Abundance, dynamics and mortality of the Delaware Bay stock of blue crabs, Callinectes sapidus.

ABSTRACT The Delaware Bay stock of blue crabs supports a bistate fishery in New Jersey and Delaware, with annual landings climbing through the 1980s and 1990s to almost 11 x [10.sup.6] pounds (4,390 metric tons) in 1995 and then declining to a recent average of 7 x [10.sup.6] pounds (2,796 metric tons) over the last 5 y. In Delaware, this fishery ranks as number one in value. Landings declines in 1996 spurred efforts to conduct a stock assessment, which is now updated annually. This assessment was based on: (1) a biomass-based minimum recruitment threshold from a Ricker stock-recruitment model fit to indices of relative abundance from a research trawl survey and (2) a catch-survey model incorporating observation and process error that produced annual estimates of absolute abundance, biomass, and fishing mortality rates from 1979 through 2002. Adult blue crab abundance estimates showed a positive trend over the period, ranging from 20 x [10.sup.6] in 1979 up to 146 x [10.sup.6] in 1993, with recent estimates between 70 x [10.sup.6] and 97 x [10.sup.6]. Estimated average exploitable stock biomass over the period was 23.43 x [10.sup.6] pounds (9,357 metric tons). Recruit abundance was highly variable, ranging from 34 x [10.sup.6] up to 631 x [10.sup.6]. Use of the log survival ratio to estimate Z showed no trend in Z, although estimates were highly variable. Estimation of the exploitable stock size was problematic due to high density-dependent recruit mortality. Because of this fact, we developed upper and lower bounds of the exploitation rate, then estimated upper and lower bounds of F from Baranov catch equation, F = [micro]/(1 - [e.sup.-Z])*Z. We also estimated the Collie & Kruse (1998) harvest rate and extended it to estimate F. The upper bound of F ranged from 0.13 up to 0.77 and averaged 0.44. The upper bound on F and the Collie-Kruse F showed a positive linear or curvilinear trend. Annual M estimates from Z - F, conditioned on an original model input value of constant M = 1.0, were erratic and showed no trend but were correlated with recruitment, supporting the hypothesis of compensatory density dependence. The relatively low estimate of F versus M and the overcompensatory and resilient stock-recruitment relationship suggest that overfishing is not occurring on this stock.

KEY WORDS: Delaware Bay, blue crabs, Callinectes sapidus, catch-survey model, density, dependent mortality, compensatory mortality

INTRODUCTION

The blue crab (Callinectes sapidus Rathbun) inhabits estuaries from southern New England to Uruguay (Williams 1984). The center of its distribution is tropical; severe winters can kill significant proportions of overwintering blue crabs from Chesapeake Bay north (Sharov et al. 2003, Kahn et al. 1998). Juveniles and adults primarily remain within one estuary for life, although tagging work has shown that adults may move into nearby coastal waters (Fischler & Walburg 1962, S. McKenna, North Carolina Division of Marine Fisheries, pers. comm.).

The peak hatching of larvae in Delaware Bay occurs in July. After hatching, larvae are carried onto the continental shelf along the North American Atlantic coast and into the Gulf of Mexico along the Gulf coast, where larval development occurs (Epifanio 1995, Perry et al. 1998, Epifanio & Garvine 2001). Stock mixing occurs at this stage; the current hypothesis for Delaware and Chesapeake Bays, however, is that larvae travel in an oval or circular trajectory on the shelf and can re-enter their estuary of origin (Epifanio 1995, Garvine et al. 1997, Epifanio & Garvine 2001). Garvine et al. (1997) concluded that the Delaware Bay stock was probably the primary source of its own recruits, agreeing with Epifanio et al. (1984). The mouth of Chesapeake Bay is 160 km to the south, so it is not likely a dominant influence on Delaware Bay recruitment. For both estuaries, larvae travel south initially along the shore, then are pushed offshore and northward via Eckman flow from the dominant southwesterly winds of late summer and early fall. If the larvae wind up off the mouth of Delaware or Chesapeake Bay, they can re-enter in large numbers when northeasterly or northerly wind events push large volumes of water into estuaries via Eckman flow, according to Garvine et al. (1997) and Epifanio (1995). Other stocks in smaller estuaries north and south of Delaware Bay contribute larvae to the Bay to a greater or lesser extent (Garvine et al. 1997). Peak settlement occurs from late summer to early fall in Delaware Bay (Kahn et al. 1998). The Delaware Bay blue crab fishery is shared between the states of New Jersey and Delaware. The Bay is split longitudinally between the two states along the shipping channel, in contrast to the division of the Chesapeake Bay between Virginia and Maryland, which assigns the lower Chesapeake to Virginia and the upper Chesapeake to Maryland.

Despite the fact that Delaware Bay is near the northern extent of the species range, blue crab catches produce the largest dockside value of any fisheries resource in Delaware. The blue crab fisheries in Maryland, Virginia, and North Carolina provide the highest value in those states as well. Reliable Delaware landings data begin in 1973, whereas reliable New Jersey landings begin with 1978 (Fig. 1). In 1977, Delaware landings dropped sharply to less than 800,000 pounds, approximately 26% of 1976 landings. The winter of 1977 was the most severe in recent decades and was followed by another severe winter in 1978. Both these winters undoubtedly caused major winterkill of mature crabs. Landings in Delaware and New Jersey have followed similar patterns. After the decline in 1977, combined landings remained low for 8 y, averaging about 1.5 million lbs. Not until 1985 did Delaware landings return to the level of the mid 1970s. Landings climbed to the peak of 10.8 million lbs. in 1995, then declined in 1996 after a severe winter, when the Delaware Division of Fish and Wildlife documented winterkill. Dredge samples in March 1996 found 47% mortality, primarily of mature females. Landings then rebounded, but have not returned to the levels of the mid 1990s.

[FIGURE 1 OMITTED]

The commercial fishery follows a seasonal pattern determined by blue crab behavior (Cole 1998, Kahn et al. 1998). Two commercial gears are deployed, crab pots from April into November, and dredges from November or December into March or April. The crab dredge fishery, which is limited in Delaware to December 15 through March 30 and in New Jersey to November 15 through April 15, focuses on overwintering aggregations of adult female crabs in lower Delaware Bay. The crab pot fishery targets adults and juveniles showing physical signs of an imminent molt, called peelers, which are a high value catch. Peelers are used for fishing bait or as a source of soft crabs if they are held until they molt. Pots account for 84 percent of the annual harvest. The pot fishery usually begins in April or May in lower Delaware Bay. Sooks (mature females) begin feeding inshore in the lower Bay in spring before the jimmies (mature males) move inshore. This is known as the "sook run," which is targeted by crabbers. In June, the females usually extrude and carry an egg mass, known as a sponge, thereby becoming legally protected from harvest. The fishery then moves further up in the Bay, concentrating on males and peelers. This movement by the fishery up the Bay has the effect of leaving mature females in a refuge in the lower Bay. Juveniles tend to remain in mid-upper Bay. Maturation and mating occur over the summer. In the fall, mature mated females migrate downbay into higher salinity waters to overwinter. In recent years, crabbers have targeted this downbay migration in the fall in response to higher market demand for females (Cole 1998).

To successfully manage this stock for a sustainable fishery, managers need to know the status of the stock. Both trends and absolute values of abundance, recruitment, and exploitation are important for development and evaluation of management strategy and tactics. We present an analysis and model that estimates these parameters for the Delaware Bay blue crab stock, focusing on methods to deal with the density-dependent mortality exhibited by juveniles (Kahn et al. 1998). Stock assessment efforts originally focused on developing indices of stock abundance, developing an index-based stock-recruitment model, and estimating recruitment thresholds as biologic reference points (Kahn et al. 1998). Helser and Kahn (1999) developed a catch-survey population dynamics model to estimate absolute abundance and fishing mortality rates. Helser et al. (2002) developed a decision-based framework for evaluating effects of uncertainty in both the estimated fishing mortality rate and the overfishing threshold for this stock. In the catch survey model context, observation error is the difference between the indices of relative abundance estimated ("predicted") by the model and the observed indices. Process error is the difference between estimated indices and the indices calculated deterministically using the central model (Eq. 5; see Methods). Here, we explore the sensitivity of model results to several different weighting approaches of these two types of error in model fitting: (1) observation error only; (2) equal weighting of process and observation error; (3) process error downweighted to 0.5 of the weight of observation error; and (4) process error downweighted to 0.1 of observation error. Our earlier assessments estimated F as Z-M, where M was a constant and Z was the log survival ratio of stock Sizes estimated via the catch-survey model. Instead of that approach, Collie and Kruse (1998) used a harvest rate to measure fishing impacts. Here we estimate exploitation rate and then convert that rate to estimates of instantaneous fishing mortality via Baranov's catch equation (Ricker 1975).

METHODS

Commercial and Recreational Catch

Both Delaware and New Jersey require mandatory reporting of landings in logbook format. In addition, the Delaware Division of Fish and Wildlife (DDFW) conducted a dockside intercept survey of shellfish landings at dockside from 1985 through 2003, which gives a second estimate of landings (Whitmore & Cole 1997, 2001, 2003). Landings are reported in bushels, which is a standard bushel basket full of crabs. Bushels must weigh at least 40 lbs before most dealers will accept them. New Jersey landings are reported in logbook format to the New Jersey Division of Fish, Game and Wildlife (NJDFGW) and are then transmitted to the National Marine Fisheries Service, which has maintained the New Jersey landing records. NMFS converts bushels to pounds based on 40 lbs/ bushel 

(W. McCowski, NMFS Cape May, NJ, pers. comm.). For the catch-survey model, we converted bushels to numbers of crabs based on the following conversion factors per market grades: 93.8 per bushel for grade number 1 (large males), 134.7 per bushel for grade number 2 (small males), and 123 per bushel for grade number 3 (females). The conversion factors were developed during an at-sea sampling program (R. V. Cole, DDFW, pers. comm.). Peeler landings are reported in number, but the New Jersey landings from NMFS have been converted to pounds. NMFS converts the number of peelers into dozens and assumes one dozen weigh 2.5 lbs., so we convert pounds back to number with this estimate.

Landings of hard crabs estimated from the DDFW intercept survey were higher than those reported in logbooks in general, but not always. We use the intercept survey estimates of Delaware landings. We see this discrepancy as an indication of underreporting in Delaware logbooks. The discrepancy was even greater for peeler landings. We obtained correction factors for underreporting for these two components by regressing Delaware logbook landings on Delaware intercept landings from 1987 through 1993. We then used these factors to correct reported New Jersey landings for underreporting, multiplying reported hard crab landings by 1.4 and reported peeler landings by 2.3. This correction assumes that the New Jersey logbooks had the same degree of under-reporting as the Delaware logbooks.

Our estimates of recreational landings are based on a 3-y survey of the Delaware recreational blue crab fishery from 1995 (pilot study) through 1997 (Cole et al. 1996). Over that period, recreational harvest averaged 2.5% of commercial harvest. We assumed this ratio was representative of the time series of landings from 1979 to 2002 used in the model for New Jersey and Delaware.

Research Trawl Survey

The DDFW Juvenile Crab and Finfish Survey was the source of our indices of relative abundance. The survey was initiated in 1978 in Delaware Bay and samples fixed stations on the western side of the bay (Michels 2003). We used data from the 26 stations that have been sampled continuously since 1978 (see Kahn et al. 1998 for map). The survey used a 4.9-m otter trawl with 38-mm mesh and a cod-end liner of 12.7 mm stretch mesh. Each station was sampled monthly from April through October for 182 tows per year. Tows were 10 min against the tide. Blue crabs were measured to 5-mm increments and sexed; subsampling of small crabs (<60 mm CW) occurred if over 30 were collected. Kahn et al. (1998) developed several indices from this survey. Two different indices of recruits were developed. For the spawning stock-recruitment model, the index of recruits is the geometric mean catch per tow (GM CPT) of medium crabs (60 mm [less than or equal to] carapace width (CW) < 120 mm) from April through August.

To calculate the index of spawning stock biomass, we use data from the survey catch of large females (>120 mm CW) in April and May. We convert the average CW of these females to average biomass per female using an equation from Rothschild et al. (1992),

weight (gm) = 3.4865 x [10.sup.-3] * cw [(mm).sup.2.11645].

We then multiply the arithmetic mean catch per tow by the average biomass per female to get the index of spawning stock biomass (ISSB).

We plotted the index of recruitment, lagged 1 y, as a function of the index of spawning stock biomass, because the recruits are defined as the medium crabs of the year following birth. Two stock recruitment models were fit to the data. One model was a null model with no density-dependent mortality (Fogarty et al. 1992). This is a simple linear regression of recruitment on spawners with a linear term, a quadratic term, and no intercept. If the quadratic term in the regression model is significant, then curvature is present in the relationship between recruitment and spawning stock, and the null model of no compensation should be rejected. We rejected the null model of no compensation, because the quadratic term in the regression analysis was highly significant (P < 0.01) and substantially improved the model fit ([r.sup.2] = 0.62; P = 0.0002) over a model based on just the linear term ([r.sup.2] = 0.38; P < 0.002). We then used PROC NLIN in SAS to fit a nonlinear Ricker model with additive error structure, R = A x SSB x exp(B x SSB), where R is the index of recruitment, SSB is the index of spawning stock biomass and A and B are model parameters. We calculated the coefficient of determination, [R.sup.2], as [R.sup.2.sub.7] of Kvalseth (1985): 1 - (residual SS / uncorrected total SS).

We derived a minimum recruitment threshold (Mace 1994) from the Ricker parameter estimates. We do not refer to it as an overfishing threshold because severe winters have apparently caused the index to fall below this threshold. This reference point, labeled as [SSB.sub.50], defines the threshold level of spawning stock biomass that will, on average, produce one-half of the maximum theoretical level of recruitment. If spawning stock biomass is reduced below this threshold, recruitment tends to decline. Because the value of the recruitment threshold estimated from the Ricker parameters is deterministic and the S-R model was based on only 25 data points, we used the bootstrap to evaluate the Ricker parameter estimates and to estimate a variance for the reference point.

For inputs to the catch survey model, we used as an index of recruits the geometric mean catch per tow of small and medium crabs (CW < 120 mm) in August and September tows. For the index of adults, we use the geometric mean catch per tow of crabs with CW [greater than or equal to] 120 mm in August and September. These months are when the new recruits appear in the survey. In October, the large crabs migrate to deeper waters of Delaware Bay for the winter and thus out of range of our survey, which focuses on the inshore areas of Delaware Bay.

Catch-Survey Analysis

We applied Catch-Survey or Collie-Sissenwine Analysis, also known as the modified Delury model (Collie & Sissenwine 1983, Conser & Idoine 1992, Collie & Kruse 1998) to Delaware Bay blue crabs to estimate absolute abundance and fishing mortality rates. We used the CSA program, version 2.0, in the National Fisheries Toolbox (NFT) provided online by the Northeast Fisheries Science Center, National Marine Fisheries Service ([e-mail]). Information for Delaware Bay blue crabs was available to meet minimum data requirements for the model, which include: (1) annual indices of population size for each life history stage [i.e., recruit and adult size] estimated by research surveys; (2) total annual fishery catch in numbers; and (3) an estimate of instantaneous natural mortality (discussed later). Other auxiliary data not endogenously required by the model are needed such as mean weights for each life stage and the relative selectivity to the survey gear.

The catch-survey model is based on the first order difference equation,

(1) [N.sub.0,y+l] = ([N.sub.0,y] + [R.sub.0,y] - [C.sub.y])[e.sup.-M]

which relates the adult stock size at the beginning of the year ([N.sub.0,y+1]), to the adult stock size at the beginning of the previous year ([N.sub.0,y]), plus recruitment in the previous year ([R.sub.0,y]), minus the catch ([C.sub.y]), all discounted for natural mortality, M.

The above equation assumes that a recruit is any animal smaller than the minimum size vulnerable to the fishery at the beginning of the survey year (120 mm CW, for purposes of the model), whose size will be greater than or equal to the adult, fully-recruited size by the end of the survey year. In other words, recruits must grow from a size not vulnerable to the fully vulnerable size within 1 y. For this analysis, we defined the "survey year" to run August and then September of each year to the next. New year-classes appear in the research survey during August to September, whereas large crabs from the previous years' cohorts are still available to the survey gear/area before they migrate into deeper Bay waters for overwintering beginning in October. We chose 120 mm CW as the demarcation point for several reasons. First, current conservation measures specify a 120 mm and 127 mm minimum CW for hard crabs in New Jersey and Delaware, respectively. Second, width frequencies indicate that the majority of recruits have grown to the fully recruited size (120 mm) by the following year. Third, Kahn et al. (1998) demonstrated a significant positive correlation between large blue crab indices (defined as >120 mm) in a given year and size crab (<60 mm) indices during the previous year. The catch-survey model is predicated on the basis that a "signal" exists between recruit and adult sizes. Because approximately 80% of the calendar blue crab catches from the fishery are taken prior to the midpoint of the August to September research survey, indices of recruit and adult relative abundance were lagged to index population abundance at the beginning of the next calendar year. We set T, the point in the year at which the catches are assumed to occur for purposes of the model, as T = 0.6, based on the timing of the midpoint of catches. Therefore, the earlier difference equation can be formulated

(2) [N.sub.0,y+l] = [(([N.sub.0y] + [R.sub.0y])[e.sup.-0.60M] - [C.sub.y])[e.sup.-0.40M]

so that blue crabs from year y ([N.sub.0,y] + [R.sub.0,y]) experience 60% of the natural mortality ([e.sup.-0.60M]), catch is removed, then the survivors [([N.sub.0,y] + [R.sub.0,y])[e.sup.-04M] - [C.sub.y]] experience the remaining 40% of natural mortality.

For the catch-survey analysis, natural mortality was assumed to be 1.0 (see below). Survey indices of abundance are related to absolute stock sizes by the catchability coefficients

(3) [n'.sub.y'] = [q.sub.n][N.sub.0y][e.sup.[eta]t]

and

(4) [r'.sub.y'] = [q.sub.r][R.sub.0y][e.sup.[delta]t]

where [r'.sub.y] and [n'.sub.y] are the observed research indices of recruit and adult blue crab, q is the catchability coefficient of the research survey gear, and [e.sup.[eta]t] and [e.sup.[delta]t] are lognormally distributed random variables, which represent survey measurement errors for the recruit and adult indices, respectively. In essence, these errors represent the difference between the observed survey indices of recruits and adult animals and the expected indices predicted within the nonlinear least squares (NLLS) framework by the DeLury difference equation. Another source of error, called process error, arises from the DeLury equation itself. Process error is the difference between indices of adults calculated from (Eq. 2) and the value of the adult indices predicted within the NLLS framework from the catch-survey model. This error can be thought of as arising from discrepancies between the dynamics as predicted by the model and the actual dynamics, presumably due to factors other than those included in the model, or to variations in the value of model parameters (i.e., natural mortality).

Substituting the earlier equations into the model difference equation and including the lognormally distributed process error ([e.sup.[epsilon]t]):

(5) [n.sub.y] = [([n.sub.y-l] + [r.sub.y-l]/[s.sub.r])[e.sup.0.60M] - [q.sub.n][C.sub.y-l]][e.sup.-0.40M] [e.sup.[epsilon]t]

where

(6) [S.sub.r] = [q.sub.r]/[q.sub.n]

is the relative selectivity of recruits to the adult blue crab. For the analysis of Delaware Bay blue crabs, the relative selectivity was set equal to unity. The research survey uses a 1.5 inch mesh with a 0.5 inch codend liner (shrimp trawl). Because the very smallest instars escape the net, available data suggests animals above 20 mm CW are captured efficiently.

Estimates of parameters ([theta]) are obtained by minimizing the least squares objective function (S):

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[lambda].sub.[epsilon]] and [[lambda].sub.[delta]] are relative weights for the process error and recruit measurement error, respectively (relative to the measurement error for indices of the adult size).

We applied 4 different models, an observation error-only model recommended by Collie and Kruse (1998) and 3 different configurations of a mixed error model, with the following weights of process error relative to observation error: 1.0, 0.5, and 0.1. The observation error model estimated q (catchability coefficient), the time series of recruit indices (except the last years'), and the adult index in the first year, for Y + 1 parameters. The other adult indices were calculated, using (Eq. 5). Recruit errors are weighted as 1, relative to adult errors. The mixed error models estimated q, all adult indices, and all recruit indices except the last years', for 2Y parameters. Process error is the difference between the adult indices calculated by (Eq. 5) without the process error term (i.e., deterministically with no error) and the indices as estimated or predicted by (Eq. 5) with the process error term included.

Given estimates of [n.sub.y] , [r.sub.y], and [q.sub.n] from the nonlinear least squares minimization, and the value of relative selectivity, [s.sub.r], population sizes for the recruit and adult blue crabs are

(8) [N.sub.y] = [n.sub.y] /q

and

(9) [R.sub.y] = [r.sub.y] / [s.sub.r] q.

Because the model assumes lognormal error, all residuals are computed as log residuals, Ln (observed) - Ln (estimated). Total instantaneous mortality is estimated by the log survival ratio,

(10) [Z.sub.R+N,y] = [log.sub.e][[N.sub.0y] + [R.sub.0y] / [N.sub.0,y+1]].

The model requires an input estimate of M, instantaneous natural mortality. Rugolo et al. (1998) suggested an M of 0.375 for Chesapeake Bay blue crab based on the formula M = (3/ maximum age) and an 8-y lifespan. This formula estimates lifespan as the age at which survival is 0.05. Currently, we have revised our lifespan estimate to 3 y in line with the majority of estimates in the literature (see Discussion), so M = 3/3 = 1.0 was the input value. By subtracting annual upper and lower bounds of F estimates from Z, we produced year-specific estimates of upper and lower bounds of M in the results, conditional on this input of M = 1.0. If we were to use a higher estimate of M, the model estimate of catchability, q, would be reduced, and the estimates of stock size would be increased, producing lower estimates of exploitation and fishing mortality.

Exploitation rate was defined by Ricker (1975) as "the ratio of the number of fish caught from a year-class to the total number present when it became vulnerable to fishing" (p. 264); this definition is expressed in relation to exploitation rate of a year-class, but it also applies to the rate for a stock as a whole. We calculated 3 estimates of exploitation rate--an upper bound, a lower bound, and the harvest rate of Collie and Kruse (1998). The mortality rate of recruits is large and variable, so correct estimation of the stock size that becomes vulnerable over the year to the fishery is difficult to determine but critical. A lower bound of exploitation was

[[mu].sub.lower] = [Catch.sub.t]/([R.sub.t] + [N.sub.t]),

where R and N are the estimated absolute abundance of the recruits and adults, respectively. The majority of recruits, in most years, die before they reach the adult size, so they never enter the exploitable stock, so this formula overestimates exploitable stock, with consequent underestimation of exploitation. To develop an upper bound, we considered the 3 fates of a crab alive at the beginning of a year. First, they will survive to the next year, by which time we assume they become adults. Second, if they are adults, or attain the adult size during the year, they may be harvested. Third, they may die of natural mortality either as recruits or as adults. If we sum the number harvested over a year and the survivors (adults) at the start of the next year, we obtain a minimum estimate of the number that became vulnerable over the year. So,

[[mu].sub.Upper] = ([C.sub.t]/([N.sub.t+1] + [C.sub.t]),

where [N.sub.t+1] = absolute number of adults at the beginning of year (t + 1). This is a minimum estimate because it assumes that those that died of natural mortality did not attain the adult size (in fact, some adults die naturally). Being a minimum bound for exploitable stock, it produces a maximum bound for exploitation. A third estimate, from Collie and Kruse (1998), was to decrement the stock size for the portion of assumed input M that would occur prior to the fishery (assuming a Type 1, relatively brief, fishery; Ricker 1975), giving

[mu] = catch/([R.sub.t] + [N.sub.t])exp(-M x T).

The F estimates in our earlier assessments were estimated by Z - M, using the input value of M = 0.8. Because M is not constant, especially for recruits, this was less than accurate (Kahn 2003b). Here we use the form of the catch equation in Ricker (1975) for a Type 2 fishery,

[mu] = AF/Z.

Where A is the total annual mortality as a percent,

[A.sub.t] = 1 - exp (-Z).

Solving for F,

F = [mu]Z/A.

Z was estimated as the log survival ratio for each year as shown earlier. Because the blue crab fishery operates almost year-round, we consider it a Type 2 fishery, sensu Ricker (1975). By using upper and lower bounds of exploitation in the catch equation, we developed upper and lower bounds of F. We also extended Collie & Kruse's (1998) harvest rate estimator to estimate a Collie-Kruse F.

We developed an index of relative F using the annual adult index, by averaging the index for each pair of years to get an average adult index, then dividing this into the catch in numbers per year,

Relative F = [C.sub.t]/(([n.sub.t] + [n.sub.t+1])/2),

where n indicates the index of relative adult abundance.

The NFT CSA output includes 1,000 bootstrap estimates for the stock size estimates of recruits and adults, recruit and adult biomass, Z, the Collie-Kruse harvest rate (exploitation rate), and the catchability coefficient, q.

When we tested for trends, we used simple linear and quadratic regression, using SAS Insight software. If the quadratic term produced a significant increase in fit, we considered the relationship to have significant curvilinearity (Sokal & Rohlf 1995, p. 665). All significance tests should be considered only suggestive rather than exact, due to the effect of a large number of tests on the Type I error rate. The stated levels of statistical significance are actually overstated, because the likelihood of attaining these levels of significance due to chance alone has been increased by the large number of tests.

RESULTS

The indices of spawning biomass and recruitment produced a highly significant fit to a Ricker stock-recruitment model (Fig. 2; [F.sub.2,21] = 22.48, P < 0.0001). The model from nonlinear least squares was,

R = 277.1 * SSB * exp (-44.639 * SSB),

where R = the index of recruits and SSB = the index of SSB. Final estimates were obtained via a bootstrap (below). The [R.sup.2] of 68% can be interpreted to mean that the model explains 68% of the variation in recruitment. However, because this model is nonlinear and has no intercept, [R.sup.2] is calculated differently from that for a standard linear model with an intercept and is not directly comparable (Kvalseth 1985). One potential mechanism for the overcompensatory pattern of a Ricker model is cannibalism (Hines & Ruiz 1995, Dittel et al. 1995). This relationship indicates that when spawning stock biomass is low, chances of low recruitment increase. Recruitment tends to be highest at intermediate levels of SSB and tends to decline at high levels of SSB.

[FIGURE 2 OMITTED]

Ricker parameter estimates were significantly different from zero in the bootstrap results. The current model with the bootstrap parameter estimates is

R = 331.1 * SSB * exp(-46.5 * SSB).

We used the median values from the bootstrap, because the empirical distribution was non-normal by at least one test for all of the following parameter estimates. The median of the alpha parameter estimates from 1,000 bootstrap runs was 331.1, with the 95% CL from 178.6 to 564.3. The beta parameter median was -46.5 (95% CI -69.21 to -24.55). The alpha parameter is interpreted as 331 recruits produced per kilogram of spawning stock biomass. On average, there are 6.6 mature females per kilogram (Kahn et al. 1998), so the alpha parameter indicates that at low densities, each female spawner can produce about 50 recruits. This is a high alpha level and indicates the stock should be very resilient to fishing. For the most useful model, we would like to calculate recruitment to the spawning stock, instead of to the recruit stage. However, losses to the fishery complicate this goal.

The level of SSB that produces only half of the maximum recruitment, on average, is termed the [SSB.sub.50], which has been recommended as a recruitment overfishing threshold (Mace 1994). If SSB declines below this level, recruitment will decline, on average. Based on the bootstrapped estimate of [SSB.sub.50], the median ISSB value corresponding to [SSB.sub.50] was 0.0051 kg SSB/tow. The 95% confidence limits were 0.003 to 0.010 from 1,000 bootstrap replicates. Median recruitment below this reference point (0.88 kg/tow) was markedly reduced, compared with that above the threshold (1.9 kg/tow; Fig. 2). In past assessments, we have termed this the recruitment--overfishing threshold. However, we now call this index value the low recruitment threshold because there have been several years in which the index of SSB has been reduced after severe winters, even to a value of zero in 1978, 1996, and 2003 among other years (2003 is not included in Fig. 2, because the corresponding estimate of recruitment is not yet available). Kahn (1996), however, did find a significant negative regression of the ISSB through 1994 on total landings per year-class. The ISSB was the only life-stage index that was not predicted by earlier indices of year-class strength in that report. The present assessment includes recovered values for early years of the index that were previously missing (1979 and 1980), plus nine additional years since 1994, so we would like to revisit the analysis.

While the index of adults was relatively stable, with some increase over the time series, the index of recruits was erratic and averaged much larger in magnitude than the adult index (Table 1, Fig. 3).

[FIGURE 3 OMITTED]

After running the 4 models, we compared the observed and estimated indices. The observation error model (OE) produced the largest positive recruit residuals in 7 y of high observed recruitment: 1981, 1986, 1990, 1995, 1998, 2000, and 2001 (Fig. 4a), consistently underestimating indices relative to the observed indices. With mixed error models, recruit residuals declined as process error was downweighted (Fig. 4a). For adult indices, the OE model had large negative residuals in most of the years following years of high observed recruitment: 1987, 1991, 1996, 1999, 2001, and 2002 (Fig. 4b). For mixed error models, for these same years, adult residuals declined as process error was downweighted (Fig. 4b). For recruit and adult indices, the PE0.1 model had the smallest residuals.

[FIGURE 4 OMITTED]

The OE model calculated adult indices after the first year solely from Eq. 5, deterministically, with no deviation (i.e., no process error), based on its estimated value of the index of recruits. In this model, the fitting process could reduce adult residuals only by adjusting the estimated or predicted index of recruits. When high observed recruitment occurred, the model calculated that the adult index would be high also, assuming constant M. These calculated adult indices tended to be higher than the observed indices; consequently, to minimize adult observation error, the model then underestimated the index of recruits relative to the observed index. Even with this adjustment, the calculated indices of adults were still higher than the observed indices, because reducing observation error of adults tended to increase it for recruits, thus constraining the fitting process. The constant M estimate had greater force in this model configuration than in those including process errors, because there is no allowance for deviation of estimated adult indices from the calculated indices.

Conversely, the mixed error models were able to compromise between the calculated and observed indices, unlike the GE model, due to their allowance of process error. The magnitude of the weight on the process error determined the discrepancy between the observed and estimated indices. If the process error weight were high (as in the EW model), then the objective function put as much stress on minimizing the process error as on minimizing the observation error. Consequently, the EW model tended to produce estimated indices that were intermediate between the calculated indices and the observed indices. When the weight on the process error was reduced (PE0.5, PE0.1), the objective function put more stress on minimizing the observation error by moving the predicted indices closer to the observed and further from the calculated indices.

We plotted recruit residuals as a function of observed recruit indices (Fig. 5). The general pattern was that the recruit residuals were correlated with the observed indices. This pattern was the most extreme in the GE model (Fig. 5a). In the mixed error models, the pattern was reduced as weighting of process error was reduced (Fig. 5b,c,d). Even for the PE0.1 model, however, which had the smallest magnitude of these residuals, a significant correlation existed between the two variables, although its effect was greatly reduced. A second aspect of the residual pattern was a correlation between observed recruit indices and residuals from the estimated adult indices the following year (Fig. 6). When observed recruit indices were high, the projected adult index was estimated as higher than the observed index, so as recruit indices increased, the lagged adult residuals trended in a negative direction.

[FIGURES 5-6 OMITTED]

As the process error was downweighted, the observation error declined as the estimated indices moved closer to the observed indices. Consequently, the residual sum of squares declined with the weight on the process error. The RSS values for the models were 6.2988 (GE), 4.3753 (EW), 3.3363 (PE0.05), and 1.1327 (PE0.01).

Given the residual patterns and the residual sum of squares, we chose PE0.1 as our preferred model. This choice indicates that the survey indices have less error than the difference equation (Eq. 5). The remainder of the paper presents results from this model (Table 2). The ranges of bootstrap CVs for parameters and derived quantifies were moderately precise. Recruit abundance CVs ranged from 30% to 42%. Adult abundance CVs ranged from 25% to 47%, with the same range for adult biomass CVs. The CV for Z ranged from 13% to 38%. The CV for the estimate of the catchability coefficient was 25%.

A major unforeseen effect of this process of selecting the model configuration was a reduction in the catchability coefficient, q, and a concomitant increase in stock size estimates (Fig. 7). In the progression from GE, through EW, PE0.5 and PE0.1, estimated q declined steadily, from a maximum of 0.0379 for GE to 0.0163 for PE0.1. Because previous assessments had used an equal weighting configuration, the stock size estimates had been significantly lower. The effect of the weighting change was compounded by the similar, but foreseen, effect of the increase in the input M of 25% from 0.8 to 1.0, which also resulted in a decrease in q.

[FIGURE 7 OMITTED]

Estimates of blue crab recruit abundance on January 1 ranged from 34 to 726 million, with an average of 284 million over the 1979 to 2003 period (Table 2; Fig. 8), with a positive linear trend ([F.sub.1,22] = 8.61, P = 0.008, [R.sup.2] = 28%). During the last 3 y, recruitment has trended down to 217 million in 2003. Adult abundance estimates were more stable and averaged 72 million, only 25% of the recruit average. Adults ranged from 19.6 million at the beginning of the time series to 146 million in 1993 and also had a positive linear trend ([F.sub.1,22] = 19.88, P = 0.0002, [R.sub.2] = 47%). On January 1, 2003, adult stock size was slightly below average at 69.8 million. The adult stock undoubtedly declined significantly due to winterkill over the next 3 mo, because the winter of 2003 was severe. The dredge fishery reported significant numbers of dead crabs and the 2003 spring index of spawning stock biomass was 0.

[FIGURE 8 OMITTED]

Total exploitable stock biomass on January 1 of each year averaged 23.4 million pounds (9345 MT) and ranged from 7 million to 45 million pounds (28,000 MT to 180,000 MT; Table 2, Fig. 9). Exploitable biomass has shown a positive linear trend, and a quadratic term was marginally significant, suggesting that the positive trend in biomass has flattened out or declined in recent years (linear regression: [F.sub.1,22] = 15.27, P = 0.0008, [R.sup.2] = 41%; quadratic term in polynomial regression P = 0.0733).

[FIGURE 9 OMITTED]

Catch in numbers fit a polynomial regression, showing catch increased significantly over the time series, peaking in 1995, then declined (Table 1, Fig. 8; [F.sub.2,22] = 30.54, P < 0.0001, [R.sup.2] = 74%). Catch in numbers was a significant linear function of adult stock size,

catch = 3.0806 + 0.2762 x adults

([F.sub.1,22] = 21.70, P < 0.0001, [R.sup.2] = 50%). The intercept was probably due to recruitment over the year into the adult stock. The regression suggests that catch tended to be 3 million plus 28% of exploitable stock size. Yield also showed a curvilinear trend (Fig. 9, [F.sub.2,21] = 32.81, P < 0.0001, [R.sup.2] = 76%). Yield showed a highly significant linear fit to estimated exploitable biomass, with

landings = 0.313 + 0.2903 x exploitable biomass

(Fig. 9; [F.sub.1,22] = 20.74, P = 0.0002, [R.sup.2] = 49%), implying that landings were 0.31 million pounds plus about 29% of adult biomass. Total survival ranged between 12% and 62%, but has not been above 30% since 1992 (Table 2, Fig. 10). Estimated Z was variable, ranging from 0.55 to 2.1, with an average of 1.45 (Table 2). There was no trend in Z (Fig. 10; [F.sub.1,21] = 2.58, ns), which was correlated with recruit abundance ([F.sub.1,21] = 30.18, P < 0.0001, [R.sup.2] = 59%). This correlation shows that total mortality increased at higher recruit abundance, indicating density-dependent mortality.

[FIGURE 10 OMITTED]

The stock size used for estimation of the lower bound estimate of exploitation increased with each peak in recruit abundance, as did the stock size for the Collie-Kruse estimate, which was simply scaled down from the lower bound estimate (Fig. 11). Stock size for the upper bound estimate, however, did not follow these peaks, because the peaks did not usually translate into proportional increases in survivors or catch numbers. The plot also reveals an increasing discrepancy between the several stock sizes estimates. Prior to 1992, the low points of all three estimates converged. Since then, however, the stock size estimates have diverged at the low points, with the upper bound estimates lower than the other two, suggesting that survival of recruits may have declined during the 1990s.

[FIGURE 11 OMITTED]

The lower bound of exploitation averaged 0.08 (Table 2, Fig. 12). For 2002, this measure was 0.09. Collie-Kruse exploitation averaged 0.14, although for 2000 to 2001 it averaged 0.11, increasing in 2002 to 0.17. The upper bound estimate of exploitation averaged 0.23. This estimate is unavailable for 2002, because it requires an estimate of adult survivors in 2003. For 1999 to 2001, it averaged 0.24. F estimates derived from the three exploitation estimates diverged considerably (Table 2, Fig. 13). The CollieKruse F was generally intermediate between the other two, except that it exceeded the upper bound estimate in 3 y.

[FIGURES 12-13 OMITTED]

The F estimate we used previously, F = Z - [M.sub.constant input], was erratic and estimated negative values for 5 y (Fig. 13). This F estimate was a highly significant function of the minimum bound of exploitation ([F.sub.1,21] = 16.25, P = 0.0006, [R.sup.2] = 44%), but had a negative slope (-7.99), contrary to the catch equation. It was only marginally significant as a function of the maximum bound of exploitation ([F.sub.1,21] = 3.32, P = 0.08, [R.sup.2] = 14%). There was also wide variation of the F = Z - [M.sub.constant] input estimate for any given value of upper bound F; for example, the former ranged between -0.07 and 0.85 when the latter was about 0.3 and ranged between 0.20 and 1.05 when the latter was approximately 0.55. In years when the year-specific M was high, F = Z-[M.sub.constant input] attributed the increased M to F, resulting in much more variable estimates of F.

The upper bound F peaked in 1995, at 0.77 (Table 2, Fig. 13). That year had the largest catch in the time series. The upper bound estimate of F averaged 0.44 over the time series and average UB [F.sub.1999-2001] = 0.52. Although the upper bound F was quite variable, there was a significant positive linear trend,

UB F = 0.2834 + 0.013 x Year,

suggesting that F increased on average by 0.013 annually (Fig. 13: [F.sub.1,21] = 6.67, P = 0.0173, [R.sup.2] = 24%). In contrast, the CollieKruse and lower bound F estimates showed a highly significant curvilinearity, but no linear trend (Fig. 13: C-K F, linear trend ns, curvilinear regression C-K F = 0.0578 + 0.0386 x Year -0.0015 x [Year.sup.2], [F.sub.2.20] = 6.51, P = 0.0066, [R.sup.2] = 39%; LB F linear trend ns, curvilinear regression LB F = 0.0313 + 0.0216 x Year -0.0008 x [Year.sup.2], [F.sub.2,20] = 6.60, P = 0.0063, [R.sup.2] = 40%). The Collie-Kruse F peaked in 1991 at 0.50 and averaged 0.24 over the time series (Table 2, Fig. 13). The 1999 to 2001 average was 0.21, less than half of the upper bound F estimate. The lower bound F averaged 0.13 and peaked in 1991 at [F.sub.1991] = 0.28. Recently, LB [F.sub.1999-200l] = 0.11. For management purposes, use of the upper bound of F would be precautionary, but in some cases could possibly result in forgoing yield if fishery restrictions were based on this estimate.

As a check on our F estimates, we estimated relative F as the catch in numbers divided by the adult index averaged over 2 y. All three estimates of F are significantly correlated with relative exploitation, but the highest significance level and steepest slope was attained by the upper bound F (Fig. 14; UB F: [F.sub.1,21] = 49.89, P < 0.0001, [R.sup.2] = 70%; C-K F: [F.sub.1,21] = 9.58, P = 0.0055, [R.sup.2] = 31%, %; LB F: [F.sub.1,21] = 9.58, P = 0.0055, [R.sup.2] = 31%).

[FIGURE 14 OMITTED]

To refine our estimate of M from the original average of M = 1.0 used for the input into the CSA, we calculated an upper bound and lower bound by subtracting the lower bound and upper bound estimates of F, respectively, from Z (Table 2, Fig. 15). The resulting estimates are highly erratic; with the upper bound estimate ranging from 0.25-2.0 (average = 1.3) and the lower bound estimate ranging from 0.2-1.6 (average 1.0). Neither estimate showed a significant trend, but both were correlated with recruitment (upper bound M: [F.sub.1,21] = 29.86, P < 0.0001, [R.sup.2] = 59%; lower bound M: [F.sub.l,21] = 20.64, P = 0.0002, [R.sup.2] = 50%).

[FIGURE 15 OMITTED]

DISCUSSION

The catch-survey model assumes that animals in the recruit size category will grow into the adult or fully-recruited size by the end of the year. Therefore, a clear understanding of growth patterns in this stock is important. Our understanding is at variance with the growth pattern presented in Rugolo et al. (1998). In general, blue crabs grow to maturity and adult size approximately one year from appearance as juvenile crabs (Hay 1905, Churchill 1921, Gray & Newcombe 1938, Pearson 1948, Van Engel 1958 and pers. comm., Fischler 1965, Tagatz 1968a, Perry 1975). To summarize the growth pattern of this stock (Kahn et al. 1998), new recruits appear in August or September samples. By October, some have grown into the medium stage (60-119 mm CW). By the following June, a few age 0 crabs have become large crabs. By July, hard crab landings increase as more animals recruit into the legal size (127 mm in DE, 120 in NJ). August and September seems to be the peak of recruitment into the adult size, and commercial landings usually peak in this period. The majority of new recruits are large crabs by the end of their second October, although a minority may over-winter as medium sized crabs. This proportion varies among years. We plan to investigate this proportion and effects of density and temperature on growth. If a significant fraction does not grow into the adult stage over a survey year, the modeling process will interpret them as dying during that year and will consequently overestimate Z and F. Tagging work revealed that a significant proportion of American lobsters failed to grow into the adult stage each year, violating the catch-survey assumption and biasing high the estimates of Z (V. Crecco, Connecticut DEP, pers. comm.).

Kahn et al. (1998) confirmed the earlier mentioned description of the growth pattern by using life-stage-specific indices of abundance to predict abundance of later life stages. Young-of-year (YOY) indices (geometric mean catch per tow (cpt) of small crabs from September to October) were significant predictors of both recruit indices (geometric mean cpt of medium crabs from April through August the following year) and adult indices (geometric mean cpt of large crabs from July through October the following year). When the YOY indices were used as predictors, curvilinear regressions were the best models, suggesting density dependent mortality of a compensatory form, where survival or production increases as density decreases (Sissenwine et al. 1988).

Rugolo et al. (1998) gave length-age conversions at variance with this pattern for the Chesapeake Bay, indicating that crabs did not attain 60 mm CW until I y of age and did not reach 120 mm CW until 2 y of age. However, they presented no data or citation for these length-age conversions. The same conversions had been used in Rugolo et al. (1997), and those authors attempted to confirm them by using what they considered age 0 crabs (<60 mm CW) to predict what they considered to be age-1 crabs (60-119mm CW) and age 2 crabs (>120 mm CW) in survey data. However, the attempt failed. Instead of predicting indices of the same year class the following year, the authors reported that the surveys showed correlations within the same year of different ages. We believe the reason was that the authors mistakenly split the same year-class into 2 ages. Instead of splitting the data into YOY, age 1 and age 2, Rugolo et al. divided the age 1 crabs into 2 ages, 0, and 1 then considered the adult age 1 crabs to be 2 y old. Consequently, they found that their putative age 0 and age 1 crabs were correlated within years because they were actually the same year-class. They interpreted the failure to predict as evidence that the several surveys they analyzed provided measures of availability among years instead of measures of relative abundance among year-classes. Smith (1997) also concluded that blue crabs in the Chesapeake do not reach adult size until their third summer. Our preliminary inspection of Chesapeake Bay monthly length-frequency distributions from the Virginia Institute of Marine Science trawl survey program (supplied by C. Bonsak, Virginia Institute of Marine Sciene, pers. comm.), however, show patterns very similar to those in Delaware Bay presented in Kahn et al. (1998).

Ju et al. (2001) estimated the Von Bertalanffy K parameter (growth rate) from pond-reared Chesapeake Bay crabs as 1.09, with L[infinity] = 240 mm CW. We used MULTIFAN to estimate Von Bertalanffy growth parameters from trawl survey length frequency distributions, analyzing 3 separate blocks of years and obtained 3 sets of estimates: K = 0.75 and L[infinity] = 234.7 mm CW, K = 0.62 and L[infinity] = 200.6 mm CW, and K = 0.93 and L[infinity] = 200.3. Rugolo et al. (1998) estimated K = 0.59, with L[infinity] = 262.5 mm CW. Our estimates were intermediate between the 2 estimates from Chesapeake Bay. Our Von Bertalanffy estimates were consistent with growth to maturity in the second summer of life at age one.

Development of the stock-recruit model allowed assessment of the status of the spawning stock biomass, even if no other assessment had been available. The index of SSB has been moderate in recent years (Fig. 2), although it declined to zero in 2003 after a severe winter, indicating that the biomass was so low that the survey did not catch any spawners. In Figure 2, the levels of the ISSB for 1999 (producing the 2000 recruitment), for 2000 (producing the 2001 recruitment) and for 2001 (producing the 2002 recruitment) were in the range that the Ricker model predicts will produce on average the highest recruitment. The resulting indices of recruits in 2000 and 2001 were below the curve, indicating relatively low survival from eggs to recruits. In contrast, the index for 2002 was above the curve, indicating high survival to the recruit stage.

Previous assessments of this stock via the catch-survey model had used a constant value for natural mortality throughout, particularly in calculation of F, which consisted of subtraction of the assumed value of M from the estimate of Z. Independent analysis of survey data indicated that recruits exhibited density-dependent mortality (Kahn et al. 1998 and updated in Kahn 2003a). This discrepancy caused process error in the estimation of population abundance, apparent in the residuals (Helser & Kahn 1999), because the equal weighting of process and observation error gave high weight to the calculated indices, based on constant M. The effect of the constant M assumption reverberated through the consequent estimates of survival, Z, exploitation rate, and F. Because blue crabs grow to maturity quickly, the indices of recruitment represent quite young animals of a few months of age. Density dependent mortality is most likely to occur in juvenile stages as Myers and Cardigan (1993) found with fishes.

Model development for assessment of the Delaware Bay blue crab stock has advanced in two areas from former assessments (Helser & Kahn 1999, Helser 2000, Helser & Kahn 2001, Bancroft & Kahn 2002, Kahn 2003a). First, we explored various error weighting alternatives, as well as an observation error only model, as alternatives to our original scheme of equal weighting of process and observation error. The goal was to reduce or eliminate a pattern in the residuals of the original model, caused by the discrepancy between the constant M in the model equation and the observed data. When process error was downweighted to 0.1 of observation error, residual pattern; though still present; were minimized. Consequently, we selected this configuration (PE0.1) as the preferred model. The selected weighting reduced the impact of calculated indices as opposed to the observed indices and so reduced the effect of the assumption of constant M. This configuration also had the unforeseen effect of estimating the smallest catchability coefficient of all the models we explored. Consequently, stock size estimates are larger than in previous assessments. Average adult stock size estimated by the preferred model PE0.1 was 72.41 million versus 41.9 million in the EW model, a 74% increase.

Second, we changed the method for estimation of F. Originally, we had subtracted the input M from estimated Z (F = Z-M). In this study, we used the catch equation and the output value of Z and an estimated exploitation rate. We were able to estimate upper and lower bounds of F, as well as an estimate of F based on the Collie and Kruse (1998) exploitation rate, which uses constant M. The catch equation in the form F = [mu]Z/A implies that F should be a positive linear function of both [mu] and Z. We tested the two F derivations to see if they met this expectation. When we regressed [F.sub.Z-M], using the input M = 1.0, on the upper bound of [mu], the regression was only marginally significant ([F.sub.1,21] = 3.32, P = 0.08, [R.sup.2] = 13.6%). Regression of [F.sub.Z-M] on the minimum bound of the exploitation rate was highly significant, but the slope was negative (F = 1.046 - 7.99 x (lower bound of exploitation rate), [F.sub.1,21] = 16.2, P = 0.0006, [R.sup.2] = 43.6%). The reason is that first, Z (and so F = Z - M) was positively correlated with stock size due to density-dependence and second, the lower bound of exploitation was negatively correlated with stock size, because an increase in stock size translated directly into an increase in the denominator of the exploitation rate, and catch did not increase proportionally. Consequently, as stock size increased, Z and F = Z - M increased, and the minimum bound of exploitation decreased, leading to the negative correlation. In contrast, the upper bound of [F.sub.CE] was a positive linear function of Ix and Z. Regressions of the upper bound of F on the upper bound of exploitation and on Z were highly significant (on exploitation: [F.sub.1,21] = 90.6, P < 0.0001, [R.sup.2] = 82%; on Z: [F.sub.1,21] = 22.4, P = 0.0001, [R.sup.2] = 52%). There was no correlation between the upper bound of F and the lower bound of exploitation. This comparison shows that the F = Z - M did not perform well here as an F estimator, but the upper bound of [F.sub.CE] did.

The estimation of exploitable stock size is difficult because we have not been able to estimate the age distribution of the harvest. Consequently, we do not know what proportion of the harvest consists of recruits versus adults, complicating the estimation of accurate mortality rates for recruit versus adult crabs. Adult crabs die of natural mortality over the year. At this point, we don't have good estimates of total survival of adults. Recruit mortality, however, is apparent from the average decline in abundance from recruits to adults. Average recruit abundance was 284 x [10.sup.6], whereas average adult abundance was 72 x [10.sup.6] (Table 2). This is 25% survival on average, even assuming all adults are products of the previous cohort of recruits (i.e., no adult survival from one year to the next), so on average, recruit Z would be [Z.sub.R] = 1.39. The average of the 23 y of total survival estimates was 0.27, and the average Z was 1.46 (Table 2).

The CSA software we used (Collie Sissenwine Analysis or Catch Survey Analysis) in the National Fisheries Toolbox (NFT) allows use of the catch equation also, but it simultaneously solves two equations, Z = F - M, and [mu] = F/(F + M) x A for F, using the original input value of M. To hold M constant, this approach implicitly varies Z from the estimated log survival ratios to simultaneously solve the two equations. To vary Z, estimates of stock size must be implicitly varied from the output values. Consequently, the NFT approach departs from the output estimated stock sizes to hold M constant as the input value. In contrast, our approach, which does not directly use the input value of M, uses the output log survival ratio for Z and implicitly lets M vary. We think of the original input value of M as the first cut in a process to estimate year-specific M = Z - F, conditional on the original input value.

The Collie and Kruse estimate of exploitation, based on decrementing stock size on January 1 or the beginning of the survey year by MT, results in a scaling down of the stock to account for M prior to the time when the stock is vulnerable to exploitation (Fig. 11, Fig. 12). For stocks where fishing is at a certain discrete point in the year and where M can be assumed to be roughly constant, this may be a satisfactory estimate. The recruits in Collie and Kruse (1998) were legal sized king crabs, so they were qualitatively different and several years older than the recruits in the present study. Because blue crabs mature in 1 y, their growth is rapid enough that recruits must be defined as juveniles for purposes of the catch-survey model. The catch-survey approach, which is essentially length-based, depends on defining a group that will grow into the adult group over the ensuing year. There is no viable alternative to defining recruits as juveniles for blue crabs. Once juveniles are included, variable and density-dependent mortality will complicate estimation of stock size and mortality rates. Currently, we have been able to bound the estimate of exploitation and F. The goal of point estimates for these rates is currently unachieved, but may be in the future. By using the upper bounds as indications of the fisheries potential impact, however, management has precautionary guidance.

Collie and Kruse (1998) did not extend their estimate of exploitation rate to F. Because the reference points developed in previous assessments of the Delaware Bay blue crab stock are in terms of F (Helser & Kahn 1999), assessments of this stock must produce F estimates for comparisons to the overfishing threshold, [F.sub.REP], and various potential targets such as [F.sub.MAX] and [F.sub.01]. These reference points must be revised, but given that the estimated upper bound of F is currently well below both the assumed input estimate of M = 1.0 and the estimated lower bound of M, which averaged 1.01 (Fig. 13), the stock is not currently overexploited. F = M has been recommended as a surrogate for [F.sub.MSY] because stock productivity should be related to M (Quinn & Deriso 1999). Deriso (1982) found that M could be viewed as an upper bound for [F.sub.MSY]. Thompson (1993) found that a fishing mortality under 0.8M should keep spawning biomass per recruit above 30% of virgin SPR. Given the high level of resilience implicit in the alpha estimate from the stock recruit model and the significant compensatory mortality, this stock should be able to sustain relatively high harvest rates (V. Crecco, Connecticut Department of Environmental Protection, pets. comm.). Although estimated Z is variable, making trends difficult to detect, there was no evidence that Z increased over the period. Although the upper bound estimate of F showed a significant increase. The lower bound of F and the C-K estimate showed evidence of flattening out or declining, however.

Our estimate of adult stock size showed a positive linear trend over the period 1979 to 2002 (Fig. 8). In contrast, the Chesapeake stock, including the spawning stock (Lipcius & Stockhausen 2002), has declined, and a recent assessment indicates that F has increased (Sharov et al. 2003). The Chesapeake stock maintains its much greater abundance relative to the Delaware stock by producing much higher recruitment. Even if juvenile survival is much higher in the Chesapeake due to eelgrass beds or other factors that the Delaware Bay lacks, higher recruitment would also seem to require a much higher density of larvae to re-enter the Chesapeake Bay from the shelf waters. The mouths of the two bays are approximately equal in width. In fact, larval density is much higher off the mouth of the Chesapeake (J. McConaugha, Old Dominion University, pets. comm.). If the Chesapeake spawning stock is reduced to the point that this higher density larval density cannot be maintained, recruitment could decline. Lipcius and Stockhausen (2002) reported that spawning stock in the Chesapeake has declined by 81% over the period from 1992 through 2000. One possibly significant difference in regulation between the two fisheries affects harvest of females carrying extruded eggs (sponge crabs). In Delaware Bay, sponge crabs are protected from harvest, and once potters begin to catch them in June, they tend to move up-bay to avoid gravid females. The result is that spawning females effectively have a refugium from harvest for the summer. In contrast, Virginia, where most spawning females are found in the Chesapeake, allows the harvest of all gravid females except those in the final stage, when the egg mass has turned black. Virginia has established a spatial refuge in the lower Bay where harvest is prohibited in the summer, however.

When assessing fast-growing animals that cannot be accurately aged, catch-survey models can be helpful, assuming a research survey exists. Density-dependent mortality may be present if fast growth requires the inclusion of juveniles. This stock lacks an extended age structure, which is one reason that indices of YOY crabs and recruits stage crabs were predictive of adult stock size. Adults were predominantly composed of the most recently recruited year class (Kahn et al. 1998, Kahn 2003a). Consequently, density-dependent mortality of recruits has a major impact on adult stock size the following year. Such variable and patterned natural mortality violates a common assumption in stock assessment work of constant natural mortality. This violation can have a serious distorting effect on estimates of abundance and fishing mortality. We have shown how adjustment of the error weighting affected the fit of the catch-survey model to the data and have provided an alternative method of estimating at least upper and lower bounds of F. The large number of recruits and their high and density-dependent mortality impeded the accurate estimate of the abundance of exploitable blue crabs in any given year. Consequently, we could not develop a precise estimate of the exploitation rate. We were able to develop upper and lower bounds on exploitation rate and F, however, which may be helpful for management.

TABLE 1.
Model inputs for the catch-survey models applied to the Delaware Bay
blue crab stock (1979-2002). Survey indices based on data collected
in August and September of the previous year, lagged to January 1 of
the fishing year. Indices are the geometric mean catch per tow. For
purposes of the modeling, recruits are defined as crabs with carapace
with < 120 mm and adults as crabs with carapace width >20 mm.

                Survey Indices (#/tow)      Survey Avg. (kg)

Fishing Year    Recruits       Adults      Recruits     Adults

    1979          2.04          0.32         0.03        0.16
    1980          1.28          0.82         0.03        0.15
    1981          5.64          0.47         0.01        0.17
    1982          2.02          1.48         0.01        0.15
    1983          0.54          0.47         0.03        0.16
    1984          0.83          0.55         0.02        0.18
    1985          3.36          0.96         0.02        0.17
    1986          7.76          0.62         0.03        0.17
    1987          1.53          0.85         0.03        0.14
    1988          3.11          0.74         0.03        0.15
    1989          4.02          1.59         0.02        0.13
    1990         10.49          1.16         0.01        0.13
    1991          0.74          1.06         0.03        0.16
    1992          4.75          1.31         0.02        0.18
    1993          3.76          2.47         0.03        0.14
    1994          8.12          1.46         0.03        0.16
    1995         11.22          1.39         0.01        0.14
    1996          2.62          1.46         0.03        0.14
    1997          4.01          1.26         0.03        0.13
    1998         12.98          1.61         0.02        0.14
    1999          5.32          1.59         0.03        0.14
    2000          9.41          1.55         0.01        0.14
    2001          8.35          1.42         0.01        0.13
    2002          3.54          1.03         0.03        0.15

  Average         4.89          1.15         0.02        0.15

                             Landings

Fishing Year        Number, Mil.    1,000 Pounds

    1979                4.38             1,423
    1980                9.96             3,318
    1981                7.10             2,329
    1982                4.67             1,595
    1983                6.49             2,198
    1984                8.35             2,794
    1985               16.04             5,350
    1986               17.74             5,731
    1987               20.52             6,690
    1988               24.89             7,829
    1989               28.84             9,358
    1990               34.53            11,163
    1991               26.96             9,120
    1992               32.01             9,936
    1993               38.81            12,748
    1994               32.50            10,285
    1995               48.53            15,080
    1996               22.29             7,132
    1997               26.62             8,633
    1998               32.90            10,445
    1999               35.65            11,690
    2000               25.52             8,561
    2001               21.40             7,295
    2002               27.19             9,074

  Average              23.08           7490.68

TABLE 2.
Model outputs and calculated variables. For details on calculations,
see Methods. Results are for the preferred model with process error
downweighted to 0.1 of the observation error. Several parameters
could not be estimated for 2002, because survival into 2003 was
required. Stock sizes and exploitable biomass estimates are for
January 1.

                         q = 0.0163

                    Stock Size (millions)

 Year        Recruits      Adults    Total    Survival     Z

 1979         124.99       19.62    144.61      0.34     1.07
 1980          76.87       49.67    126.54      0.23     1.45
 1981         334.44       29.61    364.05      0.25     1.39
 1982         118.01       90.60    208.61      0.15     1.87
 1983          34.18       32.25     66.43      0.50     0.69
 1984          53.71       33.16     86.88      0.62     0.48
 1985         193.87       53.89    247.76      0.16     1.81
 1986         430.46       40.36    470.82      0.12     2.09
 1987          94.38       57.96    152.34      0.30     1.21
 1988         195.87       45.35    241.22      0.39     0.94
 1989         239.32       93.87    333.19      0.22     1.52
 1990         582.42       72.77    655.19      0.12     2.11
 1991          48.09       79.13    127.22      0.58     0.55
 1992         298.05       73.55    371.61      0.39     0.93
 1993         227.02      146.42    373.44      0.24     1.42
 1994         466.18       90.29    556.46      0.16     1.82
 1995         631.01       90.36    721.37      0.14     2.00
 1996         160.33       97.55    257.88      0.30     1.20
 1997         245.28       77.46    322.74      0.30     1.19
 1998         725.99       97.83    823.82      0.13     2.05
 1999         317.44      106.48    423.92      0.23     1.48
 2000         534.58       96.73    631.31      0.15     1.91
 2001         467.80       93.02    560.82      0.12     2.08
 2002         217.13       69.89    287.02      0.25     1.38

Average       284.06       72.41    356.47      0.27     1.44

                            Lower          Upper         Collie-
           Exploitable      Bound          Bound          Kruse
             Biomass     Exploitation   Exploitation   Exploitation
 Year      (mill. lbs)       Rate           Rate           Rate

 1979         6.86           0.03           0.08           0.06
 1980         16.35          0.08           0.25           0.14
 1981         10.81          0.02           0.07           0.04
 1982         30.86          0.02           0.13           0.04
 1983         11.55          0.10           0.16           0.18
 1984         12.87          0.10           0.13           0.18
 1985         19.61          0.06           0.28           0.12
 1986         14.85          0.04           0.23           0.07
 1987         18.06          0.13           0.31           0.25
 1988         14.56          0.10           0.21           0.19
 1989         27.83          0.09           0.28           0.16
 1990         21.06          0.05           0.30           0.10
 1991         28.62          0.21           0.27           0.39
 1992         29.35          0.09           0.18           0.16
 1993         45.77          0.10           0.30           0.19
 1994         30.92          0.06           0.26           0.11
 1995         27.97          0.07           0.33           0.12
 1996         29.97          0.09           0.22           0.16
 1997         22.42          0.08           0.21           0.15
 1998         29.31          0.04           0.24           0.07
 1999         32.62          0.08           0.27           0.15
 2000         29.60          0.04           0.22           0.07
 2001         27.08          0.04           0.23           0.07
 2002         23.42          0.09                          0.17

Average       23.43          0.08           0.23           0.14

              Lower       Upper    Collie-    Lower     Upper
              Bound       Bound     Kruse     Bound     Bound
 Year           F           F         F         M         M

 1979          0.05        0.13     0.09       0.94     1.02
 1980          0.15        0.48     0.27       0.98     1.30
 1981          0.04        0.13     0.07       1.26     1.35
 1982          0.05        0.28     0.09       1.59     1.82
 1983          0.14        0.23     0.25       0.47     0.56
 1984          0.12        0.17     0.22       0.31     0.36
 1985          0.14        0.62     0.26       1.20     1.67
 1986          0.09        0.56     0.16       1.53     2.00
 1987          0.23        0.54     0.42       0.67     0.98
 1988          0.16        0.32     0.29       0.62     0.78
 1989          0.17        0.55     0.31       0.97     1.35
 1990          0.13        0.73     0.23       1.38     1.99
 1991          0.28        0.35     0.50       0.20     0.27
 1992          0.13        0.28     0.24       0.66     0.80
 1993          0.19        0.56     0.35       0.86     1.23
 1994          0.13        0.57     0.23       1.24     1.69
 1995          0.16        0.77     0.28       1.23     1.85
 1996          0.15        0.38     0.27       0.82     1.05
 1997          0.14        0.37     0.26       0.83     1.05
 1998          0.09        0.55     0.17       1.49     1.95
 1999          0.16        0.52     0.29       0.96     1.32
 2000          0.09        0.48     0.17       1.43     1.82
 2001          0.09        0.56     0.17       1.52     1.99
 2002

Average        0.13        0.44     0.24       1.01     1.31

ACKNOWLEDGMENTS

The authors thank Stew Michels, survey PI, Captain Leon Spence, crew members Mike Greco, Jordy Zimmerman and all who have worked on the survey through the years. Rick Cole, Shellfish and Marine Fish Program Manager, designed the survey in the late 1970s with assistance from Willard Van Engel of the Virginia Institute of Marine Science. Michael Todd conducted the commercial shellfish intercept survey of effort and landings. Bill Whitmore maintained the landings database. The authors also thank Willard Van Engel, Jeremy Collie, Steve Cadrin, Michael Fogarty and especially Victor Crecco for their helpful comments and keen insight. Much of this work was supported by an Atlantic Coastal Act Project, titled Coastal Fisheries Management Assistance from NOAA's Grant Management Division.

LITERATURE CITED

Bancroft, J. & D. M. Kahn. 2002. Stock assessment of Delaware Bay blue crab (Callinectes sapidus) for 2002. Div. Fish. Wild., 89 Kings Highway, Dover, DE.

Churchill, E. P. 1921. Life history of the blue crab. Bull. U.S. Bur. Fish. for 1917-1918. 36:91-128.

Cole, R. V., W. H. Whitmore & D. M. Kahn. 1996. Recreational blue crab effort and harvest in the Delaware estuary. Div. Fish Wild., Dover DE.

Cole, R. W. 1998. Changes in harvest patterns and assessment of possible long term impacts on yield in the Delaware commercial blue crab fishery. J. Shellfish Res. 17(2):469-474.

Collie, J. S. & G. H. Kruse. 1998. Estimating king crab (Paralithodes camtschaticus) abundance from commercial catch and research survey data. In: G. S. Jamieson, & A. Campbell, editors. Proceedings of the North Pacific Symposium on Invertebrate Stock Assessment and Management. Can. Spe. Pub. Fish. Aq. Sci 125. pp. 73-83.

Collie, J. S. & M. P. Sissenwine. 1983. Estimating population size from relative abundance data measured with error. Can. J. Fish. Aq. Sc. 40:1871-1879.

Conser, R. J. & J. Idoine. 1992. A modified DeLury model for estimating mortality rates and stock sizes of American lobster populations. NEFSC Research Document SAW 14/7, NMFS.

Deriso, R. B. 1982. Relationship of fishing mortality and growth and the level of maximum sustained yield. Can. J. Fish. Aq. Sci 39:1054-1058.

Delaware Division of Fish and Wildlife in Cooperation with New Jersey Division of Fish. Game and Wildlife. 1999. 1999 Delaware Bay blue crab fishery management plan, Div. Fish Wild., Dover DE.

Dittel, A. I., A. H. Hines, G. M. Ruiz & K. K. Ruffin. 1995. Effects of shallow water refuge on behavior and density-dependent mortality of juvenile blue crabs in Chesapeake Bay. Bull. Mar. Sci. 57:902-916.

Epifanio, C. B. 1995. Transport of blue crab (Callinectes sapidus) larvae in the waters off Mid-Atlantic states. Bull. Mar. Sci. 57:713-725.

Epifanio, C. E., C. C. Valenti & A. E. Pembroke. 1984. Dispersal and recruitment of blue crab larvae in Delaware Bay, U.S.A. Estuar. Coast. Shelf Sci. 18:1-12.

Epifanio, C. E. & R. W. Garvine. 2001. Larval transport on the Atlantic continental shelf of North America: a review. Estuar. Coast. Shelf Sci. 52:51-77.

Fischler, K. J. 1965. The use of catch-effort, catch-sampling and tagging data to estimate a population of blue crabs. Trans. Am. Fish. Soc. 94:287-310.

Fischler, K. J. & C. H. Walburg. 1962. Blue crab movement in coastal South Carolina, 1958-59. Trans. Am. Fish. Soc. 91:275-278.

Fogarty, M. J., A. A. Rosenberg & M. P. Sissenwine. 1992. Fisheries risk assessments: Sources of uncertainty. Environ. Sci. Tech. 26:440-447.

Garvine, R. W., C. E. Epifanio, C. C. Epifanio & K.-C. Wong. 1997. Transport and recruitment of blue crab larvae: a model with advection and mortality. Estuar. Coas. Shelf Sci. 45:99-111.

Gray, E. H. & C. L. Newcombe. 1938. Studies of molting in Callinectes sapidus Rathbun. Growth 2:285-296.

Hay, W. P. 1905. The life history of the blue crab, Callinectes sapidus. Reprt. Of the US. Bur. Fish. 1904:395-413.

Helser, T. E. 2000. Stock assessment of Delaware Bay blue crab (Callinectes sapidus) for 2000. Div. Fish Wild., Dover, DE.

Helser, T. E. & D. M. Kahn. 1999. Stock assessment of Delaware Bay blue crab (Callinectes sapidus) for 1999. Div. Fish Wild., Dover, DE.

Helser, T. E. & D. M. Kahn. 2001. Stock assessment of Delaware Bay blue crab (Callinectes sapidus) for 2000. Div. Fish Wild., Dover, DE.

Helser, T. E., A. Sharov & D. M. Kahn. 2002. A stochastic decision-based approach to assessing the status of the Delaware Bay blue crab stock. In: J. M. Berkson, L. L. Kline & D. J. Orth, editors. Incorporating uncertainty into fishery models. Amer. Fish. Soc. Symp. 27. pp. 63-82.

Hines, A. H. & G. M. Ruiz. 1995. Temporal variation in juvenile blue crab mortality: nearshore shallows and cannibalism in Chesapeake Bay. Bull Mar. Sci. 57:884-901.

Ju, S. J., D. H. Secor & H. R. Harvey. 2001. Growth rate variability and lipofuscin accumulation rates in the blue crab, Callinectes sapidus. Mar. Ecol. Prog. Ser. 224:197-205.

Kahn, D. M. 1996.Use of indices of relative abundance as predictors of later abundance and yield for the Delaware Bay blue crab stock. Div. Fish Wild., Dover, DE.

Kahn, D. M. 2003a. Stock assessment of Delaware Bay blue crab (Callinectes sapidus) for 2003. Div. Fish Wild., Dover, DE.

Kahn, D. M. 2003b. Estimation of fishing mortality from a catch-survey model of the Delaware Bay blue crab stock. Div. Fish Wild., Dover, DE.

Kahn, D. M., R. W. Cole, S. M. Michels & W. Whitmore. 1998. Development of life-stage specific indices of relative abundance and stock-recruitment relationships for the Delaware Bay blue crab stock. J. Shellfish Res. 17(2):529-542.

Kvalseth, T. O. 1985. Cautionary note about [R.sup.2]. Amer. Statist. 39:279-285.

Lipcius, R. N. & W. T. Stockhausen. 2002. Concurrent decline of the spawning stock, recruitment, larval abundance, and size of the blue crab Callinectes sapidus in Chesapeake Bay. Mar. Ecol. Prog. Ser. 226:45-61.

Mace, P. M. 1994. Relationships between common biological reference points used as thresholds and targets of fisheries management strategies. Can. J. Fish. Aq. Sci. 51:110-122.

Michels, S. F. & M. J. Greco. 2003. Coastal Finfish Assessment Survey. Job 1-3: Bottom trawl sampling of juvenile fishes in Delaware's estuaries. Federal Aid Report. Delaware Division of Fish and Wildlife.

Myers, R. A. & N. G. Cardigan. 1993. Density-dependent juvenile mortality in marine demersal fish. Can. J. Fish. Aq. Sci. 50:1576-1590.

Perry, H. M. 1975. The blue crab fishery in Mississippi. Gulf Res. Repts. 5:39-57.

Perry, H. M., J. Warren, C. Trigg & T. V. Devender. 1998. The blue crab fishery of Mississippi. J. Shell. Res. 17:425-433.

Pearson, J. C. 1948. Fluctuations in the abundance of the blue crab in Chesapeake Bay. Res. Rept. 14. US Fish and Wildlife Service, Washington, DC.

Quinn, T. J. II & R. B. Deriso. 1999. Quantitative Fish Dynamics. Oxford, New York.

Ricker, W. E. 1975. Computation and Interpretation of Biological Statistics of Fish Populations. Fish. Res. Board Can. Bull. 191:345-352.

Rothschild, B. J., J. S. Ault, E. V. Patrick, S. G. Smith, H. Li, T. Maurer, B. Daugherty, G. Davis, C. I. Zhang, & R. N. McGarvey. 1992. Assessment of the Chesapeake Bay blue crab stock. University of Maryland, Solomons.

Rugolo, R., K. Knotts, A. Lange, V. Crecco, M. Terceiro, C. Bonzek & C. Stagg, R. O'Reilly & D. Vaughan. 1997. Stock assessment of Chesapeake Bay blue crab. Report of the Technical Subcommittee (TSC) of the Chesapeake Bay Stock Assessment Committee of the Nation Marine Fisheries Service, NOAA.

Rugolo, L. J., K. S. Knotts, A. M. Lange & V. A. Crecco. 1998. Stock assessment of Chesapeake Bay blue crab (Callinectes sapidus Rathbun). J. Shellfish Res. 17(3):906-930.

Sharov, A. F., J. H. Volstad, G. R. Davis, B. K. Davis, R. N. Lipcius & M. M. Montane. 2003. Abundance and exploitation rate of the blue crab (Callinectes sapidus) in Chesapeake Bay. Bull. Mar. Sci. 72:543-565.

Sissenwine, M. P., M. J. Fogarty & W. J. Overholtz. 1988. Some fisheries management implications of recruitment variability. In: J. Gulland, editor. Fish Population Dynamics, 2nd ed. New York: Wiley. pp. 129-152.

Smith, S. G. 1997. Models of crustacean growth dynamics. Ph.D. Dissertation. Univ. of Maryland, College Park. Sokal, R. R. & F. J. Rohlf. 1995. Biometry. Third Edition. Freeman, New York.

Tagatz, M. E. 1968. Growth of juvenile blue crabs, Callinectes sapidus, Rathbun, in the St. John's River, Florida. Fish. Bull. 67:17-33.

Thompson, G. G. 1993. A proposal for a threshold stock size and maximum fishing mortality rate. Can. Spec. Publ. Fish. Aquat. Sci. 120: 303-320.

Van Engel, W. A. 1958. The blue crab and its fishery in Chesapeake Bay. Part 1-Reproduction, early development, growth, and migration. Comm. Fish. Rev. 20:6-17.

Whitmore, W. H. & R. W. Cole. 1997. Delaware commercial shellfish harvest and status of the fisheries 1985-1995. Del. Div. Fish and Wild. Dover, DE.

Whitmore, W. H. & R. W. Cole. 2001. Delaware commercial shellfish harvest and status of the fisheries 1998-2000. Del. Div. Fish and Wild. Dover, DE.

Whitmore, W. H. & R. W. Cole. 2003. Delaware commercial shellfish harvest and status of the fisheries 2001-2002. Del. Div. Fish and Wild. Dover, DE.

Williams, A. B. 1984. Shrimp, lobsters and crabs of the Atlantic coast of the eastern United States, Maine to Florida. Smithsonian Institution Press, Washington, DC.

DESMOND M. KAHN (1) * AND THOMAS E. HELSER (2)

(1) Delaware Division of Fish and Wildlife, P.O. Box 330, Little Creek, Delaware 19961; (2) NOAA Fisheries, Northwest Fisheries Science Center, 2725 Mont Lake Blvd. Seattle, Washington 98112

* Corresponding author. E-mail: [e-mail]