Sampling and a mortality model of a Norwegian fjord cod (Gadus morhua L.) population

Fisheries Research 63 (2003) 1–20

Sampling and a mortality model of a Norwegian fjord cod (Gadus morhua L.) population Torstein Pedersen∗ , John G. Pope The Norwegian College of Fishery Science, University of Tromsø, N-9037 Tromsø, Norway Received 18 January 2002; received in revised form 28 October 2002; accepted 28 December 2002

Abstract An assessment model was developed of a coastal cod population integrating data on catch-at-age from commercial catches and survey data and tag returns from commercial and survey trawl fisheries. Cod were tagged during the period 1990–1995 and recaptured in the commercial and research survey. Quarterly trawl surveys were conducted during 1990–1996, and cod were also sampled from the commercial catches during 1994–1996. The model was fitted to the different sets of data using a weighted least square minimisation. The model appeared to fit the data sets reasonably well, and the sensitivity analysis indicated that the major tensions tended to be between the survey and the commercial catch-at-age data. The biomass of 2+ cod was estimated to be 2–3 t km−1 and the age structure of the stock was dominated by the strong 1987 year-class. The estimated total mortality rate (Z = 0.4–0.5 per year) of 2+ cod was low compared to most other cod stocks, mostly because of a low fishing mortality rate (F ≤ 0.3 per year). This is a much lower fishing mortality than observed in coastal cod stocks in southern Norway. This low fishing mortality rate implies that VPA-type models would have poor convergence and hence give low precision. Thus, the integrated model using tagging data may be used for stocks exposed to low fishing mortality and where there is a scarcity of commercial catch-at-age time series. © 2003 Elsevier Science B.V. All rights reserved. Keywords: Coastal cod; Mortality model; Tagging data; Survey data; Integrated model

1. Introduction Cod is an important commercial specie in the fjords and coastal waters in North Norway (Bax and Eliassen, 1990). The coastal cod group is relatively stationary and inhabits the fjords and coastal areas. This contrasts with the northeast Arctic cod group that are long-range migrators inhabiting the Barents Sea and using the Norwegian coast as spawning area (Bergstad et al., 1987). The investigation area of this study, Sørfjord, is the inner part of the Ullsfjord system in Troms ∗ Corresponding author. Tel.: +47-776-44697; fax: +47-776-44792. E-mail address: [email protected] (T. Pedersen).

County, North Norway, and has been the site for an enhancement program. In order to evaluate effects of enhancement, batches of wild and hatchery-reared cod have been tagged and recaptures from fishermen and experimental trawling have been recorded (Nøstvik and Pedersen, 1999a,b). The area has also been intensively sampled with trawl, and data on cod feeding, catch-at-age from experimental trawling and from fishermen exist for a 7-year period (1989–1996). A good knowledge of the population dynamics of wild cod is needed to understand possible enhancement effects. The wild cod in Sørfjord grow more slowly and mature at a much younger age and smaller size than northeast Arctic cod and have a different recruitment pattern (Berg and Pedersen, 2001). There is only a

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minor migratory exchange of cod between Sørfjord and other adjacent areas (Nøstvik and Pedersen, 1999b), indicating that the cod could be considered to be a separate stock (Berg and Pedersen, 2001). Although retrospective catch-at-age analysis has been conducted on the pooled Norwegian coastal cod group north of 62◦ N (ICES, 2000), coastal cod is not separated from the northeast Arctic cod stock with respect to management. In order to evaluate management strategies and enhancement programme effects, estimates of natural and fishing mortality as well as abundance of cod are needed. Management strategies have to be based on sound interpretations of the dynamics of the cod stocks and the interacting populations. Traditional methods for estimation of mortality rates and stock sizes of cod populations (VPA, cohort analysis) rely heavily on long time series with catch-at-age data from commercial fishing (Pope and Darby, 1995; Quinn and Deriso, 1999). Sufficient catch-at-age data are not available for most fjord stocks, but Pope and Darby (1995) suggested that tagging data might be a possible substitute. There has been a growing interest in methods for integration of different sets of data in fish population models (Ulltang, 1998; Quinn and Deriso, 1999), and progress has been achieved using tagging data to gain information on mortality processes (Myers and Hoenig, 1997; Haist, 1998). In this paper we attempt to construct and evaluate an integrated population model that estimates natural and fishing mortality and abundance based on sets of data from tagging experiments, catch-at-age from fishermen’s catches and experimental trawling. This model will represent our interpretation of wild cod population dynamics in the area. 2. Materials and methods 2.1. Sampling and tagging from research surveys A 300 m wide and 8 m deep sill separates Sørfjord and Ullsfjord; the maximum depth in Sørfjord is 130 m and the area is 55.2 km2 . For a more detailed description of the topography of the investigation area, see Kanapathippillai et al. (1994). Bottom and pelagic trawl sampling was performed at four locations with depths from ca. 50 to 130 m within Sørfjord. The

trawling locations were sampled each quarter during the time period November 1989 to December 1996, except for the second quarter (q2 ) in 1993. The bottom trawl was set up with cocco gear (only rope; no rubber or steel bobbins) from 1989 to 1991. Another larger bottom trawl with rock hopper gear was used from 1992 to 1996. For more details on sampling, see Berg and Pedersen (2001). The bottom trawl (shrimp trawl) had a 38 mm mesh size in the cod end. A towing speed of two knots was used, with a towing time of 20–30 min. After the trawl had been hauled, length measurements were made on all cod in the haul. Measurements of total and gutted weight, sex and maturity stage and otolith sampling were made from subsamples of the catch. Wild cod (n = 14,052) were caught for tagging by bottom and pelagic trawls at depths from ca. 50 to 130 m during 1990–1995, and also with hand line, trap nets and fish pots in shallow water (<30 m depth) in 1995 (Nøstvik and Pedersen, 1999b) (Table 1). Most fish were tagged during the first two quarters of each year. The fish from trawl hauls were allowed to recover for ca. 30 min in a water tank before being tagged, and fish displaying abnormal behaviour or buoyancy problems were not tagged. Following length measurement, the fish were tagged by inserting an individually numbered yellow anchor tag (Hallprint T-tag) into the fish just behind the front of the first dorsal fin. The lengths of both the filament and the cylinder were 20 mm. 2.2. Tag recoveries and estimation of gear selection pattern Tagged cod recaptured in the experimental catches were always killed and measurements of fish length and otolith sampling were made from these fish. A form was sent to fishermen with a map and a request for information on recapture position, fish length and total weight, fishing gear, depth and date of catch. Recaptures from fishermen were rewarded with NOK 25, and information on release position, date and length of the released fish was sent to fishermen who returned tags with recapture information. A total of 1432 was recovered by fishermen and 744 by the research fishery (Table 3). The reporting rate (c2 ) was integrated in the model. Population models of exploited fish stocks appear to be sensitive to assumptions about the size-selectivity

T. Pedersen, J.G. Pope / Fisheries Research 63 (2003) 1–20

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Table 1 Overview of number tagged and the recaptures from the research sampling (Cr) and the commercial catches (Cf) Recapture year

Year of tagging (no. taggeda ) 1990 (945)

1991 (1567)

1992 (6029)

1994 (3307)

1995 (2204)

Cr

Cf

Cr

Cr

Cf

Cr

Cf

Cr

1990 1991 1992 1993 1994 1995 1996

5 7 22 5 9 2 0

4 3 8 7 8 8 3

9 37 8 36 6 2

3 24 30 19 22 14

148 62 196 30 14

122 172 127 142 108

92 26 16

98 174 94

1 11

55 88

Sum

50

44

98

115

450

700

134

396

12

177

a

Cf

Cf

No. tagged denotes the number tagged each year.

patterns of fishing gear (Myers and Hoenig, 1997), and we attempted to estimate size-selective functions directly from tagging to provide checkpoints to compare with the output of selectivity parameters of the integrated model. Myers and Hoenig (1997) used tagging and recapture data from the first year after tagging, where growth and natural mortality were assumed to be minimal, to estimate gear size-selectivity for cod. The patterns of proportion of tag recoveries in 5 cm length groups reflect the size-selectivity patterns (Myers and Hoenig, 1997). We used a similar approach and combined it with the SELECT method to fit size-selectivity functions (Millar, 1992). Myers and Hoenig (1997) showed that if the capture probability is the same for all fish of a given length, and the captures occur independently and at random, then the capture probability for a given gear g (Cprobg (l)) of a tagged fish of length l in a given year will be the product of a size-selectivity function (sg (l)), the proportion of fish that survive tagging (p1 ), the proportion of tags that is not lost (p2 ), the proportion of recovered tags that is reported in gear type g (p3 ), and the exploitation rate for gear g (p4 ). The factors p1 –p4 are combined into a nuisance parameter pg : Cprobg (l) = sg (l)p1 p2 p3 p4 = pg sg (l) This nuisance parameter is estimated analogously to the relative fishing intensity for the size group that is most vulnerable to fishing in the notation of the SELECT method (Treble et al., 1998). It is assumed that tagging mortality, natural mortality, tag loss and tag re-

porting rate are independent of the length of the fish for each gear type. The selection function (sg (l)) was fitted to recapture data from the first year after tagging. It is also assumed that the natural mortality rate and growth rate are small enough to be ignored during the analysis. Berg and Pedersen (2001) found that the cod longer than ca. 30 cm in Sørfjord grew about 5–6 cm per year. The SELECT method is used to model the proportion of the number of tagged fish in various length classes that are caught in the fishing gear to produce a size-selectivity curve (Millar, 1992; Treble et al., 1998). This method was developed for estimation of contact-selection curves (Millar and Fryer, 1999), but here we adopt it for use in modelling of population selection curves. For a given length group of individual, l, the proportion of the total the number tagged caught in the experimental gear and returned, θ g (l) is given as: θg (l) =

pg sg (l) pg sg (l) + (1 − pg )

The population size-selectivity function sg (l) describes the probability that a fish of length (l) from the population will be captured in a given gear g (Millar and Fryer, 1999). The modelled proportion θ g (l) is fitted to the observed proportions of number recaptured in gear g and length group l of the total number tagged in each length group l, under the assumption that the observed proportions are binomially distributed. The SELECT model is fitted to the data using a maximum likelihood estimation procedure (Millar and Fryer, 1999), using an EXCEL sheet described by Tokai (1997).

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We evaluated two different size-selectivity functions sg (l) to the tagging data. A classic logistical selectivity function was modified by an extra parameter (α) so that it could take both the classical logistical form typical of trawl selection curves (α = 0) and the more dome-shaped form typical of gill nets (α > 0): sg (l) =

tributed with a degree of freedom (d.f.) equal to the difference in number of parameters between the two models producing the log likelihoods L1 and L2 , respectively (Quinn and Deriso, 1999). The d.f. of each model is equal to the number of length groups used in the model minus the number of parameters in each model (Millar and Fryer, 1999). Confidence intervals (95%) for the observed proportions of tag returns were calculated using the F distribution according to Zar (1984).

exp{−αl} 1 + exp(β(δ − l))

This approach made it possible to evaluate whether the logistical (α = 0) or the bell-shaped function (α > 0) had the best fit to the tagging data. We tested for difference in model fit between the full model (dome-shape) with four parameters and the logistic model with three parameters. In the likelihood ratio test, the test statistic [LR = −2 ln(L1 /L2 )] is approximately χ 2 dis-

2.3. Sampling from the commercial fishery During 1994–1996, 24,288 cod heads from the fishermen’s catches were sampled in order to estimate the number, size and age composition in those catches.

Table 2 Overview of number of cod caught by the research trawl fishery and the effort (in units of 30 min trawl time) useda Year

1989 1990

1991

1992

1993

1994

1995

1996

a

Q

4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

Age of cod 0

1

2

3

4

5

6

7

8

9

10

11

12

13

14+

0 0 0 0 6 0 0 0 1 0 0 11 320 0 0 0 26 0 0 1 77 0 0 0 3 0 0 0 31

32 22 3 2 9 24 8 3 17 15 14 60 127 82 0 16 57 4 3 16 31 0 4 10 153 30 6 6 31

104 130 37 16 20 111 22 18 64 334 294 225 62 21 0 74 203 8 124 325 137 25 93 103 164 110 163 66 133

40 743 186 89 54 168 71 63 91 681 169 167 100 50 0 106 179 39 485 286 144 8 113 133 112 364 256 129 108

14 276 110 87 29 489 115 141 324 617 539 28 197 40 0 75 80 110 337 393 105 7 59 61 68 332 255 132 93

6 58 17 5 6 436 17 30 142 2810 1788 520 420 6 0 129 117 52 276 372 69 25 211 36 50 313 195 114 97

11 63 0 0 3 79 7 6 18 303 356 280 204 162 0 107 282 105 551 158 63 16 109 8 18 152 72 80 46

12 35 16 5 1 0 2 3 13 235 19 47 38 119 0 4 224 450 1879 73 38 23 145 16 38 179 71 42 13

2 6 7 0 0 0 1 3 1 8 24 11 6 16 0 0 23 195 197 16 21 36 234 27 70 75 5 36 28

1 6 0 0 0 0 0 0 0 25 17 0 0 1 0 4 6 30 41 0 0 8 50 8 28 93 29 40 47

0 1 0 0 0 0 1 1 2 19 13 0 8 0 0 2 11 0 11 0 0 0 0 0 0 5 0 25 11

0 0 0 0 0 0 0 1 0 0 10 0 0 0 0 2 1 0 17 0 0 0 0 1 6 5 0 4 5

0 0 2 0 0 0 0 0 0 0 0 0 2 0 0 0 4 3 7 0 0 2 5 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 11 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Q denotes quarter of a year. Age groups 14–17 in the model are pooled into the 14+ group.

Sum

Effort

222 1341 378 204 128 1307 243 270 672 5046 3245 1359 1483 497 0 518 1213 997 3927 1640 685 151 1023 403 709 1658 1052 675 642

1 9 2 1.5 2 3 1.5 2.5 3 21 13 9 7 4 0 4 4 8 15 13 14 6 18 2 13 10 4 5 4

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Table 3 Overview of the number caught by commercial fishing gear in Sørfjorda Year

1994

1995

1996

a

Q

1 2 3 4 1 2 3 4 1 2 3 4

Age of cod

Sum

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14+

0 0 0 0 0 0 0 0 0 0 0 0

36 7 0 21 8 0 0 0 0 0 0 0

0 0 8 2075 56 0 83 0 6 5 0 0

58 51 63 1261 257 12 3 9 46 48 3 0

401 177 52 201 1550 149 143 102 886 829 162 28

1579 464 899 485 487 535 783 481 2009 1206 261 206

1914 611 851 1550 1910 972 722 783 2134 1032 39 272

1821 465 909 2406 3570 796 376 935 2178 847 56 329

127 126 130 0 3604 211 47 179 612 738 36 197

0 52 36 0 230 7 0 0 421 903 139 590

0 14 0 0 0 0 0 0 0 86 0 74

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0

5937 1967 2948 7998 11672 2682 2159 2489 8292 5694 696 1696

Q denotes quarter of a year. Age groups 14–17 in the model are pooled into the 14+ group.

The heads were cut from the vertebral column in the junction between the skull and the first vertebrae. A regression relating fish length to head length was established (total fish length = 4.448 × head length, r 2 = 0.998, n = 279), and was used to estimate the total fish lengths from head lengths. It was noted how many tags were recorded in each catch where heads were sampled. Otoliths for age determination were sampled from subsamples of heads stratified over area and fishing gear (hand line and long line (hook) or gill net). The numbers caught by hook or gill net were estimated for each 5 cm length interval by assuming that the tag density in the catches sampled was equal to that of the catch that was not sampled. For a given gear, quarter and 5 cm length interval, the total number of cod caught was estimated by adding the numbers of cod sampled to the (number of heads sampled × number of tags returned from fish not sampled)/number of tags in the sampled catches. It was estimated that 44% of the total catch was sampled during 1994–1996. 2.4. Ageing and age–length keys Ages and type of cod (either coastal cod (CC) or northeast Arctic cod (NEAC)) were determined from otoliths in subsamples of fish according to methods described by Rollefsen (1933). The typing of otoliths to CC or NEAC is based on differences in the form and

sizes of the two inner opaque (summer) and hyaline (winter) zones. Only data on coastal cod were used in the model. January 1 has been used as zero age. Age–length keys prepared for separate quarters were used to estimate numbers at age from the length distribution of the whole catch of cod. Separate age–length keys were prepared for the trawl research catches and for the hook or gill net catches. For commercial catch data, total length was estimated from head length. The resulting length distributions for each quarter were then used together with an age–length key to estimate age distributions for the commercial catch. Tables 2 and 3 provide an overview of the data on research trawl catches and the commercial catches.

3. Model 3.1. Model structure and populations To estimate population size, fishing mortality rates and natural mortality rates from the data, a simple model of the Sørfjord cod stock was developed in EXCEL. Because of the relative sparsity of data from Sørfjord, it was considered appropriate to adopt a model with the parameterisation reduced as much as possible. Thus, compared to models of larger systems, the model is constrained to tight functional forms for describing fishing and other mortality processes. The model is fitted to the data sets using a basic

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Table 4 Overview of key model parameters: y = year, q = quarter, a = age Parameter Key parameters c1 c2 c3 c4 α β δ k φ γ M

Explanation

Unit

Efficiency ratio small/large trawl Reporting percentage Neo-tagging mortality Tag loss rate Shape of the commercial selectivity functions (right-hand slope) Shape of the commercial selectivity functions (left-hand slope steepness) Shape of the commercial selectivity functions (length at 50% selectivity when α = 0) Catchability term Shape of the trawl selectivity functions (steepness) Shape of the trawl selectivity functions (length at 50% selectivity) Natural mortality rate

% % % per year cm−1 cm−1 cm

0.5 80 10 10 0.015

cm−1 cm−1 cm Per year

Other parameters R(y) Po(89, 4, a) Fcom(y, q, a) Fc(y) I(q) fs(y, q) Fs(y, q, a)

Number of 2-year-old recruits in quarter 1 Initial numbers in quarter 4 in 1989, a = 2, . . . , 13 Quarterly commercial fishing mortality rate for age a Annual commercial fishing mortality rate Quarterly commercial fishing intensity Measured survey trawling time (effort) Quarterly research fishing mortality rate for age a

1000 individual 1000 individual Per year Per year

Variables Ps(y, q, a) Ptag(y, q, a, #)

Number at age a in quarter q and year y Numbers at age in quarter q, age q, tag batch #

1000 individual individual

least squares minimisation approach using the EXCEL SOLVER routine. The fjord population Ps(y, q, a) and the populations of the separate batches of tagged fish Ptag(y, q, a, #) (for year y, quarter q, age a and tag batch no. #) are modelled in year-classes by quarter year. Recruitment estimated as numbers at age two in q1 in all years and numbers at age 2–13 in q4 in 1989 are model parameters for the wild population. For the various batches of tagged fish, initial populations are estimated as numbers at age at the beginning of the year of quarter in which they were tagged, from age–length keys applied to the length distribution of tagged fish at the time of their tagging. These estimates are modified by a parameter describing neo-tagging mortality (c3 ). The populations of each year-class in subsequent years and quarters were then calculated using modelled estimates of fishing and natural mortality rate in a standard exponential survival model. The natural mortality rate (M) was assumed to be constant over age (2–17 years) and years. In the case of tagged fish, these mortality rates are augmented by a modelled yearly tag loss rate (c4 ).

Key run value

30 min−1 Per year

The difference in catchability of the two trawls used for the periods 1989–1991 and 1993–1996 was described in the model by the efficiency ratio (c1 ) of the small versus the large trawl (Table 4). 3.2. Commercial fishing mortality rate Commercial fishing is predominantly by gill net and by hooks, and reliable effort data were not available. Consequently, commercial fishing mortality rate Fcom(y, q, a) for each year y, quarter q and age a was modelled as: Fcom(y, q, a) = Fc(y)I (q)Sel{L(y, q, a)} where the model parameter Fc(y) is the annual fishing mortality rate, the model parameter I(q) is a quarterly intensity rate, and Sel{L(y, q, a)} is a length-based selection pattern. In the above formulation, L(y, q, a) is the average length-at-age in each year and quarter, which is available from age–length distributions of the research catch and consequently were not modelled.

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Given the nature of the commercial fishing gear, Sel{L(y, q, a)} was chosen to have a dome-shaped selection form and scaled so its maximum value at age was 1: exp{−αL(y, q, a)} 1 + exp(β(δ − L(y, q, a))) where α, β and δ are model parameters describing the shape of the selection curve. 3.3. Research fishing mortality rate Research fishing mortality rate Fs(y, q, a) was believed to be significant in Sørfjord. It was modelled in the form: Fs(y, q, a) = fs(y, q)k Sels{L(y, q, a)} where fs(y, q) was the measured survey trawling time (or effort), the model parameter k is a catchability term, and Sels{L(y, q, a)} the selectivity at length L(y, q, a) of the survey gear available. Since the survey gear was a trawl, an asymptotic form of selection curve was chosen: Sels{L(y, q, a)} =

1 1 + exp(φ(γ − L(y, q, a)))

where the model parameters φ and γ describe the shape of the curve. 3.4. Model fitting The model structural equations are used together with initial values of the model parameters to provide estimates as follows: 1. Commercial catch-at-age data from 1994 q1 to 1996 q4 . 2. Survey catch-at-age data from 1989 q4 to 1996 q4 . 3. Commercial recaptures at age for each year from each of the five tagging experiments initiated in 1990, 1991, etc. 4. Survey recaptures at age for each year from each of the five tagging experiments initiated in 1990, 1991, etc. Model estimates of items 1–4 together with the sampling-based estimates of the same factors are used to construct a least squares function which is summed

7

for all available ages and year-quarters to provide a total sum of squares for the factor. For the commercial and research catch-at-age data, two different assumptions about the error distribution (normal and log-normal) were used. First, we describe the model fitting using the assumed normal error distribution of the catch-at-age data. The form of sum of squares function used for each factor was as follows: 1. Catch = SSq(Com.Catch) = w1 (observed − expected)2 2. Survey = SSq(Surv.Catch) = w2 (observed − expected)2 3. TagCatch = SSq(Com.Tag.Catch) = w3 (observed − expected)2 /expected 4. TagSurvey = SSq(Surv.Tag.Catch) = w4 (observed − expected)2 /expected The weighting factors (w1 –w4 ) are introduced in order to weight the different data sets and to be able to investigate the effects of different weighting of the different data sets. Since some tag returns lacked information about the precise date of capture, these tags could not be grouped into quarterly intervals and hence we used yearly intervals when the sum of squares for the tagging data was calculated. The factor sum of squares was then combined into an overall model sum of squares: Total sum of squares = Catch + Survey + TagCatch + TagSurvey EXCEL SOLVER is used to adjust model parameters to minimise this function. Resulting model parameters and the consequential estimates of population numbers, fishing mortality rate and natural mortality rate then form the output of the model. In order to ensure that the minimisation resulted in global minima, we used several starting values to check that this resulted in the same final estimates. A key model run 1 was estimated using c1 = 0.5, c2 = 80%, c3 = 10% and c4 = 10% per year, and the right-hand slope of the gear selection curve (α) of the commercial fishery was held fixed at a low value (0.015 cm−1 ) during this run. The value of catch efficiency ratio (c1 ) used in the key run was set to 0.5 based on considerations of the different sizes of the two trawls used. Kristiansen et al. (2000) suggested that the proportion of tags reported was about 38–72%

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in western and southern Norway. Since our tagging area was relatively small, the tagging experiment was well announced and the tagging density was high; we used a tag reporting percentage (c2 ) equal to 80% in the key run. Based on experiences catching wild cod for rearing experiments (Pedersen and Jobling, 1989) where about 80–90% of the fish survived the first 2 months after being caught and tagged, we used a value of 10% for neo-tagging mortality (c3 ) in the key run. Otterå et al. (1998) estimated the yearly tag loss to be about 10% during the first year after tagging for cod tagged using similar but slightly larger tags than were used in our study. Barrowman and Myers (1996) estimated initial tag loss of 10% followed by a yearly tag loss rate of 7% per year for an 8.5 cm long tag. We chose to use a tag loss rate (c4 ) equal to 10% per year in the key run. The sum of squares of the four sets of data was weighted so that the contribution of each to the total sum of squares was approximately equal. The model residuals of all sets of data were inspected visually. In one model run, the right-hand slope of the commercial selectivity function (α) was left free to be estimated by the model. In other model runs, we investigated the effects of changing the fixed coefficients (c1 –c4 ) in the model. In addition, sensitivity analysis was performed by increasing one key parameter at a time by 10% and fixing this parameter, while new estimates of all other parameters were obtained by model minimisation. We also investigated the effects of increasing the weight by a factor of two (double weighting) on each data set in turn, and by fitting the model with weights of tagging data (TagCatch and TagSurvey) changed to 16, 8, 4, 2, 1/2, 1/4 and 1/8 times the weighting used in the key run. 3.5. Runs with log-normal error assumption Since there may be uncertainty about which error distribution may be most appropriate for catch-at-age data, we also made a separate set of runs with the assumption that these data had a log-normal error distribution. This was implemented in the model by replacing the term (observed − expected)2 with [ln(observed + 0.001) − ln(expected + 0.001)]2 in the factor sum of squares factors for the catch and survey catch-at-age data. The model with the log-normal error assumption was used to perform a key model

run (key run 2) and sensitivity analysis similar to that under the assumption of normal distributed errors.

4. Results There were few fish younger than 2 and 3 years in the trawl and commercial catches (Tables 2 and 3). Three- to eight-year-old fish made up most of the survey catch, while 5- to 9-year-olds were most abundant in the commercial catch. The yearly number of tags returned decreased relatively slowly in the years following each tagging, and tags from the tagging in 1990 and 1991 were still being returned in 1996 (Table 1). The gear selection patterns estimated directly from the tagging data indicate that for hooks (long line and hand line), selectivity increases with size without any clear maximum (Fig. 1A). This contrasts to the gill net pattern, which has a clear maximum for fish of about 55 cm in length (Fig. 1B). However, the combined gear selection pattern for commercial gear has a slight decrease in selectivity for fish longer than 55 cm (Fig. 1C). The trawl has higher selectivity for smaller fish, and the selectivity levelled off for fish of about 45 cm in length, with indications of a slight decrease in selectivity for larger fish (Fig. 1D). The SELECT model fits to the tagging data from the commercial fishery had a significant decrease in the log likelihood (P = 0.03) when the α parameter was set to 0 and the curve shape shifted from dome-shaped to logistic (Table 5). Thus, the dome-shaped function was the best model describing tagging data from the commercial fishery. There was not a significant decrease in the log likelihood for the research tagging data when the α was set to zero (P = 0.09), and we thus accept the logistic curve as the best model describing these data (Table 5). 4.1. Key model run 1 The key run 1 output indicated that the modelled gear selectivity for the trawl corresponded quite well with gear selectivity patterns derived directly from the tagging data (Fig. 1E). The estimated selection function for commercial gear had higher selection for fish smaller than approximately 40 cm in length than the function derived directly from tagging data, but the maximum selectivity appeared at the same fish size

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Fig. 1. Selection patterns for commercial gear (hook and gill net) and sampling trawl estimated from tagging experiments. Proportions of tagged fish recaptured during the year following tagging shown in A–D. Selectivity functions of research trawl estimated directly from tagging data ( ) and from the integrated model (—). Selectivity function for commercial gears estimated from key run 1 (䊏- - -䊏), estimated directly from tagging data (䊏—䊏) and from the integrated model with no restrictions on α (䊐- - -䊐). Line C and D shows proportions fitted by the SELECT model.

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Table 5 Estimates of gear size-selectivity functions directly from tag recapture dataa Data set

Commercial Commercial Research Research

Shape

Dome-shaped Logistic Dome-shaped Logistic

Parameter estimates α

β

δ

pg

0.15

0.32 0.18 0.23 0.23

56.9 51.8 39.2 32.9

0.14 0.15 0.05 0.05

0.05

Likelihood

d.f.

−27.61 −30.00 −24.52 −25.98

6 7 6 7

LR

P

4.78

0.03

2.92

0.09

a

LR denotes the likelihood ratio statistic for a test of whether the full (dome-shaped) or reduced (logistic) model is the best model. The degrees of freedom for each models (d.f.) are given.

(Fig. 1E). Also, the selectivity function derived both directly from tagging and from the integrated model run with unrestricted α had very steep right-hand slopes (Fig. 1E). The model estimates at q4 1989 indicated that there were very few fish older than 2 years in the population, but the 1987 year-class (age two) was very strong, being two to three times the size of the preceding year-classes (Fig. 2). The 1987 year-class remained strong throughout the investigation period (Fig. 3). Both the biomass of 2-year and older cod and the spawning stock biomass increased during 1990–1993, and then levelled off (Fig. 4). The natural mortality rate (M) was estimated to 0.26 per year. The annual commercial fishing mortality rate rose from about 0.06 per year during 1990–1992 to about 0.16 per year in 1993–1996 (Fig. 5). Inspection of model residuals revealed that the modelled and observed number of tags returned corresponded well for most tagging groups, years of recapture and gear (Fig. 6). However, the number of tags returned from commercial fishery from the 1990

Fig. 2. Model estimates of numbers at age in the population at the start (q4 in 1989, black bars) and at the end (q4 in 1996, white bars) of the investigation period.

Fig. 3. Model estimates of the numbers at age (2–11 years) at quarter 1 from key run 1 during the period (1990–1996). Circle area is proportional to numbers in thousands.

Fig. 4. Model estimates of total stock biomass, spawning stock biomass and catch yield during the investigation period. Results from key run 1.

T. Pedersen, J.G. Pope / Fisheries Research 63 (2003) 1–20

Fig. 5. Estimates of the annual commercial fishing mortality rate during the investigation period 1990–1996. Results from key run 1.

tagging group was less than predicted for the period 1992–1996, and many fewer than predicted were caught in the trawl fishery (Fig. 6) for the 1995 group. The residuals for the survey data reveal that the 7-year old fish (1987 year-class) in q3 and q4 in 1994 appear to be outliers (Fig. 7). For the commercial catch data, there is a tendency towards a skewed distribution with a few points having large positive residuals (Fig. 8). 4.2. Sensitivity analysis When α was left free to be estimated by the model, the result was a selection function for commercial gear with a very pronounced maximum and a very steep right-hand slope (Fig. 1E). The values of δ, β and α were large (Table 6), and for lengths from 20 to 50 cm below the maximum selection length, the curve had higher selectivity than the gear selection curve estimated from first year tagging returns (Fig. 1E). The average fishing mortality (F) in 1990–1992 decreased by 15% relative to the key run value, but other parameters did not change much, except for a low recruitment in 1993–1996 (−36%) (Table 6). Commercial catch tagging data were better fitted when α was estimated by the model (Table 6). The model run where the gear selection function (α, β, δ) for commercial gear was fixed at values estimated directly from the tagging data resulted in small changes in output values except for a large increase in

11

F of 240% and 84% for the periods 1990–1992 and 1993–1996 relative to the key run (Table 6). Catch data sum of squares increased by 15%, and the total sum of squares increased by 3% (Table 6). The rest of the sensitivity analysis was performed with ␣ fixed at 0.015 cm−1 , and β and δ were left to be estimated by the model. An increase in the assumed trawl efficiency ratio (c1 ) to 1.0 corresponding to equal efficiency of the two trawls used resulted in about a 10% increase in recruitment and a 13% increase in length at 50% selectivity by the trawl (γ ). A decrease in the assumed reporting percentage from 80 to 60% affected mostly estimates of yield and fishing mortality, which increased by about 40 and 20% relative to the key run (Table 6). There was also a decrease in M (8%). An increase in the assumed tag loss rate (c4 ) to 0.20 per year resulted in a corresponding decrease in M to 0.19 per year, a decrease in initial numbers of about 20%, and changed recruitment. The F values, however, were not greatly affected (Table 6). There was also an approximate 10% decrease in total and spawning biomass and yield for 1990–1992, but these values increased for the period 1993–1996. The 10% changes in key parameters mainly affected k, φ, β and the F values. When either M, k or γ was increased, this resulted in increased F values (Table 6). Decreased φ and increased k were observed as the result of increases in M and γ . An increase in δ resulted in decreased β (25%), and correspondingly, the 10% increase in γ resulted in a 23% decrease in φ. All model solutions described thus far resulted in small changes in total sum of squares, and in less than 20% changes in sum of squares for individual data sets. Double weighting of the survey and commercial catch-at-age data had opposite effects on most estimates, especially recruitment, F and biomasses. The biomass and catch estimates were most affected by double weighting of commercial catch data (about 17% increase). Double weighting of the commercial catch data sum of squares also resulted in a decrease in φ and δ, an increase in β and recruitment for most years, and an approximate 20% decrease in F (Table 6). In contrast, double weighting of survey catch data increased φ and β, decreased recruitment, and increased F by about 15%. Double weighting of commercial tagging data caused an increase in F of about 20%. There were very small effects of double

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T. Pedersen, J.G. Pope / Fisheries Research 63 (2003) 1–20

Fig. 6. Comparison of model predicted (black bars) and observed (white bars) tag returns for commercial (left column) and survey catches (right column). Results from key run 1.

weighting of survey tagging data except for an effect on the catchability (k). The effects of relative low weights of tagging and correspondingly high weight on catch-at-age data was an increase in biomass, yield, recruits, and initial num-

bers relative to the key run and vice versa for the runs with relatively high weights on the tagging data (Table 7 ). All key parameters except for M and β changed monotonically with increasing weight on the tagging data (Table 7).

T. Pedersen, J.G. Pope / Fisheries Research 63 (2003) 1–20

13

Fig. 7. Residuals for survey data. Results from key run 1. Circle area is proportional to absolute value of residual. Residual units are the number of individuals. White circles denote negative and black circles positive residuals.

4.3. Log-normal error distribution The key run 2 using a log-normal error distribution for the catch-at-age data resulted in estimates of initial numbers in 1989, total and spawning biomass and yield that were about 30% lower than for key run 1

with normal error assumption (Table 8). The M and F values were not greatly affected, except for higher F values during 1990–1992 (Table 8). Most of the effects resulting from the double weighting of data sets and the 10% changes in key parameters were similar, but mostly smaller than in the output from key run 1. An exception to this pattern was that decreased reporting percentage (c2 ) and doubling in tag loss rate (c4 ) in key run 2 caused a relatively larger increase in F and larger changes in other parameters than in key run 1 (Table 8).

5. Discussion

Fig. 8. Residuals for commercial catch data. Results from key run 1. Circle area is proportional to absolute value of residual. Residual units are the number of individuals. White circles denote negative and black circles positive residuals.

Our model approach has similarities with that used in catch-at-age models with auxiliary information (Deriso et al., 1985; Haist, 1998; Quinn and Deriso, 1999). The choice of objective function to minimise in such models should ideally reflect the underlying error structure of the data (op. cit.). For catch-at-age data in which ageing errors dominate the measurement error, the multinomial distribution is appropriate (Fournier and Archibald, 1982). However, since our commercial catch-at-age data were the result of

14

Table 6 Sensitivity analysis for model runs assuming a normal error distribution for catch-at-age dataa

Key run 1 Double weighting Catch Survey TagCatch TagSurvey Sensitivity fixing parameters M k γ φ δ β α estimated by model c1 (trawl efficiency ratio) doubled c2 (reporting percentage) decreased c4 (tagloss rate) doubled Gear selection parameters (γ , φ, δ, β, α) fixed

M

γ

k

φ

δ

β

1.2 × 10−3

28.9

−1 4 2 0

−7 2 0 6

5 −1 0 0

−19 12 1 2

−7 1 5 0

10 3 5 −1 0 0 −5 1

7 10 10 −1 0 0 −3 3

4 4 10 −1 0 0 −2 13

−12 −18 −23 10 −1 −1 4 7

−8

−2

2

−28 15

3 17

0 6

0.26

0.25

48.9

α

0.13

Initial numbers in q4 in 1989

Recruits age 2

F

N2

1990– 1993– 1992 1996

1990– 1992

N3

Biomass

0.02

205

34

68

37

0.06

11 9 −2 3

0 0 0 0

10 −1 1 −5

−3 6 1 −4

36 −16 0 −5

11 −3 0 0

−22 21 24 4

4 2 2 0 10 −2 29 1

9 3 6 −2 −25 10 278 1

0 0 0 0 0 0 2821 0

5 −1 4 0 −1 0 −2 0

5 −1 4 0 0 0 −6 −1

6 −1 8 −1 0 0 −1 9

3 −1 8 −6 0 0 −36 6

−13

−2

1

0

−2

−6

12

−7 18

0 16

11 145

0 900

−24 −16 2 −3

−13 3

1993– 1996 0.16

Yield

Sum of squares

1990– 1993– 1990– 1993– Total 1992 1994 1992 1996

Catch Survey Tag catch

Tag survey

95

157

3

16

1890

665

485

418

322

−18 10 15 1

16 −4 0 −5

22 −10 −1 −5

12 −2 0 −5

21 −8 −1 −5

30 24 21 16

49 14 5 3

28 75 −1 0

10 −1 90 1

19 −2 1 87

35 14 15 −3 −8 4 −15 3

20 9 8 −1 8 −2 −1 0

2 −2 3 0 0 0 2 −1

−5 −5 1 0 0 0 3 1

1 −3 1 0 −3 0 6 −4

−2 −4 2 0 −1 0 −1 1

1 0 1 0 1 0 −3 1

2 2 −3 0 2 0 −2 −5

−1 1 5 0 0 0 3 14

1 0 1 0 −1 1 −16 −2

2 −4 2 0 0 0 0 −1

−11

15

21

4

7

36

43

9

9

13

11

2

12 11

5 240

1 84

−11 −1

14 −12

−5 −4

25 −11

4 3

−11 15

22 −4

5 −3

5 −4

a Each row gives results from a separate run. Bold numbers indicate the values for parameters fixed at a 10% increase relative to the key run. Absolute values are given for the key run 1, and the percentage changes relative to the key run 1 are given for the other runs. Numbers for recruits, F, biomass and yield are given as averages for the respective time periods.

T. Pedersen, J.G. Pope / Fisheries Research 63 (2003) 1–20

Data set weighted/parameter fixed

Data set weighted/ parameter fixed

Key run 1

M

0.26

γ

k

1.2 × 10−3

φ

28.9

δ

0.25

Weighting of tag catch and tag survey sum of squares relative 1/8 16 −23 4 −12 1/4 6 −18 2 −8 1/2 1 −9 1 −5 1 0 0 0 0 2 1 7 −1 7 4 3 10 −2 22 8 4 12 −3 98

β

48.9 to the −9 −9 −7 0 5 5 4

0.13

α

0.02

key run 1 59 0 44 0 18 0 0 0 1 0 13 0 29 0

Initial number in q4 in 1989 N2 N3

Recruits age 2

F

1990– 1992

1993– 1996

1990– 1992

205

34

68

37

42 24 10 0 −6 −9 −11

37 22 9 0 −5 −8 −11

49 29 13 0 −8 −14 −19

41 30 13 0 −4 −7 −11

Biomass

0.06 1 −12 −14 0 31 64 86

1993– 1996 0.16 −12 −18 −15 0 19 30 29

Yield

Sum of squares

1990– 1992

1993– 1994

1990– 1992

1993– 1996

Total

Catch

Survey

Tag catch

Tag survey

95

157

3

16

1890

665

485

418

322

38 24 10 0 −7 −11 −14

25 21 11 0 −8 −14 −19

35 22 9 0 −6 −10 −13

24 20 11 0 −9 −15 −20

36 21 10 0 −6 −11 −14

36 25 12 0 −8 −14 −18

−40 −33 −21 0 38 111 254

−27 −21 −12 0 11 19 27

−81 −68 −45 0 90 271 633

a Each row gives results from a separate run with different weighting of the tagging data sum of squares. Absolute values are given for the key run 1, and the percentage changes relative to the key run 1 are given for the other runs. Numbers for recruits, F, biomass and yield are given as averages for the respective time periods.

T. Pedersen, J.G. Pope / Fisheries Research 63 (2003) 1–20

Table 7 Sensitivity analysis for model runs assuming a normal error distribution for catch-at-age dataa

15

16

Data set weighted/ parameter fixed

M

γ

k

φ

δ

β

1.4 × 10−3

27.4

1 2 −1 0

2 2 −2 −1

2 0 −1 0

−4 −9 9 7

−1 0 2 0

Sensitivity fixing parameters M 10 k 2 γ 2 φ 0 δ 4 β 0 α estimated by model −1 c1 (trawl efficiency ratio) doubled 0 c2 (reporting percentage) decreased 4 c4 (tagloss rate) doubled −33 Gear selection parameters 9 (γ , φ, δ, β, α) fixed

8 10 7 −1 7 0 −2 4 10 7 −1

7 7 10 −1 0 0 −1 13 10 0 −6

−25 −37 −28 10 −42 2 4 −20 −33 −27 −24

4 1 1 0 10 −2 13 1 3 0 17

Key run 2 Double weighting Catch Survey TagCatch TagSurvey

0.28

0.29

48.5

α

0.21

Initial numbers in q4 in 1989

Recruits age 2 F

N2

1990– 1993– 1992 1996

N3

Biomass

1990– 1992

0.02

125

5

89

22

0.09

2 3 −3 0

0 0 0 0

8 −19 −5 40 −1 −2 0 −1

18 −18 2 2

4 −3 6 5

−6 3 4 0

−3 −1 0 0 −24 10 25 0 13 6 53

0 0 0 0 0 0 573 0 0 0 900

11 10 2 1 2 1 0 0 3 −1 −1 0 −3 2 −2 −3 3 0 −38 −38 14 10

5 0 1 0 1 0 −2 −1 2 −25 9

−3 −1 0 0 −3 7 −2 2 20 14 −8

22 6 4 0 2 −1 5 5 103 7 −14

1993– 1996 0.16

Yield

Sum of squares

1990– 1993– 1990– 1993– Total 1992 1994 1992 1996

Catch Survey Tag catch

Tag survey

65

120

1

12

1960

783

464

420

292

−5 0 3 0

13 −11 0 0

11 −13 2 2

8 −6 0 −1

12 −13 1 1

38 22 21 15

84 8 4 0

9 80 1 1

10 1 88 0

0 0 0 100

20 6 3 0 25 −6 3 3 43 1 8

5 0 2 0 0 0 −1 −2 2 −27 10

−5 3 −1 0 0 1 0 0 −3 −3 1 −3 −1 3 −1 −3 −1 27 5 −25 −1 −18

−2 −1 1 0 −6 1 1 −2 32 11 0

1 0 0 0 3 0 −2 −2 4 4 6

0 0 0 0 4 0 2 −1 5 4 6

−2 0 0 0 1 −1 −3 −4 0 7 −4

5 1 1 0 5 1 −8 3 13 2 21

1 1 1 0 1 0 0 1 −3 1 2

a Each row gives results from a separate run. Bold numbers indicate the values for parameters fixed at a 10% increase relative to the key run. Absolute values are given for the key run 2, and the percentage changes relative to the key run 1 are given for the other runs. Numbers for recruits, F, biomass and yield are given as averages for the respective time periods.

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Table 8 Sensitivity analysis for run assuming a log−normal error distribution for catch-at-age dataa

T. Pedersen, J.G. Pope / Fisheries Research 63 (2003) 1–20

calculations using data from various commercial gear and age–length keys, we did not use the multinomial distribution, but rather used least squares similar to Deriso et al. (1985). We assumed that sampling errors for the survey and commercial catch-at-age data were either normally or log-normally distributed. Other alternative statistical distributions used for similar data include chi-square or gamma distribution (Guldbrandsen Frøysa et al., 2002). The results from the model run with an assumed log-normal error distribution were similar to those using simple least squares (Tables 6 and 8), except for lower biomass and yield estimates and higher F values. In maximum likelihood, estimation of tagging data, multinomial or Poisson distributions are often assumed (Hilborn, 1990; Quinn and Deriso, 1999). We assumed chi-square distribution, which may be a proxy for the multinomial distribution (Quinn and Deriso, 1999; Guldbrandsen Frøysa et al., 2002). There are few examples of use of tagging data in assessment models for cod, and our results indicate that they may relatively easily be integrated in a model also containing catch-at-age data. Our approach for modelling mortality of tag batches is similar to that used by Haist (1998), except that we did not estimate migration rates. The ability to estimate natural mortality rate separately from the fishing mortality rate in our model may be attributed to the extensive use of tagging data, since natural and fishing mortality are often confounded with the gear selection patterns in catch-at-age models (Pope and Shepherd, 1982; Quinn and Deriso, 1999). The fact that short time series and few years (3 years) with commercial catch-at-age data were available made a model with relatively few parameters feasible. The disadvantage of such low parameterisation was that we had to assume constant natural mortality rate and length-dependent gear selection patterns over the time period. However, since length-at-age generally increased during the modelled time period and the selection functions were length-based, the selectivity of given age groups changed during the time period. The consistently lower number of tag returns from the 1990 tagging batch than expected by the model (Fig. 6) may have been due to either a higher natural mortality rate in 1990–1991 or higher immediate tagging mortality for this group. The low number of parameters in the model made possible a sensitivity analysis and also fastened calcu-

17

lation speed, and may have been advantageous with regard to avoiding model misspecification and confounding of parameters. A problem in fitting the model appeared to be estimating the selectivity pattern for commercial gear within the model. The relative high selectivity of small fish estimated by the integrated model in the unrestricted fit where all selectivity parameters (α, β, δ) were estimated may be due to the lack of commercial catch-at-age data for the period from 1990 to 1993. The integrated model uses mean length-at-age to calculate the selectivity for each year-class. Thus, an overweight of large, fast growing individuals of each year-class could have been caught in commercial gear in the length interval 20–55 cm, and this may have also contributed to the relative higher selectivity values from the integrated model compared to the selectivity curve derived directly from tagging data (Fig. 1E). The very steep right-hand slope of the selectivity functions estimated both directly from the tagging data and by the unrestricted fit of the integrated model may be an artefact, since there were few fish larger than about 60 cm sampled during the investigation period. The strong 1987 year-class had an average length of ca. 56 cm in 1996, which corresponds to the length of maximum selectivity. Thus for commercial gear, the selectivity pattern derived directly from the tagging data (Fig. 1C and E) contains sparse information about the right-hand slope of the dome-shaped curve, and we therefore chose to fix the right-hand slope at a small value during the key runs. The sensitivity analysis did however indicate that the uncertainty about the exact gear selectivity pattern for the commercial gear did not appear to influence the main results from the integrated model (Tables 6 and 8). The trawl selectivity pattern was not greatly affected by any changes in model parameter assumptions (Table 6), and this may be due to the fact that the trawl caught fish over a broader length and age range than did the commercial data. The factors used to weight the sum of squares of each set of data should reflect the relative variance of the different sets of data (Quinn and Deriso, 1999). It appeared that the model outcome was not very sensitive to weightings different from those used in the key run. The observations that double weighting of commercial catch data and the survey data had opposite effects on gear selection parameters, recruitment and fishing mortality estimates indicate that these data had

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different information content and hence were balancing each other. The increase in recruitment, biomass and yield when weighting of the commercial catch was increased (Tables 6 and 8) indicates that the commercial catch data scale the abundance of fish in the model. In view of the large effort expended in sampling the commercial catch-at-age data, we believe that these data may be more precise and have lower variance than the survey catch-at-age data, and thus the commercial catch-at-age were given higher weight, i.e. producing a larger sum of squares in the key run than the other sets of data. The fact that double weighting of the commercial tagging and of the commercial catch data set had an opposite effect on commercial F indicates a counter-balance also between these two sets of data. The effects of changing the assumed relative efficiency of the small trawl (c1 ) were generally small, indicating that the choice of c1 was not critical for the model results. Different assumptions about the reporting percentage (c2 ) and tag loss rate appear to have more impact (Tables 6 and 8). The estimates of fishing mortality and yield were quite sensitive to a change in reporting percentage (c2 ). It is very difficult to measure the percentage of recaptured tags that was actually reported, but the tagging experiment was well announced and the majority of returned tags came from fishermen that returned more than 20 tags per year. We believe that it is more probable that the reporting percentage is closer to 80 than 60, but we cannot exclude that the reporting rate was lower during 1990 and 1991 than in later years, since the number of fish tagged and thus tagging density in 1990 and 1991 were low (Table 1). If this was the case, the fishing mortality for the early years may have been underestimated. The main uncertainty associated with the estimate of the natural mortality rate (M) appears to be due to the choice of the annual tag loss rate (c4 ). The fact that an increase in tag loss rate (c4 ) from 10 to 20% per year caused a decrease in M of about 0.10 per year (Tables 6 and 7) was expected, since the tag loss rate was modelled as an instantaneous loss rate in the model equations for the tagging data and would thus be confounded with M. We expect that the tag loss rate in our experiments has most probably been less than 10% per year. Our estimate of natural mortality rate of 2+ cod ranging from 0.19 to 0.30 per year obtained by different assumptions about input parameters and

error distributions (Tables 6 and 8) is close to most values used in VPA for cod stocks in the northeast Atlantic (ICES, 2000). However, the assumption of a constant natural mortality rate for all 2+ ages may be questionable, since 2- and 3-year-old cod would be exposed to some cannibalism mortality, and other predators such as cormorants (Phalacrocorax carbo carbo) and otters (Lutra lutra) may prefer cod in the length range of 20–50 cm (Heggberget, 1995; Johansen et al., 1999, 2001). Thus, the actual M may be higher for 2to 3-year-old cod than for older cod. The increase in exploitable and spawning biomass in Sørfjord during 1990–1994 was mainly due to growth and maturation of the strong 1987 year-class (Figs. 3 and 4). The low fishing mortality rate values in the period 1990–1993 compared to values for 1994–1996 and to what is observed in other cod stocks, was probably because very few cod were large enough to be caught. Most fishermen in the fjord may also alternatively fish outside Sørfjord if cod biomass inside the fjord is low, as was the case during 1990–1994. Low fishing mortalities similar to those in Sørfjord have been estimated for coastal cod in the Malangen Fjord not far (ca. 60 km) from Ullsfjord (Larsen and Pedersen, 2002). In contrast, much higher fishing mortalities in the range of 0.6–1.4 per year have been recorded for age 2+ coastal cod in southwestern Norway and for the northeast Arctic cod (Svåsand et al., 2000; ICES, 2000). The reason for the low fishing mortality for the cod stock in Sørfjord may be due to the fact that fishermen get their quota in cod regardless of whether it is coastal cod or northeast Arctic cod. Thus, since the minimum legal size is 47 cm, which corresponds to about 6 years of age for cod in Sørfjord, the most active fishermen may prefer fishing the larger and highly priced northeast Arctic cod in outer coastal areas. The cod in Sørfjord is exploited by small boats, and the fishermen use mainly gill net, jigs or long lines (unpublished material). Alternative assessment tools to the kind of model developed in this paper would be to estimate parameters separately from each set of data, or to use a VPA or cohort-type model which assumes no error in the commercial catch-at-age data (Salvanes and Ulltang, 1992; Quinn and Deriso, 1999). The low commercial fishing mortality in Sørfjord (Fig. 5) implies that it is difficult to use VPA or cohort analysis, since these models require long time-series of commercial

T. Pedersen, J.G. Pope / Fisheries Research 63 (2003) 1–20

catch-at-age data and also a relatively high (about 1.0) accumulated fishing mortality over the lifespan of each cohort in order to achieve precise estimates (Quinn and Deriso, 1999). Another advantage of the type of model developed in our paper is that the model could easily be expanded to integrate feeding data and estimate cannibalism mortality and multispecies effects such as estimation of predation mortality from cod on other prey species. There is however a need for further work on modelling of precision estimates (standard errors) of the parameters in the model. In other similar models, such estimates have been obtained by bootstrap methods (Quinn and Deriso, 1999).

Acknowledgements We thank Erik Berg, Hilde Mo, Stein Erik Eilertsen, Tove Jacobsen and Olav Nordgård for help with the field collection of data and age reading. Linda Bennet is thanked for help with the English language. This study has been partly financed by the Research Council of Norway (NFR Project T-04, PUSH).

Appendix A Overview of major equations used in the population modelling; see Table 4 for explanations of variables and parameters. The numbers at age the next quarter Ps(y + 1, q, a) were calculated as: Ps(y, q + 1, a) = Ps(y, q, a) exp(−Z(y, q, a)) where Z(y, q, a) is the quarterly total instantaneous mortality rate: Z(y, q, a) = Fcom(y, q, a) + Fs(y, q, a) + 0.25M The commercial fishing mortality rate is: Fcom(y, q, a) = Fc(y)I (q)Sel{L(y, q, a)}

19

and the research fishing mortality rate is: Fs(y, q, a) = fs(y, q)k Sels{L(y, q, a)} where fs(y, q) is the measured survey trawling time and k is the catchability coefficient for research trawling. The research trawling fishery selection function is given as: Sels{L(y, q, a)} =

1 1 + exp(φ(γ − L(y, q, a)))

The number caught by commercial gears (Ccom(y, q, a) and by research trawling (Cres(y, q, a) is calculated as: Ccom(y, q, a) Fcom(y, q, a)Ps(y, q, a){1 − exp(−Z(y, q, a))} = Z(y, q, a) Cres(y, q, a) Fs(y, q, a)Ps(y, q, a){1 − exp(−Z(y, q, a))} = Z(y, q, a) The number at age Ptag(y, q, a, #) in the tagged batches number # in the quarter after tagging (qtag) were calculated as: Ptag(y, qtag + 1, a, #)


c3  = Ptag(y, qtag, a, #) 1 − 100   0.25c4 × exp −Ztag(y, q, a, #) − 100

and for later quarters as: Ptag(y, q + 1, a, #) = Ptag(y, q, a, #)   0.25c4 × exp −Ztag(y, q, a, #) − 100 The catch equation was used to calculate the expected numbers caught of each tagging batch by treating the tag loss rate as a mortality rate.

where Fc(y) is the annual fishing mortality rate and I(q) is the quarterly fishing intensity rate. The commercial fishery selection function is:

References

exp{−αL(y, q, a)} Sel{L(y, q, a)} = 1 + exp(β(δ − L(y, q, a)))

Barrowman, N.J., Myers, R.A., 1996. Estimating tag shedding rates for experiments with multiple tag types. Biometrics 52, 1410–1416.

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