ICES Journal of Marine Science: Journal du Conseil Advance Access originally published online on May 7, 2007
ICES Journal of Marine Science: Journal du Conseil 2007 64(4):661-670; doi:10.1093/icesjms/fsm016
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Harvest control rules for the Western horse mackerel (Trachurus trachurus) stock given paucity of fishery-independent data
Cefas Lowestoft Laboratory, Pakefield Road, Lowestoft, Suffolk, NR33 0HT, UK
Correspondence to B. A. Roel: tel: +44 1502 524358; fax: +44 1502 524511; e-mail: beatriz.roel{at}cefas.co.uk
Roel, B. A., and De Oliveira, J. A. A. 2007. Harvest control rules for the Western horse mackerel (Trachurus trachurus) stock given paucity of fishery-independent data. – ICES Journal of Marine Science, 64: 661–670.The Western horse mackerel stock, widely distributed in EU waters, is characterized by spasmodic recruitment. Currently, the strength of a year class cannot be confirmed before it is fully recruited to the fishery and has reached the age of 5 years. The only fishery-independent information available is an estimate of egg abundance made every third year. The state of the stock is considered uncertain, and there is no agreed management plan. Following EU requests, a set of harvest control rules (HCRs) that allows for the increased proportion of juvenile fish taken by the fishery in recent years was tested by simulation. The proposed HCRs are based either on the results of a full assessment (Constant-proportion strategy) or simply on the egg estimate used as an indicator of stock status (Slope strategy). Biological risk is compared for scenarios where: (i) uncertainty regarding stock dynamics and in the relationship between egg data and spawning-stock biomass is high (current situation); (ii) variable fractions of the total allowable catch (TAC) are taken in the areas occupied by juveniles and adults; and (iii) there is an implementation error. Results suggest that taking a larger portion of the TAC in the area occupied by juveniles increases the risk of impaired recruitment. Comparison of the Constant-proportion and Slope strategies suggests that the former is more conservative, if the assessment is unbiased or if bias is low. Given the paucity of fishery-independent data, a strategy resulting in an approximately constant catch may be appropriate for this stock.
Keywords: biological risk, data poor, harvest control rules, horse mackerel
Received 14 July 2006; accepted 2 January 2007; advance access publication 7 May 2007.
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Introduction |
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 Top Introduction MethodsResultsDiscussionAppendix (adapted from De...References |
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The Western horse mackerel is one of three widely dispersed stocks of Trachurus trachurus (ICES, 1990, 1991) in the Northeast Atlantic, extending from the Gulf of Cadiz to the Norwegian Sea (Figure 1). The spawning-stock biomass (SSB) of the Western stock has been dominated by an outstanding 1982 year class (
18 times the long-term average), a situation not unlike that of the South African horse mackerel, which produced two extraordinary year classes that dominated the fishery during the period 1950–1971 (Geldenhuys, 1973). Such stocks, characterized by strong, low-frequency components without clear periodicities, can be classified as "spasmodic", sensu Spencer and Collie (1997). Caddy and Agnew (2004) classify a species as spasmodic if there is good recruitment only once in a generation, a classification that would also include the Western horse mackerel.
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The strong 1982 year class has been gradually fished down, whereas recruitment has remained low, resulting in a steady decline of SSB since its peak in 1988. The ongoing decline makes it difficult to provide advice on sustainable management, so in response to a joint EU–Norway request to ICES to "advise on appropriate management systems including management strategies, objectives and ecosystem considerations" for the stock (ICES, 2005b), we evaluated simple stock assessment approaches and management based on a simulation study.
The only fishery-independent information available is from triennial egg surveys based on the Annual Egg Production Method (ICES, 2006a). In the past, this application resulted in an estimate of SSB based on an estimate of total fecundity, that was also obtained during the survey. Similar to other stocks of the genus Trachurus (Karlou-Riga and Economidis, 1997), the Western horse mackerel stock is almost certainly an indeterminate spawner. Therefore, total fecundity is unknown. Recent analytical assessments, based on the Separable ADAPT (SAD) assessment method, which combines a separable virtual population analysis (VPA) with an ADAPT model structure (ICES, 2002), have used the triennial estimates of egg abundance as indices of SSB.
The fishery on the Western stock can be separated spatially into two components, one mainly exploiting juveniles and the other targeting adults (ICES, 2003). Therefore, any potential management evaluation would need to include an analysis of the juvenile fishery, as well as of stock boundaries and misreporting (ICES, 2005b). Consequently, we defined two "fleets": one operating in the "juvenile area" (ICES Divisions VIIe, f, g, h, VIIIa, b, d) and the other operating in the "adult area" (Divisions IIa, IIIa-west, IVa, VIa and b, VIIb, c, j, k). From approximately 1994, the fishery shifted from a fishery on adults towards a fishery on juveniles. This may have been caused by the lack of older fish (decline of the 1982 year class) and the development of a market for juveniles. The percentage of the total catch (by weight) in the juvenile area increased gradually to peak at 60% in 2001, then declined.
Harmonizing stock distribution areas with total allowable catch (TAC) management areas is an important issue, if the stock is to be managed rationally. In this respect, two issues have to be considered: (i) that the North Sea and Western stocks mix in the English Channel, where they are exploited by a single fishery, and (ii) the mismatch between the TAC applying to EU waters only and the Western stock distribution area extending into Norwegian waters. Implementing the changes required to address these problems could be a long process, but in the meantime, the stock needs to be managed.
ICES (2006b) suggested that for stocks that cannot be assessed properly, a small TAC might be set with small annual adjustments based on trend indicators. If fecundity were assumed constant (even if unknown in absolute terms), such a trend indicator might be based on egg abundance from the triennial egg survey. However, fecundity is unlikely to be constant over time. In a recent study, De Oliveira et al. (2006) suggest that batch fecundity in the stock is stable in relation to fish size. An analysis of the raw standing-stock fecundity data revealed a weak but positive relationship between fecundity and fish weight (ICES, 2002). We account for this relationship in the operating model (OM) by making fecundity a function of the age structure of the stock. Such an approach seems sensible for a stock characterized by strong year classes passing through incidentally.
Management strategies can only be tested for robustness by simulation (Kell et al., 2005a). We present the results of simulation testing of several possible harvesting strategies considered for Western horse mackerel. In essence, our objectives are to: (i) evaluate simple multi-annual (3-y) TAC harvest control rules (HRCs) based on the existing information; (ii) investigate the effect of an increasing exploitation rate in the juvenile area; and (iii) analyse the impact of implementation error in the strategies considered.
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Methods |
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TopIntroductionMethodsResultsDiscussionAppendix (adapted from De...References |
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The (Fortran-coded) simulation framework accounts for the key features of the stock, in particular the main sources of uncertainty such as observation and process error (see Appendix), estimation error, and implementation uncertainty (Rosenberg and Restrepo, 1994).
Operating model
Biological parameters
The OM is based on the parameters estimated in the latest exploratory assessment (ICES, 2006c). There is a scaling problem in the numbers-at-age estimated by the SAD assessment, which is linked to the uncertainty in realized fecundity. In an attempt to overcome the problem, we assume that the slope of the relationship of standing-stock fecundity (fecundity measured before the onset of spawning) per female against fish weight (both in gramme; ICES, 2002) would apply to total realized fecundity, but we estimated the intercept in a modified version of the SAD assessment (ICES, 2006c). The intercept was estimated by assuming lognormal error and by setting a penalty term in the likelihood function, a frequentist equivalent of a Bayes prior (Butterworth et al., 2003). The penalty term gave low weight to values of the intercept, which could result in a total realized fecundity > 2080 oocytes per gramme females. This upper limit was based on results from the application of the Daily Egg Production Method (Eltink, 1991). Perfect knowledge was assumed for the weights-at-age in both stock and catch, age-at-maturity, and natural mortality. The values were based on historical data.
Estimates of egg abundance and SSB were based on the numbers-at-age generated by the OM and on the fecundity-at-age relationship explained above. The assessment was simulated by introducing bias and error to the SSB (see Appendix).
Exploitation
Given the recent development of a fishery on juveniles (consisting of fish 1–3 years old) and its likely impact on sustainable exploitation, separate HCRs for the juvenile and adult areas should be considered. In the absence of a reliable recruitment index for the next year's catch, the juvenile fishery can only be regulated by a fixed catch or by limiting effort. Effort control on pelagic shoaling species is generally considered ineffective (Hilborn and Walters, 1992), so might have to be combined with area closures. Because our OM cannot take spatial distribution into account, area closures cannot be evaluated, and we only consider HCRs that result in a TAC.
Fixed exploitation patterns estimated for the juvenile and the adult areas separately were based on ICES (2003). Selectivity of juveniles is greater in the juvenile area than in the adult area (Figure 2a). In the absence of information on annual variability, the simulations assume a constant exploitation pattern for each fleet. Although this assumption may be unrealistic, the HCRs were tested for scenarios allowing for different fractions of the overall TAC being taken in the juvenile area. Given that no area management is proposed, this approach effectively results in a range of selectivity patterns.
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Another question is whether an annual or a multi-annual TAC is more appropriate in this case. Currently, the TAC is adjusted every year based on the analytical assessment performed with the SAD model. Conversely, a 3-y TAC could be considered, based on an assessment every third year, when the results of the new egg survey become available. Some arguments in favour of multi-annual TACs for Northeast Atlantic mackerel (Scomber scombrus; Simmonds et al., 2003) also apply to Western horse mackerel: (i) tuning data are restricted to one point estimate of SSB every third year; (ii) the noise in the SSB data is carried over to the assessments made between survey years; and (iii) low recruitment variability (besides the infrequent occurrence of exceptionally large year classes) and the absence of clear indication of changes in weight and/or maturity over time.
Simmonds et al. (2003) concluded that over-parameterization may be a problem in the mackerel assessment, and that the most reliable assessments are those with an egg survey in the terminal year. Similarly, the egg production estimate available for horse mackerel every third year substantially affects the tuning and causes large revisions in the perception of recent stock development. Therefore, testing a 3-y TAC regime would seem appropriate.
We model the mismatch between TAC area and the area where the catch is taken as implementation error. Examination of catch trends suggests that the EU TAC was largely exceeded as long as the strong 1982 year class was present. In recent years, when this year class had virtually disappeared, total catches were close to or slightly below the EU TAC. Based on historical information (Figure 2b), we estimate the overshoot as a function of the EU TAC, with random variation added. Notwithstanding stronger enforcement in recent years, the 2005 TAC was exceeded by 20%.
Choice of simulation period
To account for spasmodic recruitment (see Appendix for detailed formulation), the simulation period needs to be of sufficient duration for an average of one major recruitment event to be included. Because managers may wish to know how to make the best use of a strong year class, ideally the simulation period should see such a year class passing through the fishery. Given that the SAD models ten formal age groups, the simulation period was fixed at 20 y. Because the results are contingent on the recruitment assumptions used, results are also shown for the case when recruitment fluctuates around the average, excluding the 1982 year class (Table 1: no spasmodic recruitment).
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TAC strategies tested
Results are presented for three types of 3-y TAC strategies:
- Slope strategy 1: The TAC equals last year's TAC, adjusted by a function of the trend over the last three egg abundance estimates, f(slope), but subject to some minimum value (TACmin):
This formulation ensures a minimum TAC of TACmin, unless the stock is depleted and unable to support it, in which case Equation (9) and the condition stated thereafter applies. The function of the slope results in values between 0 and 1.4 (Figure 2c). However, the HCR that incorporates this function [Equation (1)] caps upward adjustment of the TAC, so that the TAC cannot increase from one year to the next by > 40%. Conversely, the TAC cannot be decreased to zero unless TACmin equals zero. Two slope functions (Figure 2c) were considered: f1(slope)—the TAC is decreased quickly for slope values < 0 and is increased slowly for values > 0; and f2(slope)—TACs are adjusted at a constant rate irrespective of whether slopes are positive or negative [which means slower for slope values < 0 than for f1(slope)].
(1)
- Slope strategy 2: The TAC is a weighted average between some reference TAC (TACref) and last year's TAC adjusted by f(slope):
where w is a weighting factor (for w = 1, this strategy reduces to a constant TAC regime). The slope function used in this case is the f2(slope) (see above).
(2)
For the two slope strategies considered, both the TACmin and the term that includes TACref ensures that fishery closures are kept to a minimum. The actual levels of both TACmin and TACref are defined based on the results from simulation testing of a range of possible values.
- Constant-proportion strategy: The TAC is computed as a fraction (
) of the estimated SSB:
Table 1 provides an overview of the HCRs and sensitivity tests presented. The base case for the fraction of the total TAC taken in the juvenile area is
(3) 
= 0.5. Results using other values ( = 0.3 and 0.7) are shown only for Slope strategy 2 and the Constant-proportion strategy, and the same applies for effects of overshooting the TAC. The spasmodic recruitment option was turned off in runs for the slope strategies.
Performance statistics
The performance statistics below, based on 500 simulation runs, should allow for an informed choice of the most appropriate TAC strategies presented.
Risk SSB < Bthresh: the probability of the SSB falling at least once within the simulation period below a biomass reference point Bthresh equated to the biomass that produced the extraordinary 1982 year class, but should be kept consistent with the assessment results. The biomass in 1982 is considered a good proxy for Blim, but can only be interpreted in relative terms because of scaling problems in the assessment (ICES, 2006c).
Frequency SSB < Bthresh: the average number of times SSB fell below Bthresh during the 20-year projection period.
Mean catch: median value of the average annual catch over 20 y.
Terminal SSB: median values of the biomass at the end of the 20-y projection period.
Mean interannual catch variability: mean value of the average 20-y interannual catch variability (ICV):
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| (4) |
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Results |
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TopIntroductionMethodsResultsDiscussionAppendix (adapted from De...References |
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The results from implementing four HCRs derived from Slope strategy 1 (cf. Table 1) are compared in Figure 3a. The curves represent the Risk SSB < Bthresh associated with the mean catch taken during the 20-y projection. Each successive point on a curve (from bottom left) results from taking an increasing fraction of the stock biomass (increasing
or ß).
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Slope strategy 1 performs best for the f2(slope) option, particularly for mean catch > 200 kt. The f1(slope) option can only produce such large catches on average by imposing a greater exploitation rate (ß), resulting in greater risk and a larger ICV. The value of TACmin hardly influences these results.
Based on these results, f2(slope) was used to investigate the effects of applying Slope strategy 2 in terms of mean catch and associated risk for various values of w (Table 1). The TACref was fixed at 150 kt, which has been the agreed TAC for Western horse mackerel since 2004 (including ICES Division VIIIc). The risk curves (Figure 3b) show little difference, and apparently this strategy is not very sensitive to whatever weighting is applied. The Constant-proportion strategy performs substantially better than Slope strategy 2 as long as the mean catch is < 300 kt (Figure 3b). Beyond that level, Slope strategy 2 leads to less risk.
The terminal SSB and the mean ICV are compared in Figure 4 for the four values of w used with Slope strategy 2 and for three scenarios corresponding to a low (
150 kt), medium (250 kt), and high (370 kt) mean catch. The constant catch strategy (w = 1) appears to be more conservative for high-catch regimes because it results in a bigger SSB at the end of the projection period.
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Mean ICV is a direct function of the weighting factor (w) and of the level of exploitation. Consequently, catch variability increases rapidly at higher levels of exploitation, particularly when w is low. This shows that, when the slope term of the HCR has more weight [Equation (2)], the TACs may reach high levels, but those may often be followed by low TACs and stock collapse. This can be explained by the fact that the TAC is kept constant during 3 y, and that, subsequently, the TAC cannot be reduced fast enough.
The risk of SSB < Bthresh was computed by year for Slope strategy 2 with w = 0.5, for the Constant-proportion strategy with and without assessment bias, and for two levels of catch. For a mean catch of 200 kt, Slope strategy 2 is more risk-prone than the Constant-proportion strategy, even when the bias in the assessment is as high as 20% (Figure 5a). For a mean catch of 300 kt, the former performs better initially, but is soon outperformed by the latter (Figure 5b). The initially better performance of Slope strategy 2 is related to the starting conditions of the projections, which assume a low TAC in 2004. For a mean catch of 150 kt, all options resulted in annual risks < 5% (not shown).
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Figure 6 shows the risk of SSB < Bthresh for different proportions of the TAC caught in the juvenile area (
= 0.3 and 0.7) for the Constant-proportion strategy and Slope strategy 2. Although higher proportions result in somewhat greater risks, the differences are relatively small for either strategy.
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A TAC overshoot comparable with the level seen in the past increases the risk of SSB < Bthresh substantially for both the Constant-proportion strategy and Slope strategy 2 (Figure 7) for any given level of exploitation (
and ß, respectively).
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Switching off the spasmodic recruitment option reduces the productivity of the stock; as a result, the exponential increase in biological risk is triggered at lower rates of exploitation than with the case where chance events of high recruitment are taken into account (Figure 3).
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Discussion |
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TopIntroductionMethodsResultsDiscussionAppendix (adapted from De...References |
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The arguments for a 3-y TAC regime given by Simmonds et al. (2003) for Northeast Atlantic mackerel also apply to Western horse mackerel. However, given the paucity of fishery-independent information and the uncertainty in the assessment, any 3-y TAC rule needs to be conservative. The slope functions that are proposed limit the maximum increase in TAC from one year to the next to 40% at most for ß
1, which in itself appears cautious. However, the slope strategies considered potentially result in very large TACs when exploitation is heavy, and this can reduce the stock substantially, because they are set for three years at a time. If the two slope functions investigated are compared for the same ß, the associated risk is higher for the constant rate of change option [f2(slope)], because if the TAC has accidentally been set too high, it is only reduced slowly 3 y later. Nevertheless, for the same mean catch, strategies based on f1(slope) result in greater associated risk (Figure 3a). Although the difference only becomes apparent for a mean catch > 200 kt, the result seems counter-intuitive. The reason is that such high yields can only be achieved by implementing comparatively high rates of exploitation, and the stock cannot sustain the associated large increases in TAC if they are kept constant for 3 y in succession. Subsequently, the TAC cannot be reduced fast enough to prevent stock decline. The differences between slope strategies are more apparent in terms of ICV. If stability in yield is a management objective, Slope strategy 2 is more appealing because variability can be regulated by adjusting the weight (w) given to the previous year's TAC, once an acceptable long-term reference or target level has been agreed. Other criteria could be used to decide on an appropriate value for w. For example, Johnston and Butterworth (2005) found that for the South African West Coast rock lobster (Jasus lalandii) fishery, a value of 0.5 leads to smooth and usually monotonic trends. Similarly, minimum annual yields required for the industry to remain viable may dictate the choice of a minimum TAC.
Notwithstanding the above and central to this investigation is the comparison between HCRs based on survey trends only and the one based on the results from a full stock assessment. The Constant-proportion strategy would be expected to result in smoother stock trajectories and consequently TACs (Hilborn and Walters, 1992) and to perform better than slope strategies because it responds directly to all information available at the time of adjusting the TAC. The weak link between the estimates of egg abundance and the SSB in the simulations might contribute to the poor performance of the slope strategy. Although preliminary trials to strengthen this link did not result in substantial improvement in the performance of the slope strategy, our results suggest that, if the assessment overestimates the SSB, the slope strategies are likely to perform better, particularly if the exploitation rate is high. At low rates of exploitation, all the options compared result in annual risks < 5%. Therefore, if catches could be kept relatively low on average, a slope strategy may be preferable because of its simplicity.
To conclude, given the uncertainties in the dynamics of the Western horse mackerel stock and the paucity of fishery-independent data, HCRs that are based either on an analytical assessment or on a close-to-constant catch strategy may perform better both in terms of maximizing catch and reducing biological risk than HCRs based on indicators such as egg abundance. However, because of the scaling problem mentioned above, it is difficult to specify the catch level that is sustainable (ICES, 2006a). Based on empirical evidence, ICES (2006a) advises that, if recruitment remains at the level observed in recent years, catches of 150 kt appear sustainable. Managers would still have to choose between HCRs that will result in that level of catch and agree on issues of interannual variability and minimum catch that would allow the industry to remain viable. Further, given the possibility of a pulse of exceptionally good recruitment, any HCR for this stock should retain enough flexibility to increase the TAC in response to an increase in the SSB (Caddy and Agnew, 2004).
The associated risk under conditions of TAC overshoots and/or increased pressure on the juvenile area increases for all strategies investigated. If no exceptional year classes recruit to the stock, productivity and the sustainable catch level will be lower. For example, for an average catch of 150 kt, the associated risk for both Slope strategy 1 and the Constant-proportion strategy is 8% higher when spasmodic recruitment is switched off. In short, if the conditions assumed for the base case are not met, both strategies result in similar increases in risk to the stock. The simulation approach used here has been applied before to assess and compare the performance of selected management strategies for the Thames (Blackwater) herring (Clupea harengus; Roel et al., 2004), but it may pose problems when assessing the merits of a particular strategy against management objectives expressed in absolute terms, e.g. attaining a specific catch level over a period. Kell et al. (2005a, b) point to a need to incorporate full assessment feedback in the framework. Future work could address this issue by performing a SAD assessment in FLR (Kell et al., 2007).
Signatories to the World Summit on Sustainable Development (WSSD, 2002) are committed to maintaining or restoring stocks to levels that can produce the maximum sustainable yield (MSY). Kell and Fromentin (in press) highlight the difficulty in defining and estimating MSY in a biological sense and in implementing MSY management strategies, because of the inherent uncertainty of the systems being managed, limits in knowledge, and our ability to assess stocks. MSY is usually calculated based on models that assume that survival rates of eggs and larvae do not change substantially along with changes in the age structure of the population. Recently, this assumption has been challenged repeatedly, because larger, older spawners appear to make disproportionately greater contributions to production of viable offspring (Longhurst, 2006). This effect may be even more important for long-lived species (Birkeland and Dayton, 2005). Therefore, given the life history characteristics of horse mackerel, resulting in sporadic events of exceptionally large year classes that drastically change the age structure of the population, defining MSY with any precision is likely to be extremely difficult.
We propose managing the Western horse mackerel stock by means of HCRs that allow for monitoring stock responses such as age structure or SSB at the end of the implementation period. HCRs are not necessarily precautionary (Kirkwood and Smith, 1996), but we formally evaluate the likelihood that they achieve objectives that, a priori, have to be agreed among all players: politicians, economists, scientists, fishers, and managers. The performance statistics presented here allow us to evaluate HCRs against objectives underlying the MSY concept, such as maximizing catch and ensuring sustainability.
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Appendix (adapted from De Oliveira et al., 2006) |
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TopIntroductionMethodsResultsDiscussionAppendix (adapted from De...References |
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Spawning-stock biomass
The SSB in the underlying model, referred to as the "true" SSB, is calculated as:
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| (5) |
Recruitment
Recruitment is generated using a combination of the Ricker stock–recruit function with parameters a and b (Figure 8) estimated from a fit to stock–recruit estimates derived from the SAD assessment and a process that allows the influx of very large recruitment with a frequency of roughly 1 in 20 y [as long as SSB is at least one million tonnes, to prevent large recruitment being generated from SSB values lower than the lowest estimated in the most recent assessment; Equation (6)]. The recruitment variation and serial correlation parameters, 
R and 
ser [Equations (7) and (8)], are derived from this fit.
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| (6) |
is independently drawn from a U[0; 1] distribution, and
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| (7) |
A cap was placed on the level of recruitment generated in the most likely case [top row of Equation (6)], so as not to exceed the maximum estimated in the most recent assessment.
A cumulative probability distribution of the recruitment values used in the simulations and of the historically observed time-series (excluding the 1982 year class) is shown in Figure 9. Simulated values of recruitment, based on the Ricker curve, larger than the 95th percentile of the distribution, were omitted in the simulations.
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Numbers-at-age
The underlying age-structured deterministic model is based on a separable assumption with regard to fishing mortality and selectivity and assumes a plus-group at age 11. Uncertainty in the starting numbers-at-age is taken into account.
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| (8) |
Catch and fishing mortality
The TAC provided by the management strategy is split between the juvenile and adult areas using as follows: TACju,y = TACy and TACad,y = (1 – )TACy. The fishing mortality that results from applying TACju,y and TACad,y is calculated by solving for Ff,y from the following (for f = ju and ad):
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| (9) |
This results in a pair of non-linear simultaneous equations that require a numerical solution. An upper limit is placed on catching efficiency. To achieve this, Ff,y is restricted to be 11, which results in
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TACf,y, even when Ff,y < 11.
Generating egg-abundance observations
The "true" egg abundance (EGGtrue) is obtained from SSBtrue [Equation (5)] and modelled based on the relationship between egg abundance and SSB estimated from the SAD model (ICES, 2002), which has been extended to account for the fecundity–weight relationship referred to in the main text. To incorporate different components of variance into this relationship, the total variance can be apportioned into a "process" error component (
egg) linking EGGtrue to SSBtrue (which could result, in part, from variability in fecundity), and an "observation" error component (cvegg) linking observed egg abundance (EGGobs) to EGGtrue:
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| (10) |
egg represents the process error component in log-terms, 
yN[0;1], and SSBW is calculated in the same way as SSB, but replacing wastock with (wastock)2 in Equation (5), reflecting the dependence of fecundity on mean weight (q + bwastock), with parameter b derived empirically (see description of the OM) and with q estimated from the extended SAD model (which accounts for this relationship); and
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| (11) |
y N[0; 1]. |
Acknowledgements |
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We thank Mike Armstrong and Carl O'Brien for their encouragement and advice, without which this manuscript would not have been written, and members of the ICES Working Group on the Assessment of Mackerel, Horse Mackerel, Sardine, and Anchovy for stimulating discussions which allowed us to refine our views. Finally, we are very grateful to guest editor Niels Daan, Colm Lordan (Marine Institute, Ireland), and Mike Hammill for their helpful comments during manuscript finalization.
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References |
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