© 2005 International Council for the Exploration of the Sea
Investigating the use of proxies for fecundity to improve management advice for western horse mackerel Trachurus trachurus
a CEFAS Lowestoft Laboratory Pakefield Road, Lowestoft, Suffolk NR33 0HT, England, UK
b Animal Sciences Group, Wageningen UR, Netherlands Institute for Fisheries Research PO Box 68, 1970 AB IJmuiden, The Netherlands
*Correspondence to J. A. A. De Oliveira: tel: +44 1502 562244; fax: +44 1502 524511. e-mail: j.deoliveira{at}cefas.co.uk.
Observations of fecundity from the 2001 western horse mackerel spawning-stock biomass survey suggest that the species is an indeterminate spawner. Therefore, estimates of fecundity based on biological analyses and until recently used in the calibration of the stock assessment are now questioned. The stock is assessed by fitting a linked Separable and ADAPT VPA-based model to the catch-at-age data and to the egg production estimates. Currently, the assumption is that egg production and spawning-stock biomass are linked by a constant but unknown fecundity parameter, estimated within the model. In this study, the effects of introducing relationships linking biological indicators of fecundity, such as lipid content or feeding intensity during the spawning season, to actual fecundity are examined within a simulation framework. Simulations suggest that when the underlying relationships between fecundity and the proxy are poorly described, weak, or based on a relatively short time-series of data, the assumption of constant fecundity will result in better management advice than using the proxy.
Keywords: egg production, fecundity index, management advice, simulation testing, spawning-stock biomass
Received 7 July 2004; accepted 23 July 2005.
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Introduction |
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 Top Introduction MethodsResultsDiscussionAppendixReferences |
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There is growing interest in the role of condition and fat content in the population dynamics of commercially exploited stocks (Marshall et al., 2000; Marteinsdottir and Begg, 2002; Blanchard et al., 2003; Morgan, 2004). Most recent studies have concentrated on the link between spawning potential and subsequent recruitment. However, as egg production is often used as a proxy for SSB in stock assessments through the use of ichthyoplankton surveys, investigators are beginning to suggest that fat, or condition, be incorporated as a proxy within the assessment as well (ICES, 2003a). Attempting to reduce the unexplained variance in the relationship between SSB and eggs may benefit the quality of a stock assessment. For example, incorporating a proxy for fecundity could well reduce process error (although it could also introduce other sources of error such as measurement error). A case in point is western horse mackerel Trachurus trachurus.
For a number of years questions about the spawning strategy of western horse mackerel have been asked (Abaunza et al., 2003), and in 2003 growing evidence led scientists to conclude that it was an indeterminate spawner (ICES, 2003a). The only fishery-independent data for calibration of the assessment model is a time-series of triennial egg survey estimates. Prior to 2002, this time-series was combined with fecundity estimates to create an SSB time-series. However, in 2002, given concern about the fecundity estimates, a total egg production time-series was used instead (ICES, 2003b). The western horse mackerel stock is dominated by a few strong cohorts, with the extremely strong 1982 and less abundant 1987 year classes constituting the bulk of recent catches. Additionally, in recent years there has been a change in the selection pattern towards increasing exploitation of younger fish. To account for some of these problems, the stock is now assessed by fitting a linked Separable VPA and ADAPT VPA-based model (SAD model; ICES, 2003b) to the catch-at-age data and to the egg estimates. An assumption is made that the egg production estimates are based on a constant, but unknown, fecundity that is estimated in the assessment model.
Total egg production time-series can be used in assessments in the same way as larval abundance series as indicators of SSB (ICES, 2003c), but if large-scale trends in fecundity occur (as seen in NE Atlantic mackerel: ICES, 2003a; and noted in horse mackerel: Macer, 1974; Karlou-Riga and Economidis, 1996), they may introduce bias into the assessment and negatively affect the management of the stock. Furthermore, if interannual variability in fecundity is ignored and an empirically estimated variance parameter for the relationship between eggs and SSB is not available, uncertainty in the assessment would likely be underestimated, the more so if this interannual variability is substantial.
A proxy could be used to account for variability in fecundity (Marshall et al., 1999; Blanchard et al., 2003). Such a proxy may be condition factor, lipid content, or feeding intensity during the spawning season (SchÜlein et al., 1995; Girish and Saidapur, 2000; Kreiner et al., 2001; Henderson and Morgan, 2002). However, it is likely that the assessment would only improve if the predictive power of an index is substantial. Also, to remain biologically plausible, it is important to understand the functional relationship between the index of the proxy and fecundity prior to its use. Another concern is the effect of changes in the age-structure of the population on total fecundity and on fecundity proxies. Although there is no evidence, for horse mackerel, of an increase in fecundity per body weight as body weight increases (fecundity per gramme appears stable with size: Macer, 1974; Eltink and Vingerhoed, 1989; Karlou-Riga and Economidis, 1997; Abaunza et al., 2003), it is considered an indeterminate spawner, and older females could, for example, spawn for longer periods or more often than younger ones. Therefore, changes in age-distribution could potentially lead to systematic trends in the relationship between SSB and mean eggs. However, in the absence of evidence to the contrary, in this study we assume that if total fecundity is affected by the age-structure of the stock, then this will also be reflected in the fecundity proxies considered.
This study makes use of a simulation framework, which is a useful way to analyse the sensitivities to assumptions within assessment methods, the robustness of stock dynamics to management measures, and the ability to detect real changes in stock dynamics, given the uncertainties inherent in the stock assessment process (Punt, 1992, 1993; Kirkwood, 1992, 1997; Rosenberg and Restrepo, 1994; Kell et al., 1999; McAllister et al., 1999). The use of simulations also provides a tool to investigate the impact of new techniques in assessments prior to their implementation (Cochrane and Starfield, 1992; Basson, 1999; De Oliveira and Butterworth, 2005). A simulation framework is used here to investigate the effects of different assumptions about the relationship between an index of fecundity (the proxy) and true fecundity on the sustainable exploitation of the stock. We consider the strength and nature of this relationship within the context of management advice. The underlying stock dynamics assume three possible functional forms for the relationship between the index and true fecundity, one linear, and two non-linear. The exploitation is regulated on the basis of an annual TAC calculated as a fraction (
) of the perceived spawning-stock biomass (SSBperc). This study evaluates and compares the consequences of assuming fecundity constant, the current practice within ICES (ICES, 2003b), and alternatively making use of an index of fecundity with its associated potential problems.
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Methods |
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Simulations are based on a deterministic age-structured model with recruitment generated on a stochastic basis; these serve as the "true" underlying dynamics of the western horse mackerel stock (Appendix). The stock-recruit function used is a mixture between a Ricker function and a process that overrides the Ricker function to allow for the influx of a very large recruitment roughly once in 20 years (Macer, 1977; Eltink and Kuiter, 1989). There is little clear evidence for the choice of a Ricker model for Trachurus trachurus other than the depressed recruitment after the exceptionally large 1982 year class in the western horse mackerel stock. However, when considering stock to biomass relationships alone, Ricker models appear to fit other Trachurus species better than other simple models (Daskalov, 1999; Zhang and Lee, 2001).
Management is based on the perception of the spawning-stock biomass (SSBperc), which is obtained from an estimate of egg abundance (the observed egg abundance, EGGobs) by either using a fecundity index, or not. The fundamental question asked in this simulation study is whether and to what extent management of western horse mackerel can be improved, in terms of minimizing the risk of spawning-stock biomass falling below Bpa (the threshold SSB value that constrains ICES advice; ICES, 1998), and maximizing catches, by incorporating a fecundity index to obtain SSBperc (compared with the current practice of not using a fecundity index for SSBperc). In order to investigate this question, a model is required linking SSBperc to the true spawning-stock biomass, SSBtrue, through the following chain:
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"True" egg abundance
The True egg abundance is modelled on the basis of the relationship between egg abundance and spawning-stock biomass estimated from the SAD model (ICES, 2003b). To incorporate different components of variance into this relationship, the total variance can be apportioned into a process error component (
egg) linking true egg abundance to true spawning-stock biomass (where fecundity plays a role), and an observation error component (cvegg) linking observed egg abundance to true egg abundance through the sampling CV of egg abundance estimates. The values used in this study for egg and cvegg are 0.6 and 0.3, respectively. These values were selected to reflect a moderate sampling CV for the egg abundance estimates (the 2004 western horse mackerel egg survey was recently confirmed as at least 36%; Enrique Portilla, pers. comm., FRS Marine Laboratory, Aberdeen), but more uncertainty (double the observed sampling CV) about the process linking egg abundance to spawning-stock biomass. These values were not available from the SAD model, because sampling CVs were not computed at that stage, and because the SAD model is currently not structured to estimate variances as part of the assessment model fit (ICES, 2003b). A simulation run with a lower egg value of 0.3 is also attempted to investigate the effect of a smaller process error component.
EGGtrue is derived from SSBtrue with process error, as follows:
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| (1) |
y models the process error component. There are two sources of noise, A and B, so that
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| (2) |
, representing the total variance of the egg abundance vs. spawning-stock biomass relationship (in log-terms), which could in part be due to variability in fecundity. Assuming A and B are independent, can been re-expressed in terms of variances for A and B, namely 
and 
, where 0 
r 1, with r representing the split of the variance between A and B.
As fecundity is difficult to measure directly, a proxy for fecundity, represented by fi(Iy), is used, where fi is a functional form relating a (normalized) fecundity index Iy to fecundity. Assuming that is entirely due to fecundity, and that A in Equation (2) represents the noise component of y explained by the fecundity index, Iy, and B the unexplained component, then Equation (2) can be re-written as
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| (3) |
y is an independently generated random variable. A high value for r2 therefore indicates a very strong relationship between fecundity and the fecundity index Iy, which are linked by the function fi.
Functional forms for fi
Three functional forms for the relationship between the index and fecundity are considered.
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| (5) |
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| (6) |
y] = 0 and 
The quadratic model f3 is perhaps unrealistic for an index–fecundity relationship (f3(Iy) is positive for 
), otherwise it is zero or negative, so the index behaves in a similar way to temperature, where successful egg incubation, say, is only possible within a narrow temperature window, but it is nevertheless included to provide a greater contrast between the underlying functional form (in this case quadratic), and its perceived form (always assumed linear) for the "estimated index" model (see section Perceived SSB models below).
Observed egg abundance
The observed egg abundance is generated from EGGtrue, with observation error as follows:
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y 
N[0;1], and cvegg represents the sampling CV related to observed egg abundance estimates.
Perceived SSB models
Three perceived SSB models are considered relating to how SSBperc is derived from EGGobs.
No index: model (a)
This model omits the use of an index, and assumes a constant linear relationship between egg abundance and SSB.
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| (8) |
Perfect index: model (b)
This model assumes perfect knowledge about function fi (i.e. function fi is known), and assumes no measurement error for the index. It is therefore unrealistic, but it is nevertheless shown to indicate the best one could do by incorporating a fecundity index.
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| (9) |
Estimated index: model (c)
This model selects among several indices (Iy, Jy, Ky), only one of which (Iy) is related to fecundity. Selection is done through stepwise regression assuming a linear model (function g), with measurement error included for all the indices and y.
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| (10) |
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| (11) |
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| (12) |
For each simulation run, a new set of parameters 
for the estimated index model was derived prior to the 20-year projection of that simulation run. This was achieved by generating a 10- or 30-year time-series of "data" 
using Equations (3) and (12) (with additionally X = in Equation (12) to generate 
), and applying a stepwise regression technique using Equation (11) and forward selection, with a p value of less than 5% serving as a criterion for inclusion of a parameter in the model. Therefore, a situation where (incorrectly) 
(i.e. spuriously correlated indices are selected, and not the index that is actually related to recruitment) is possible. Once a set of parameters for Equation (11) has been derived for a simulation run, they are used for the entire duration of the projection period for that simulation run. A point to note with model (c) is that it effectively assumes the availability of independent estimates of SSB (becauthorse of how is generated), which is necessary in order to evaluate problems associated with estimating parameters from a time-series of data. In the case of western horse mackerel, such independent estimates of SSB are not currently available.
Setting the TAC
A simple constant proportion harvesting strategy is used to set the TAC, similar to the approach used to set TACs for South African sardine (Sardinops sagax) and anchovy (De Oliveira and Butterworth, 2004). Although this is not the procedure currently followed by ICES to facilitate TAC advice for western horse mackerel (see for example ICES, 2003b), it is nevertheless considered in this study for simplicity and ease of presentation.
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| (13) |
Summary performance statistics
Summary performance statistics are used to compare the overall performance of one perceived SSB model relative to another. Three summary performance statistics are considered:
- "Proportion < Bpa" calculates the number of cases (out of 20 years x 500 simulations) for which SSBtrue is below Bpa = 500 000 t.
- "Median Catch" calculates the median catch from 10 000 (=20 years x 500 simulations) catch realizations.
- "Average Catch Variation" calculates the average variation in annual catch Cy as follows:
(14)
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Results |
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Results were obtained for the three perceived SSB models: model (a), which corresponds to the current assumptions in the SAD model, i.e. constant fecundity; model (b), which assumes perfect knowledge of the relationship between the fecundity index and fecundity; and model (c), where the parameters of the relationship between the index and fecundity are estimated using a stepwise linear regression model (even if the underlying form of the index–fecundity relationship is non-linear) with measurement error included. Each of the plots shown (Figures 1
–5) provides a comparison between these three perceived SSB models, given values for r2 (the "strength" of the index–fecundity relationship), the number of years of data used to estimate the parameters for perceived SSB model (c), and the underlying form of the index–fecundity relationship.
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Assuming a linear underlying functional form for the index–fecundity relationship with 10 years of data used for estimating parameters for model (c), Figure 1 provides a comparison of the three perceived SSB models for three different r2 values. Each point along the "Proportion < Bpa" vs. "Median Catch" curves reflects a particular
value, with values of zero at the origin, and increasing in fixed steps (0.01) towards the top of each curve. Performance of a perceived SSB model is better than another if its associated curve lies more towards the bottom right of the plot. The numbers in the bottom right of each plot show "Average Catch Variation" for the three models (the average across all positive values considered), the top of these indicating the value for model (b), and the middle and bottom of these the values for models (a) and (c), respectively, each shown as a percentage increase relative to model (b). Results in Figure 1 indicate poorer performance of model (c) relative to model (a) for r2 = 0.5 in terms of "Proportion < Bpa" and "Median Catch". Only under conditions of a very strong index–fecundity relationship (r2 > 0.5) will model (c) do better than model (a) in terms of these two summary statistics, and in terms of "Average Catch Variation". As expected, model (a) performs the same irrespective of the level of r2. This is because it does not use an index, and because the process error structure (Equation (2)) maintains the same level of variance (and statistical distribution), regardless of the r2 value and given a linear underlying functional form for the index–fecundity relationship.
Figure 2 compares the three perceived SSB models for r2 = 0.5, for a linear underlying functional form for the index–fecundity relationship, and for the case where both 10 and 30 years of data are used to estimate the parameters for model (c). An improvement for model (c), in terms of "Proportion < Bpa", "Median Catch", and "Average Catch Variation", is apparent when a longer time-series of data is used.
Figure 3 compares the three perceived SSB models for r2 values of 0.5 and 1 and for three underlying functional forms for the index–fecundity relationship, assuming only 10 years of data available for model (c). Comparing the performance of a given perceived SSB model for different r2 values is difficult for the non-linear underlying functional forms (particularly the quadratic model), because the distribution of y becomes less Normal (and more asymmetrical in the case of the quadratic underlying functional form) as r2 is increased (Equation (3)). Therefore, the performance of model (a) apparently improves with an increasing r2 for the quadratic underlying form, even though it does not actually make use of an index, but this is artificial because it is purely due to the changing distribution of y. The same effect (i.e. apparent improvement in models (a) and (c)) is evident when moving from a linear to a quadratic underlying functional form. The overall effect of the distributional change in y is that SSBperc is more conservative for the quadratic underlying functional forms compared with the other functional forms, so once again, the apparent improvement across functional forms is artificial. Therefore, in the case of Figure 3 (and 4) the performance for a given perceived SSB model should not be compared across r2 values for the non-linear underlying functional forms, and should not be compared across underlying functional forms. Comparisons between perceived SSB models for a given r2 value and underlying functional form are nevertheless still valid.
Based on 10 years of data for model (c), Figure 3 shows poorer performance of this model than for model (a) for r2 = 0.5, regardless of the underlying functional form for the fecundity index–fecundity relationship. When there is a very strong relationship between the fecundity index and fecundity (r2 = 1), the reverse is true (model (c) outperforms model (a)) for a linear underlying functional form, but not for the other two functional forms (logistic and quadratic). This is essentially because there is a mismatch between the underlying functional form (link 1) and the perceived functional form (link 3), i.e. the estimated index model mis-specifies the index–fecundity relationship.
As shown in Figure 2, performance of model (c) improves throughout when it is based on 30 years of data (Figure 4) rather than 10 years (Figure 3). This improvement means that model (c) outperforms model (a) for both the linear and logistic underlying functional forms for r2 values of 0.5 and above (Figure 4). In the case of the logistic underlying functional form, this is probably possible because it is near-linear in a range of fecundity index values of highest probability (Equations (3) and (5)), so is well estimated by a linear model (link 3) if the time-series of data used in the estimation process is long enough. This is not the case, however, for the quadratic underlying functional form, for which a linear model is not a good approximation, even if a long time-series of data is available for estimating parameters.
Figure 5 explores the possibility of both lower levels of process (egg = 0.3) and measurement (m = 0.25) error (half the levels in Figure 2 for these quantities). Comparing Figure 5 with Figure 2, all three models show improved performance, and the differences between the models are reduced (almost negligible for lower "Proportion < Bpa" levels). Both of these are to be expected given the lower levels of noise. However, as for Figure 2, model (c) in Figure 5 is unable to outperform model (a) in the case of 10 years of data, and hardly so for 30 years of data.
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Discussion |
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For simplicity, the length of the time-series referred to throughout this study (10 or 30 years) relates to the number of data points used to both develop the hypothesis about the index–fecundity relationship, and estimate the regression coefficients of this relationship (performed simultaneously through the stepwise regression procedure for model (c)). Strictly, these two processes (hypothesis generation, and estimation of coefficients) should have been handled separately (Myers, 1998), so time-series would need to be longer than those used in this study. In contrast, the estimated parameters for the index–fecundity relationship are not updated as potentially new data become available in future to estimate this relationship. Such updating would improve parameter estimation over time and have an overall positive effect on "risk of falling below Bpa" and "Median Catch", as is evidenced by comparisons between options for model (c) based on 10 and 30 years of data.
As with all simulation studies, the results are conditioned upon the assumptions of the model. An analysis of sensitivity to other plausible values for egg, cvegg, and m warrants further investigation, as does the effect of assessment uncertainty (not currently accounted for) on results. Furthermore, the harvest strategy rule (constant proportion of SSBperc; Equation (13)) relies on the availability of an egg estimate from an annual egg survey, which is "corrected" to account for fecundity (models (a)–(c)). This model-free approach (similar to that used for South African sardine and anchovy; De Oliveira and Butterworth, 2004) is attractive due to its simplicity, but one could potentially do better than this by incorporating some form of shrinkage (e.g. by using an assessment model) to estimate SSBperc rather than just making a "point correction" to an egg survey estimate (the approach used for the South African hake stocks is model-based; Geromont et al., 1999). Simulation work has shown that harvest strategy rules based on a shrinkage approach (i.e. model-based) tend to outperform simpler model-free approaches in terms of reducing variability in performance statistics (McAllister et al., 1999). Nevertheless, the simpler approaches tend to be more robust to uncertainty (Hilborn et al., 2002).
Model (c) relies on stepwise regression, but this is not necessarily a good tool for prediction, because its criterion for variable selection is geared more towards error rates rather than predictive power. Alternative methods exist, such as those using "shrunken" estimators (as in ridge regression), or those using orthogonal (or non-orthogonal) linear combinations of the predictor variables (Miller, 1990). Also, this study only investigates the scenario of random variation in fecundity. Trends in fecundity related to changes in the productivity of the ecosystem have been observed in stocks such as cod (Kjesbu et al., 1998) and Northeast Atlantic mackerel (ICES, 2003a) and are likely to result in trends in the residuals from the assessment model fit. Scenarios where a trend in fecundity was introduced were not simulated in this study but evaluating their effects should be the subject of future work.
The main conclusion of this study, given levels of process and observation error incorporated, is that the use of a proxy for fecundity to estimate spawning-stock biomass from egg abundance estimates is likely to be detrimental for management of western horse mackerel unless a sufficiently large (and possibly unrealistic) amount of data is available. Results suggest that the use of a proxy can only improve management if a strong relationship exists between the index used as a proxy for fecundity and fecundity itself, a relatively long time-series of data is available to estimate the parameters of this relationship, and the underlying functional form of this relationship is well understood. The strength of the relationship required for improved management if a fecundity proxy is used depends on the length of the time-series available for estimating the parameters of the index–fecundity relationship. For example, results show that improvements are not possible for an r2 of 0.5 when only 10 years of data are available for estimating these parameters, but may well be possible for a longer time-series. In reality, very few studies show relationships as strong as r2 = 0.5, and few have very long time-series (Arctic cod, Gadus morhua, being a notable exception; Marshall et al., 2000). Further, if the underlying functional form of the index–fecundity relationship is not well understood (i.e. the perceived form does not match the true underlying form), poorer performance is likely when using the proxy for fecundity than when not using a proxy and assuming fecundity is constant, even if the relationship between the index and fecundity is very strong. Thus, in the particular case of western horse mackerel and despite the limitations discussed above, introducing time-series of proxies to predict fecundity is likely to result in lower yields and higher associated risk for the stock if the underlying relationships between the two variables are poorly described, weak, or based on few data.
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Appendix |
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Spawning-stock biomass
The spawning-stock biomass in the underlying model, referred to as the true spawning-stock biomass, is calculated as follows:
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| (A1) |
is the mean weight of fish aged a in the stock/catch; sa is the selectivity at age a; Fy is the fishing mortality in year y; Ma is the natural mortality at age a; pF is the proportion of fishing mortality that occurs before spawning; and pM is the proportion of natural mortality that occurs before spawning.
Recruitment
Recruitment is generated using a combination of the Ricker stock-recruit function with parameters a and b estimated from a fit to stock-recruit estimates derived from the SAD model (ICES, 2003b), and a process that allows the influx of very large recruitment with a frequency of roughly one in 20 years (Equation (A2)). The recruitment variation and serial correlation parameters, 
R and 
ser (Equations (A2) and (A3)), are derived from this fit.
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| (A2) |
is independently drawn from a U[0;1] distribution, and
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Numbers-at-age
An age-structured deterministic underlying model is used, and is based on a separable assumption with regard to fishing mortality and selectivity, and assumes a plus group at age 11. The only stochastic part of the underlying model is recruitment (Equations (A2) and (A3)).
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| (A4) |
Calculating the fishing mortality and catch
The fishing mortality that results from applying TACy is calculated by solving for Fy from the following:
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An upper limit is placed on catching efficiency (Fy 20, which results in 
for any age group, given the values used for sa and Ma). As long as Fy < 20, it follows that Cy = TACy (assuming that the TAC is always taken). However, when Fy is restricted to a value of 20, this is no longer the case and Cy is calculated by solving Equation (A5) (with Fy = 20) after replacing TACy with Cy.
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Acknowledgements |
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We thank the ICES Working Group on the Assessment of Mackerel, Horse Mackerel, Sardine and Anchovy for providing a forum for presentation and discussion of results. We also acknowledge suggestions by Doug Butterworth and Chris Darby regarding the approach used, as well as two anonymous reviewers for comments on an earlier draft. Participation of the first two authors was funded by the Department for Environment, Food and Rural Affairs, United Kingdom, and the third by the Netherlands Institute for Fisheries Research (RIVO).
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References |
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