Use of length-based models to estimate biological parameters and ...

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Use of length-based models to estimate biological parameters and conduct yield analyses for male Dungeness crab (Cancer magister) Z. Zhang, W. Hajas, A. Phillips, and J.A. Boutillier

Abstract: Length-based models were developed for the male Dungeness crab (Cancer magister) population on the Fraser delta near Vancouver, British Columbia. The models incorporate the probability of moulting, moult increments, natural mortality during moulting and non-moulting periods, direct fishing mortality, and handling mortality that occurs when sublegal-sized crabs are caught and released. The models were used to investigate how long-term yield might be affected by the combination of handling mortality and an intensive fishery. The models were calibrated to survey data, and key biological parameters were estimated. The probability of moulting is near one for male crabs in the 130- to 150-mm carapace width range and decreases as crabs get larger. There is a 70.1% probability a crab will survive the 1-month period beginning with a moult. The non-moulting natural mortality rate is 0.97 year–1. When handling mortality is incorporated into the model, yield per recruit increases with the exploitation rate until it reaches approximately 94%. F0.1 is equivalent to 70%. An approach was developed to calculate the threshold ratio of discarded to retained crabs beyond which fishing would reduce the long-term yield. Résumé : Nous avons mis au point des modèles basés sur la longueur que nous avons appliqués à une population de crabes dormeurs mâles (Cancer magister) du delta du Fraser, près de Vancouver. Les modèles tiennent compte de la probabilité de la mue, de la croissance à la mue, de la mortalité naturelle durant les périodes de mue et en dehors de ces périodes, de la mortalité directe due à la pêche et de la mortalité due à la manipulation, lorsque des crabes de taille inférieure à la taille réglementaire sont capturés et relâchés. Les modèles ont servi à déterminer comment le rendement à long terme peut être affecté par la combinaison de la mortalité due à la manipulation et d’une intensité de pêche élevée. Nous avons calibré les modèles aux données d’inventaire et nous avons estimé les paramètres biologiques. La probabilité de la mue chez les crabes de largeur de carapace 130–150 mm est près de 1 et elle diminue en fonction de la taille chez les crabes plus grands. La probabilité qu’un crabe survive à la période d’un mois qui suit la mue est de 70,1 %. La mortalité naturelle est de 0,97·année–1 en l’absence de mue. Si la mortalité due à la manipulation est ajoutée au modèle, le rendement par recrue augmente en fonction du taux d’exploitation jusqu’à atteindre approximativement 94 %. F0,1 équivaut à 70 %. Nous avons développé une méthodologie pour calculer le seuil du rapport des crabes rejetés sur les crabes retenus au-dessus duquel la pêche réduit le rendement à long terme. [Traduit par la Rédaction]

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Introduction This study was motivated by concerns about the very intensive Dungeness crab (Cancer magister) fishery in the Fraser delta near Vancouver, British Columbia. Dungeness crabs are commercially harvested using traps. Only male crabs of carapace width excluding spines (CW) ≥ 155 mm may be retained. On the Fraser delta, mature male moulting crabs primarily moult in the spring (Jamieson et al. 1998). The commercial fishery is opened when the crabs are sufficiently hard, usually by late June or mid-July on the Fraser delta. Legal-sized male crabs are rapidly fished down over a period of about 6 weeks from the fishery opening. Nearly all legal-sized

male crabs have been removed from the stock by the end of the fishing season at the end of November. Despite escape rings in the traps, sublegal-sized crabs are also captured and then discarded back into the water. As the season progresses, the ratio of sublegal- to legal-sized crabs appearing in the traps increases, and by the end of the season the ratio is often greater than 10. The expected rate of handling mortality is largely unknown. A higher rate of injured crabs has been observed in areas where there is an intensive harvest (our unpublished data). Tegelberg (1971) did some experiments to try to determine handling rates. The lowest value was 5%. The highest value of 57% was achieved by dropping soft-shell crabs on the deck of a ship.

Received 29 May 2003. Accepted 2 July 2004. Published on the NRC Research Press Web site at http://cjfas.nrc.ca on 26 January 2005. J17551 Z. Zhang,1 W. Hajas, A. Phillips, and J.A. Boutillier. Shellfish Stock Assessment Section, Pacific Biological Station, 3190 Hammond Bay Road, Nanaimo, BC V9T 6N7, Canada. 1

Corresponding author (e-mail: [email protected]).

Can. J. Fish. Aquat. Sci. 61: 2126–2134 (2004)

doi: 10.1139/F04-155

© 2004 NRC Canada

Zhang et al.

Some of the sublegal-sized crabs that suffer handling mortality would otherwise survive, grow into legal-sized crabs, be harvested by the fishery, and contribute to future yields. There is an optimum fishing effort where the long-term yield is a maximum. We evaluate changes in yield per recruit with various exploitation rates, and we estimate the amount of gain or loss in long-term yield for continuing fishing at various ratios of discarded to retained crabs. The goal of these analyses is to determine the impact of such an intensive fishery and how the handling mortality affects the optimum fishing effort. These analyses require models to describe the growth and mortality of male Dungeness crabs. Crabs grow by moulting, so it is necessary to model both the probability of moulting and the size increments. The resulting growth model is more complex and difficult to implement than the length-at-age models commonly used for fin-fish. The mortality of crabs is complicated by the period immediately after moulting when the mortality rate is presumably high. The models require parameter values. Natural mortality rates (M) have been estimated for Dungeness crabs. For example, Butler and Hankin (1992) review the literature and conclude that M for male Dungeness crabs may lie in the range of 0.8–1.2 year–1, although lower values are quite plausible. Morado et al. (1999) found that natural mortality is especially high immediately after moulting. However, moulting and non-moulting periods are generally not considered when natural mortality rates are estimated. We not only estimate the natural mortality rate during the non-moulting period but developed, for the first time, a method for estimating the moulting survival fraction. Smith and Jamieson (1989) provide suitable parameter values for moult increments, but the probability of moulting is a more difficult issue. Previously, probabilities of moulting have been estimated from tagging studies (Hancock and Edwards 1967; Bennett 1974; McCaughran and Powell 1977) or based on analyses of size–frequency distributions and moult increments (Weber and Miyahara 1962; Hankin et al. 1989; Mohr and Hankin 1989). However, the results for both of these methods are confounded with moulting-related mortality. To overcome this problem, we develop a new method that is not affected by moulting-related mortality. With suitable models and corresponding parameter values, we evaluate the impact of fishing intensity and handling mortality on long-term yield.

Materials and methods The models for these analyses were calibrated to data collected from Vancouver Harbour. The harbour is part of the Fraser Delta, but because of navigational concerns, it is closed to the crab fishery. The traps used for the scientific surveys are similar to commercial traps except that the escape ports are wired shut. We assume that Vancouver Harbour is representative of the entire delta. The assumption that crabs do not migrate into or out of the harbour is supported by our unpublished tagging data and a study by Smith and Jamieson (1991), who reported that male Dungeness crabs undergo only limited net migration. The data used in these analyses were collected from 1994 to 2000. Surveys were performed in June and October of

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each year. It is convenient to consider the October surveys as “pre-moult” and the June surveys as “post-moult”. There was also a survey done in February 2001 to investigate the impact of soak time on crab catch rates. Traps were set in strings of 3–22 traps. Entire strings were deployed or retrieved at the same time. We assume that traps of the same string do not interfere with each other. Traps were retained after spending 1–24 h in the water in the June and October surveys. In the February 2001 survey, soak time ranged from 1 to 46 h. Each time a set was pulled, the number of traps and the soak time were recorded. For each captured crab, the sex, size, shell condition, and sometimes weight were recorded. The shell condition was used to determine if the crab was new shell or old shell (moulted or not moulted during the latest moulting season). Size was measured by the interspine carapace width excluding spines (CW). The minimum size limit for the crab fishery corresponds to a size of 155 mm. It is rare for a female crab to reach the minimum size limit. Male crabs reach maturity at approximately 130 mm. Immature males are rarely observed in the traps (Fig. 1). Neither females nor immature males were considered in the analyses. In this study, we use catch-per-unit-effort (CPUE) as a measure of abundance. We standardize CPUE using the data collected in February 2001. We estimate natural mortality rates based on the changes in standardized CPUE between the post- and pre-moult surveys. We estimate proportions of moulting based on the estimated relative abundances of new and old shell crabs at the beginning of the moulting season. After estimating size-dependent vulnerability to the traps, we estimate the survival fraction for newly moulted crabs through the moulting season. These estimated parameters are used in models to assess the impact of fishing intensity and handling mortality of sublegal-sized crabs on long-term yield. Standardisation of fishing effort For trap fisheries, fishing effort is usually measured in trap hours. There is a saturation effect for Dungeness crabs caught in the traps (Fig. 2). To better measure the abundance of crabs, catch per trap (CPT) was modelled as (1)

CPT = CPT∞(1 − exp(−k τsoak ))

where CPT∞ is the maximum number of crabs likely to appear in a trap, k measures the rate at which the catch-rate declines, and τsoak is the soak time. CPT∞ and k were estimated from February 2001 survey data by minimizing the sum of squares of error. The standardized soak time, τ*soak , is the time required to catch the same number of crabs if every hour was as effective as the first: (2)

CPT (τsoak ) τ*soak = CPT (1)

or 1 – exp(−k τsoak ) τ*soak = 1 − exp(−k) The standardized effort is (3)

E* = τ*soak NT © 2004 NRC Canada

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Fig. 1. Size–frequency distribution for male Dungeness crabs (Cancer magister) caught in all the scientific surveys from Vancouver Harbour.

Fig. 2. Catch per trap (CPT) with soak hours (t) for male Dungeness crabs (Cancer magister): CPT = 14.62(1 – exp(–0.19t)); 䊉, observed data.

where NT is the number of traps in a string. The standardized CPUE is (4)

where C is the number of crabs caught in a string of traps. CPUE* is assumed to be proportional to the density of the crab population. Natural mortality rate during non-moulting period The natural mortality rate was estimated by comparing CPUE* from two different surveys taken during the same intermoult period in Vancouver Harbour. The expected relationship between the survey results and the natural mortality rate is (5)

⎛ CPUE* (tβ ) ⎞ −1 M = ln ⎜ ⎟ tβ − t α ⎝ CPUE* (t α ) ⎠

where tα and t β are the times at which the surveys occurred and CPUE*(t) is the standardized CPUE at time t. M is the non-moulting, natural, instantaneous mortality rate. There is the assumption that catchability is the same during both surveys. Only legal-sized crabs (CW > 155 mm) were used to estimate natural mortality. M was estimated for the June–October period from 1994 to 2000. The survey in June 1994 was conducted with only six trap hauls in contrast to 20 or more trap hauls in the other survey occasions, and the estimated value for 1994 is much higher than for other years (Table 1). Therefore, 1994 was excluded from the estimate of the mean value of M. The mean natural mortality rate was calculated with the values weighted according to fishing effort in the surveys: 2000

(6)

2000

∑

CPUE* = C/E*

M =

∑

M y E*y

y =1995 2000

∑

E*y

y =1995

where My is the estimated natural mortality rate for the year and E*y is the smaller of E* for June and October of that year. The standard error (SE) for M

(7)

f y (M y − M ) 2

y =1995

SE (M ) =

n (n − 1)

where n is the number of years where data were used and fy =

nE * y 2000

∑

E *y

y =1995

The estimated value of M is applied all year except for the 1-month period beginning when a crab moults. M is termed the non-moulting natural mortality rate in this paper. Probability of moulting The probability of moulting is an important aspect of growth for Dungeness crabs. On the Fraser delta, solf-shell crabs are primarily observed during a brief period in the spring. To simplify calculations, all moulting is assumed to occur at the same time t0. During a post-moult survey, shell condition can be used to identify old-shell crabs that did not moult during the previous moulting season. The probability of moulting can be determined by estimating the number of crabs (old-shell crabs + new-shell crabs) just before t0 and the number of old-shell crabs just after t0. Let t–1 and t1 be the times of the pre- and post-moult surveys. Variations of eq. 5 were used to estimate abundances immediately before and after moulting. The relative abundance of crabs immediately before moulting is CPUE* (t 0) = CPUE* (t −1) exp(– M (t 0 – t –1)) The relative abundance of old-shell crabs immediately after moulting is * CPUE * O(t 0) = CPUE O(t1) exp(– M (t 0 – t1)) where CPUE * O(t) is the CPUE* of old-shell crabs at time t. The fraction of crabs that moult is © 2004 NRC Canada

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2129 Table 1. Fishing effort and catch-per-unit-effort (CPUE) in June and October surveys and estimated instantaneous annual natural mortality (M) for the legal-sized male crabs in Vancouver Harbour in years 1994–2000. Number of traps

Standardised effort

Standardised CPUE

Year

June

October

October

June

October

1994 1995 1996 1997 1998 1999 2000

6 27 33 30 60 62 60

20 21 41 29 52 69 69

June 34.14 153.09 183.64 168.30 336.13 308.73 275.37

111.80 120.09 229.19 162.40 293.84 219.84 221.42

3.75 0.88 1.14 1.42 1.07 1.89 1.78

0.75 0.62 0.94 0.57 1.11 1.21 1.27

Weighted meana a

(8)

p=

M 4.83 1.05 0.58 2.74 –0.11 1.34 1.01 0.97 ± 0.39

Weighted mean of estimated natural mortality rates in 1995–2000 ± standard error.

CPUE* (t 0) − CPUE *O(t 0) CPUE* (t 0)

Fig. 3. Size-specific moulting probability for male Dungeness crabs (Cancer magister): 䊉, estimated proportions of moulting; 䉫, weighted mean proportion of moulting.

or p =1−

CPUE * O(t1) exp(M (t − t )) −1 1 CPUE* (t −1)

For the Vancouver Harbour surveys, the probability of moulting was estimated for 5-mm size intervals from 130- to 195-mm CW. Mean probabilities were calculated with values weighted as they were for the mean natural mortality rate. There appears to be a discontinuity at CW ≈ 150 mm (Fig. 3). One regression line was fitted for the 130- to 150mm size range and another for the 150- to 185-mm size range. Crabs larger than 185-mm CW were not considered because the corresponding sample size was too small to achieve reliable estimates.

Uj

(10) Moult increment The moult increment is another aspect of growth that could affect the impact of fishing intensity and handling mortality on yield. Previous studies indicate that moult increment is correlated with pre-moult size of crabs (Butler 1961; Collier 1983; Warner 1987). Combining the data collected from British Columbia and California, Smith and Jamieson (1989) formulated the following equation for the post-moult carapace width, PW, based on the pre-moult carapace width, MW: (9)

PW = 1.069MW + 18.07 + ε MI

where ε MI is a normally distributed random variable with a mean of zero and a standard deviation of 3.29 mm. The equation is applicable when the pre-moult size is in the range 80 ≤ MW ≤ 174 mm. Equation 9 can be rewritten to express the probability distribution of the post-moult size. When a crab of the ith size class moults, the probability that it ends up in the jth size class is

ri, j =

∫

Lj

1 2π ⋅ 3.29 ⎛ (x − 1.069MWi − 18.07) 2 ⎞ × exp ⎜ − ⎟ dx 2 ⋅ 3.292 ⎝ ⎠

where MWi is the mid-width of the ith size and Lj and Uj are the lower and upper bounds of the jth size class, respectively. They are defined as (11)

⎧L j = 130 + 5( j − 1) ⎪ ⎨MWj = L j + 2.5 ⎪U = L + 5 j ⎩ j

With the parameter values used in these analyses, there is a negligible probability of negative growth. Size-dependent vulnerability Size-dependent vulnerability is important in two ways. Firstly, it is a further refinement in establishing the relationship between abundance and CPUE. Also, low vulnerability will protect a size class from the traps and fishing-related mortality. An approximation of size-dependent vulnerability © 2004 NRC Canada

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is derived from the size–frequency distribution of crabs appearing in the surveys. The size–frequency distribution for male Dungeness crabs caught in the scientific traps exhibits a peak at approximately 157.5 mm CW (Fig. 1). Few crabs in the traps are larger than 200 mm or less than 130 mm CW. The lack of large crabs likely reflects issues of mortality. The lack of smaller crabs likely reflects lower vulnerabilities. We assume that all legal-sized crabs are fully vulnerable. For sublegalsized crabs, we assume vulnerability declines exponentially with size. The equation used for vulnerability is (12)

⎧v i = 1 ⎨ ⎩v i = exp(−a (157.5 − MWi))

(i ≥ 6) (1 ≤ i ≤ 5)

where vi and MWi are the vulnerability and mid-width size class for the ith size class, respectively, and a is the model parameter to be estimated. To estimate a, the vulnerability model (eq. 12) was combined with the probability of moulting (eq. 8) and the growth increments (eq. 10) and applied to pre-moult data to predict the size–frequency distribution of new-shell crabs during the post-moult surveys. The value of a was chosen to give the most accurate size–frequency distribution for crabs in the 165- to 195-mm CW range (8 ≤ i ≤ 13). Crabs below this range are not considered because they may have been smaller than the minimum size (130-mm CW) considered in the analysis. Crabs larger than this range are just not commonly observed. Survey results were adjusted for size-specific vulnerability: (13)

N i(t) =

CPUE*i (t) vi

where Ni(t) and CPUE*i (t) are the corrected (or absolute) abundance and CPUE* for the ith size class at time t, respectively. At the time of the post-moult survey, the predicted size frequency of new-shell crabs in the 8th to 13th size ranges is predicted to be j −1

(14)

Γ$N, j (t1) =

∑ v j N i(t−1) pi ri, j

i =1 13 k −1

(8 ≤ j ≤ 13)

∑ ∑ vk N i(t−1) pi ri, k

k = 8 i =1

Γ$N , j (t1) was calculated on a year-specific basis with yearspecific values of pi. The expected catches of new-shell crabs in the jth (8 ≤ j ≤ 13) size classes are 13

(15)

C$ N , j = Γ$N , j (t1)∑ C N , i (t1) i= 8

where C N , i(t1) is the year-specific actual catch of new-shell crabs in the ith size class in the post-moult survey. The expected catches were then compared against the actual catches using a χ 2 statistic: 13

(16)

χ2 =

∑∑

year j = 8

(C$ N , j (t1) − C N , j (t1)) 2 C$ N , j (t1)

A grid search was used to find the value of a that minimized χ 2. Moulting survival fraction Dungeness crabs are believed to suffer high mortality during and shortly after they moult. This is modelled by replacing M by a survival fraction, S, that applies for a time period, τmoult , that begins with moulting. As was the case for a, S was estimated by using pre-moult surveys to predict the results of the post-moult surveys. In order to estimate S, we consider absolute abundances instead of just a normalized size–frequency distribution. The amount of time between the pre- and post-moult surveys is t1 – t–1. If the crab moults, the moulting survival fraction, S, applies for a period of τmoult and the natural mortality rate, M, applies for a period of t1 – t–1 – τmoult . Therefore, the probability that a crab in the ith size class in the pre-moult survey becomes a new-shell crab in the post-moult survey is pi · S · exp(–M(t1 – t–1 – τmoult )). Equation 10 gives the size distribution in the post-moult survey. The predicted abundance of new-shell crabs of the jth size class (8 ≤ j ≤ 13) predicted for the post-moult survey is

(17)

N$ N , j (t1) =

j

∑ N i (t−1) exp(−M(t1 − t−1 − τmoult ) i =1

× S pi ri, j Year-specific values of N$ N , j (t1) were calculated using year-specific values of pi, and τmoult was given a value of 1 month. The predicted values were compared with survey values on the basis of ⎛ * , j (t1) ⎞ CPUE N ⎟ SSD = ∑ ∑ ⎜ N$ N , j (t1) − ⎟ ⎜ vj year j = 8 ⎝ ⎠ 13

(18)

2

where CPUE * N , j (t1) is CPUE* for new-shell crabs in the jth size class in the post-moult survey. As when the vulnerability coefficient was estimated, only crabs with a post-moult size in the 165- to 195-mm size range (8 ≤ j ≤ 13)) were considered. A value of S was chosen to minimize the value of SSD. Yield-per-recruit analysis The growth and mortality models can be used to estimate yield per recruit for crabs on the Fraser delta. When handling mortality is included in the models, we can estimate its effect on yield per recruit and determine which levels of fishing effort result in the maximum yield per recruit. For these analyses, a recruit is defined to be a male crab with 130 ≤ CW < 155 mm. Smaller crabs have very low vulnerability to the traps. Most of the recruits will be of legal harvest size after their next moult. The recruit represents a size range. At year 0, the beginning of the simulation, the size frequency corresponding to a recruit is based on survey data and the following vulnerability equation: © 2004 NRC Canada

Zhang et al.

(19)

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⎧ C i / vi ⎪N 0, j = 5 ⎪ ⎨ ∑ C j /v j j =1 ⎪ ⎪⎩N 0, j = 0

16

(1 ≤ i ≤ 5)

(24)

(i ≥ 6)

N y , j = N y −1, j exp(−Z j τharvest ) exp(−M (1 − τharvest ) × (1 − p j ) +

j −1

∑ N y −1, i exp(−Zi τharvest ) i =1

× exp(−M (1 − τharvest − τmoult )) S pi ri, j (y = 1, 2, 3, ...) where Zi is the size-specific instantaneous mortality rate during harvest, and τharvest is the length of the harvest in years. The first term in eq. 20 represents old-shell crabs and the summation represents new-shell crabs. Because crabs greater than 210-mm CW are rarely observed in Vancouver Harbour, we set 1 ≤ j ≤ 16. Any crabs destined to grow larger than 210-mm CW are assumed to suffer senescent mortality instead. Equation 20 is applied until mortality has reduced the original recruit to a very small ( 150 mm), there is a highly significant (p = 0.01) linear trend between size and the probability of moulting (Fig. 3). The estimated average probability of moulting decreases from 54.7% to 11.5% when the size of crabs increases from 152.5-mm to 182.5-mm CW. For crabs in the 185- to 195-mm CW range, estimated proportions of moulting have large variations and do not appear to be reliable, probably because of the relatively small size of catch data for these crabs (Table 2). For smaller crabs, the linear relationship is not statistically significant (p = 0.65). Almost all of these smaller crabs would moult in the spring (Fig. 3). The estimated coefficients are given in Fig. 3. The parameter a for the vulnerability model (eq. 12) was estimated to be 0.076. The estimated vulnerability coeffi© 2004 NRC Canada

Zhang et al.

cient decreases exponentially from 1.0 for legal-sized crabs to 0.15 for crabs of 130- to 135-mm CW. When the handling mortality rate is 5%, yield per recruit increases with the exploitation rate until it reaches 94%. An exploitation rate greater than 94% would result in a decline in yield per recruit (Fig. 4). F0.1 was estimated to be equivalent to approximately 70%. Although the retention rate for barely legal-sized crabs has very little influence on this trend, a higher retention rate results in higher yield for a given exploitation rate (Fig. 4). As the ratio of sublegal- to legal-sized crabs in the traps increases, the gain in long-term yield decreases at an approximately linear trend (Fig. 5). At handling mortality rates of 5%, 10%, 15%, or 20%, the net gain is zero when the ratio of sublegal- to legal-sized crabs in the catch reaches approximately 40:1, 20:1, 13.5:1, or 10:1, respectively. Fishing above these threshold ratios would result in a net loss in long-term yield.

Discussion These analyses comprise the development of suitable models for the growth and mortality of Dungeness crabs, the estimate of model parameters for the models, and the application of the models to address concerns of a fishery. The experience gained from working with these models suggests further research that could be done. The growth models developed in these analyses are sizedependent. The probability of moulting and the distribution of the moult increments depend on the size of the crab. Numerically, size-dependent growth models such as eq. 20 are generally more complex to implement than length-at-age models such as the von Bertalanffy growth equation as described in Quinn and Deriso (1999). The parameter values necessary to predict moult increments were taken from the literature, but a new approach was used to estimate the probability of moulting. These analyses have contributed a method for estimating the probability of moulting that is unaffected by circumvents of moultrelated mortality. The two common methods for estimating the probability of moulting are based on the analysis of mark–recapture data (Bennett 1974; McCaughran and Powell 1977; Hankin et al. 1985) or size–frequency data (Weber and Miyahara 1962; Hankin et al. 1989; Mohr and Hankin 1989). The weakness of these methods is that they either ignore moult-related mortality completely or have had to guess at its extent. Because of variations in natural mortality rates and possible year-to-year variations in the vulnerability of crabs to the traps and in the accuracy of using CPUE* as a measure of abundance, the estimated moulting proportion for individual years could be under- or over-estimated. However, the mean proportions of moulting would be more accurate, as underand over-estimates of individual moulting proportions would cancel each other out to some extent. As in Koeneman (1985), the probability of moulting is found to be size-dependent. Smaller crabs (CW < 150 mm) are almost certain to moult during the season. The probability of moulting declines as crabs get larger. Generally, our analyses show a higher probability of moulting for large crabs than those reported by Koeneman (1985).

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Both moulting-related and non-moulting-related natural mortality are considered. It is important to differentiate between these two natural mortalities as Morado et al. (1999) concluded that newly moulted crabs are more susceptible to disease-related infections. The non-moulting natural mortality rate was estimated from estimates of abundance and standard methodology. The estimated mean non-moulting natural mortality rate is 0.97 year–1. When the moulting and nonmoulting mortalities are pooled, the effective instantaneous rate becomes 1.25 year–1. This mortality rate is lower than the natural mortality rates for male Dungeness crabs reported by Gotshall (1978) and Smith and Jamieson (1991). Gotshall (1978) estimated that the natural mortality rate varied from 1.08 to 3.56 year–1 in Northern California using commercial CPUE data. Smith and Jamieson (1991) found that the natural mortality rate was around 2.5 year–1 near Tofino on the west coast of Vancouver Island using tagging information. However, our estimated rate is more consistent with the mortality range of 0.8 to 1.2 year–1 proposed by Butler and Hankin (1992). Moulting-related mortality was estimated by fitting models to survey data. Our result quantitatively confirms that newly moulted crabs have a lower survival fraction than non-moulted crabs through the moulting season. Moulting-related mortality has been incorporated through the moulting-survival fraction, S, which is used instead of the M during the moulting season. Handling mortality of sublegal-sized crabs is a concern for the fishery that is addressed through the model parameter, h, which is used in eq. 21. The analyses also used a model for the time dependence of CPUE. This model makes it possible to combine data where a range of soak times has been used. The vulnerability model used in the analyses is useful to make estimated abundances comparable among different size ranges. The size-specific vulnerabilities estimated in these analyses are consistent with the observation that relatively few small crabs appear in the traps. The yield-per-recruit analysis shows that long-term yield peaks when the exploitation rate reaches 94%. Handling mortality does reduce the potential long-term yield. With 5% handling mortality, long-term yield will eventually start to decrease as fishing effort increases further. Although the yield-per-recruit analysis seems to show that such a high exploitation rate is needed to produce the maximum amount of yield, F0.1 was estimated to be equivalent to 70%, suggesting that this high fishing level should be reduced. Deriso (1987) showed that for a broad range of models of stock dynamics, F0.1 policy does not unduly reduce the spawning abundance. The F0.1 measure is also, in a sense, a bioeconomic criterion in that a marginal yield of less than 10% was felt to be close to the point at which most fisheries administrators would consider further increases in fishing mortality or effort to be no longer economically worthwhile (FAO 1993). Fishing with a high intensity would also increase the mortality on sublegal-sized crabs, as they are frequently caught and released in search for legal-sized crabs, especially after a large fraction of legal-sized crabs have been removed. Continuing to fish would result in a net loss in the long-term yield, when the ratio of discarded to retained crabs is higher © 2004 NRC Canada

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than the threshold level. Determination of the threshold ratio relies on the handling mortality rate, which is yet to be investigated for the Dungeness crab fishery. This analysis provides the fishery managers another measure for regulating the fishery, once the handling mortality rate is known. The fishing season should be closed when the threshold level is reached. These analyses suggest further work that could be done. One of the most useful things to do would be to consider the impact of uncertainty on our conclusions. For example, there was much uncertainty in our estimate of the non-moulting mortality rate and that will have direct impact on the estimated value of the probability of moulting, the moulting survival fraction, and the application of the models. An integrated approach to parameter estimation would give more confidence in the estimated parameter values. The results of these analyses depend heavily on CPUE* as an indicator of abundance. Any investigation of this assumption would be beneficial. In short, the paper describes a new way of estimating sizespecific moulting probability and a methodology for determining the survival fraction for newly moulted crustaceans through the moulting season. The methods should be applicable to any crustaceans having a well-defined moulting season and a reliable postmoult indicator. A length-based yield-perrecruit model was developed to assess the impact of fishing intensity on yield. The paper also presents an approach for calculating the amount of gain or loss in long-term yield for fishing at various ratios of discarded to retained crabs. This approach can be applied to other fisheries where sublegalsized animals can be released and the ratio of discarded and retained animals is known.

Acknowledgements We thank two anonymous reviewers and an Associate Editor, Dr. G.T. Evans, for constructive comments, which greatly improved the quality of the paper.

References Bennett, D.B. 1974. Growth of the edible crab (Cancer pagurus L.) off southwest England. J. Mar. Biol. Assoc. U.K. 54: 803–823. Butler, T.H. 1961. Growth and age determination of the Pacific edible crab Cancer magister Dana. J. Fish. Res. Board Can. 18: 873–889. Butler, T.H., and Hankin, D.G. 1992. Comment on mortality rates of Dungeness crabs (Cancer magister). Can. J. Fish. Aquat. Sci. 49: 1518–1525. Collier, P.C. 1983. Movement and growth of post-larval Dungeness crabs, Cancer magister, in the San Francisco area. Calif. Dep. Fish Game Fish Bull. 172: 125–133. Deriso, R.B. 1987. Optimal F0.1 criteria and their relationship to maximum sustainable yield. Can. J. Fish. Aquat. Sci. 44(Suppl. 2): 339–348.

Can. J. Fish. Aquat. Sci. Vol. 61, 2004 Food and Agriculture Organization. 1993. Reference points for fisheries management: their potential application to stradling and highly migratory resources. FAO Fish. Circ. No. 864. Gotshall, D.W. 1978. Catch-per-unit-effort studies of northern California Dungeness crabs, Cancer magister. Calif. Fish Game, 64: 189–199. Hancock, D.A., and Edwards, E. 1967. Estimation of annual growth in the edible crab (Cancer pagurus). J. Cons. Int. Explor. Mer, 31: 246–264. Hankin, D.G., Diamond, N., Mohr, M., and Ianelli, J. 1985. Molt increments, annual molting probabilities, fecundity and survival rates of adult female Dungeness crabs in northern California. In Proceedings of the Symposium on Dungeness Crab Biology and Management. Alaska Sea Grant Rep. No. 85-3. pp. 189–206. Hankin, D.G., Diamond, N., Mohr, M., and Ianelli, J. 1989. Growth and reproductive dynamics of adult female Dungeness crabs (Cancer magister) in northern California. J. Cons. Int. Explor. Mer, 46: 94–108. Jamieson, G.S., Phillips, A., and Smith, B.D. 1998. Implications of selective harvests in Dungeness crab (Cancer magister) fisheries. In North Pacific Symposium on Invertebrate Stock Assessment and Management. Edited by G.S. Jamieson and A. Campbell. Can. Spec. Publ. Fish. Aquat. Sci. No. 125. pp. 309–321. Koeneman, T.M. 1985. A brief review of the commercial fisheries for Cancer magister in southeast Alaska and Yakutat waters, with emphasis on recent seasons. In Proceedings of the Symposium on Dungeness Crab Biology and Management. Alaska Sea Grant Rep. No. 85-3. pp. 61–76. McCaughran, D.A., and Powell, G.C. 1977. Growth model for Alaska king crab (Paralithodes camtschatica). J. Fish. Res. Board Can. 34: 989–995. Mohr, M.S., and Hankin, D.G. 1989. Estimation of size-specific molting probabilities in adult decapod crustaceans based on postmolt indicator data. Can. J. Fish. Aquat. Sci. 46: 1819–1830. Morado, J.F., Giesecke, R.H., and Syrjala, S.E. 1999. Molt related mortalities of the Dungeness crab Cancer magister caused by a marine facultative ciliate Mesanophrys pugettensis. Dis. Aquat. Org. 38: 143–150. Quinn, T.J., and Deriso, R.B. 1999. Quantitative fish dynamics. Oxford University Press, New York. Smith B.D., and Jamieson, G.S. 1989. Growth of male and female Dungeness crabs near Tofino, British Columbia. Trans. Am. Fish. Soc. 118: 556–563. Smith, B.D., and Jamieson, G.S. 1991. Movement, spatial distribution, and mortality of male and female Dungeness crabs (Cancer magister) near Tofino, British Columbia. Fish. Bull. U.S. 89: 137–148. Tegelberg, H.C. 1971. Condition, yield, and handling mortality studies on Dungeness crabs during the 1969 and 1970 seasons. In 23rd Annual Report of the Pacific Marine Commission for the Year 1970. Pacific Marine Commission, Portland, Oregon. pp. 42–47. Warner, R.W. 1987. Age and growth of male Dungeness crabs, Cancer magister, in northern California. Cal. Fish Game, 73: 4–20. Weber, D.D., and Miyahara, T. 1962. Growth of the adult male king crab Paralithodes camtschatica (Tilesius). U.S. Fish. Wild. Ser. Fish. Bull. 200: 53–75.

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