An Individual-based Modeling Approach to Spawning-potential Per-recruit Models- An Application to Blue Crab (Callinectes Sapidus) in Chesapeake Bay

8/9/2019 An Individual-based Modeling Approach to Spawning-potential Per-recruit Models- An Application to Blue Crab (Callin

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An individual-based modeling approach to

spawning-potential per-recruit models: an

application to blue crab (Callinectes sapidus) in

Chesapeake BayDavid B. Bunnell and Thomas J. Miller

Abstract: An individual-based modeling approach to estimate biological reference points for blue crabs ( Callinectes

sapidus) in Chesapeake Bay offered several advantages over conventional models: (i) known individual variation in size

and growth rate could be incorporated, ( ii) the underlying discontinuous growth pattern could be simulated, and

(iii) the complexity of the fishery, where vulnerability is based on size, shell status (e.g., soft, hard), maturity, and sex

could be accommodated. Across a range of natural mortality (M) scenarios (0.3751.2year1), we determined the ex-

ploitation fraction () and fishing mortality (F) that protected 20% of the spawning potential of an unfished population,the current target. As M increased,20% and F20% decreased. Assuming that M = 0.9year1, our models estimated

20% = 0.45, which is greater than field-based estimates of in 64% of the years since 1990. Hence, the commercialfishery has likely contributed to the recent population decline in Chesapeake Bay. Comparisons of our results with con-ventional per-recruit approaches indicated that incorporating the complexity of the fishery was the most important ad-

vantage in our individual-based modeling approach.

Rsum : Une mthodologie de modlisation base sur lindividu pour estimer les points de rfrence biologique du

crabe bleu (Callinectes sapidus) dans la baie de Chesapeake prsente plusieurs avantages par rapport aux mthodes

conventionnelles : (i) on peut incorporer les variations individuelles connues de taille et de taux de croissance, ( ii) on

peut simuler le patron sous-jacent de croissance discontinue et (iii) on peut tenir compte de la complexit de la pche

dans laquelle la vulnrabilit dpend de la taille, de ltat de la coquille (par exemple, molle, dure), de la maturit et

du sexe. Sur une gamme de scnarios de mortalit naturelle (M= 0,3751,2 an1), nous avons dtermin la fraction de

lexploitation () et la mortalit due la pche (F) qui protgent 20 % du potentiel de ponte de la population non

affecte par la pche, ce qui est lobjectif actuel. Lorsque Maugmente, 20 % et F20 % diminuent. En assumant que

M= 0,9 an1, nos modles estiment 20 % 0,45, ce qui est plus que les estimations de terrain de dans 64 % des

annes depuis 1990. La pche commerciale a donc vraisemblablement contribu au dclin rcent de la population dansla baie de Chesapeake. La comparaison de nos rsultats ceux des approches conventionnelles, qui font les calculs par

recrue, indique que lincorporation de la complexit de la pche est lavantage le plus significatif de notre mthodo-

logie de modlisation base sur lindividu.

[Traduit par la Rdaction] Bunnell and Miller 2572

Introduction

Crustacean fisheries offer unique challenges for stockassessment modelers (Smith and Addison 2003). First, crus-taceans grow discontinuously through molting rather thancontinuously as finfishes. Hence, models that use a vonBertalanffy growth subroutine, which assumes continuous

growth, do not reflect the underlying biology of crustaceans.

Second, similar to many other organisms, the components ofthe crustacean growth process (i.e., the intermolt period andthe growth per molt) can be highly variable among individu-als. This variability, however, is ignored in many conven-tional fisheries models. Third, crustacean ages are difficultto estimate because they do not have scales or otoliths thatare generally used to age finfishes (but alternatives may ex-

ist, see Ju et al. 2001). As a result, developing age-structuredmodels is problematic. Fourth, crustacean fisheries oftenhave regulations that are dependent on the sex, maturity, andshell status (e.g., hard or soft) of the individual. Thus, anideal stock assessment model for crustaceans would be sizebased and allow for individual variation in discontinuousgrowth and for sex-, shell- and maturation-dependent harvestregulations.

Individual-based models are perfectly suited for this task,and this modeling approach has been used in previous stockassessment models to accommodate complexities that are illsuited for conventional models. Previous per-recruit assess-

Can. J. Fish. Aquat. Sci. 62: 25602572 (2005) doi: 10.1139/F05-153 2005 NRC Canada

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Received 8 July 2004. Accepted 2 May 2005. Published onthe NRC Research Press Web site at http://cjfas.nrc.ca on14 October 2005.J18213

D.B. Bunnell1,2 and T.J. Miller. Chesapeake BiologicalLaboratory, University of Maryland Center for EnvironmentalScience, P.O. Box 38, Solomons, MD 20688, USA.

1Corresponding author (e-mail: [email protected]).2Present address: USGS Great Lakes Science Center,

1451 Green Road, Ann Arbor, MI 48105-2807, USA.


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ment models have used an individual-based approach toincorporate spatial variation in fishing mortality (F) (Hart2001), multiple spawning within a season (Lowerre-Barbieriet al. 1998), size-selective mortality among individuals ofvarying growth rate (Kristiansen and Svsand 1998), and themolting growth process of crustaceans (Fogarty and Idoine1988). In this paper, we build on the seminal work of

Fogarty and Idoines (1988) individual-based per-recruitmodels for American lobster (Homarus americanus) by de-veloping individual-based per-recruit models for blue crab(Callinectes sapidus) in Chesapeake Bay.

Per-recruit models are generally used to evaluate howchanges in fishing mortality or size limits will influence theyield or spawning potential (e.g., eggs produced, spawningstock biomass) of a cohort of recruits. Beverton and Holt(1957) developed a yield-per-recruit (YPR) model that usedthe von Bertalanffy growth parameters to find the biologicalreference point Fmax, or the level of fishing mortality thatmaximizes yield. Because YPR models do not considerwhether resulting fishing mortality reference points are sus-tainable, spawning potential per recruit (SPPR) models (also

called egg-per-recruit models) were developed later (seeGoodyear 1993). SPPR models produce biological referencepoints, Fx%, which represent the fishing mortality rate thatreduces the SPPR of the population to x% of the SPPR of avirgin, unfished population (see Goodyear 1993). Re-sulting reference points have been recommended to be aslow as F10% for American lobsters (Anonymous 1993) andas high as F50% for rockfishes (Sebastes spp.) in westernNorth America (Clark 2002).

In Chesapeake Bay, SPPR models have been used to setbiological reference points for the blue crab fishery, whichhas averaged more than $45 million in annual market valuesince 1981 (range = $25$72 million; US Department ofCommerce, NOAA Fisheries, http://www.st.nmfs.gov/st1/

commercial/index.html). Maryland, Virginia, and the Poto-mac River Fisheries Commission have management author-ity for their respective waters in the Bay. Although each

jurisdiction has its own unique and complex set of regula-tions, blue crabs in all waters of the Bay generally recruitinto and out of different fisheries based on their shell status(hard shell, soft shell, or peeler), size, sex, and maturity. Inresponse to declining population sizes and harvests, the Bi-State Blue Crab Advisory Committee agreed on F20% as atarget reference point in 2001 (Chesapeake Bay Commission2001). Their SPPR model estimated F20% = 0.7 (ChesapeakeBay Commission 2001), which was slightly lower than apreviously published estimate of 0.8 (Rugolo et al. 1998).Both of these estimates, however, derived from age-basedmodels that assumed continuous von Bertalanffy growth andallowed recruitment to the fishery to be a function of onlycrab age. Hence, these modeling approaches used a biologi-cally inaccurate growth subroutine and did not incorporatethe complexity of the blue crab fishery.

Developing an accurate and biologically realistic SPPRmodel for the blue crab fishery in the Chesapeake Bay is in-creasingly important given the blue crab population declinein recent years. Although population abundance over thepast 50 years has varied considerably, abundances in the late1990s and early 2000s have remained consistently low(Lipcius and Stockhausen 2002; Sharov et al. 2003). Both

spawning stock biomass and larval abundance are estimatedto be below the long-term average (Lipcius and Stockhausen2002). In addition, the average size of males (Abbe 2002)and mature females (Lipcius and Stockhausen 2002) has de-clined. Finally, rates of fishing mortality and exploitationappear to have increased in recent years (Rugolo et al. 1998;Sharov et al. 2003). Recent matrix-based modeling efforts

have indicated that current exploitation rates are not sustain-able and should be reduced (Miller 2001, 2003).

In this paper, we develop an individual-based model thatincorporates individual variation in growth per molt (GPM)and the intermolt period (IP) and incorporates the complex-ity of the fishery regulations. First, we provide a validationof the growth subroutine of the model by comparing sizedistributions of modeled crabs with size distributions ofcrabs sampled during the year from the Chesapeake Bay.Under a range of natural mortality (M) scenarios, we thenuse our model to estimate F20% and20%, where is the ex-ploitation fraction. Finally, we perform a sensitivity analysisto determine whether our individual-based reference pointestimation is influenced by (i) individual variation in size

and growth, (ii) the type of growth subroutine used (i.e.,molting or von Bertalanffy growth), and (iii) the type offishery regulation that is modeled (i.e., one based on size,sex, shell status, and maturity or one based only on putativeage). In short, we explored whether adding the complexityand reality in our individual-based model provided differentresults from more simplified models. Taken together, our re-sults will provide specific reference points for the Chesa-peake Bay blue crab fishery, but they also will inform othercrustacean stock assessment modelers as to whether incorpo-rating complexities such as discontinuous growth and the ap-propriate fishing regulations is in fact necessary.

Methods

Blue crab life historyThe blue crab ranges from South America to Nova Scotia

in the Atlantic Ocean and its estuarine tributaries. In Chesa-peake Bay, blue crab larvae, termed zoea, are released by fe-males from high-salinity waters during late spring and earlysummer. Larvae are transported offshore but return to theChesapeake Bay as megalopae (i.e., the last larval stage) insummer and autumn (Olmi 1995). Juvenile crabs, initially 23 mm carapace width (CW), settle in structured habitat inthe lower bay. Thereafter, blue crabs undergo a series ofmolts in which their size is increased between 8% and 50%(Tagatz 1968; Leffler 1972; Fitz and Wiegert 1991). Prior toeach molt, blue crabs are termed peelers as their hard shellbegins to soften and crack. After shedding the hard shell, thesoft-shelled crabs are highly vulnerable to predators while anew hard shell hardens over the next 2448 h (Ryer et al.1997). As juvenile blue crabs increase in size, they dispersethroughout the Bay. In the Chesapeake Bay, molting ceasesduring winter with the onset of cold temperatures and bluecrabs burrow into the sediment. Crabs emerge from sedi-ments in late spring and recommence growth. While matur-ing females are in their last soft-shell stage, males couplewith them and deposit sperm into female oviducts. After thismaturity molt, females are believed to cease molting (Hineset al. 2003). Males, however, continue to molt after maturity.

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Throughout the Chesapeake Bay, mating occurs in late sum-mer (Hines et al. 2003). These females can store the spermover the winter and fertilize and release their eggs the fol-lowing summer. In the lower Bay, mating also is believed tooccur in late spring (Hines et al. 2003), and these femaleslikely fertilize and release their eggs during summer of thesame year. Between one and three broods are produced by

females each summer (Hines et al. 2003). Fecundity varieslinearly as a function of body size and typically ranges be-tween 1 and 8 million eggs (Prager et al. 1990).

Individual-based per-recruit modelWe developed a sex-specific, individual-based per-recruit

model that monitored the fate of individual blue crabs on adaily time step. To accommodate the complexity of the fish-ery, we monitored the shell status (i.e., peeler, soft shell, orhard shell) and maturity of individuals as they grew accord-ing to a molting interval that depended on temperature andblue crab size. Each individual blue crab in the simulationwas characterized as a super-individual in that each repre-sented some larger number of individuals (herein referred to

as their internal amount) in the population (sensu Scheffer etal. 1995). The use of super-individuals allowed the totalnumber of individuals in the simulation to remain suffi-ciently large in the face of relatively high rates of fishingand natural mortality.

GrowthIn the model, blue crabs grew discontinuously by molting,

the magnitude and frequency of which were governed bytemperature (Tagatz 1968; Leffler 1972) and crab size(Tagatz 1968; Fitz and Wiegert 1991). Empirical estimatesof the size increase associated with each molt (i.e., GPM)have ranged from 1.08 to 1.50 times premolt CW (Tagatz1968; Leffler 1972; Fitz and Wiegert 1991). Our model re-

lied on the data from Tagatz (1968) because he provided thewidest size range of blue crabs. We modeled GPM to in-crease with size among females and to be constant acrosssizes for males (Tagatz 1968). For females, the GPM wasdrawn from a normal distribution with a premolt, CW-dependent mean (1.218 + (7.09 104)CW) and a standarddeviation of 0.07. For males, the mean GPM also was drawnfrom a normal distribution with a mean of 1.25 and an stan-dard deviation of 0.06 (Tagatz 1968).

IP, or the time between molts, was represented by degree-days (degrees Celsius) and also was dependent on blue crabCW. Each day in the model, blue crabs accumulated degree-days until some threshold had been reached, which, in turn,resulted in molting. Degree-days were calculated by sub-tracting 8.9, the physiological minimum temperature forblue crab growth (Smith 1997), from the mean daily watertemperature. Hence, molting does not occur when watertemperatures are less than 8.9 C (i.e., winter in the Chesa-peake Bay) because degree-days will not accumulate. Incrustaceans, IP has been modeled to increase either linearly(e.g., Smith 1997) or exponentially (e.g., Hoenig andRestrepo 1989) as a function of size. A plot of IP versusblue crab CW from several studies indicates that most havefocused on blue crabs less than 80 mm CW and that there ishigh variation in IP among blue crabs of similar sizes(Fig. 1). Once again, we based our growth model on the em-

pirical results of Tagatz (1968) because he provided datawith the broadest size distribution. For blue crab of a givensize, IP was drawn from a shifted exponential density func-tion (Smith 1997):

(1) f( ) ( / ) ,IP e IP

IP

=

1

where represents the required or physiological minimumphase and represents an additional and variable phase. Inthis distribution, the expected value,+, is relatively closeto the minimum value (i.e.,) but the distribution has a longtail of larger values, which corresponds to the data from allof the studies (Fig. 1). Assuming that the minimum IP re-ported by Tagatz (1968) represents the required phase (),we modeled to increase exponentially with blue crab CWas (Fig. 1):

(2) = 69.70 1.0149)CW(

To ensure that the variable phase did not intersect with therequired phase, we also modeled to increase exponentiallywith blue crab CW based on the mean IP from Tagatz(1968) (Fig. 1):

(3) = ( ( ) )166.39 1.0115 CW

In the model, IP was determined for each individual crabat day 0 and at each molt. To do so, the model calculated acumulative exponential probability function for possible val-ues of IP ranging from to the maximum IP value, whichwas calculated as 2.11 times the expected IP (maximummultiplier observed from Tagatz 1968). A random numberbetween 0 and 1 was generated, and the first IP for whichthe value of the cumulative exponential probability functionexceeded the random number provided the IP for that indi-vidual. At each daily time step, degree-days were added todegree-day exposure until the sum exceeded the IP. On that

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Fig. 1. Intermolt period as a function of blue crab (Callinectes

sapidus) size from several published studies. Because of the

wide size distribution, we relied on Tagatz (1968) to generate

size-dependent estimates of the two phases of the growth process

model (sensu Smith 1997): (broken line) represented the re-quired or physiological minimum phase and represented thevariable phase (see eqs. 4 and 5). The sum of and (solid

line) represented the average intermolt period for a blue crab ofa given size.


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day, individual blue crabs molted to their new size, theirdegree-day exposure was reset to 0, a new IP was deter-mined (as a function of their new size), and their shell statuswas changed to soft.

We monitored the status of the blue crab shell (i.e., hardshell, soft shell, or peeler) because the fishery regulationschange with shell status. Blue crabs were assumed to be

soft-shelled for the day of molting and the following dayonly (i.e., individuals returned to hard-shell status 2 days af-ter molting). Blue crab shells were classified as peelers forapproximately 1 week before molting. Because the modeldoes not calculate the day of molting in advance (i.e., futuretemperature is unknown), we used a preliminary simulationrun to estimate the predicted proportion,, of the IP that isobtained 1 week prior to molting:

(4) = degree-day exposure1 week before moltingIP

In general, increased linearly with blue crab CW, but theslope and intercept of theversus CW relationship differedacross months in which degree-day exposures were reset

(i.e., the month when molting occurred). During the warmermonths of MaySeptember, = 0.60 + (1.81 103)CW.During October, because individual blue crabs carry a rela-tively high degree-day exposure with them during winter,was the highest:= 0.8722 + (2.94 104)CW. Finally, dur-ing the cold months of NovemberApril, = 0.7355 +(9.59 104)CW. On the day when degree-day exposure ex-ceeded IP, the shell status of blue crabs was changed topeeler until the crab molted to a soft shell.

MaturityMaturity of individual blue crabs was assigned only to fe-

males in this model, as maturity was assumed not to influ-ence growth of males. Female maturity was a function of

size and time of year. Using data from the Chesapeake BayWinter Dredge Survey from 1990 to 1998 (for details, seeSharov et al. 2003), we calculated a female maturity ogivebased on 5-mm size bins (N= 23 610 female blue crabs). Alogistic function best described the maturity ogive, given as

(5) Pr(maturity) 0.9994

1 CW

117.981

0.928.51

2=

+

= , r 9

Maturation was a function of time: females could mature be-tween 1 April and 1 June or between 1 July and 1 Octobercorresponding to the two mating periods in the Bay (Hines

et al. 2003). During each molt within those periods, animmature female reached maturity if a randomly generatednumber between 0 and 1 was greater than her size-dependentprobability of maturing. Once a female matured, she re-mained a hard-shell crab and ceased molting for the remain-der of her lifetime.

MortalityNatural and fishing mortality rates were applied separately

in the model. Because there is considerable uncertainty sur-rounding estimates of natural mortality of blue crabs, we ranour models with four different values of natural mortality. Tobe consistent with previous blue crab population models, we

used an M= 0.375year1, which is based on the assumptionthat 5% of blue crabs lived to a maximum age of 8 years(Rugolo et al. 1998). To address recent concerns that M =0.375year1 is an underestimate, we used life history invari-ant theory to provide alternative estimates of natural mortal-ity.

Owing to evolutionary trade-offs among rates of mortality,

growth, and maturity, constant relationships (i.e., invariants)among these life history parameters have emerged acrosstaxa (see Charnov 1993). Three caveats are worth notingwhen applying life history invariants to blue crabs. First, un-certainty in blue crab aging also leads to uncertainty in esti-mates of age at maturity and growth rates. Second, invariantsare derived largely from fishes rather than from aquatic in-vertebrates owing to the larger sample size of fishes. Al-though the trade-offs that produce the invariants in fishesalso should produce invariants among aquatic invertebrates,the values of invariants can differ (see Charnov 1993). Third,estimates of total mortality (Z) derived from the ChesapeakeBay Winter Dredge Survey (for details, see Sharov et al.2003) constrain the upper value of natural mortality. These Z

estimates ranged from 0.34 to 1.47year1 (mean of 0.99) be-tween 1990 and 2002 (Lynn Fegley, Maryland Departmentof Natural Resources, 580 Taylor Avenue, Annapolis, MD21401, USA, personal communication).

Given these caveats, we estimated natural mortality withthree different life history invariants. The product of naturalmortality and age at maturity is equal to 1.65 (Jensen 1996)or 2.0 (Charnov and Berrigan 1990) in fishes. Female bluecrabs are believed to mature sometime between age 1 and2 years (Hines et al. 2003); in our model, 50% of femalesfrom the initial cohort matured at age 13 months. Con-sidering the range of invariants and age at maturity, Mshould be between 0.83 and 2.00year1 for female bluecrabs. A second invariant revealsMdivided by k(the Brody

growth coefficient in the von Bertalanffy equation) to equal1.50 (Jensen 1996) or 1.65 (Charnov 1993). Estimates of kfor blue crab have ranged from 0.49 to 1.09 (Rugolo et al.1998; Ju et al. 2001), which, in turn, results in an Mbetween0.74 and 1.80year1. Finally, in a meta-analysis, Pauly(1980) found natural mortality to be a function of mean an-nual temperature as well as Linf and k from the vonBertalanffy equation. Using previous ranges ofk, a range ofLinf (i.e., CWinf) from 18.1 cm (Ju et al. 2001) to 26.3 cm(Rugolo et al. 1998) and an annual mean temperature of16.5 C (grand mean of average annual temperature from1991 to 2002 at the Virginia Institute of Marine Science(VIMS) pier, Gloucester Point, Virginia) reveal M to rangefrom 1.02 to 1.57year1. Comparisons across the three esti-mates of natural mortality indicate a range of M between0.74 and 1.8year1; given that Zhas been estimated to be nogreater than 1.5year1, estimates of Mgreater than 1.3 ap-pear highly unlikely. Nonetheless, the early age at maturityand fast growth of blue crabs lead to considerably higher es-timates of natural mortality using life history invariants thanthose generated from the maximum age approach in previ-ous blue crab models.

To span the range of potential natural mortality estimates,we used four values ofM(0.375, 0.6, 0.9, and 1.2year1) inour models. In an individual-based model, mortality is mod-eled as a probability. The annual probability of surviving

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sources of natural mortality was equal to eM. Thus, the dailyprobability of natural mortality was equal to 1 eM/365. In-dividuals with soft shells were modeled to have twice theprobability of natural mortality as similar-sized peelers orhard-shelled crabs owing to their greater vulnerability with-out the hard shell (Ryer et al. 1997). In the model, mortalitywas represented by a reduction in the internal amount of

each super-individual. The internal amount (a) was reducedby some number d, which represented the number of indi-viduals within each super-individual that died. The value dwas drawn from a normal distribution when the product ofaand p was greater than 5, with

(6) mean = ap

standard deviation = [ap(1 p)]0.5

where p is the daily probability of dying (Scheffer et al.1995). Otherwise, dwas drawn from a binomial distribution(details in Scheffer et al. 1995). Given the internal amount aof each super-individual blue crab and its shell-status-dependent daily probability of mortality p, our model esti-mated the number of individuals d within the super-individual that died using eq. 6.

We based the vulnerability to fishing mortality on Mary-land fishing regulations, for which the season is 1 Aprilthrough 15 December. Male blue crabs and immature femaleblue crabs with a hard shell were vulnerable when theyreached 130 mm CW; this size is a compromise between the127 mm CW limit between 1 April and 14 July and the133 mm CW size limit between 15 July and 15 December.Mature females can be harvested regardless of their size. Formale and female crabs that were in peeler or soft shellstages, the minimum harvestable size was set at 89 mm CW.Note that because blue crabs can only have one shell status(i.e., peeler, hard, or soft), each individual was vulnerable to

only one fishery on any given day.In the model, the probability of daily fishing mortalityequaled 1 eF/259, where F represented a nominal annualrate of fishing mortality and 259 is the number of days in thefishing season. As with natural mortality calculations, themodel used eq. 6 to reduce the internal amount a for eachsuper-individual blue crab by some number d according tothe probability of daily fishing mortality. The nominal fish-ing mortality that was used to set the probability of dailyfishing mortality was not the realized fishing mortality onthe entire population because individuals in the model be-came vulnerable to the fishery at different times based ontheir growth rate and shell status. As a result, we estimatedthe realized fishing mortality by solving for F in Baranovs

catch equation (Quinn and Deriso 1999):

(7) FN Z

=

C Z

021( )e

where Cequaled the total number of crabs harvested overthe 2 years of the simulation, Z equaled the total mortalityrate, and N0 equaled the number of blue crabs alive on1 April, year x+ 1 (see Fig. 2), when the fishery begins. To-tal mortality Zwas estimated as

(8) Z N N= ln( ) ln( )0

2f

where Nf equaled the number of blue crabs alive at the endof the 2 years of the simulation (Quinn and Deriso 1999).

Reference points also can be presented in terms of, theexploitation fraction. We calculated the exploitation fraction

as the number of crabs harvested divided by N0 (Quinn andDeriso 1999). In our view, reference points for blue crabs inChesapeake Bay based on the exploitation fraction ratherthan fishing mortality have two advantages. First, compari-sons with empirical estimates are possible because both in-puts to the exploitation fraction are measured reliably: thewinter dredge survey estimates N0 for each year and man-agement agencies monitor the harvest data throughout theyear. Second, calculation of the exploitation fraction makesno assumptions regarding natural mortality, over which thereis considerable uncertainty for blue crab. Because currentreference points rely upon fishing mortality, however, wewill report reference points in terms of both and fishingmortality.

Validation of growth parametersTo validate the growth subroutine of the individual-based

model, we compared size distributions of blue crab gener-ated from our model with size distributions of blue crabsampled from the Chesapeake Bay during 19971999. Ches-apeake Bay (herein field) distributions were derived fromtwo surveys: (i) the VIMS Juvenile Fish and Blue CrabTrawl Survey, which occurs monthly in the Virginia watersof the Chesapeake Bay (including the James, York, andRappahannock rivers), and (ii) the Chesapeake Bay WinterDredge Survey, which occurs between December and Marchthroughout the entire Chesapeake Bay. We used the dredgesurvey for winter field distributions because it samples crabs

burrowed into the sediments more effectively than a trawl.The trawl survey more effectively captures smaller crabs as aresult of its 6.4-mm liner compared with the 13-mm liner ofthe dredge (for more survey details, see Montane et al.(2003) for the trawl survey and Sharov et al. (2003) for thedredge survey). In both surveys, all blue crabs were mea-sured (nearest millimetre CW), sex was determined, and fe-male blue crabs were assessed for maturity. Within eachyear, we assumed that crabs smaller than 60 mm CW cap-tured in September or later represented a new cohort (Sharovet al. 2003) and those individuals were deleted from the fielddistributions.

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Fig. 2. Timeline for the individual-based per-recruit model, which

began on 1 January of year x + 1 and ended on 31 December of

year x + 2, where xis the year of settlement.


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For each year, we seeded the individual-based model withthe sex, maturity (for females), and CW of all blue crabscaptured in the winter dredge survey, which resulted in dif-ferent numbers of crabs for each modeled year (1997: 5587,1998: 3236, 1999: 1970). For each year, the internal amountaof each individual crab equaled 400 million divided by thetotal number of crabs modeled (i.e., we modeled a popula-

tion of 400 million crabs, which is within the range of totalcrab abundance in the 1990s; Sharov et al. 2003). Using themean daily temperature recorded at the VIMS pier, we simu-lated the growth and survival of those blue crabs during1 year. We used an intermediate value ofM, 0.75year1. Weused two different daily probabilities of fishing mortality toproduce exploitation fractions that bounded the minimum(30%) and maximum (70%) exploitation fractions observed inthe Chesapeake Bay during the 1990s (Sharov et al. 2003). Wethen compared size distributions predicted for May, July, Sep-tember, and January from the model with size distributionsfrom the same months of blue crabs sampled in the field.

SPPR models and estimation of biological referencepoints

For each SPPR model, we simulated a cohort of 2000 ju-venile super-individual blue crabs through 2 years followingthe year of initial settlement. The model began on 1 Januaryof year x + 1 and continued through 31 December of yearx+ 2, where xis the year of settlement (see Fig. 2). By as-signing each super-individual an internal amount of 150 000at the start of each simulation, we modeled a cohort of 300million individuals, which is within the range (95540 mil-lion) of new recruits estimated in the Chesapeake Bay dur-ing the first winter of life (Sharov et al. 2003). Sizes foreach super-individual in the cohort were drawn from alognormal distribution with a mean of 27.2 mm CW and a

standard deviation of 10.3, which is reflective of the sizedistribution sampled in the winter dredge survey. We as-sumed an initial 1:1 sex ratio for each cohort. Water temper-atures were equal across both years and equaled the meandaily water temperature at the VIMS pier from 1991 to2002. As with conventional per-recruit models (Bevertonand Holt 1957), we varied rates of natural mortality andnominal fishing mortality to determine their effects onspawning potential.

To estimate20%and F20%, we estimated the spawning po-tential per recruit under various mortality regimes. Spawningpotential equaled the sum of the total numbers of eggs pre-dicted to be spawned by females in the second year of thesimulation (see Fig. 2); no females were large enough to ma-

ture and spawn by 15 September of the first year of themodel. Mature females were randomly assigned a spawningday between 15 May and 15 September. The spawning po-tential, or the number of eggs spawned, for each super-individual was the product of its size-based fecundity (mil-lions of eggs = 2.248 + 0.337(CW); Prager et al. 1990) andits internal amount a (i.e., the number of individuals thatwere living) on the day of spawning. For each combinationof fishing and natural mortality, we summed the spawningpotential of each super-individual. We assumed that the vir-gin, unfished spawning potential occurred when or F =0.0year1.

Sensitivity to added model complexities

Our individual-based model was able to accommodateseveral realistic complexities that most conventional length-or age-based per-recruit models do not, including (i) individ-ual variation in crab sizes and growth, (ii) discontinuousgrowth, and (iii) vulnerability to harvest that is a function ofcrab size, shell status, maturity, and sex. To determine the

impact of including these complexities on reference point es-timation, we systematically removed one facet of complex-ity, replaced it with the more simplified facet, and comparedthe resultant reference points. Specifically, using a factorialdesign approach, we crossed three different growth subrou-tines ((i) discontinuous growth with individual variation ininitial sizes and growth (original model), (ii) discontinuousgrowth without any individual variation, and (iii) continuousgrowth without any individual variation) with two differentharvest subroutines ((i) function of size, sex, shell status,and maturity (original model) and (ii) function of age-dependent recruitment to the fishery). We ran these modelsat one level of natural mortality (M= 0.375year1). For thecontinuous growth subroutine, we calculated the average

daily size across a low and high level of fishing mortalitywhen M= 0.375year1 in the original model and used thosevalues to estimate the von Bertalanffy growth equation(CW = 263.1(1 e0.6942(t0.007))). For the age-dependent har-vest subroutine, vulnerability to the fishery was 0.75F forthe first year (i.e., age 1) and 0.95Ffor the second (i.e., age2), which corresponds to previous age-based SPPR modelsfor blue crabs (Rugolo et al. 1998). For the models with thevon Bertalanffy growth subroutine, we could not use the ma-turity subroutine that was based on molting events at a givensize. Hence, maturity became a function of age: 10% of age-1 females and 90% of age-2 females matured (Rugolo et al.1998). As in the original model, spawning by mature fe-males occurred in the second year of the simulation.

Results

Validation of growth parameters

Our intent with the validation was not to perfectly matchthe field distributions by going through iterations of varyingrates of fishing pressure or natural mortality. Rather, wewanted to demonstrate that the literature-derived growth pa-rameters could effectively capture the general trajectory ofblue crab size frequencies, and hence growth rates, in theChesapeake Bay over the course of 1 year, within the rangeof reasonable estimates for fishing and natural mortality. In

each year, the modeled size distributions were broadly simi-lar to the field distributions, with the low fishing pressuresimulations performing slightly better than the high fishingpressure ones (Fig. 3). As might be expected, models withthe low (0.33 in 1997, 0.46 in 1998, and 0.39 in 1999)predicted more large crabs during each month of each yearthan was observed in the field. Conversely, models with thehigh (0.61 in 1997, 0.71 in 1998, and 0.65 in 1999) pre-dicted considerably fewer blue crabs larger than 130 mmCW than was observed in the field. Having validated thisgrowth subroutine, we subsequently used this individual-based population model as the basis of our per-recruit models.

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SPPR model results

In our SPPR model, a cohort of newly recruited blue crabswas simulated for 2 years beginning on 1 January. To de-scribe the size distributions of the cohort through the2 years, we present results from four representative modelruns of varying levels of natural (M) and fishing mortality(exploitation fraction ): M = 0.375year1 and = 0.32(Figs. 4a4c), M= 0.375year1 and = 0.65 (Figs. 4d4f),M = 0.90year1 and = 0.30 (Figs. 4g4i), and M =0.90year1 and = 0.48 (Figs. 4j4l). Initial size distribu-tions were the same for all models and reflected size distri-

butions of presumed newly recruited blue crabs in theChesapeake Bay (see Figs. 4a, 4d, 4g, and 4j and comparewith the first mode of fig. 4 in Sharov et al. 2003). Size dis-tributions in the next two winters were somewhat influencedby exploitation rate. By the second winter of life(31 December, year x+ 1) or the end of the first year of themodel run, the average surviving blue crab had recruited tothe hard-shell fishery, independent of the natural or fishingmortality regime (Fig. 4), but within a level of natural mor-tality, the mean size of crabs was larger when exploitationfractions were lower (e.g., compare Figs. 4b and 4h withFigs. 4e and 4k). By the third winter of life (31 December,

year x + 2), the average crab had grown to more than188 mm CW, but only a small percentage of the original co-hort had survived (24% in Fig. 4c, 4% in Fig. 4f, 4% inFig. 4i, and 0.4% in Fig. 4l). Comparing survival betweenthe sexes, more males survived than females in all mortalitytreatments, likely because all mature females were vulnera-ble to the fishery, independent of size.

Under all natural mortality regimes, the large majority ofblue crabs harvested were hard-shell ones. Within a level ofnatural mortality, however, the percentage of soft-shell orpeeler crabs in the total harvest (by number) increased with

exploitation fraction. For example, whenM= 0.9year

1

, thesoft-shell and peeler crab harvest increased from 13.5% ofthe harvest when = 0.19 to 46.6% of the harvest when =0.60. Because soft-shell and peeler blue crabs are vulnerableto the fishery at smaller sizes than in the hard-shell fishery,higher exploitation fractions in the model increased the num-ber and overall percentage of soft-shell and peeler crabs thatwere harvested.

For each rate of natural mortality, we estimated20% (andthe associated F20%), which equaled the harvest rate that pro-tected 20% of the spawning potential of an unfished popula-tion. Our20% estimate decreased with increasing levels of

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Fig. 3. Validation of the growth parameters in the individual-based per-recruit model. Comparison of size frequency distributions of

blue crab (Callinectes sapidus) captured in the Chesapeake Bay (vertical shaded bars) with distributions of simulated blue crabs from

the model (lines) during May, July, September, and January of (a) 1997, (b) 1998, and (c) 1999. The broken line represents modeled

size distributions when exploitation fractions were low (i.e., 0.33 in 1997, 0.46 in 1998, and 0.39 in 1999), and the solid line repre-

sents modeled size distributions when exploitation fractions were high (i.e., 0.61 in 1997, 0.71 in 1998, and 0.65 in 1999).


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natural mortality:20% = 0.67, 0.57, 0.45, and 0.36 whenM= 0.375, 0.6, 0.9, and 1.2year1, respectively. These ex-ploitation fractions correspond to F20% ranging as high as1.24year1 when M = 0.375year1 to as low as 0.90year1

when M= 1.2year1. We interpolated between these modelruns to create a contour plot that reveals the proportion ofthe virgin, unfished spawning potential that is protected for a

given level of natural mortality and fishing pressure (Fig. 5).For example, should Mbe determined to equal 1.0 and man-agers choose to protect 20% of the virgin spawning potential(sensu Chesapeake Bay Commission 2001), then the plot re-veals a corresponding allowable of 0.43 or less.

We also sought to determine whether adding replicatemodel runs and changing the initial cohort size would influ-ence our results. We ran four replicate simulations at thelowest (M = 0.375year1 and = 0.18) and highest (M =1.2year1 and = 0.50) mortality regime, expecting thatvariability would be greater at the high-mortality regimegiven the more frequent use of the uniform random number

generator subroutine (i.e., the source of the stochasticity). Asexpected, a 1% difference between the smallest and largestegg productions among the replicates existed within the low-mortality regime, whereas a 7% difference between thesmallest and largest egg productions among the replicatesexisted within the highest mortality regime. Even this 7%difference, however, had a small impact on the proportion of

maximum spawning potential that was calculated for eachmodel simulation and used to find 20%. For example, in-creasing and decreasing the spawning potential by 7% whenM= 0.9 and = 0.46 changed the proportion of maximumspawning potential from 0.18 to either 0.17 or 0.19. As a re-sult, we were convinced that running several replicate simu-lations for each level of fishing mortality would not changethe overall results. Next, we ran our model with an initialabundance of only 150 million crabs rather than 300 millioncrabs, given that the abundance of new recruits in the Chesa-peake can be highly variable (Sharov et al. 2003). Per-recruit estimates versus fishing mortality revealed the bio-

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Fig. 4. Size frequency distributions of modeled blue crabs ( Callinectes sapidus) under different mortality scenarios in the individual-

based per-recruit model. Across each suite of panels (left to right), the size distributions during each of the three winters for each mor-

tality scenario are provided: the leftmost panel is the distribution at the initiation of the model in the first winter (1 January, year x+ 1),

the middle panel is the distribution in the second winter (1 January, year x+ 2), and the rightmost panel is the distribution in the third

winter (31 December, year x + 2). (ac) = 0.32 and M= 0.375year1; (df) = 0.65 and M= 0.375year1; (gi) = 0.30 and M =0.90year1; (jl) = 0.48 and M= 0.90year1. The mean on each panel represents the mean blue crab CW and N represents the num-ber of blue crabs alive in the model on that day.


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logical reference points to be the same, indicating that our

results are independent of the number of newly recruitedblue crabs for any given year.

Sensitivity to added model complexitiesExploitation-based reference points were sensitive to the

harvest subroutine but were insensitive to the growth subrou-tine (discontinuous with individual variation, discontinuousgrowth without individual variation, continuous growth with-out individual variation). When we maintained the complexfishery subroutine (i.e., harvest is a function of size, sex,shell status, and maturity) from the original model,20% =0.67 independent of the growth subroutine (Table 1). How-ever, when we changed the harvest subroutine to make it age

dependent, as has been used in previous SPPR models forblue crabs, 20% increased from 0.67 to 0.73 (Table 1);again20% was independent of the growth subroutine. Refer-ence points based on fishing mortality also were influencedby the harvest subroutine, with higher values of F20% underthe complex harvest subroutine than under the age-dependent harvest subroutine for a given growth subroutine

(Table 1). In addition, F20% was dependent on the growthsubroutine when the complex harvest subroutine was used.When the simple harvest subroutine was used, F20% wasequal across all growth subroutines (Table 1).

Discussion

Using estimates of IP and GPM from the literature, we de-veloped an individual-based model for blue crabs that simu-lated crustacean discontinuous growth and the complexity ofthe Chesapeake Bay fishery and allowed for individual vari-ation in blue crab sizes and growth. We used the model toestimate the level of fishing that would protect 20% of thespawning potential of an unfished Chesapeake Bay popula-

tion, the target reference point (Chesapeake Bay Commis-sion 2001). To validate the growth parameters, we firstseeded the model with size distributions of blue crabs in thefield, allowed blue crabs to grow in the model under fieldtemperatures, and compared size distributions of crabs in themodel with those in the field over the course of 1 year. Weexpected some differences between the predicted and fieldsize distributions, given uncertainty in natural mortality andthe simplifications inherent in our model (i.e., lack of tem-poral variation in fishing mortality or spatial variation intemperature, growth, and fishing mortality). However, giventhese limitations, our predicted distributions compared favor-ably with the field distributions, even under extremely lowand high scenarios of harvest. Applying our model to a hy-

pothetical cohort of newly recruited blue crabs, under arange of natural mortality regimes (M = 0.3751.2year1),we determined that protecting at least 20% of the spawningpotential of an unfished population required a managementregime that limited to range between 0.36 (when M =1.2year1) to 0.67 (when M = 0.375year1). Finally, wefound that20% was influenced by the harvest subroutine ofour model but was not influenced by the growth subroutineor the presence of individual variation. Our estimate ofF20%was influenced by both the harvest subroutine and thegrowth subroutine.

A few caveats to our model should be noted. First, thespawning potential calculation in our model uses only fe-

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Fig. 5. Contour plot of the spawning potential per recruit as a

function of (a) natural mortality and exploitation fraction and

(b) natural mortality and fishing mortality.

Harvest subroutine

Growth subroutine Complex fishery Age-dependent fishery

Discontinuous, with individual variation in initial size and growth 0.67 (1.24) 0.73 (1.17)

Discontinuous, without individual variation 0.67 (1.24) 0.73 (1.17)

Continuous, without individual variation 0.67 (1.33) 0.73 (1.17)

Note:Discontinuous growth simulates the realistic molting growth pattern of blue crabs. Continuous growth uses the von Bertalanffy growth equation,where the parameters were derived from the average daily size of crabs in the discontinuous growth subroutine, with individual variation. In the complexfishery, vulnerability to the fishery is a function of crab size, sex, shell status, and maturity. In the age-dependent fishery, vulnerability to the fishery was0.75F in the first year (i.e., age 1) and 0.95Fin the second year (sensu Rugolo et al. 1998). The reference points provided elsewhere in the Results arebased on a discontinuous growth subroutine (with individual variation) and a complex fishery in the harvest subroutine.

Table 1. Estimates of20% (F20% in parentheses) when different subroutines were used in the individual based model and when M =0.375year1.


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male size and abundance to predict egg production and thusassumes that males are not limiting. Although a high per-centage of females are mated in the Chesapeake Bay, recentevidence suggests that the quantity of sperm delivered bythe male is declining, which may limit egg fertilization(Hines et al. 2003). Second, we modeled natural mortality asindependent of blue crab size, despite evidence that among

crabs less than 70 mm CW, smaller crabs are more likely todie from predation or cannibalism than larger crabs (Wilsonet al. 1987; Hines and Ruiz 1995). We completed simula-tions with a size-dependent natural mortality for blue crabsless than 70 mm CW, but the high rate of mortality amongsmall crabs resulted in sustainable between 0.02 and 0.05,as relatively few individuals from the cohort recruited to thefishery. We removed this size dependency from our modelwhen we compared these exploitation fractions with thosemeasured in the winter dredge survey, which are between0.40 and 0.70 (Sharov et al. 2003). Thus, even though size-dependent mortality among juvenile crabs likely occurs inthe field, including it in the model yielded unrealistic results.

Individual-based models were first used in fisheries ecol-

ogy to explain the survival outcomes of fish cohorts. Thesemodels revealed that consideration of the initial variationamong individual sizes and inclusion of time- or size-varying relationships could lead to a different distribution ofsurvivor sizes than if a population of homogenous averageindividuals with time- or size-independent relationshipswere simulated (e.g., DeAngelis and Coutant 1982).Individual-based models have enjoyed much broader appli-cation in recent years, including per-recruit models for fish-eries stock assessment. Previous individual-based per-recruitmodels have used the flexibility offered by this approach toincorporate multiple spawning by weakfish (Cynoscionregalis) within a season (Lowerre-Barbieri et al. 1998), al-low for size-selective mortality among juvenile Atlantic cod(Gadus morhua) that have individually varying growth rates(Kristiansen and Svsand 1998), and apply spatially explicitrates of fishing mortality for sea scallops (Placopectenmagellanicus) in the Atlantic Ocean (Hart 2001).

We chose an individual-based modeling approach for bluecrab for three reasons. First, we wanted to include variationboth in initial sizes within the cohort and in the GPM and IPthat has been observed in field and laboratory studies. Thesensitivity model runs revealed that including this individualvariation yielded the exact same reference point as anindividual-based model that included all blue crabs of thesame average size and the same average growth rate. Be-cause harvest and IP were size dependent, we had expected

that including this individual variation would lead to differ-ent results than if we ignored it. Specifically, we had hypoth-esized that small individuals in the cohort might survive at ahigher rate because of a later recruitment to the fisheries,and this asymmetrical survival would influence spawningpotential. Considering that spawning potential is a functionof blue crab size and the internal amount a (i.e., number stillalive) at the time of spawning, this hypothesis was incorrectfor two reasons. First, there was no relationship between ini-tial size and size at maturity, even though large individualsmatured earlier than small ones. Larger individuals in the co-hort accumulated more harvest mortalities, whereas smallerindividuals accumulated more natural mortalities. The latter

is explained by smaller individuals having larger internalamounts during the period when small individuals had notyet recruited to the fishery. As a result of no differences intotal mortality and blue crab sizes at spawning among ini-tially small and larger individuals, the differences in spawn-ing potential were minimized.

Second, an individual-based approach permitted us to

simulate the true biology of the crustacean molt process.However, the sensitivity analyses revealed20% to be insen-sitive to either growth model (i.e., discontinuous growth ver-sus continuous growth). Our estimate of F20% was slightlyinfluenced by the growth subroutines when the complex har-vest subroutine was used: F20%was 6% higher in the contin-uous growth subroutine than in the discontinuous subroutine.When the simple harvest subroutine was used, F20% was in-sensitive to the growth subroutines. Overall, these resultsimply that blue crab per-recruit models that rely on a vonBertalanffy growth subroutine should provide referencepoint estimates that are as reliable as results from a per-recruit model that more accurately models blue crab growthas discontinuous. This is an important finding because many

conventional stock assessment models rely on a vonBertalanffy growth subroutine and these results providegreater credibility for using this growth subroutine for crus-taceans.

Third, we used an individual-based modeling approachbecause it enabled us to simulate the complexity of theChesapeake Bay blue crab fishery, where regulations are afunction of crab size, sex, shell status, and maturity. Here,the sensitivity results revealed that both 20% and F20% areinfluenced by the harvest subroutine: within a growth sub-routine,20%was 9% greater with the age-based harvest sub-routine than with complex harvest one, whereas F20% was6%14% greater with the complex harvest subroutine than

with the age-based one. Before explaining this discrepancy,it is important to note that the numbers and sizes of bluecrabs alive to spawn were roughly similar in the two harvestsubroutine models, which resulted in the equal spawning po-tentials between the two models. Thus, the reference pointdifferences are explained by differences in natural mortalityand harvest that occurred in the two models. In other words,how can the two models have the same number of spawningfemales when more females were harvested from the age-based harvest model than from the complex harvest model?In the first year of the model, the number of crabs harvestedin the age-based harvest model was considerably higher.With the complex harvest subroutine, most individuals didnot recruit to the hard-shell fishery until October of the first

year, meaning that individuals were only briefly vulnerableto the fishery in earlier months (i.e., as peeler or soft-shelledcrabs). Conversely, in the age-dependent harvest model, allage-1 crabs in the same year received 75% of the nominalfishing mortality during the fishing season (sensu Rugolo etal. 1998). Over the 2 years of the simulation, this first-yeardifference resulted in many more crabs being vulnerable tonatural mortality in the complex harvest simulation. Hence,despite the higher harvest in the age-based model, the higherlevels of natural mortality in the complex harvest model re-sulted in near equal numbers of spawning females betweenthe two models. The higher natural mortality in the complexharvest model also resulted in a higher estimate of total mor-

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tality than in the age-based one. Because total mortality in-fluenced fishing mortality more than the catch (see eq. 7),the higher total mortality explains the higher estimate ofF20% in the complex harvest subroutine. This examinationreveals how the timing of recruitment to the fishery can in-fluence the reference point estimate. Because of the com-plexity of the Chesapeake Bay fishery, assessment models

that maintain this complexity should be favored over previ-ous models that simplify the fishery into one that is basedonly on age.

Our model produced an estimate of the target biologicalreference point that is considerably higher than for previousmodels. Currently, the Chesapeake Bay fishery is managedwith a target F20% = 0.7 (Chesapeake Bay Commission2001), which is based on M = 0.375year1. This value isslightly smaller than the 0.8 estimate from Rugolo et al.(1998) and considerably smaller than our estimate of 1.24,both assuming M= 0.375year1. To understand these differ-ences, we used the NOAA Fisheries Toolbox Yield-Per-Recruit model, an age-based model similar to the approachof previous models. We should note that this model uses bio-

mass of mature crabs to estimate spawning potential ratherthan egg production as we used. Because egg production andblue crab size are linearly related (Prager et al. 1990), how-ever, these results should be much more similar than a situa-tion where egg production is nonlinearly related to size.

First, we used the age-based model to attempt to duplicateour results. We used the mean crab weight at age from ourmodel output, the same age-dependent harvest schedule usedin the sensitivity analysis, and the same proportion of F(0.635) and M(0.75) before spawning reported in Rugolo etal. (1998). For maturity schedule, we assigned 0% for age-1crabs and 100% for age-2 crabs because eggs were producedonly in the second year of the simulation. These inputs yieldedan F

20%

= 1.19, well within the range of our individual-based modeling results (baseline F20% = 1.25, F20% fromsensitivity analyses = 1.171.33). This exercise provided avalidation of our individual-based modeling approach.

Second, we used Rugolo et al. (1998) and data that wereused to generate the Chesapeake Bay Commission (2001)target reference point to recreate their respective model re-sults and then attempt to understand why their results dif-fered from ours. Compared with our inputs, those modelshad different mean sizes at ages (generally larger than ours),included more age classes (eight versus two in ours), andhad a different maturity schedule (several age classes withmature females compared with only one age class in ours).By changing only their maturity schedule to match ours, we

were able to generate F20%= 1.11 for both models, consider-ably higher than their original results and much closer to ourestimate.

This exercise revealed maturity schedule to have a clearinfluence on spawning potential based reference points forblue crabs in Chesapeake Bay. In the Rugolo et al. (1998)model, they assumed the following schedule of maturity:10% of age 1, 90% of age 2, 100% of age 3, 50% of age 4,10% of age 5, and 0% of ages 68. Rugolo et al. (1998) ac-knowledged that a maturity schedule that more quickly re-duced the percentage of mature females after the first fullyear of spawning (i.e., 0% maturity of ages 48) was more

biologically realistic than the one they chose. Nonetheless,they chose their maturity schedule because it providedsmaller estimates ofF20%, making it risk averse. Since thoseanalyses, however, aging blue crabs by the accumulation oflipofuscin in their eyestalks has revealed that 90% of bluecrabs larger than 120 mm CW in Chesapeake Bay are age 1and younger (Ju et al. 2003). The remaining 10% were age

2, although there was a statistically indistinguishable modeof crabs within this percentage that may have been age 3(Ju et al. 2003). Because of the demographic results of Ju etal. (2003) and the conventional wisdom that female maturityoccurs sometime between age 1 and age 2 (Hines et al.2003), we structured our model so that females mature atage 1 and spawn at age 2, and at age-3 females are not in-cluded. Although our estimates of F20% and20% are higherthan for previous models because of our truncated maturityschedule, this schedule most accurately reflects currentknowledge of blue crab demography and maturity.

Finally, it is not surprising that our biological referencepoint estimates were dependent on the assumed rate of natu-ral mortality (sensu Beverton and Holt 1957). As natural

mortality increased, fewer blue crabs were available to beharvested, which effectively led to an inverse relationshipbetween20% and Mas well as between F20% and M. Previ-ous blue crab modeling efforts have assumed an M =0.375year1, which was based on an assumed maximumcrab age of 8 years (Rugolo et al. 1998). Recently, Hewittand Hoenig (2005) suggested that even were the maximumblue crab age to be 8 years, an estimate ofM= 0.375year1

remains too low because of methodological concerns. As aresult of continued uncertainty in maximum age, we ex-plored the use of life history invariants for finfishes as an al-ternative estimate for blue crab natural mortality. Assummarized in the Methods, the invariants provided a rangeof M between 0.74 and 1.8year1, but we did not considerestimates of M greater than 1.2year1 because field-basedestimates of Z have never exceeded 1.5year1 from 1990 to2002. A revised Chesapeake Bay stock assessment is cur-rently underway, and they have recommended that managersuse M= 0.9year1 for two primary reasons (T.J. Miller, per-sonal communication). First, nearly all life-history invariantestimates ofM include 0.9year1. Second, the recent popula-tion decline observed in the Chesapeake Bay necessitated ahigher estimate of natural mortality than previous stock as-sessment models have used.

Assuming that a 20% spawning potential ratio is sufficientto sustain the blue crab population and M = 0.9year1, ourmodel reveals20%= 0.45 and F20% = 1.02. We advocate us-

ing the20% reference point because annual estimates of ex-ploitation factor can be calculated directly from the absoluteabundance data measured in the winter dredge survey andfrom harvest monitoring data by the state management agen-cies. Hence, managers can measure the impact of the fisheryusing exploitation factor directly and avoid using fishingmortality, which would require some estimate of naturalmortality. From 1990 through 2003, has ranged from 0.34to 0.71 (L. Fegley, Maryland Department of Natural Re-sources, 580 Taylor Avenue, Annapolis, MD 21401, USA,personal communication), and exceeded20% in 64% ofthe years assuming M= 0.9year1. With lower estimates of

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natural mortality, our model predicts thatexceeded20%infewer years (29% of years when M= 0.6year1, 7% of yearswhen M= 0.375year1).

In conclusion, our individual-based modeling approachprovided estimates ofF20% and 20% for a range of levels ofnatural mortality. For a given level of natural mortality, wefound that our estimate ofF20% was at least 55% larger than

previous estimates based on an age-based SPPR. This differ-ence was explained by differences in the maturity scheduleof blue crabs. Limiting the percentage of females that repro-duce in older age classes, which appears realistic given thecurrent knowledge of blue crab age structure (Ju et al. 2003),leads to higher reference point estimates (sensu Rugolo et al.1998). We found that some advantages of this individual-based approach, including incorporation of individual varia-tion in blue crab size and growth and simulation of the realmolting growth process, did not provide reference point esti-mates that differed from a more simplified model where allindividuals were of average size and grew at the same aver-age rate through a continuous growth process. Conversely,the capacity to simulate the complexity of the Chesapeake

Bay blue crab fishery in the individual-based model did re-sult in reference point estimates that were about 10% differ-ent than a more simplified model where recruitment to thefishery was simply age dependent. Finally, a growing bodyof evidence suggests that M is ~0.9year1 rather than0.375year1 used in previous modeling efforts. With this as-sumption, the model predicts that 0.45 is necessary toprotect at least 20% of the unfished spawning potential.With available field-based estimates of exploitation factor,we found that the commercial fishery has exceeded 20% in64% of years since 1990. Hence, these results join others(Miller 2001, 2003) that have implicated the commercialfishery as a likely contributor to the recent blue crab popula-tion decline in Chesapeake Bay.

Acknowledgments

We thank the numerous participators of the VIMS Juve-nile Fish and Blue Crab Trawl Survey and the ChesapeakeBay Winter Dredge Survey for collecting these valuabledata. We thank Kenny Rose for advice on modeling super-individuals. The comments of Dave Hewitt and three anony-mous reviewers greatly improved this paper. This is contri-bution No. 3862 of the University of Maryland Center forEnvironmental Science Chesapeake Biological Laboratory.This work was supported by grants from Maryland SeaGrant (R/F-93B) and by the NOAA Coastal Ocean Program

(NA17OP265).

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Documents

An Individual-based Modeling Approach to Spawning-potential Per-recruit Models- An Application to Blue Crab (Callinectes Sapidus) in Chesapeake Bay