Application of the sex-specific age-structured assessment method
for swordfish, Xiphias gladius, in the North Pacific Ocean
Sheng-Ping Wang
a,1, Chi-Lu Sun
a,∗, Andr´e E. Punt
b, Su-Zan Yeh
a aInstitute of Oceanography, National Taiwan University, Taipei 106, TaiwanbSchool of Aquatic and Fishery Sciences, University of Washington, Seattle, WA 98195, United States Received 29 May 2006; received in revised form 28 October 2006; accepted 3 November 2006
Abstract
Swordfish are known to be sexually dimorphic. However, previous assessments of the status of swordfish in the North Pacific Ocean have ignored this. A sex-specific age-structured assessment model was therefore constructed and fitted to catch, catch-rate and length–frequency data for the swordfish fisheries that operate in the North Pacific Ocean. Except if natural mortality is lower than its “best” estimate, the results indicate that the spawning stock biomass in 2002 was at a high fraction of its unfished level and that the fishing intensity in 2002 was less than FMSY. Therefore, the
swordfish stock in the North Pacific Ocean appears to be relatively stable at the current level of exploitation. However, the results of the assessment model are sensitive to the values for natural mortality and the steepness of the stock-recruitment relationship. Forward projections based on samples from a Bayesian posterior distribution indicate that there is negligible risk of the stock dropping below 40% of the unfished spawning stock biomass if fleet-aggregated fishing intensity remains at the current level. However, the risk of population depletion is higher if natural mortality is lower than the “best” estimate.
© 2006 Elsevier B.V. All rights reserved.
Keywords: Sex-specific age-structured assessment method; Stock assessment; Swordfish; North Pacific
1. Introduction
Swordfish (Xiphias gladius, Linnaeus 1758) is a cos-mopolitan species found in tropical, subtropical, temperate, and sometimes cold waters of all oceans and adjacent seas (Nakamura, 1985). In the North Pacific Ocean (defined, for the purposes of this paper, to be the area of north of the equator;
Fig. 1), the bulk of the swordfish catch has been taken by Japan, the United States, Taiwan and Mexico, with very small catches by Korea and China, whose swordfish catch is estimated to be less than 4% of the total swordfish catch in the North Pacific (Anon., 2002, 2004).
The biomass of swordfish in the North Atlantic Ocean is esti-mated to be close to that corresponding to Maximum Sustainable Yield (MSY) while the fishing mortality and catch are estimated to be slightly less than those at MSY (Anon., 2005a). Catches of
∗Corresponding author. Tel.: +886 2 23629842; fax: +886 2 23629842.
E-mail address:chilu@ntu.edu.tw(C.-L. Sun).
1 Present address: Department of Environmental Biology and Fisheries Sci-ence, National Taiwan Ocean University, Keelung 202, Taiwan.
swordfish in the South Atlantic Ocean increased rapidly between 1988 and 1995. However, the population status of swordfish in the South Atlantic remains unknown because of uncertainty about fishing effort (Anon., 2005a). The results of assessments of swordfish in the Indian Ocean based on surplus production models indicate that exploitation rates are presently likely above those at which MSY is achieved and hence that overfishing is likely to be occurring (Anon., 2006).
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) (e.g.Bartoo and Coan, 1989; Di Nardo and Kwok, 1998; Kleiber and Bartoo, 1998; Nakano, 1998; Uosaki, 1998; Anon., 2002, 2004) and the results of production model assessments (Anon., 1999), although attempts have been made 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 relatively stable at current levels of exploitation.
Swordfish are known to be sexually dimorphic (females grow faster and to a larger size than males (Sun et al., 2002), females mature later than males and the sex-ratio varies with length
0165-7836/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fishres.2006.11.027
S.-P. Wang et al. / Fisheries Research 84 (2007) 282–300 283
(DeMartini et al., 2000; Wang et al., 2003)). All previous stock assessments have ignored the possible impact of sexual dimor-phism although the results of assessments may be sensitive to this. For example,Wang et al. (2005)showed using simulation that ignoring sex-structure when conducting population model-based stock assessments can lead to substantial bias.
The absence of information about the age-structure of the catches precludes the use of standard age-structured methods of fisheries stock assessment such as Virtual Population Analysis. In contrast, it is possible to assess swordfish using ‘Length-based Synthesis’ (e.g.Methot, 2000; Wang et al., 2005). This approach to fisheries stock assessment involves developing an appropriate (and biological realistic) population dynamics model and then estimating the values for its parameters by maximizing an objec-tive (or likelihood) function. In this case, the data used when fitting the model include the length–frequency of the catches, along with information on relative abundance inferred from catch-rates.
This study therefore explores the current status of the sword-fish population in the North Pacific Ocean by fitting a variant of the age- and sex-structured population dynamics model tested using simulation byWang et al. (2005). The uncertainty associ-ated with the assessment is evaluassoci-ated using sensitivity analyses and by summarizing model outputs in the form of Bayesian posterior distributions. The analyses also examine whether the common assumption that selectivity is logistic for longline fish-eries is valid for this particular stock of swordfish.
2. Materials and methods
2.1. The fisheries
Swordfish were targeted by the Japanese longline fleet along with albacore in some areas of the North Pacific during 1952–1962 when they were fishing at 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; Sakagawa, 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 mid-1963 of Taiwanese vessels into the fishery, and increased catches by the US in the late 1960s and early 1970s. The pro-portion of the total catch taken by Japan declined even further after 1990 (to about 42–61% of the total) because of the rapid expansion of the Hawaii-based swordfish fisheries (Di Nardo and Kwok, 1998; Skillman, 1998; Anon., 2002, 2004). How-ever, swordfish catches by the Hawaii-based longline fishery dropped off after 2000 because of restrictions to reduce sea turtle interactions. Hawaii-based longline vessels that targeted swordfish were also active in the California longline fishery. This fishery grew in the late 1990s and has been the largest U.S. fishery for swordfish since 2001 (Anon., 2004, 2005b). The development of swordfish fisheries in Mexico began in 1964 (longliners) and 1986 (gillnets). China and Korea developed longline fisheries in the North Pacific in recent years, but their catches of swordfish have been much less than those by other fisheries.
The time-series of historical catches (1952–2003) used in the assessment (Fig. 2) was based on reports to the Interim Scientific Committee for Tuna and Tuna-like Species in the North Pacific Ocean (ISC) (Anon., 2004, 2005b).
2.2. Data used
Consistent with several past assessments, the North Pacific is divided into four fishing regions at 30◦N and 160◦E (Fig. 1). This is primarily because the US fisheries are then restricted to the northeast and southeast regions (Table 1).
Catch in number and effort data are available for the Japanese high-seas longline fisheries (1952–2002) and for the Hawaii-based longline fisheries (1990–2002). The data for each catch-effort record includes the catch in number of fish, the catch-effort in hooks, and the operating type. Operating type is defined as the number of hooks between each float (i.e. hooks per basket) and is available for the Japanese longline fisheries (1975–2002) and for the Hawaii-based longline fisheries (1995–2000). The data for the Japanese longline fisheries are aggregated by 10◦× 20◦ block, year, and month, and are divided into three categories [surface operations (3–4 hooks per basket), regular operations (5–9 hooks per basket), and deep operations (more than 10 hooks per basket)]. The data for the Hawaii-based longline fisheries
Fig. 2. Annual catches of swordfish in the North Pacific Ocean reported to the Interim Scientific Committee for Tuna and Tuna-like Species in the North Pacific Ocean (1952–2003).
are aggregated by 5◦× 5◦block, year, and 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-ing (GLM) approach (Gavaris, 1980; Kimura, 1981; Hilborn and Walters, 1992), with the assumption that the errors are log-normally distributed (Fig. 3; seeWang (2004)for details).
Length–frequency data (eye fork length, EFL) are avail-able for the Japanese longline fisheries since 1975 and for the Hawaii-based longline fisheries since 1994. Information on the sex-structure of the catch by Japan is, however, only available after 1984. The sample sizes for the JPNW, JPSW and JPSE fleets are too small to construct reliable sex-specific length–frequency distributions so the length–frequency data for these three fleets are aggregated over sex when treated as input to the assessment.
2.3. The population dynamics model
The population dynamics model that forms the basis for the assessment is described inAppendix A. This model is sex- and age-structured and considers both sexes from age 0 to 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 devia-tions about this reladevia-tionship are log-normally distributed. Owing to lack of data, the recruitment deviations prior to 1971 and those thereafter are treated differently. 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 whereas those for the years after 1970 are treated as free parameters of the assessment model.
The logistic curve, which assumes that the vulnerability of a fish increases monotonically to an asymptote with increasing
Table 1
Definition of the fleets that have operated in the North Pacific Ocean, the data available for each, and how fleet-specific selectivity is modeled
Fleet Gear Fleet code Catch data Length data Catch-rate Selectivity
Japan High-seas longline JPNW 1952–2002 (Number) 1975–2000 1952–2002 Estimated
JPSW Estimated
JPNE Estimated
JPSE Estimated
U.S. Hawaii-based longline HINE 1990–2002 (Number) 1994–2000 1990–2000 Estimated
HISE Estimated
Taiwan High-seas and offshore longline TWLL 1967–2002 (Weight) 1997–2001 – Estimated
Japan Other gears JPOTH 1952–2002 (Weight) – – Assumed the same as JPNW
U.S. Other gears USOTH 1970–2002 (Weight) – – Assumed the same as HINE
Mexico All gears MEX 1979–2002 (Weight) – – Assumed the same as HISE
Korea Longline KOR 1995–2002 (Weight) – – Assumed the same as JPNW
S.-P. Wang et al. / Fisheries Research 84 (2007) 282–300 285
Fig. 3. Annual catch-rate indices of swordfish for the six fleets (the nominal and standardized catch-rate indices are scaled to the value in 1975 for the Japanese fleets and to the value in 1996 for the Hawaii-based fleets).
length, is used most commonly in fisheries stock assessment models to represent selectivity for longline gears (e.g. Erzini et al., 1998; Sousa et al., 1999). However, the assumption that selectivity-at-length follows a logistic curve might be inadequate to mimic the bimodal length–frequency data for the JPSE fleet (Figs. 4 and 5). Three alternative ways of modeling selectivity are therefore considered:
Model 1: Selectivity for all six fleets is logistic.
Model 2: Separate selectivity parameters are estimated for each age and sex for the JPSE fleet, and selectivity is assumed to be logistic for the remaining fleets.
Model 3: Selectivity for the JPSE fleet is modeled by the combi-nation of logistic and dome-shaped components, and selectivity is assumed to be logistic for the other fleets.
2.4. Parameter estimation
The parameters of the model can be divided into those for which auxiliary information is available (Table 2) and those which need to be estimated from the monitoring data (Table 3). The values for the parameters related to natural mortality (M), the steepness of 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
fit-Table 2
The values assumed for the parameters of the relationships between length and weight, length and age, and maturity and age
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
Standard deviation of length-at-age,σas
Age 1 25.00 25.00 Age 2 18.63 15.92 Age 3 10.79 9.59 Age 4 10.27 10.82 Age 5 12.26 10.28 Age 6 13.39 11.46 Age 7 9.79 11.53 Age 8 14.41 13.94 Age 9 12.64 11.47 Age 10 15.45 11.21 Age >11 10.00 10.00
Fig. 4. Observed (histograms) and three MPD model-predicted length-frequencies for each of seven fleets for the base-case analyses.
ting the model to the data (results not shown) and must therefore be pre-specified. In this study, the base-case value for M is taken to be 0.25 year−1based onPauly’s (1980)empirical equation,
h is assumed to be 0.9 (Anon., 1997; Punt et al., 2001), and
σvis assumed to be 0.4 (Punt et al., 2001). Given the lack of length–frequency data, the selectivity patterns for some of the fleets had to be assumed to the same. The fleets used to deter-mine selectivity for those fleets without length–frequency data (Table 1) were chosen based on how and where the various fleets operate.
The objective function minimized1 to find the estimates of
the ‘free’ parameters of the model includes two components (the data available for assessment purposes and the constraints based on a priori assumptions) (Appendix B). The data available
1 Using AD Model Builder (version 5.0.2) (Otter Research Ltd, 2000).
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. Constraints are imposed on the extent to which the number of 0-year-olds can deviate from the underlying stock-recruitment relationship and the smoothness of the selectivity patterns (when selectivity-at-age is estimated separately by selectivity-at-age).
The factors used for weighting the various data sources cannot be estimated when fitting the model to the data, and must instead be pre-specified. The values for the weighting factorsρs1,ρs2,
ρs3andρs4are all set equal to 0.001,ψ is set equal to −1 and ρs5is set equal to 0 (i.e. selectivity is not constrained to increase
with age) for the base-case analyses. The weight assigned to the sex-ratio data,ρs, is set equal to 0.1, and the overall weight
S.-P. Wang et al. / Fisheries Research 84 (2007) 282–300 287
Fig. 5. The spatial (aggregated by 20◦× 10◦block) length-frequencies of swordfish caught by the JPSE fleet.
assigned to the length–frequency data,ρL, is set equal to 0.01. The values for these weighting factors are semi-arbitrary and reflect qualitative (a priori) considerations. The sensitivity of the model results to the values for these weighting factors is examined by changing the values forρs,ρL,ρs1,ρs2,ρs3,ρs4 andψ by ±25% and by setting ρs5to 1, 10 and 100. Sensitivity is also examined to the value assumed for natural mortality M (0.2 year−1 and 0.3 year−1 – based on the assumptions made by previous researchers –Skillman, 1998), that assumed for the steepness of the stock-recruitment relationship h (0.5, 0.8 and 0.95), and that assumed for the extent of variation in recruitment
σv(0.2 and 0.6).
2.5. Key model outputs
The model outputs examined are the key quantities of man-agement interest.
(1) S0, the spawning stock biomass at unfished equilibrium;
(2) S2002, the spawning stock biomass at the start of 2002;
(3) SPRF2002, the spawning stock biomass-per-recruit
corre-sponding to the 2002 exploitation rates by fleet (see
Appendix A.7for the definition of this quantity) relative to its unfished level;
Table 3
The parameters of the population dynamics model not known from auxiliary information
Parameter No. of parameters
Estimated
Unfished recruitment, R0 1
Process errors,vt 1 per year
Selectivity Model 1
Length-at-50%-selectivity,Lf50 1 per fleet Length-at-95%-selectivity,Lf95 1 per fleet Model 2
Length-at-50%-selectivity,Lf50 1 per fleet, expect for the JPSE fleet
Length-at-95%-selectivity,Lf95 1 per fleet, expect for the JPSE fleet
Age- and sex-specific selectivity,ss,fa 1 per age and sex for the JPSE fleet
Model 3
Length-at-50%-selectivity,Lf50 1 per fleet Length-at-95%-selectivity,Lf95 1 per fleet Length-at-mean-selectivity,Lfmu JPSE only Standard deviation of selectivity,Lfsd JPSE only Weight assigned to the dome-shaped
ogive,ϕf JPSE only Pre-specified Natural mortality, M 1 Steepness, h 1 Variation in recruitment,σv 1
(5) SMSY, the spawning stock biomass at which MSY is
achieved;
(6) SPRFMSY, the spawning stock biomass-per-recruit under the
exploitation rate at which MSY is achieved (i.e. FMSY)
rel-ative to its unfished level;
(7) S2002/S0, the ratio of S2002to the spawning stock biomass at
unfished equilibrium;
(8) S2002/SMSY, the ratio of S2002to the spawning stock biomass
at which MSY is achieved;
(9) SPRF2002/SPRFMSY, the ratio of SPRF2002to SPRFMSY. Quantities (1), (2) and (5) relate to the estimates of spawning stock biomass at unfished, current and MSY levels, while quan-tities (3) and (6) evaluate the exploitation rate at present and at MSY. A fleet-aggregated measure of exploitation rate is used in quantity (3) to reduce the volume of results. Quantities (7), (8) and (9) relate to relative measures of spawning stock biomass and exploitation rate and can be used to evaluate the health of the population and its exploitation status.
2.6. Bayesian posterior distribution
The Markov Chain Monte Carlo (MCMC) method, based on the Metropolis-Hastings algorithm (Hastings, 1970; Gelman et al., 2004), is used to develop Bayesian posterior distribu-tions for the parameters of the model and the key quantities of management interest. The posterior distributions are constructed
based on samples generated by conducting 1,020,000 cycles of the MCMC algorithm, ignoring the first 20,000 cycles as the “burn in” period, and selecting every 1000th parameter vector thereafter. A potential drawback associated with the application of the MCMC algorithm is how to conclude whether conver-gence to the posterior distribution has actually occurred. In this study, whether convergence of the MCMC algorithm has been achieved satisfactorily is examined using the diagnostic methods developed byGeweke (1992),Heidelberger and Welch (1983)
andRaftery and Lewis (1992) as implemented in the CODA package of R (Version 1.9.1) (The R Foundation for Statistical Computing, 2004) (Plummer et al., 2004).
2.7. Risk analysis
The objective of the risk analysis is to examine the future status of the population for a variety of situations and potential future management actions. In this study, risk is quantified by the probability of the spawning stock biomass dropping below 40% of the virgin spawning stock biomass. The choice of 40% is relatively arbitrary, but 0.4S0has been used as a reference level in
other contexts for other relatively long-lived marine fishes (e.g.
Griffiths, 1997; Kirchner, 2001; Sun et al., 2005). Projections are conducted for various levels of future fishing intensity, and results are shown for projection periods of 5, 10 and 20 years.
The exploitation rates used in the projections for a given assessment scenario (the base-case analysis or one of the sen-sitivity tests) are specified as a fraction of the posterior median for FMSY. Each evaluation of a level of fishing intensity is based
on 500 simulations, each of which involves randomly selecting a parameter vector from the posterior distribution to account for uncertainty about model parameters, and generating future devi-ations about the stock-recruitment reldevi-ationship (vt ∼ N(0; σv2)) to allow for uncertainty in future recruitment.
3. Results
3.1. Model selection for selectivity
The values for the parameters related to natural mortality, steepness, the extent of variation in recruitment, the weight assigned to the sex-ratio data, and the weight assigned to the length–frequency data are set to their base-case values (i.e.
M = 0.25 year−1, h = 0.9,σv= 0.4,ρs= 0.1 andρL= 0.01) for the purposes of selecting among selectivity models. The results for selectivity model 2 are insensitive to the values assumed forψ,
ρs1,ρs2,ρs3 and ρs4 (results not shown) so that conclusions
regarding the choice among the three selectivity models should be robust to the values assumed for these weighting factors.
The sex- and age-specific selectivity ogives for the seven fleets for each of the three selectivity models are shown in
Fig. 6. The selectivity patterns for females and males are very similar for the JPSW, HINE, HISE and TWLL fleets whereas males are slightly less selected than females for the JPNW, JPNE and JPSE fleets. The selectivity patterns for the JPSE fleet differ substantially depending on the choice of the selectivity model. The selectivity of the JPSE fleet on animals of age 1 is
S.-P. Wang et al. / Fisheries Research 84 (2007) 282–300 289
Fig. 6. Base-case model-predicted selectivity ogives for female and male swordfish in the North Pacific Ocean for the seven fleets.
much higher for selectivity models 2 and 3 than for selectivity model 1.
The observed and model-predicted length–frequencies for female, male and sex-aggregated animals are illustrated for each
of the three selectivity models inFig. 4. The results are aggre-gated across years for ease of presentation. This is not the form in which the data enter the likelihood function, but is rather a convenient way of assessing the overall fit of the model to
the length–frequency data for each fleet and selectivity model. The model-predictions for selectivity models 2 and 3 mimic the observed data much better than those for selectivity model 1 for the JPSE fleet (particularly males and sex-aggregated animals).
Selectivity model 1 clearly provides the worst fit to the data of the three models. Although the model fits for selectivity model 2 are similar to those for selectivity model 3, the number of esti-mated parameters for the former is much larger than for the latter (the number of estimated parameters is 75 for model 2 and only 48 for model 3). Models 2 and 3 cannot be compared using, say, AIC because the objective function for model 2 includes some penalty terms that the objective function for model 3 does not. However, selectivity model 3 is nevertheless selected as “best”
because it provides fits that are as good as those for selectivity model 2, but with fewer parameters. Also, the selectivity pattern for selectivity model 3 is much smoother than that for selectivity model 2 (Fig. 6) and is hence seems more biologically realis-tic. The remaining analyses of this paper are therefore based on selectivity model 3.
3.2. Base-case assessment
The model-estimates of the exploitation rates correspond-ing to the maximum of the objective function (i.e. Maximum Posterior Density, MPD, estimates) are shown inFig. 7. The exploitation rates for the four Japanese offshore fleets were rel-atively high before the early 1960s and declined substantially
S.-P. Wang et al. / Fisheries Research 84 (2007) 282–300 291
Fig. 8. Time trajectories of MPD estimates, posterior medians and posterior 95% intervals for the spawning stock biomass relative to its unfished level (S/S0), the spawning stock biomass relative to that at which MSY is achieved (S/SMSY), and the spawning stock biomass-per-recruit corresponding to the annual exploita-tions rates relative to that at FMSY, SPR/SPRFMSY, for base-case analysis.
thereafter as a result of reduced catches. Except perhaps for the JPSW fleet, the exploitation rates by the Japanese offshore fleets have stabilized since the early 1960s. The exploitation rates by the Hawaii-based fleets decreased after 1991–1993 when the catches by these fleets dropped. The exploitation rates for Japanese ‘other gears’ and Korean longline fisheries fluc-tuated before 1990 and, similarly to those for the Taiwanese longline, Chinese and American other gears fisheries, increased thereafter.
Fig. 8shows the time trajectory of MPD estimates for the ratio of the spawning stock biomass to S0, the ratio of the spawning
stock biomass to SMSY, and the spawning stock
biomass-per-recruit corresponding to the annual exploitation rates relative to SPRFMSY. The spawning stock biomass declined due to the high
catches from 1952 until the early 1960s when it was about 40% of S0, recovered to about 75% of S0in 1978, before declining
slightly again to about 50% of S0. The spawning stock
biomass-per-recruit corresponding to the annual exploitation rate has been about twice SPRFMSYduring recent years.
3.3. Sensitivity analysis
Table 4examines the sensitivity of the objective function cor-responding to the maximum of the posterior distribution and the MPD estimates of the nine quantities of management interest to changing the values for the pre-specified parameters (M, h,σv,
ρsandρL), to changing the selectivity model, and to ignoring sex
when applying the stock assessment method. The fit of the model to the data, as reflected by the value of the objective function, is sensitive to the choice of selectivity model. This is not surprising becauseFig. 4suggests that the ability to fit the length–frequency data differs among selectivity models. The results for selectiv-ity model 2 are more similar to those for the base-case analysis, which is based on selectivity model 3. The fit of the model to the data is totally insensitive to the weight assigned to sex-ratio data, but somewhat sensitive to that assigned to length–frequency data. This is not unexpected because the length–frequency data provide the largest contribution to the objective function.
The management quantities are very sensitive to the value assumed for M; increasing M from 0.25 to 0.3 year−1 leads to the much larger estimates of all quantities except SPRFMSY.
The opposite pattern is evident when M is reduced from 0.25 to 0.2 year−1. Increasing M leads to better fits to the data (an objective function value of 810.97 for M = 0.3 year−1compared to 814.01 for the base-case analysis). Most of the model outputs are not very sensitive to the value assumed for h in the range 0.8–0.95, but the model outputs change noticeably when h is set equal to 0.6. The model outputs are relatively insensitive to the values forσv,ρs andρL. The relative quantities (SPRF2002, SPRFMSY, S2002/S0, S2002/SMSY, and SPRF2002/SPRFMSY) are the
less sensitive to the values for the pre-specified parameters. In contrast, the estimates of S0, S2002, SMSY, and MSY vary quite
markedly depending on the values assumed for the pre-specified parameters.
The factor that impacts S0, S2002, SMSY, and MSY to the
great-est extent is whether the model is sex-specific or sex-aggregated. For example, the estimate of MSY is only 63% of the base-case value when the model is sex-aggregated. It should be noted, however, that the fit of the sex-aggregated model, as reflected by the value for the objective function, is much poorer than that of the sex-specific model.
3.4. Bayesian analysis
The values for the selectivity parameters were assumed known when conducting the Bayesian analyses. This is because the values for these parameters are highly correlated, which pre-vents the MCMC algorithm from converging. The parameters considered uncertain when developing the Bayesian posteriors were the unfished recruitment (R0) and the annual recruitment
deviations (νt).
It is not feasible to list the values for all of the convergence statistics for all of the parameters. Instead,Fig. 9illustrates some of the convergence statistics for three key quantities of manage-ment interest (S2002/S0, S2002/SMSYand SPRF2002/SPRFMSY) for
the base-case analysis. The convergence statistics inFig. 9are the trace of the posterior samples, the posterior density function
Table 4
The value of objective function and the MPD estimates of nine quantities of management interest for the base-case analysis and the sensitivity analyses
Specification S0 S2002 SPRF2002 MSY SMSY SPRFMSY S2002/S0 S2002/SMSY SPRF2002/SPRFMSY Objective function Base-case 110547 54804 0.47 13151 23704 0.24 0.50 2.31 1.97 814.01 M 0.2 115845 43864 0.33 10548 28298 0.27 0.38 1.55 1.22 842.70 0.3 145314 101811 0.68 21588 29786 0.23 0.70 3.42 2.98 810.97 h 0.6 136286 73893 0.55 10928 43072 0.44 0.54 1.72 1.25 819.92 0.8 116123 58418 0.49 12321 29805 0.31 0.50 1.96 1.58 815.09 0.95 108553 53705 0.46 13671 20954 0.21 0.50 2.56 2.26 813.72 σv 0.2 114330 59577 0.50 13625 25062 0.24 0.52 2.38 2.07 846.89 0.6 108908 52324 0.45 12964 23285 0.24 0.48 2.25 1.90 802.52 Selectivity Model 1 102084 46137 0.42 12188 22782 0.25 0.45 2.03 1.71 905.98 Model 2 108245 51183 0.44 12863 24247 0.25 0.47 2.11 1.79 784.59 Sex-aggregated model 40918 11923 0.23 8310 8214 0.23 0.29 1.45 1.00 886.66 ρs −25% 110547 54804 0.47 13151 23704 0.24 0.50 2.31 1.97 814.01 +25% 110547 54804 0.47 13151 23704 0.24 0.50 2.31 1.97 814.01 ρL −25% 113697 57566 0.48 13583 25384 0.25 0.51 2.27 1.95 554.746 +25% 108515 53057 0.46 12885 24305 0.25 0.49 2.18 1.86 1070.96
The unit of biomass is metric tonnes.
(estimated using a normal kernel density estimator), the autocor-relation at different lags, and the cumulative patterns (running mean and running 95% probability intervals). The trace and the cumulative patterns do not exhibit obvious signals, and the pos-terior density functions are not only smooth, but also unimodal. In addition, the autocorrelations are close to zero for all lags and quantities. The results inFig. 9therefore provide no evidence for convergence problems when applying the MCMC algorithm. Similarly, the convergence statistics for S2002/S0, S2002/SMSY
and SPRF2002/SPRFMSY for the sensitivity tests than involve changing the values for M, h and σvprovide no evidence for convergence problems for these three management quantities (diagnostic statistics not shown).
Fig. 8shows the posterior median and posterior 95% intervals for the time-trajectories of St/S0, St/SMSYand SPRFt/SPRFMSY. 3.5. Risk analysis
Fig. 10shows the probability that the spawning stock biomass will exceed 40% of S0when the, future exploitation rate is set to
various percentages of FMSY. Results are shown inFig. 10for
projection periods of 5, 10 and 20 years, and for the base-case analysis and for various assumed values for M, h andσv.
Except when M = 0.2 year−1, the probability that the spawn-ing stock biomass will drop below 40% of S0 is negligible if
fishing intensity remains at the current level (the dots inFig. 10). The projection results are sensitive to the values for some of the assumed parameters. For example, the results are more pes-simistic whenσv= 0.6 and h = 0.95 because current spawning stock biomass is lower (Table 4). The stock is most resilient
when h = 0.6 (Fig. 10, upper panel). This result may seem counter-intuitive because stock status becomes more pessimistic when h = 0.6. However, the current spawning stock biomass and spawning stock biomass-per-recruit are largest when h = 0.6 (Table 4).
For M = 0.2 year−1, the probability that the spawning stock biomass will be above 40% of S0is close to 20% if the
exploita-tion rate remains at the present level. The spawning stock biomass will, however, rebuild over the next 10–20 years if the exploitation rate was reduced to 50–60% of FMSY.
4. Discussion
4.1. Population status
Catch-rates are often assumed to be relative indices of popu-lation abundance. The standardized and MPD model-predicted catch-rates for Japanese fleets indicate that the swordfish popu-lation in the North Pacific has varied without major trend over 1952–2002 (Fig. 3). However, this inference is predicated on the GLM standardization removing the impact of changes over time in, for example, fishing technology, and the temporal and spatial distribution of effort. Technological changes have occurred in these fisheries. However, the ability to quantify such changes is limited owing to lack of operational data beyond fishing location and hooks per basket.
The catch-rates can be divided into three fairly distinct periods: a period of declining catch-rates during the 1960s, a period of generally increasing catch-rates from the 1960s until the 1980s, and a period of generally declining
catch-S.-P. Wang et al. / Fisheries Research 84 (2007) 282–300 293
Fig. 9. MCMC convergence diagnostics for three quantities of management interest for the base-case analysis.
rates thereafter. Although the standardized catch-rates appear to have declined somewhat from the mid 1980s, they nevertheless remain higher than those during the 1960s. The standardized and MPD model-predicted catch-rates for the Hawaii-based fleets are not particularly consistent (this is perhaps not very surpris-ing given the short length of the time-series concerned), but also suggest a lack of trend in population size from 1996–2000 (Fig. 3).
The lowest level of spawning stock biomass occurred dur-ing 1962 (MPD estimate 0.41S0, posterior median and 95%
probability interval 0.41S0[0.36–0.48S0]). Therefore, although
some depletion of the spawning stock biomass did occur during 1952–1962, and exploitation rates are highest for these years (Fig. 7), the spawning stock biomass was not depleted sub-stantially even in 1962. The spawning stock biomass gradually recovered once the Japanese high-seas longline fleets changed their fishing practices to target tunas rather than swordfish. In the late 1970s, the spawning stock biomass increased to about 0.75S0 (95% probability interval 0.68–0.84S0), but declined
slightly thereafter due to increased catches and then increased again until 1990. The spawning stock biomass has declined again since 1990 owing to the catches by the Hawaii-based longline
Fig. 10. Probability of the spawning stock biomass being larger than 40% of S0for various levels of fishing intensity for projection periods of 5, 10 and 20 years for various assumed values for M, h andσv(the dot corresponds to the current (2002) fishing intensity).
fleets. Nevertheless, the spawning stock biomass is currently (2002) estimated to be at a fairly high fraction of its unfished level (0.50S0, 95% probability interval 0.42–0.61S0).
The least optimistic appraisal of stock status occurs when M is assumed to be 0.2 year−1 when S2002/S0 is estimated to be
about 0.39. For the other assumed values for M, h andσv, the MPD estimates and the posterior medians indicate that S2002
is 0.49–0.73S0. The MPD estimates and the posterior medians
suggest that the current spawning stock biomass is much higher than SMSY; in general S2002/SMSYis at least 2 (Table 4).
The exploitation rates for the Japanese high-seas fleets during 1952–1962 were much higher than those thereafter, particularly for the JPNW and JPNE fleets (Fig. 7). The exploitation rates for the JPNW and JPNE fleets were higher than those for the JPSW and JPSE fleets for all years. However, the exploitation rates for the Japanese high-seas longline fleets appear to have decreased to about half of those during 1960s in recent years. The Hawaii-based longline fleets have targeted swordfish in the North Pacific since 1990, and their exploitation rates are now similar to those for the JPSW and JPSE fleets during 1960s. The exploitation rates for the remaining fisheries (except for those by Mexico) have increased in recent years.
The spawning stock biomass-per-recruit corresponding to the fleet-aggregated exploitation rates (Fig. 8, lower panel) evalu-ates the overall impact of all fleets; the value of spawning stock biomass-per-recruit for a given year should be interpreted as the spawning stock biomass-per-recruit that the stock would equili-brate at if exploitation rates remained equal to those for that year indefinitely. The MPD estimates of spawning stock biomass-per-recruit for the base-case analysis fluctuated over recent years, with the current spawning stock biomass-per-recruit being about twice that corresponding to MSY. The MPD estimates and the posterior medians for the current spawning stock biomass-per-recruit for the sensitivity analyses (Tables 4 and 5) are 122–298% of its MSY level (95% probability intervals of 100–352% of its MSY level).
No management regulations exist for swordfish in the North Pacific, except for the prohibition on targeting of swordfish by Hawaii-based longline vessels to protect sea turtles (Anon., 2002). Therefore, a range of 0–150% of FMSYwas examined
in the projections. The spawning stock biomass is estimated to remain above 0.4S0 if fishing intensity remains at its current
level, expect if M = 0.2 year−1. In fact, unless fishing inten-sity increases to about 50–120% of FMSY(i.e. an increase by
S.-P. Wang et al. / Fisheries Research 84 (2007) 282–300 295 Table 5
The posterior medians and posterior 95% intervals (values in parentheses) for three quantities of management interest for the base-case analysis and the sen-sitivity analyses S2002/S0 S2002/SMSY SPRF2002/SPRFMSY Base-case 0.51 (0.42, 0.61) 2.31 (1.92, 2.78) 1.64 (1.97, 2.35) M 0.2 year−1 0.39 (0.31, 0.45) 1.65 (1.34, 1.98) 1.33 (1.00, 1.54) 0.3 year−1 0.73 (0.61, 0.87) 3.53 (2.95, 4.21) 3.08 (2.63, 3.52) h 0.6 0.73 (0.61, 0.87) 1.73 (1.45, 2.05) 1.25 (1.07, 1.46) 0.8 0.51 (0.43, 0.61) 2.00 (1.67, 2.42) 1.63 (1.36, 1.91) 0.95 0.51 (0.42, 0.60) 2.61 (2.16, 3.11) 2.31 (1.92, 2.74) σv 0.2 0.53 (0.45, 0.62) 2.45 (2.06, 2.87) 2.14 (1.79, 2.47) 0.6 0.49 (0.40, 0.58) 2.22 (1.85, 2.71) 1.90 (1.54, 2.28)
50–100%), there appears to be negligible risk that the spawn-ing stock biomass will drop below 0.4S0during next 10 or 20
years (Fig. 10). This implies that swordfish in the North Pacific is probably under-exploited at the current level of fishing intensity. The fairly optimistic results of this study should be interpreted with some caution. Specifically, trends in standardized catch-rates were assumed to be related linearly to abundance, and this may not be the case. Furthermore, the Bayesian analyses under-estimate the true extent of estimation uncertainty because they assume that selectivity, M and steepness are known exactly, and they are based on pre-specified values for the standard deviations of the various data sources (seeAppendix Band Section3.1).
4.2. Selectivity model
The logistic curve is used very commonly in fisheries stock assessments to represent selectivity for longline gear. However, the logistic curve is obviously inappropriate for the JPSE fleet because this selectivity pattern is unable to mimic the bimodal length–frequency distributions for this fleet (Fig. 4). Therefore, a selectivity function that consists of the combination of a logistic and a dome-shaped component is used to better mimic these data. The selectivity function with two components is preferred to estimating separate selectivity parameters for each sex and age. This is commonly done when applying complex statistical catch-at-age or catch-at-length models (Fournier et al., 1998; Ianelli, 2002; Maunder and Watters, 2003). However, large numbers of parameters are needed for this approach (32 for each fleet in this case, 16 age groups for each of two sexes). In contrast, only five parameters are needed to implement the selectivity model with logistic and dome-shaped components. The fits to the data are not markedly better with the larger number of parameters (Table 4;
Fig. 4). However, estimating additional selectivity parameters could lead to much less precise estimates of the model outputs. The bimodal length–frequency distributions for the JPSE fleet were unexpected given that this fleet uses longline gear.Fig. 5
shows the spatial length–frequency distributions for the JPSE fleet. A large number of fish smaller than 100 cm (particularly males) are caught by the JPSE fleet, and these fishes are likely
to be age 1 (Sun et al., 2002). A high proportion of the catch of small fishes by the JPSE fleet occurs between 180◦and 160◦W near Hawaii. However, spatial patterns in fishing effort cannot be the sole reason for the JPSE fleet catching small fish because the HISE fleet also fishes in this area and does not catch small fish. The differences in length–frequency distributions could be a consequence of targeting. The Hawaii-based longline fleets target swordfish whereas swordfish is a bycatch of the Japanese longline fleets. The latter fleets might change operating type (e.g. size of hooks, fishing depth, bait, operating time) when targeting different species of tuna, which could lead to the capture of a range of sizes of swordfish. Unfortunately, there is no informa-tion on target type for the Japanese fleets which could be used to examine this issue in detail.
4.3. General discussion
Recent stock assessments of swordfish in the North Pacific indicate that this stock is not over-exploited and that it has been relatively stable at current levels of exploitation (Anon., 1999, 2002, 2004). However, these assessments ignored sexual dimor-phism, and instead focused on the results of sex-aggregated analyses. The results of this study indicate that the values for the quantities related to the absolute measures (S0, S2002, and
MSY) are substantially different when sex-structure is taken into account. This result is consistent with the simulations con-ducted by Wang et al. (2005) who found that estimates of absolute measures for swordfish will be substantially biased if a sex-aggregated population dynamics model is fitted to the data for swordfish in the North Pacific. However, the results of the models that take sexual dimorphism into account still suggest that the spawning stock biomass of swordfish in the North Pacific is currently at a fairly high fraction of its initial level and that the spawning stock biomass-per-recruit under cur-rent exploitation rates is higher than that corresponding to the MSY level.
Acknowledgements
We thank Drs. Yuji Uozumi and Kotaro Yokawa of the National Research Institute of Far Seas Fisheries, Shimizu, Japan, and Dr. Pierre Kleiber of the Pacific Islands Fisheries Science Center, NMFS, Honolulu, Hawaii, USA, for providing the Japanese and the Hawaii-based longline swordfish catch, effort and length–frequency data, respectively. An earlier ver-sion of this manuscript was improved by the comments of two anonymous reviewers. This study was in part supported finan-cially by the National Science Council, Taiwan through grant NSC92-2313-B-002-082 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 middle 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− (Cst−1,λ−1+ Cst−1,λ)]e−M/2, if a = λ (A.1)
whereNt,as is the number of fish of age a and sex s (m/f) at the start of year t,Rst the recruitment (at age 0) of fish of sex s at the start of year t,Ct,as is the catch in number of fish of age a and sex s during year t:
Cst,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 is 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:
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 = sas,fFtf (A.4)
Ftfis the exploitation rate by fleet f during year t on fully-selected animals: ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ Ftf = Cnf,obst asss,fa Nt,as e−M/2
, if the catches are reported in numbers
Ftf = Cwf,obst
asss,fa Nt,as e−M/2wsa
, if the catches are reported in weight
(A.5)
Cnft is the observed total catch (in number) by fleet f during year
t, Cwft the observed total catch (in weight) by fleet f during year
t, andss,fa is 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, andLsais the expected length of an animal of sex s and age a:
Lsa= Ls∞[1− e−ks(1−ms)(a−as0)]1/(1−m
s)
(A.7)
Ls
∞, ks,as0, msare the parameters of the generalized von
Berta-lanffy growth equation (Richards, 1959; Ehrhardt et al., 1996).
The expected length (lower jaw fork length, LJFL) is con-verted to eye fork length (EFL) based on the LJFL-EFL conversion equation ofSun et al. (2002).
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 mature, Lm
the length-at-50%-maturity for females, and rmis the maturity
slope parameter.
A.4. Recruitment
Recruitment (number of animals of age 0) is assumed to be related to spawning stock biomass by means of the Beverton–Holt stock-recruitment relationship, parameterized in terms of the ‘steepness’ of the stock–recruitment relationship, h (Francis, 1992), and the recruitment at unfished pre-exploitation equilibrium, R0, i.e.:
Rst = 0.5 4hR0St/S0
(1− h) + (5h − 1)St/S0
evt−σ2v/2 (A.9)
where S0 is the spawning stock biomass at unfished
pre-exploitation equilibrium and Stis the spawning stock biomass at the start of year t, i.e.:
St= λ a=1
φawfaNt,af (A.10)
vtis a normally-distributed process error,vt ∼ N(0; σv2), andσv2 is the (assumed) variance of the process error in recruitment.
The sex-ratio at age 0 is assumed to be 1:1 in the absence of evidence to the contrary.
A.5. Selectivity A.5.1. Logistic curve
This age-specific selectivity pattern is based on the assump-tion that selectivity is length-specific and governed by a logistic function, i.e.: ss,fa = 1+ exp − ln 19Lsa− Lf50 Lf95− L f 50 −1 (A.11)
S.-P. Wang et al. / Fisheries Research 84 (2007) 282–300 297
whereLf50is the length-at-50%-selectivity for fleet f, andLf95is the length-at-95%-selectivity for fleet f.
A.5.2. Separate parameters selectivity
The “separate parameters selectivity” approach of this paper follows the method used in the “Age-Structured Statistical
Catch-at-Length Analysis” (A-SCALA) method (Maunder and Watters, 2003). The selectivities for each age and sex are treated as separate parameters, but the amount that selectivity can change from age to age is penalized using smoothness penal-ties. These penalties (seeAppendix B.4.1) therefore avoid the model being over-parameterized.
A.5.3. Combination of logistic and dome-shaped components
Selectivity as a function of age is determined using a general functional form that combines a logistic and a dome-shaped component: ss,f a = s 1,s,f a + ϕfs2a,s,f max a (s 1,s,f a + ϕfs2a,s,f ) (A.12)
wheres1a,s,fis the logistic component (Eq.(A.11)) ands2a,s,f is the dome-shaped component:
s2,s,f a =√ 1 2πLfsdexp ⎡ ⎣−(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, andϕfis, for fleet
f, the weight assigned to the dome-shaped component of the
selectivity function.
A.6. Initial conditions
The initial conditions correspond to a population at its unfished equilibrium level at the start of the first year for which catches are available, i.e.:
Ns t1,a= ⎧ ⎪ ⎨ ⎪ ⎩ 0.5R0, ifa = 0 Ns t1,a−1e −M, if 0< a < λ Ns t1,λ−1e −M/(1 − e−M), if a = λ (A.14)
where t1is the first year for which catch data are available, i.e. t1= 1952.
A.7. Fleet-aggregated exploitation rate
A measure of overall exploitation rate that accounts for mul-tiple fleets that each have a different selection pattern is the
spawning stock biomass-per-recruit under a fleet-aggregated fishing intensity (R.D. Methot, NOAA-NMFS, personal com-munication). The spawning stock biomass–per-recruit for year
t is derived under the assumption that the population is in
equi-librium under the exploitation pattern that applied during year t, i.e.: Nt,as = ⎧ ⎪ ⎨ ⎪ ⎩ 0.5, ifa = 0 Nt,a−1s e−M(1− Ft,a−1s ), if 1≤ a < λ Ns t,λ−1e−M(1− Ft,λ−1s )/(1 − e−M(1− Ft,λs )), if a = λ (A.15)
whereFt,as is the exploitation rate on animals of age a and sex s during year t, i.e.:
Ft,as = f
Ft,as,f (A.16)
The spawning stock biomass-per-recruit can be calculated using Eq. (A.10) and the initial spawning stock biomass-per-recruit by assuming Ft,as = 0. The measure of overall exploitation rate for year t is the ratio of the spawning stock biomass-per-recruit for year t relative to the spawning stock biomass-per-recruit at the unfished level. Unlike most defini-tions of exploitation rate, higher values of this ratio correspond to lower (rather than higher) exploitation levels.
Appendix B. Contributions to the objective function
B.1. The sex-ratio data
In principle, the sex ratio data (the fraction of the catch that is female) should be treated as being binomially distributed when fitting the model. However, the sampling process is such that the data are highly overdispersed relative to the binomial distribution. Therefore, 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 (fraction of females) for fleet f during year t, andQft is the model-predicted sex-ratio for fleet f during year t:
Qft = aCf,ft,a a2s=1Ct,as,f (B.2)
ρsis the weight assigned to the sex-ratio data. B.2. The length-frequency data
The likelihood function assumed for the length-frequency data is the robust normal formulation ofFournier 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. Washington, personal communication). 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 func-tion (ignoring constants independent of the model parameters) is: − ln L3= 0.5 t l 2 s=0 f ⎧ ⎨ ⎩ρtf ⎡ ⎣ (Pt,ls,f,obs− Pt,ls,f) 2 2(Pt,ls,f,obs[1− Pt,ls,f,obs]+ (1/Nl))(τs,ft )−2 + 0.01 ⎤ ⎦ ⎫ ⎬ ⎭ (B.3)
wherePt,ls,f,obsis 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 andPt,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:
Pt,ls,f = C s,f t,l lCt,ls,f (B.4)
Cs,ft,l 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: Cs,ft,l = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ λ a=1 Λsa,lCs,ft,a, ifs = m/f s λ a=1
Λsa,lCs,ft,a, otherwise
(B.5)
Nlis the number of length-classes,τts,f is the ‘effective’ sample size for year t, fleet f and sex s (the maximum of the number of animals measured and 50 andΛsa,lis the fraction of animals of age a and sex s that are in length-class l:
Λsa,l= Ll+ L Ll− L 1 √ 2πσasexp −(L − Lsa)2 2(σas)2 dL (B.6) σs
ais the standard deviation of the length of a fish of sex s and age a, Llthe mid-point of length-class l, L half the width of a length-class (the length-frequency data were aggregated to 5 cm length-classes owing to the way the original data were collected),
ρtf is the weight assigned to the length–frequency data for year
t and fleet f:
ρtf = ρL Cf,obst tCtf,obs
/nf (B.7)
ρL is the overall (pre-specified) weight assigned to the length–frequency data, and nfis the number of years for which length-frequency data are available.
B.3. The catch-rate data
The catch-rate-based indices of abundance are assumed to be lognormally distributed about the corresponding model predic-tions. Ignoring constants independent of the model parameters,
the contribution of the catch-rate data to the negative of the log-likelihood function is:
− ln L4= f t ⎧ ⎨ ⎩ln ˜σf + (lnItf,obs− ln Itf)2 2( ˜σf)2 ⎫ ⎬ ⎭ (B.8)
whereItf,obsis the observed catch-rate index for year t and fleet
f andItf is the model-estimate corresponding toItf,obs:
Itf = qfBtf (B.9)
qfis the catchability coefficient for fleet f andBft is the number of exploitable animals during year t corresponding to fleet f:
Bft = s λ a=1 ss,fa Nt,as e−M/2−C s,f t,a 2 (B.10) ˜
σfis the standard deviation ofIf
t , assumed to be time-invariant.
B.4. Contributions of the constraints to the objective function
B.4.1. Constraint on the separate parameters selectivity function
Constraints are added to the objective function when selectivity-at-age parameters are estimated for each age and sex. These constraints are based on difference equation approxima-tions to the first, second, and third derivatives of the selectivity curve. The first difference constrains the selectivity curve to be constant, the second difference constrains the selectivity curve to be linear, and the third difference constrains the selectivity curve to be quadratic (Maunder and Watters, 2003).
First difference: ρs1 f s λ−1 a=1 [Lsa+1− Lsa+ 0.01]ψ[ln(ss,fa+1)− ln(sas,f)]2 (B.11) whereρs1is the weighting factor for the first difference, andψ
S.-P. Wang et al. / Fisheries Research 84 (2007) 282–300 299 Second difference: ρs2 f s λ−2 a=0 [ln(ss,fa )− 2 ln(ss,fa+1)+ ln(ss,fa+2)]2 (B.12)
whereρs2is the weighting factor for the second difference.Third difference: ρs3 f s λ−3 a=0
[ln(ss,fa+3)− 3 ln(sa+2s,f )+ 3 ln(ss,fa+1)− ln(ss,fa )]2 (B.13) whereρs3is the weighting factor for the third difference.
In addition, the selectivity parameters for each fishery are penalized to average one to reduce the correlation among the selectivity parameters and that between the selectivity parame-ters and the other parameparame-ters of the model.
ρs4 f s ln λ a=0ss,fa λ + 1 2 (B.14)
whereρs4 is the weighting factor for the penalty on average selectivity.
In addition, except for the JPNE and JPSE fleets, the selec-tivity parameters are constrained so that selecselec-tivity increases monotonically with age, i.e.:
ρs5 f s λ−1 a=0 [ln(ss,fa )− ln(ss,fa+1)]2, if ss,fa > ss,fa+1 (B.15)
whereρs5is the weighting factor.
B.4.2. Constraint contribution to the objective function
A constraint is placed on the deviations about the stock–recruitment relationship: 1 2σ2 v t v2 t (B.16) References
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