© 2004 by ICES/CIEM International Council for the Exploration of the Sea/Conseil International pour l'Exploration de la Mer
Estimation of the age-specific rate of natural mortality for Shetland sandeels
FRS Marine Laboratory PO Box 101, 375 Victoria Road, Aberdeen AB11 8DB, UK
*tel: +44 1224 295393; fax: +44 1224 295511. e-mail: cookrm{at}marlab.ac.uk.
It is generally difficult to obtain reliable direct estimates of natural mortality, M, from conventional fisheries data and stock assessments. However, as a result of the closure of the Shetland sandeel (Ammodytes marinus) fishery from 1991 to 1994 and in the absence of any significant fishery in other years, research vessel survey data offer a rare opportunity to obtain estimates of M directly. A model is described that assumes that M can be decomposed into an age effect and year effects. Application of the model to the survey data produces values of M that decline from 2.1 for 0-group fish to 0.6 at age 2. There is some indication of an increase for ages 4 and older. Although there does not appear to be an overall trend in the mean value of M for the period 1985–1999, the annual values change by up to 50%. The values calculated from the model are in line with estimates obtained for the North Sea from multispecies virtual population analysis (MSVPA).
Keywords: natural mortality, research vessel survey, sandeel, separable model
Received 29 July 2003; accepted 24 November 2003.
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1 Introduction |
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 Top 1 Introduction 2 Analytical model for...3 Data4 Methods5 Results6 DiscussionReferences |
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Many analyses in stock assessment require an estimate of the natural mortality, M, of populations. This applies to basic population estimates from methods such as virtual population analysis (VPA; Gulland, 1965), as well as estimates of yield and biomass derived from yield-per-recruit or maximum sustainable yield calculations. As most populations subject to assessment were fished before scientific observations were made on them, it is often problematic to partition the total mortality, Z, between the mortality attributable to fishing, F, and M. This is because the deaths caused by predation, a major component of M, are inherently difficult to measure. Analysts, therefore, are inclined to rely on methods that estimate M indirectly from the biological characteristics of the species, such as its longevity and growth rate (Pauly, 1980; Hoenig, 1983). It is sometimes possible to estimate M within stock assessment models, provided data are sufficient. For example, stock synthesis (Methot, 1990) can estimate M given data on catch and survey abundance indices. Such estimates are rarely age dependent, and are often affected by missing catch components (such as discards) or are confounded with estimates of survey catchability.
In some cases, it has been possible to estimate M from the analysis of predator stomachs. For certain North Sea stocks, multispecies VPA (Helgason and Gislason, 1979) has made use of such data to calculate natural mortality (e.g. ICES, 1997), and these estimates are used routinely in ICES fish stock assessments. While such values probably represent the most realistic estimates available, they too are subject to modelling assumptions, such as the availability of food and the food requirements of the various stocks included in the model. Where possible, therefore, it would be useful to validate such estimates from independent data that estimate M directly. In the case of the Shetland sandeel (Ammodytes marinus) stock, recent closures in the fishery have provided an opportunity to estimate the total mortality in the stock from research vessel surveys, and hence to obtain an estimate of natural mortality.
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2 Analytical model for RV data |
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Top1 Introduction2 Analytical model for...3 Data4 Methods5 Results6 DiscussionReferences |
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One of the major potential problems of surveys is that the sample size is generally small, making abundance estimates inherently noisy. To attempt to reduce this problem, a simple model is used here to try to remove some of the noise. The model is a modification of the commonly used separable model often applied in analysis of catch-at-age data (Pope and Shepherd, 1982; Deriso et al., 1985; Gudmundsson, 1986). Unlike the other models, however, rather than F being separable, the underlying assumption is that the rate of natural mortality, M, can be decomposed into an age-specific effect, m, and year effects, k. The values of m effectively express the vulnerability of the fish to predation mortality, while the k values express the scale of predation that changes as predator populations evolve over time. This allows M to be modelled as
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| (1) |
Using the approach adopted by Cook and Reeves (1993), if populations decay exponentially over time, the number of fish, N, at the start of the year from a particular cohort with an initial number of recruits, R, is given by
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| (2) |
Now, for an abundance index, u, we may assume the relationship
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| (3) |
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| (4) |
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| (5) |
If catchability is constant for all age classes, this ratio will be unity and can be ignored. However, it is possible that it will not be constant for one or more of the youngest age classes, so in that case, estimates of the ratio will be required in order to obtain unbiased estimates of the rate of mortality. In the present study, it has been assumed that catchability is constant for all age classes because the sampling gear has a very fine mesh (16 mm) and should sample all age classes equally, provided their availability to the gear is not age dependent. There is evidence that the availability of 0-group fish differs from that of older age classes (Reeves, 1994), in which case the estimates of M obtained for that age class may be biased. This subject is discussed below.
From Equations (1) and (4), any abundance index, u, can be described in terms of the initial cohort size, ur, the age effect, m, and the year effects, k. Now, let the observed abundance index, û, be measured with lognormal error such that
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| (6) |
Given A age groups and Y years of data it is now possible to estimate the parameters ur, m, and k by minimizing the sum of squares:
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| (7) |
In practice, the estimates of k obtained by minimizing Equation(7) are sensitive to noise in the data. An alternative objective function has been used which restrained the estimates using a penalty function, i.e.
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| (8) |
Because the m and k parameters are multiplicative, it is necessary to constrain the estimates of the parameters. This has been done by setting k equal to 1 in the first year, effectively scaling all the other parameters to this value.
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3 Data |
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Top1 Introduction2 Analytical model for...3 Data4 Methods5 Results6 DiscussionReferences |
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For many years, FRS Marine Laboratory has conducted an August survey of the sandeel fishing grounds around Shetland, from which an abundance index is derived. ICES routinely uses this index in the assessment of the Shetland sandeel stock (e.g. ICES, 2002). For the current analysis, the data used were restricted to the period when the fishery was closed or taking place at a very low level. Table 1 shows the indices from 1985 to 2000 with the corresponding estimate of F from the last ICES assessment (ICES, 1998). Where F is not zero, it is well below the conventional values of natural mortality, which range from 1.2 at age 1 to 0.6 for the older age classes. For the purposes of this analysis, F was assumed to be zero for all these years.
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There are two potential problems with the survey data. First, the catchability of sandeels declines during the year (Reeves, 1994), meaning that fish aged 1 and older in August are less available than those available earlier in the year. This will not matter when estimating annual mortality rates provided that the catchability at the time of the survey is similar over the time-series of observations. Second, there may be age reading problems with older fish that will affect estimates of year-class strength. This should not have a major impact on estimates of m because these will be driven by the average decline in abundance between successive age classes and should not be unduly affected by age mis-specification.
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4 Methods |
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Top1 Introduction2 Analytical model for...3 Data4 Methods5 Results6 DiscussionReferences |
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The model described above was fitted to the survey data in Table 1. The value of
was set to 1, a fairly arbitrary choice which implies that the residual error associated with the abundance data is the same as that for the interannual change in M. Trial model fits indicated that this represented relatively weak smoothing, so should not introduce much additional bias. Estimated trends in recruitment and spawning stock biomass derived from the model fit were not sensitive to the choice of . Trial runs of the model suggested that the residuals associated with the youngest and oldest age classes were much larger than those associated with the middle age classes, and were therefore down-weighted. The choice of weight is somewhat arbitrary, but was intended to produce homoscedastic residuals and did not, in any case, have a large effect on the parameter estimates. The values used were 0.1, 1.0, 1.0, 1.0, 0.1, 0.1, 0.1, for ages 0–6, respectively.
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5 Results |
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Top1 Introduction2 Analytical model for...3 Data4 Methods5 Results6 DiscussionReferences |
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The total number of model parameters estimated is 42 with 112 observations. Adjusting for the number of parameters, the coefficient of determination after fitting the model is 0.87. Figure 1 shows the model residuals plotted against the fitted index values. Overall, the adjusted coefficient is high and the residuals show no obvious pattern, although there is some indication of a negative trend for the 0-group fish and a positive trend for fish aged 1.
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Figure 2 shows the estimated age effects, which are effectively the values of M when k=1. They indicate a high value of M for the youngest ages, decreasing to lower values at ages 2–3, then rising again for the oldest ages. Those values can be compared with the conventional values of M for the North Sea derived from MSVPA. The latter are adjusted values to account for the survey year, which runs from August to July, as opposed to a calendar year in MSVPA. Figure 3 shows the estimate of the year effects, k, over time; it fluctuates by a factor of 3 between the lowest and the highest estimates. There appears to be little or no long-term trend.
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Figures 4 and 5 show the estimated trends in spawning stock biomass (SSB) and recruitment from the model. The trends of these quantities from the last ICES assessment (ICES, 1998) are shown for comparison. The ICES assessment uses catch-at-age data, and is therefore theoretically an absolute measure of abundance, whereas the survey provides a relative index. Therefore, it is only possible to compare trends.
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6 Discussion |
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Top1 Introduction2 Analytical model for...3 Data4 Methods5 Results6 DiscussionReferences |
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The estimates of stock trends resulting from the model fit to survey data are consistent with the last ICES assessment; that assessment includes commercial catch data and used fixed values of M. While the two approaches are not entirely independent, because they both use survey data and make comparable statistical assumptions, the similarity does provide some grounds for confidence in the results, given the fundamentally different assumptions about M. Furthermore, the estimated values for natural mortality compare well with the traditional values calculated from MSVPA, especially for the intermediate age classes (Figure 2). If the survey data were seriously undermined by problems of sandeel availability in August, it would be unusual to obtain such good agreement between the three different analyses. This would suggest that the estimates of M obtained here are representative.
At the youngest ages, the current analysis implies higher values of M than the conventional ones. Some care must be taken with this comparison. First, the survey analysis assumes that the catchability, q, for all age classes is the same. Second, the survey assessment assumes that the population is closed, i.e. that there are no losses or gains through migration. If the smallest fish (i.e. the 0-group) have a higher catchability, as suggested by Reeves (1994), then the estimated M would be higher than expected. However, the error bound on the estimate contains the MSVPA value, which implies that any bias is not substantial. For older ages, the increasing rate of natural mortality with age is consistent with ecological theory about natural mortality rates, but it would be unwise to attach too much significance to the results here in view of the high uncertainty associated with them.
The survey data are inherently noisy and will be subject to annual changes in the availability of sandeels to the survey resulting from their variable behaviour (Reeves, 1994). These changes are likely to be random and would be expected to manifest themselves in the estimated year effects, k. These values are plotted in Figure 3 and do show some variability, but it is not extreme. Very large changes in annual availability, for example, would be expected to result in negative values. It seems reasonable to conclude, therefore, that catchability effects do not appear to be a major problem in the analysis.
While the time-series of observations is relatively short, the mean value of M appears to be more or less stationary, because there is no clear overall trend emerging from the year effects, k (Figure 3). At face value this is encouraging, because it would mean that the assumption of constant M in many assessments is reasonable in so far as the effect of long-term trends is concerned. Nevertheless, fluctuations in M will affect assessments that assume constant M by distorting annual fishing mortality and population estimates. How this impacts the assessment will depend not only on the magnitude of the true fluctuations, but also on the assessment model used. If the range of fluctuations estimated here is realistic, then the effect on an assessment would be expected to be large, unless fishing mortality substantially exceeds M.
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Acknowledgements |
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The author is grateful to the many staff at FRS who contributed to the collection of the sandeel survey data and for the comments of two anonymous referees. The work was funded by the Scottish Executive as part of project MF01p.
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References |
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