© 2004 International Council for the Exploration of the Sea
An ecosystem element added to the assessment of Norwegian spring-spawning herring: implementing predation by minke whales
a Institute of Marine Research PO Box 1870, Nordnes, N-5817 Bergen, Norway
b Institute of Marine Research PO Box 6404, N-9294 Tromsø, Norway
*Correspondence to U. Lindstrøm: tel: +47 77609728; fax: +47 77609701. e-mail: ulf.lindstroem{at}imr.no.
Predation by minke whales is incorporated in the assessment model of the Norwegian spring-spawning herring stock (SeaStar) used by the ICES Working Group. Three assessment scenarios are performed and evaluated: (1) Default historic assessment where the natural mortality (M) is fixed, (2) assessment where natural mortality is estimated both for adult and juvenile herring, (3) assessment where consumption by minke whales is modelled and the predation and residual natural mortalities are estimated. The annual consumption of juvenile herring in the Barents Sea is estimated exogenously using diet data and a bioenergetic model. The estimated consumption is included in the objective function and the parameters determining the modelled consumption are estimated together with other free parameters of the model in a single operation. The estimated total natural mortality of juvenile herring is lower than the value assumed by the working group (M = 0.9) when either minke whales are included in the model (M = 0.49) or the parameter is estimated directly (M = 0.48). Assessment 3 generates 19% and 34% lower adult and juvenile stock sizes, respectively, than assessment 1, whereas assessments 2 and 3 generate relatively similar stock size estimates. The predation mortality constituted 45% and 10% of the total natural mortality of adult (M = 0.15) and juvenile herring (M = 0.49), respectively.
Keywords: assessment, herring, minke whales, northeast Atlantic, predation
Received 10 March 2004; accepted 14 December 2004.
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Introduction |
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Since industrialized fishing started about 50 years ago, marine ecosystems around the world have undergone profound changes in resource composition and abundance due to overfishing (e.g. Pauly et al., 1998; Jackson et al., 2001; Myers and Worm, 2003). Depletion of predatory fish in continental shelf areas, great whales in the southern ocean and Norwegian spring-spawning herring in the 1960s in the northeast Atlantic are examples of overfishing (e.g. Toresen and Østvedt, 2000; Myers and Worm, 2003). They emphasize the importance of managing marine ecosystem in a more holistic or ecosystems orientated context where inter- and intra-specific interactions are considered. Inclusion of predatory interactions in management of marine resources may therefore shed light into the causes of changes in marine ecosystems.
Scientists involved in the management of fish populations have traditionally relied on single species models where the interactions with other species are not explicitly modelled but rather treated as an exogenous influence. Thus, single species models cannot be used to answer questions such as "How do top predators impact prey populations?". A common approach to answer such questions is to build multispecies models, and then run simulations to experiment with various scenarios. These models, however, often become very complicated; estimation of n x n interaction terms is problematic for even modestly dimensioned systems (e.g. Bax, 1998). As a result, the answers given by such models may to a large extent depend on unsubstantiated assumptions. An alternative approach may be to build these interactions directly into assessment models (see Hollowed et al., 2000). This approach is feasible for a wide range of assessment models. It should be emphasized, however, that it is only possible to study first order effects with these models, i.e. the direct impact of predators on prey populations. To study second order effects that arise when predators forage on several prey populations and trophic levels, ecologically orientated models or multispecies models are indispensible tools.
Two recent workshops arranged by the IWC (International Whaling Commission) and NAMMCO (North Atlantic Marine Mammal Commission) in 2002 (Anon., 2002; IWC, 2002) emphasized the importance of understanding marine mammal–fishery interactions. Previous attempts to study marine mammal–fishery interactions in the Barents Sea have been made using minimum realistic multispecies models (Bogstad et al., 1997; Schweder et al., 1998), and this line of approach should be continued.
This study presents a framework for including predation from important predators into standard fish stock assessment models. The framework is demonstrated by incorporating predation by northeast Atlantic minke whales into the stock assessment model of herring (SeaStar). These species were selected mainly because: (i) they are ecologically important in the northeast Atlantic ecosystem (e.g. Dragesund et al., 1997; Schweder et al., 1997; Lindstrøm et al., 2002; Skaug et al., 2002), (ii) there appears to be a strong predator–prey relationship between them (Bogstad et al., 1997), and (iii) the abundance of both populations appears to have changed much both in time and space (Røttingen, 1990; Schweder and Volden, 1994; Dragesund et al., 1997; Toresen and Østvedt, 2000).
The Norwegian spring-spawning herring stock is one of the largest pelagic fish stocks in the northeast Atlantic and may exceed 10 million tonnes during periods of good recruitment (e.g. Dragesund et al., 1997). They spend the winter in fjords in northern Norway, from where they start their southward spawning migration in January (Figure 1). Spawning takes place along the coast south of the wintering areas in early April. After spawning, the adult herring migrate to their feeding areas in the Norwegian Sea. The herring larvae drift northwards into the Barents Sea, where they reside for 3–4 years until they join the adult stock in the Norwegian Sea (e.g., Røttingen, 1990; Dragesund et al., 1997).
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A combination of reduced recruitment, partly environmentally driven, and high fishing mortality caused the stock to collapse in the 1960s (Figure 2). The stock remained small until the end of the 1980s when the strong 1983 year class recruited to the spawning stock. Two consecutive large year classes emerged in 1991 and 1992 as a result of favourable recruitment conditions and, probably also, an increased spawning stock. The spawning stock reached more than 7 million tonnes in 1998 (ICES, 2003). Since then the stock has declined and the present (2003) spawning stock is about 5 million tonnes.
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The Norwegian spring-spawning herring is managed by the coastal states of the EU, Faroe Islands, Iceland, Norway, and Russia. These countries have adopted a long-term management strategy, which implies that:
- The spawning stock shall exceed 2.5 million tonnes,
- The fishing mortality of adults shall not exceed 0.125,
- The fishing mortality shall be reduced if the spawning stock falls below 5 million tonnes.
The choice of fishing mortality (F = 0.125) stems from a precautionary argument. Simulation experiments performed by the ICES Northern Pelagic and Blue Whiting Fisheries Working Group (NPBWWG) suggest optimal fishing mortalities in the range 0.13–0.15 (Røttingen, 2000). Previous simulation experiments of this stock have been based on assessments where the natural mortality of adult herring, irrespective of age and year, has been set to 0.15, as an informed guess. In 2002, NPBWWG estimated an adult natural mortality of 0.14 (ICES, 2003).
Minke whales undertake extensive seasonal migrations from subtropical breeding areas to temperate and boreal regions in spring where they exploit the seasonally high biological productivity. A combination of being numerous (Schweder et al., 1997; Skaug et al., 2002) and their opportunistic feeding habits make them one of the most conspicuous high trophic-level predators in northeast Atlantic ecosystem.
Minke whale consumption of herring and other prey in the northeast Atlantic has been assessed and has contributed to the recent scientific debate concerning the management of the northeast Atlantic ecosystem (Folkow et al., 2000; Olsen and Holst, 2001; Lindstrøm et al., 2002). However, no practical use of this knowledge has been made for the management of the resources in this ecosystem until this study.
To sum up, the following issues will be investigated in this paper: (i) the assessment of Norwegian spring-spawning herring with and without predation from minke whales, (ii) an estimation of natural mortality (M) in the above cases. A detailed description of the assessment model SeaStar will also be given.
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Material and methods |
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TopIntroductionMaterial and methodsResultsDiscussionAppendixReferences |
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Assessment of Norwegian spring-spawning herring
The assessment of Norwegian spring-spawning herring is made using the assessment model SeaStar (Stock Estimation with Adjustable Survey observation model and Tag–Return data). SeaStar, which has evolved gradually over a long time in the NPBWWG, is a statistical model using maximum likelihood parameter estimation. The model is based upon traditional assessment methods (see Skagen and Hauge, 2002; Ulltang, 2002).
In addition to traditional acoustic survey data, the use of tag–return data (mark–recapture data) has always been a keystone in the assessment of the Norwegian spring-spawning herring (see Appendix Table 1A). After the stock collapse in the 1960s, a moratorium on fishing was enforced (Hamre, 1990), during which tag–return data were the only quantitative source of information that could be used to monitor the development of the stock. In recent years a larval index series has been added, as a proxy for the spawning stock. For the juvenile component in the Barents Sea the 0-group index is used in the tuning (Table 1A).
None of the existing assessment packages could provide an assessment based on a combination of acoustic surveys and tag–return data. As a result WG members constructed a simple spreadsheet model. The model was then translated into a Fortran-based package in 1996, where the uncertainty was evaluated from Bayesian posteriors derived with the Markov Chain Monte Carlo method in 1997 (Patterson, 1999). The present version of the assessment model SeaStar is a direct continuation of the former models, but is run in Mathematica, and the uncertainty is now evaluated using bootstrapping instead of a Bayesian analysis. During bootstrapping, new input data are drawn from the same distributions as are assumed when the likelihood is constructed (parametric bootstrap). The catch data are assumed exact, and new catch data are drawn assuming that the uncertainty stems from erroneous age reading by 1 year, where an estimated transfer probability of 0.3 when the catch in adjacent age groups are equal is assumed. This probability is decreased as the difference in catch between adjacent age groups increases.
Estimating prey consumption by minke whales from stomach data
A bioenergetic model, proposed by Folkow et al. (2000) and the modelling framework by Lindstrøm et al. (2002), was used to estimate the annual consumption of prey i by the northeast Atlantic stock of minke whales in the Barents Sea over the period 15 April–15 October:
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| (1) |
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Owing to lack of diet samples in time and space we assume that the dietary importance of all prey in the Barents Sea is constant throughout the season. Additionally, because the movements of whales in the Barents Sea is unknown, we assume that the number of whales in the study area as a function of time is "bell-shaped", with a peak abundance (Npeak) in mid summer:
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| (2) |
N(µ,
2). To estimate the total mean consumption of prey, and to construct 95% confidence intervals, 500 Monte Carlo simulations were performed with respect to whale abundance and diet. This matrix is denoted Cobs,i.
Modelling predation in SeaStar
Minke whales readily switch between herring, capelin, and krill in their diet (Haug et al., 2002; Lindstrøm et al., 2002). To capture this density-dependent predation behaviour the following model is used:
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| (3) |
Because there is no feedback from herring to minke whales or to alternative prey, exogenous time-series of both minke whales and alternative prey are needed to run the model. For capelin, we use the annual biomass estimates from the joint Russian–Norwegian capelin surveys, which have been conducted since September 1972. Prior to 1972, the biomass estimates of capelin were taken from Marshall et al. (2000), who based their estimates on reconstructed stomach content data of cod. The biomass of cod by age is taken from the assessment made by the ICES Arctic Fisheries Working Group, whereas the biomass of krill is estimated.
We assume that minke whales in the southern Barents Sea feed on juvenile herring, capelin, krill, cod, and other food, whereas minke whales outside the Barents Sea feed on adult herring and other food (Olsen and Holst, 2001; Haug et al., 2002). Because whale abundance estimates were only available for 1988, 1989, and 1995 and partly for the period 1996–2001, we had to make some assumptions for the other years (Figure 3). The average whale abundance estimates from sighting surveys in 1988 and 1989 (Schweder et al., 1997) were used as abundance estimates over the period 1983–1989, whereas the abundance estimates over the period 1990–2001 are interpolated between abundance estimates from surveys 1988, 1989, 1995, and 1996–2001. Prior to 1983, the relative abundance calculated by Schweder and Volden (1994) using the time-series from the method referred to by the authors as the ACD method was used. This time-series was scaled to give the same abundance in 1983 as the abundance obtained from the sighting surveys. Although there is uncertainty as regards the absolute abundance estimates, this is less critical than the temporal trend in predation. Reducing the number of whales preying on herring with a constant proportion would simply result in a similar increase in consumption for each individual whale as total herring consumption is fixed.
A multivariate normal distribution on log-scale was assumed for the likelihood of prey consumption inside the Barents Sea:
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| (4) |
is the covariance matrix. The superscript T represents the transpose of an array, and the notation | | denotes the determinant of a matrix. Because consumption estimates are only available for the Barents Sea, the estimation of herring consumption outside the Barents Sea, i.e. in the Norwegian Sea, is based upon its effect on the time trend of the cohorts only.
Parameter estimation
The following parameter groups are estimated: terminal Fs, residual mortalities (adult and juvenile), prey-specific suitabilities, herring suitability at age in the Barents Sea, survey catchabilities and survey variance or CV (see Appendix), tagging survival, prey-switching coefficients. The estimation is performed in two steps. First, an assessment of the adult herring in the Norwegian Sea is made. The youngest year class with a free terminal F was set to the 1999 year class by NPBWWG in 2004. In this step, the terminal F for age groups younger than the 1999 year class is linearly interpolated down to zero at age –1. Next, the estimated parameters are kept fixed and an assessment of juvenile herring in the Barents Sea is made. In addition to the acoustic surveys in the Barents Sea, the 0-group index is also used as tuning data in the second step. The estimate of residual natural mortality of juvenile herring in the Barents Sea is independent of the parameters that determine the adult herring because separate data time-series are used (see Appendix). During the second step, all year classes are represented with free terminal F-values. Therefore, this estimation step will update the previous estimate of year classes younger than the 1996 year class, i.e. also some age groups that are not actually found in the Barents Sea.
The log-likelihood of the parameter vector (Pv) is given by:
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For each population group, adults and juveniles, the estimation procedure can be summarized in four steps:
- Give starting values to all free parameters.
- Run a VPA and calculate the likelihood of the tuning data (survey indices, tagging data) conditional on the current parameter values and the VPA estimate (NVPA).
- Repeat steps (i)–(ii) until the likelihood function reaches its maximum.
- Obtain the final stock estimates by inserting the free parameter values estimated in step (iii) and run a new VPA. The VPA results then become the assessment of the adult stock.
- Steps (i)–(iv) are repeated for juvenile herring, i.e. the Barents Sea component.
For comparative purposes, the standard working group assessment (assessment 1), as well as an assessment where, natural mortality rates were estimated but not predation (assessment 2), was carried out in addition to the model including predation (assessment 3). The catchabilities and tagging survival in assessment 3 were set equal to those of assessment 2, i.e. we assume that incorporating minke whales does not affect our perception of how much herring is observed in the surveys.
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Results |
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TopIntroductionMaterial and methodsResultsDiscussionAppendixReferences |
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Figure 4 illustrates the annual mean consumption of prey by minke whales in the period 1992–2001. The estimates are plotted with 95% confidence intervals determined by 500 bootstrap replicates. The figure shows that juvenile herring was by far the most heavily exploited prey by minke whales in the southern Barents Sea in the period 1992–2001. Considering the total average prey consumption in this decade, minke whales may have consumed 413 x 103 t (CV = 0.81) juvenile herring, 268 x 103 t (CV = 0.94) capelin, 212 x 103 t (CV = 0.94) krill, 195 x 103 t (CV = 1.08) other fish (haddock, saithe, and sandeel), and cod (122 x 103 t). The average estimated prey consumption varies much between years: herring (1–95 x 103 t), capelin (0–76 x 103 t), krill (0–67 x 103 t), other fish (2–68 x 103 t), and cod (0–25 x 103 t). Part of the substantial inter- and intra-annual variation in prey consumption is a result of few whale samples and great variation in diet composition between individual whales.
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Figure 5a illustrates the total biomass and Figure 5b the natural or residual plus predation mortality of the adult and juvenile Norwegian spring-spawning herring in three standard assessment scenarios: (1) standard historic or baseline assessment (Madult = 0.15, Mjuvenile = 0.9), (2) assessment where natural mortality is estimated both for adult and juvenile herring, and (3) assessment where consumption by minke whales is modelled and the residual natural mortality estimated. The estimates are plotted with 95% confidence intervals determined by bootstrapping. The standard assessment, i.e. the assessment conditions set by the NPBWWG, yields a larger perceived stock (Badult = 7.0 x 106 t, CV = 0.14, Bjuvenile = 2.5 x 106 t, CV = 0.24) than assessment 2 (Badult = 5.8 x 106 t, CV = 0.15, Bjuvenile = 1.2 x 106 t, CV = 0.27) which again yields a slightly larger adult stock but a smaller juvenile stock than assessment 3 (Badult = 5.6 x 106 t, CV = 0.14, Bjuvenile = 1.7 x 106 t, CV = 0.25). The uncertainty decreases when the natural mortality is estimated and decreases somewhat more when the consumption by minke whales is included.
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Figure 5b shows the natural mortality estimates of adult and juvenile herring in the three assessments. The stacked bars filled with circles under assessment 3 illustrate the predation mortality whereas the non-filled stacked bars illustrate the residual natural mortality. The natural mortality of the adult and juvenile stock is estimated as 0.11 and 0.48 (CVadult = 0.11, CVjuvenile = 0.18), respectively, in assessment 2. By including minke whale predation, the total natural mortality (residual + predation) of adult herring (Madult = 0.15, CVadult = 0.12) increases noticeably compared with assessment 2, whereas the natural mortality of juveniles (Mjuvenile = 0.49, CVjuvenile = 0.25) is similar. The median of the yearly age-averaged predation mortalities generated by minke whales was 0.083 (CVadult = 0.06) for adult herring and 0.048 (CVjuvenile = 0.04) for juvenile herring.
The switching coefficient (k) in the functional response model was estimated at 1.94 (CV = 0.10), 3.79 (CV = 0.05), 6.76 (CV = 0.10), and 3.15 (CV = 0.09) for herring, capelin, cod, and krill, respectively. Using a common k for all prey species, the estimated value becomes 3.83 (CV = 0.08). Using a common k for all prey, the log-likelihood decreased from –622 to –642.
The suitability of herring age was set to 1 for 1-year-old herring and estimated as 0.26 for 2-year-old herring. Initial estimation runs gave estimated values of 0 for 3- and 4-year-old herring, so these ages were omitted from further analysis.
Increasing the time period when the number of whales in the Barents Sea equals Npeak from 1 to 60 days has negligible influence on the results, i.e. the assumption about the time dependence of minke whales in the Barents Sea is not critical.
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Discussion |
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The present study demonstrates that incorporating predation by high trophic-level predators such as the minke whale in standard assessment models is feasible and can be a valuable tool in fish stock assessment, assuming there is a weak feedback from fish to the abundance of marine mammals, i.e. a decreased abundance of the fish species in question does not lead to increased predator mortality. We know little about the numerical response of high trophic-level predators with long lifespan such as minke whales owing to their slow response to changes in prey availability. Although there may be some indications that the abundance of minke whales in the northeast Atlantic has fluctuated over the past five decades both in time and space, it is difficult to say whether these space–time fluctuations in stock size are human-driven, predator–prey driven, or both.
In general, individuals should distribute themselves in a fashion that equalizes returns among habitats (ideal free distribution, Fretwell and Lucas, 1970), implying that minke whales should be less abundant in areas of low food availability. Recent ecological studies (Bogstad et al., 1997; Haug et al., 2002) and sighting surveys (Schweder et al., 1997; Skaug et al., 2002) appear to support this theory; changes in minke whales' use of prey and habitat during the past decade correlate well with changes in the abundance of their favoured prey (capelin and juvenile herring) (Haug et al., 2002). Minke whales foraging in the southern Barents Sea switched to less profitable prey such as krill and cod, resulting in lower body condition (Haug et al., 2002).
Predation thresholds, which have been observed in minke whales (e.g. Piatt and Methven, 1992; Reid et al., 2000) along with other marine mammal species (e.g., Piatt et al., 1989; Kann and Wishner, 1995), may have important ecological consequences via their stabilizing effect on predator–prey systems and coexistence (e.g. Rosenzweig and MacArthur, 1963; Fryxell and Lundberg, 1994; Abrams and Ginzburg, 2000).
In this study, the prey-switching behaviour of minke whales was modelled using a switching coefficient in the predation equation. By including this coefficient as a free parameter in the likelihood function permitted simulation of either a Type I, II or III functional response for the whales, depending on the fit to the data. This is in contrast to most predator–prey studies (e.g. Mackinson et al., 2003), where the modellers set the parameters in the functional response equation. This study suggests that minke whales exhibit a Type III functional response. However, these functional responses are estimated on a population level and are not necessarily related to the predators' reaction to ambient food. Nonetheless, our results do to some extent substantiate the study by Lindstrøm et al. (2002), who showed that the dietary importance of juvenile herring increased non-linearly as a function of herring availability.
The three assessment models generated statistically similar stock biomass estimates. Despite that, the baseline model (assessment 1) seems to generate higher stock biomass estimates compared with the two other models. The natural mortality of juvenile herring in the baseline assessment, set by the NPBWWG (Myoung = 0.9), is probably too high considering the mortality estimates generated in assessments (2) and (3). The assumption of a constant M-value across years, which implies that predators exert a Type I functional response (linear), is dubious because a density-dependent natural mortality will generate a different time trend of cohorts, and hence a different assessment despite small changes of the overall M. Despite that, no definite management-related conclusions can be drawn until the standard harvesting control rule for this stock has been re-evaluated using the new model.
Despite the small difference in the assessment of Norwegian spring-spawning herring generated by assessments 2 and 3, there may be a danger of not fully exploring the model structure if not incorporating minke whale predation on herring.
In an ecosystem perspective, it is important to separate the total natural mortality into different mortality components. This study suggests that the minke whales caused about half of the total natural mortality of adult herring, whereas for juvenile herring the minke whales' predation constituted 10% of the total natural mortality. The latter partly confirms a previous study (Lindstrøm et al., 2001), i.e. that minke whales can inflict very high mortality on juvenile herring (2–50%) during their residence in the Barents Sea depending on the residence time and availability of herring.
The international trend in the perception of how management of fish should be conducted is that of an ecosystem-based approach. Frid (2003) suggested that "Models that consider fish stocks in isolation from their ecosystem have clearly had their day, and fisheries science is moving on". Apart from the works by Mohn and Bowen (1996), who studied grey seal predation on eastern Scotian Shelf cod by combining two alternative predation models and a VPA model, and Hollowed et al. (2000), who included predation by three predators in the assessment of walleye pollock, there are very few examples of assessment models that consider ecosystem aspects other than those directly related to the fish stock in question in the ICES area.
We believe that traditional assessment models in which ecosystems aspects are incorporated will be of great importance in future management of marine resources. As such the present paper illustrates an example of how this could be done. Future management of fish will probably rely on a suite of models with different purposes. Models that address wider aspects of the ecosystem at the expense of valid short-term predictions crucial in the annual management will be of great importance. Such models might be Ecopath and Ecosim (e.g. Christensen and Walters, 2002; Mackinson et al., 2003), Gadget (Begley, 2003), Systmod (Hamre and Hattlebakk, 1998), or Scenario C (Schweder et al., 1998).
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Appendix |
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TopIntroductionMaterial and methodsResultsDiscussionAppendixReferences |
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General model description
The model is basically a VPA using Pope's approximation, where terminal Fs are estimated by maximum likelihood using six time-series of acoustic data, larval data as a proxy for spawning-stock biomass, tagging data, and 0-group data.
Surveys
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It is assumed that the surveys only give trends. Proportionality constants between survey indices and historic stock size must therefore also be estimated, adding as many free parameters as there are surveys. Data from acoustic surveys of both juveniles and mature herring are used in the tuning because the survey series in the Barents Sea as well as the 0-group data are needed to estimate the residual natural mortality of juvenile herring (Table A1). It is, however, assumed that once a year class has recruited to a survey, catchability is the same for all fully recruited ages. The fact that size-dependent selectivity and natural mortality estimates will be confounded (Thompson, 1994) makes it more sound to assume a flat selection than to try to estimate a selection pattern. An age-varying selection pattern might also affect the estimation of terminal F-values.
We assume the likelihood of survey observations P(SY; b, c) to follow a gamma distribution with a CV common for all surveys except the Barents Sea surveys. The scaling parameter b and shape parameter c of this Gamma distribution are estimated using the following relationships:
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For the Barents Sea surveys, a lognormal error distribution is assumed. When a lognormal distribution is assumed for the surveys outside the Barents Sea or a gamma distribution is assumed for the Barents Sea, quantile–quantile plots deviate strongly from a straight line, indicating that these assumptions are less appropriate.
Tag–return data
The number of tag returns x in a given year is assumed to be Poisson distributed, with mean 
equal to the proportion of fish carrying tags times the number of screened fish:
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The reliability of detectors is assumed to be 100%. Nonetheless, multiplying S accounts for detector failure with a probability constant. The number of tag returns has a two-dimensional structure since both the tagging year and the recovery year is recorded.
Larval data
A survey designed to assess the abundance of herring larvae along the Norwegian coast is conducted annually in April. These data, which are regarded as a proxy for the spawning stock, are used as a tuning series in SeaStar. The larval data are modelled analogously to the survey data, i.e., a gamma distribution with a constant CV is assumed.
0-group data
As part of the tuning series' for estimating juvenile herring abundance in the Barents Sea, the logarithmic 0-group index from the joint Russian–Norwegian 0-group survey is used (ICES, 2003). These data are assumed to be normally distributed with a constant standard deviation.
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Acknowledgements |
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We thank Dr Rob Barrett at the University of Tromsø, Norway, for correcting the language. Special thanks to the two referees and editor Verena Trenkel who contributed for improvement of the last version of the manuscript.
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Footnotes |
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1 Authorship equal.
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References |
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