQRs are small (2.5–11.3%) for all quantities for sample sizes larger than 50 (Fig. 6). There is little discernable change in performance once the sample size reaches 100.
3.2. Impact of factors in combination
Distributions of relative error are shown for three quantities of interest for three levels for the extent of ob-servation error for the catch-rate data (σI= 0.1, 0.25 and 0.5) and three length–frequency sample sizes (n = 50, 100 and 250) inFig. 7. The extent of observation error
Fig. 6. Box plots of the relative errors for the quantities of management interest corresponding to various sample sizes for the length–frequency data. The centerline represents the median and the box represents the quartiles. The whiskers extend 1.5 times the inter-quartile range.
for the catch data is the same for all of the simula-tions in Fig. 7 (σC= 0.1). Fig. 7 is restricted to the three quantities which performed ‘best’ for the sim-ulations in which only one data source is subject to error. Relative biomass is generally determined better than relative exploitation rate (MREs between−14.1 and−11.6% (n = 50 and 100) and of −3.7 and −4.8% (n = 250) for relative biomass and of 31.0, 35.8, and 7.8% (n = 50, 100 and 250, respectively) for relative ex-ploitation rate). As before, less catch-rate observation error and larger length–frequency sample sizes lead to improved performance, although the impact of, for ex-ample, increasing the sample size from 100 to 250 is much more substantial inFig. 7than was the case in
Fig. 6.
3.3. Impact of fixing parameters to the wrong values
The results inFigs. 4–7are based on the assump-tion that natural mortality, stock–recruitment steep-ness, and the extent of variation in recruitment are known exactly. This is clearly unrealistic so Fig. 8
explores the implications of the estimation model as-suming incorrect values for these parameters (M = 0.2 and 0.3 year−1; h = 0.6, 0.8 and 0.95;σv= 0.2 and 0.6). The values for these parameters in the operating model were set equal to their base values (i.e. M = 0.25, h = 0.9,
σv= 0.4). Results are shown inFig. 8for the same three quantities as inFig. 7and forσC= 0.1,σI= 0.25, and
Fig. 7. Box plots of the relative errors for three quantities of management interest for various length–frequency sample sizes and catch-rate observation error standard deviations. The centerline represents the median and the box represents the quartiles. The whiskers extend 1.5 times the inter-quartile range.
The results inFig. 8are somewhat counter-intuitive because the best performance does not occur when the values for h, M and σv assumed when conduct-ing the assessment equal their ‘true’ values. For ex-ample, the estimates were more accurate (less biased) when M = 0.3 year−1 and σv= 0.2. The results for h show no consistent pattern; setting h = 0.6 leads to more biased estimates of S2000/SMSY and F2000/FMSY but
less biased estimates of S2000/S0. The impact of the
values assumed for h and M on bias substantially ex-ceeds that of the impact of the value assumed forσv. The sensitivity of the results to changing the values for the weights assigned to the various data sources (seeAppendix B) was explored, but found to be fairly minor.
3.4. Impact of sex factor of the model
The estimation model considered in Figs. 4–8 is based on a sex-structured population dynamics model. A set of simulations were therefore conducted in which the estimation model was based on sex-aggregated rather than sex-specific growth curves (length-at-age was set equal to the average of that for males and fe-males). The values for the parameters M, h andσvwere set to their base values (i.e. assumed to be known ex-actly) while the data were generated based onσC= 0.1,
σI= 0.25, and n = 100.Fig. 9compares the relative er-ror distributions for all nine management-related quan-tities for the sex-specific and sex-aggregated estima-tion models. The estimates of S0, S2000, MSY and
Fig. 8. Box plots of the relative errors for three quantities of management interest for analyses in which the values for natural mortality, stock–recruitment steepness, and the extent of variation in recruitment differ from those in operating model. The centerline represents the median and the box represents the quartiles. The whiskers extend 1.5 times the inter-quartile range.σc= 0.1,σI= 0.25, and n = 100 for all of the analyses in this figure.
SMSYare substantially more biased (MREs of−70.2, −80.7, −53.8 and −75.5%, respectively) when based
on the sex-aggregated estimation model. The estimates of F2000, FMSYand S2000/SMSYare also more biased
(and S2000/SMSYless biased) when sexual dimorphism
is ignored when conducting the assessment.
4. Discussion
4.1. Impact of the factors considered
Of the three data sources considered (catches, catch-rates, and length–frequency), uncertainty about the re-lationship between catch-rate and abundance has the greatest impact on performance. This is reflected not only in reduced precision but also in increased bias (Fig. 5). Although the biases are relatively small for
σI= 0.1, they can be fairly substantial (e.g.∼30% for
F2000) whenσIis larger than 0.25. Fitting the age- and
sex-structured model to the catch-rate indices for North Pacific swordfish suggests thatσImay, in fact, be 0.1 or less (Wang, 2004).
For swordfish in the North Pacific, catches should be known (at least for recent years) with a coeffi-cient of variation of 10%. The impact of increased length–frequency sample sizes is fairly small when there is no uncertainty associated with the catches or catch-rates. In contrast, larger (effective sample sizes > 250) sample sizes do improve estimation per-formance when there are several sources of uncer-tainty (Fig. 7). Unfortunately, the range of effective sample sizes inferred by fitting the age- and sex-structured model to the data for North Pacific sword-fish is 1–513 (average 83). This suggests that the actual length–frequency sample sizes are not quite large enough for some fleets and there may be value in increasing the number of fish measured to improve
Fig. 9. Box plots of the relative errors for the quantities of management interest for the sex-specific and sex-aggregated estimation models. The centerline represents the median and the box represents the quartiles. The whiskers extend 1.5 times the inter-quartile range.σc= 0.1,σI= 0.25, and n = 100 for all of the analyses in this figure.
the accuracy and imprecision of the estimates from the model.
The results are clearly sensitive to the values as-sumed for M and h (Fig. 8). Moreover, the best per-formance did not occur when the values assumed for these parameters were the true values. This is surpris-ing, but reflects the complicated nature of the estima-tion method. In principle, estimates for M, h and σv could be obtained from research (such as tagging stud-ies to determine M—Hampton, 2000). However, in the short- to medium-term, it seems likely that the values for these parameters will have to be based on the results of studies for other stocks of swordfish. Unfortunately, the data for all swordfish stocks are sparse (Punt et al., 2001) and it may be inappropriate anyway to infer the values for the biological parameters for a stock in one ocean basin from those for stocks in other ocean basins.
The estimates of spawning stock biomass, MSY, and (to a lesser extent) fishing intensity are substan-tially more biased when the assessment ignores sex-ual dimorphism (Fig. 9). Therefore, if control rules based on, for example, FMSYor MSY are to be used
to manage swordfish in the North Pacific Ocean, an assessment approach which allows for sexual dimor-phism (such as that outlined inAppendices A and B) needs to be applied to reduce uncertainty. It is per-haps somewhat reassuring from a conservation view-point that the estimates of the biomass-related man-agement quantities are negatively biased while those of the fishing mortality-related quantities are pos-itively biased when the assessment model ignores sex-structure (Fig. 9). However, the extent to which this result is general and applies to other stocks of large pelagic fish is unclear, although this could be
determined (Figs. 7–9) even though it is a ratio. The reasons for this are unclear, but may relate to the greater dependence of fishing intensity on selectivity (see Eqs.
(A.15) and (A.16)).
4.3. General discussion
The impact of noisy catch-rate data can be off-set by larger length–frequency sample sizes (Fig. 7). There is a trade-off between cost and precision when deciding on samples sizes for length–frequency and age–composition data (e.g. Chen, 1996). Neverthe-less, attempts should be made to develop a sampling program that aims to achieve a random sample of at least 250 animals from the catch of each sex for each fleet. However, about 60% of the effective sam-ple sizes computed by fitting the model to the actual length–frequency data for North Pacific swordfish are not only less than 250 but also less than 83 (the aver-age effective sample size). It is not possible at present to relate effective sample size to the actual number of fish measured. However, given the non-random nature of fishing operations, collecting 250 animals randomly from the catch (e.g. using observers) could necessitate the collection of a large number of length measure-ments for some fleets. The level of 250 animals as be-ing sufficient should, however, not to be taken as an
model, although it is computationally highly intensive. To date, applications of this integrated approach have been restricted to age-structured and biomass dynamics models. Extension of the approach to models based on length–frequency data is needed although testing such an assessment approach by means of Monte Carlo sim-ulation may be computationally prohibitive at present.
Kleiber and Yokawa (2002, 2004)attempted to as-sess the stock of swordfish in the North Pacific Ocean using MULTIFAN-CL. However, the results of this study suggest that estimates of some of the quantities of interest to management (including estimates of cur-rent and historical spawning stock biomass) will be substantially biased and that an assessment approach takes sexual dimorphism into account should lead to less biased estimates of these quantities.
Acknowledgements
We thank Drs. Yuji Uozumi and Kotaro Yokawa of the National Research Institute of Far Seas Fish-eries, Shimizu, Japan, and Dr. Pierre Kleiber of the Pacific Islands Fisheries Science Center, NMFS, Hon-olulu, Hawaii, USA, for providing the Japanese and the Hawaii-based longline swordfish catch, effort and length–frequency data, respectively. We also thank two
anonymous reviewers for their comments on an ear-lier version of this manuscript. This study was in part supported financially by the National Science Coun-cil, Taiwan through grant NSC91-2313-B-002-330 to Chi-Lu Sun.
Appendix A. Population dynamics model
A.1. Basic sex-specific population dynamics
The population dynamics are governed by an age-and sex-structured model with an annual time-step. Natural mortality occurs continuously throughout the year and the fishery occurs instantaneously in the mid-dle of the year, i.e.:
Ns t,a= Rs t ifa = 0 (Nt−1,a−1s e−M/2− Ct−1,a−1s ) e−M/2 if 0< a < λ [(Nt−1,λ−1s + Nt−1,λs ) e−M/2− (Ct−1,λ−1s + Ct−1,λs )] e−M/2 ifa = λ (A.1)
whereNt,as is the number of fish of age a and sex s (m/f) at the start of year t,Rstthe recruitment (at age 0) of fish of sex s at the start of year t,Ct,as the catch in number of fish of age a and sex s during year t:
Cs
t,a=
f
Cs,ft,a (A.2)
Cs,ft,a is the catch in number of fish of age a and sex s by fleet f during year t, M the instantaneous rate of natural mortality on fish of both sexes, andλ the maximum age (treated as a plus-group).
A.2. Catches
The number of fish of age a and sex s caught by fleet
f during year t can be calculated using the equation, i.e.:
Cs,ft,a = Ft,as,fNt,as e−M/2 (A.3)
whereFt,as,f is the exploitation rate on fish of age a and sex s during year t by fleet f:
Ft,as,f = ss,fa Ftf (A.4)
Ftf is the exploitation rate by fleet f during year t on fully selected animals:
Ftf = Cf,obst
assas,fNt,as e−M/2
(A.5)
Ctf,obsis the observed total catch (in number) by fleet f
during year t, andss,fa the selectivity for fleet f on fish of age a and sex s.
A.3. Growth and maturity
The relationship between length and weight is given by the equation:
wsa= As(Lsa)Bs (A.6)
where As, Bsare the parameters of the length–weight relationship for animals of sex s, andLsathe expected length of an animal of sex s and age a:
Ls
a= Ls∞(1− e−ks(1−ms)(a−as0))1/(1−m
s)
(A.7)
Ls
∞, ks, as0, ms are the parameters of the generalized
von Bertalanffy growth equation.
Maturity as a function of age is modeled by means of a logistic curve, i.e.:
φa= 1
1+ exp[rm(Lfa− Lm)]
(A.8)
whereφais the fraction of females of age a that are ma-ture, Lm the length-at-50%-maturity for females, and
rmthe maturity slope parameter.
A.4. Recruitment
Recruitment (number of animals of age 0) is assumed to be related to the spawning stock biomass by means of the Beverton and Holt (1957)
stock–recruitment relationship, parameterized in terms of the ‘steepness’ of the stock–recruitment relation-ship, h (Francis, 1992) and the recruitment at unfished
a general functional form which combines a logistic and a dome-shaped component:
ss,fa = s 1,s,f
a + ϕfs2a,s,f
maxa(sa1,s,f+ ϕfs2a,s,f )
(A.11)
wheres1a,s,f is the logistic component:
s1,s,f a = 1+ exp −ln 19Lsa− L f 50 Lf95− Lf50 −1 (A.12)
Lf50 is the length-at-50%-selectivity for fleet f for the
logistic component of the selectivity ogive, Lf95 the length-at-95%-selectivity for fleet f for the logistic component of the selectivity ogive, s2a,s,f the dome-shaped component: s2,s,f a = 1 √ 2πLfsd exp −(Lsa− Lfmu) 2 2(Lfsd)2 (A.13)
Lfmuis the mode of the dome-shaped component of the
selectivity ogive for fleet f,Lfsdthe standard deviation of the dome-shaped component of the selectivity ogive for fleet f, andϕf, for fleet f, the weight assigned to the dome-shaped component of the selectivity function.
Ft =
fCtf,obs
asssaNt,as e−Ms/2 (A.15) Ftis the fleet-aggregated fishing intensity during year t,
andssathe ‘fleet-aggregated’ selectivity for fish of age a and sex s, calculated by weighting the fleet-specific se-lectivity patterns by the fleet-specific exploitation rates:
sas = ftFtf ss,fa ftFtf (A.16)
where the summations over year relate to the last 11 years (1990–2000).
Appendix B. Contributions to the objective function
B.1. The sex-ratio data
The sex-ratio data are included in the likelihood function by assuming that the observed estimates of the sex-ratio of the catch are normally distributed about the model predictions. The contribution of the sex-ratio data to the negative of the logarithm of the likelihood function (ignoring constants independent of the model
parameters) is therefore: L2= t f ρs(Qf,obst − Qft )2 (B.1)
whereQf,obst is the observed sex-ratio for fleet f during year t: Qf,obst = C f,f,obs t 2 s=1Cs,f,obst (B.2)
Cs,f,obst is the observed catch of fish of sex s by fleet
f during year t,Qft the model-predicted sex-ratio for fleet f during year t:
Qft = aCft,a,f a2s=1Cs,ft,a (B.3)
ρs is the weight assigned to the sex-ratio data (set to
0.1).
B.2. The length–frequency data
The likelihood function assumed for the length– frequency data is the robust normal formulation of
Fournier et al. (1990), adjusted so that the variance of the predictions is based on the observed rather than the model-predicted fractions (B. Ernst, Univ. Washing-ton, pers. commun.). Sex-specific and sex-aggregated length–frequency data are available. Denoting the sex-aggregated data (i.e. the length–frequency data for the fleets JPNW, JPSW and JPNE for all years and for the fleet JPSE before 1987) as sex s = 0, the contribution of the length–frequency data to the negative of the log-likelihood function (ignoring constants independent of the model parameters) is:
−ln L3= 0.5 t l 2 s=0 f ρ f t (P s,f,obs t,l − Pt,ls,f) 2 2 Pt,ls,f,obs[1− Pt,ls,f,obs]+N1 l (τts,f)−2 + 0.01 (B.4)
where Pt,ls,f,obs is the observed fraction that fish in length-class l made up of the catch (in numbers) by fleet f of animals of sex s during year t,Pt,ls,f the model-estimate of the fraction that fish in length-class l made up of the catch (in numbers) by fleet f of animals of sex
s during year t: Pt,ls,f = C s,f t,l lCs,ft,l (B.5)
Ct,ls,f is the model-estimate of the fraction that fish in length-class l made up of the catch (in numbers) by fleet f of animals of sex s during year t:
Ct,ls,f = λ a=1Λ s a,lCt,as,f ifs = m/f s λ a=1
Λsa,lCs,ft,a otherwise
(B.6)
Nlis the number of length-classes,τts,f the ‘effective’
sample size for year t, fleet f and sex s (the maximum of the number of animals measured and 50),Λsa,l the fraction of animals of age a and sex s that are in length-class l: Λs a,l= Ll+(L Ll−(L 1 √ 2πσsaexp −(L − Lsa)2 2(σas)2 dL (B.7) σs
ais the standard deviation of the length of a fish of
sex s and age a (Table 2), Llthe mid-point of
length-class l, (L the width of half of a length-class (the length–frequency data were aggregated to 5 cm length-classes owing to the way the original data were col-lected),ρft the weight assigned to the length–frequency data for year t and fleet f:
ρft = ρL Ctf,obs
tCf,obst
/nf (B.8)
ρLis the overall (pre-specified) weight assigned to the
length–frequency data (set to 0.01), and nfthe number of years for which length–frequency data are available.
fleet f: Bft = s λ a=1 ss,fa Nt,as e−M/2− Cs,ft,a/2 (B.11) ˜
σfis the standard deviation ofIf
t , assumed to be
time-invariant.
B.4. Constraint contribution to the objective function
A constraint is placed on the deviations about the stock–recruitment relationship: 1 2σ2 t ν2 t (B.12) References
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