effect of nonlinear predation rates on rebuilding the georges bank haddock (melanogrammus...

Effect of nonlinear predation rateson rebuilding the Georges Bank haddock(Melanogrammus aeglefinus) stock

Paul D. Spencer and Jeremy S. Collie

Abstract: The collapse of several northwest Atlantic groundfish stocks, including Georges Bank haddock (Melanogrammusaeglefinus), necessitated new fishing regulations and generated interest in the role of predation on stock productivity. Thesharp break between prolonged periods of high (pre-1965) and low (post-1965) haddock abundance suggests differing levelsof stock productivity, consistent with a surplus production model incorporating a nonlinear predation rate (Steele andHenderson’s (1984. Science (Washington, D.C.), 224: 985–987) model). This model and a Schaefer (1957. Inter-Am. Trop.Tuna Comm. Bull. 2: 245–285) model without a predation term were fit to haddock data to evaluate various rebuildingstrategies with two performance measures: the sums of discounted yield and discounted revenue. Steele and Henderson’smodel provided plausible parameter estimates for the entire data set (1931–1993) whereas Schaefer’s model providedplausible estimates only for years of low productivity (1976–1993). The presence of multiple equilibria in Steele andHenderson’s model resulted in minor shifts ofF, potentially producing large shifts in projected future biomass. For eithermodel, levels ofF that maximize either yield or revenue were lower than the recently adopted target level ofF0.1= 0.24.Recent low production provides impetus for managers to consider a variety of plausible stock production models, and theuncertainty of production dynamics, in choosing rebuilding strategies.

Résumé: Suite à l’effondrement de plusieurs stocks de poissons de fond de l’Atlantique Nord-Ouest, notamment de l’aiglefin(Melanogrammus aeglefinus) du banc Georges, de nouveaux règlements de pêche ont été formulés, et on s’est intéressé aurôle de la prédation dans la productivité du stock. La brusque rupture entre les périodes prolongées de grande abondance(avant 1965) et de faible abondance (après 1965) de l’aiglefin laisse supposer différents taux de productivité du stock,compatibles avec un modèle de production excédentaire intégrant un taux de prédation non linéaire (modèle de Steele etHenderson (1984. Science (Washington, D.C.), 224 : 985–987)). Ce modèle et un modèle de Schaefer (1957. Inter-Am. Trop.Tuna Comm. Bull. 2 : 245–285) sans terme de prédation ont été ajustés aux données sur l’aiglefin pour évaluer différentesstratégies de rétablissement comprenant deux mesures de la performance : les sommes du rendement actualisé et du revenuactualisé. Le modèle de Steele et Henderson offrait des estimations plausibles des paramètres pour l’ensemble complet desdonnées (1931–1993), tandis que le modèle de Schaefer n’en a produit que pour les années de faible productivité(1976–1993). La présence d’équilibres multiples dans le modèle de Steele et Henderson s’est traduite par des changementspeu importants de la valeurF qui pourrait produire des changements importants de la biomasse future projetée. Dans les deuxmodèles, les taux deF qui maximisent le rendement ou le revenu étaient plus faibles que la valeur cible adoptée récemment deF0,1= 0,24. La faible production enregistrée ces dernières années incite les gestionnaires à prendre en compte divers modèlesplausibles de production du stock et l’incertitude de la dynamique de la production lorsqu’ils choisissent les stratégies derétablissement.[Traduit par la Rédaction]

Introduction

Stock abundances of several exploited, demersal fish speciesin the northwest Atlantic, including the Georges Bank haddock(Melanogrammus aeglefinus), have collapsed in recent years.Haddock biomass fluctuated around 150 kt from 1931 to 1960,

after which a period of strong recruitment occurred in the early1960s (Fig. 1a). Extensive exploitation in the mid-1960s rap-idly reduced the stock to low levels, and the 1993 abundancewas the lowest on record (NEFSC 1995). In response to over-exploitation, Amendment 7 to the Northeast MultispeciesFishery Management Plan has recently been enacted, with thegoal of reducing fishing mortality to rebuild the spawningstock above a minimum threshold (80 kt for haddock).Clearly, an adequate rebuilding strategy will require substan-tial reductions in fishing effort, but how quickly, and to whatlevel, the stock will rebuild is less clear.

The evaluation of rebuilding strategies will require an as-sessment of the productivity of the haddock stock, which hasgenerally been accomplished with traditional single-speciesmodels. Overholtz et al. (1986) found that with an age-structured model, in which recruitment was related to stocksize via an empirically derived probability schedule (Getz andSwartzman 1981), rebuilding was not likely to occur unless

Can. J. Fish. Aquat. Sci.54: 2920–2929 (1997)

Received November 6, 1996. Accepted May 5, 1997.J13741

P.D. Spencer1 and J.S. Collie.Graduate School ofOceanography, University of Rhode Island, Narragansett,RI 02882, U.S.A.

1 Author to whom all correspondence should be addressed.Present address: National Marine Fisheries Service,Southwest Fisheries Science Center, Tiburon Laboratory,3150 Paradise Dr., Tiburon, CA 94920, U.S.A.e-mail: pauls@melanops.nmfs.tib.gov

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the instantaneous fishing mortality rateF was lowered below0.40. More recently, Overholtz et al. (1995) applied a similarmodel to several northwest Atlantic groundfish stocks andconcluded that reducing effort to 30–60% of the 1990 levelwould produce substantial gains in biomass, yield, and reve-nue. Using the recent catch and effort data from 1976 to 1989,Edwards and Murawski (1993) applied an age-aggregated sur-plus production model to the Gulf of Maine and Georges Bankhaddock stocks and concluded that economically efficient har-vesting could produce annual yields of 45 kt.

Predation is generally regarded as a major force structuringmarine fish communities (Bailey and Houde 1989; Bax 1991),and low production by haddock in recent years has promptedresearch on the role of predation in community dynamics. TheGeorges Bank food web is tightly coupled, with most of theproduction of juvenile fish being consumed by other fish spe-cies (Sissenwine et al. 1984). Annual surplus production, esti-mated as the change in biomass during the year plus catch, wasrelatively high in 1976 and 1979, but was very low during the1980s and early 1990s (Fig. 1b). The Georges Bank fish com-munity structure has also changed greatly since the mid-1970s,as pelagic fishes that prey upon the early life history stages ofother fish (e.g., herring, mackerel) and demersal piscivores

(e.g., skates, spiny dogfish) have greatly increased (Sissen-wine 1986). Haddock recruitment on Georges Bank is in-versely correlated with mackerel biomass (Koslow et al.1987). Predation plays a major role in pelagic community dy-namics (Overholtz et al. 1991) and may be equally importantin structuring the demersal fish community.

A simple age-aggregated model incorporating predation de-veloped by Steele and Henderson (1984) may offer insight tostocks with periods of low production such as the haddock. Inthis model, the rate of production is constrained by predationat low stock sizes and intraspecific, density-dependent inter-actions at high stock sizes. A nonlinear rate of predation canlead to multiple equilibria, and gradual changes in fishing mor-tality and (or) predation may rapidly move the stock from oneequilibrium to another. This model incorporates depensationat low stock sizes and provides an intuitive explanation forwhy some stocks may exist at low levels despite moderatefishing intensities. Despite the theoretical appeal of the model,a statistical fit to an exploited stock has not been obtained,although qualitative fits were applied to the Georges Bankhaddock and the Sitka Sound herring (Collie and Spencer1993). Although the statistical significance of depensatoryproduction dynamics has proven difficult to confirm with typi-cal marine fisheries data (Myers et al. 1995), the Georges Bankhaddock would appear ideal for application of the model, asthe sharp contrast in the data before and after the mid-1960sintuitively suggests periods of differing stock productivity.

Rebuilding strategies under the Steele and Hendersonmodel may differ considerably from those with previous as-sessment models. If a stock is constrained at low stock sizesby some combination of fishing and predation, it may be nec-essary to reduce fishing mortalities to very low levels to re-build to higher stock sizes (Collie and Spencer 1993, 1994).Policies that maximize yield or economic rent generally at-tempt to maintain the stock at an “optimal” size and thus re-semble an escapement policy and have large variation in catchand fishing effort (Spencer 1997). For resources as economi-cally valuable as Georges Bank haddock, it is prudent to evalu-ate rebuilding strategies under a variety of harvest strategies,performance measures, and assumptions of stock productivity.

The purpose of this paper is to obtain a statistical fit of theSteele and Henderson model to the Georges Bank haddockdata and compare this fit with that obtained from the Schaefer(1957) model. An estimation procedure incorporating bothprocess and observation errors was applied, and simulationswere conducted to evaluate parameter variances and correla-tions. For each of the model fits, rebuilding of the stock issimulated under a variety of harvest policies and performancemeasures.

Methods

Data sourcesCatch and estimated population biomass (ages 2+) were used as inputdata for fitting the biomass dynamics models. Recent estimates ofbiomass (1963–1993) were computed from the virtual populationanalyses (VPA) in NEFSC (1995) and the January 1 weight-at-agetable estimated by Hayes and Buxton (1992). Earlier estimates ofage 2+ biomass (1931–1978) were obtained from Clark et al. (1982).A linear regression was made for the two data sources for the period

Fig. 1. (a) Estimated biomass (solid line; from NEFSC 1995; Clarket al. 1982) and catch (dotted line) and (b) estimated annual surplusproduction and biomass of Georges Bank haddock.

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of overlap (1963–1978), and the biomass data from 1931 to 1962were scaled to the NEFSC estimates.

Biomass dynamic models of the type examined here are often fitwith catch per unit effort (CPUE) serving as the index of abundance.Due to the management history of the haddock (catch constraints, triplimits, seasonal closures, etc.), it is suspected that CPUE does notreflect true population abundance (Hayes and Buxton 1992; NEFSC1995). For this reason, catch data and biomass estimates based onVPA (calibrated to research trawl surveys) were used as input data.

Population models and estimation procedureTwo surplus production models were fit to the Georges Bank haddockdata. In each case, fishing was assumed to occur in a short (i.e., sea-sonal) period. The general form of the model assumes lognormallydistributed errors and is

(1) Bt+1 =

∫t

t+1

f(Bt − Ct)dt

eµt

whereBt is the biomass at the beginning of yeart, Ct is the catchduring yeart, f(⋅) is a biological production function, andµt is theresidual process error representing variability in biological produc-tion. Note that the assumption of a short fishing season differs fromprevious continuous-time models of marine fish populations (Schae-fer 1957; Pella and Tomlinson 1969; Spencer 1997) in which catch ismodeled as occurring continuously throughout the year. In the ab-sence of representative data on fishing effort by which a rate of re-moval could be estimated, the removal of observed catch was viewedas the most straightforward approach to parameter estimation.

The continuous-time form of the production functionf(⋅) was re-tained to provide consistency with the models of Schaefer and Steeleand Henderson; the specific formulations are as follows:Schaefer model

a) f(B) = rB 1 − B

K

Steele and Henderson model

(2b) f(B) = rB 1 − B

K

− DB2

A2 + B2

wherer is an intrinsic growth rate andK is an intraspecific density-dependence term. The second term in the Steele and Henderson modelis a sigmoidal, type III predator functional response (Holling 1965)with a maximum rate of predationD (the product of an individual rateof predationd and the predator population size) and a saturation pa-rameterA. Certain parameter combinations may produce three equi-librium stock sizes (high and low stable equilibria, separated by anunstable, intermediate equilibrium point) in the Steele and Hendersonmodel, a situation referred to as the triple-value region. The propertiesof the Steele and Henderson model are discussed further in Collie andSpencer (1993) and Spencer (1997).

The “total least squares” method of Ludwig et al. (1988) was usedto estimate model parameters; this method includes both process er-rors and measurement errors in the objective function. In the presenceof both measurement and observation errors, the total least squaresmethod provides estimates with lower bias than a process error model(Ludwig et al. 1988). Measurement errors refer to differences be-tween the VPA-derived biomass data (bt) and the actual, but un-known, biomass (Bt); these differences are assumed to be lognormallydistributed:

(3) bt = Btevt

whereνt is the residual measurement error. The estimation procedureis similar to assuming only process errors, but has the modificationthat an estimate of the “nuisance parameter” (Ludwig et al. 1988)νtis made at each time step (thus providing an estimate ofBt that is usedin eq. 1). The sum of squares criterion to be minimized is

(4) TSS= 11 − λ ∑

t=1

n−1

µt2 + 1

λ ∑t=1

n

vt2

whereλ is the ratio of the measurement error variance (σv2) to the total

variance (σv2 + σµ

2) andn is the length of the data set. For example,in the Steele and Henderson model the parametersr, K, D, andA,ν1,...,νn, would be chosen such that TSS is minimized. The estimatedvalues ofr, K, D, andA were constrained to be less than 5.0, 5000,500, and 500, respectively. Note that the relative importance of thetwo error types is affected byλ, which is not likely to be known. Forthe Steele and Henderson production model, fits to the data wereevaluated with three levels ofλ (0.25, 0.50, 0.75). As will be seenlater, reasonable parameter estimates for the Schaefer model were notobtained under these assumptions ofλ; reasonable estimates wereobtained by using only the later years of the data set (1976–1993) andsettingλ to 0.01 (essentially a process-error model).

Standard deviations of parameters in the Steele and Hendersonmodel were estimated with bootstrapping. For the fits obtained witheach of the three levels ofλ, data were simulated with eqs. 1, 2b, and3 with the parameters set to their estimated values. Catch was re-moved by applying an exploitation rateut to the beginning yearbiomass; the ratioCt/Bt was taken as an estimate ofut that capturesreasonably well the exploitation history of the haddock. Residual er-rors were generated by allowingµt andνt to have the general formεt–σ2/2, whereεt is a random normal variable with mean 0 and varianceσ2. The subtraction of one half the variance ensures that the lognormalerrors have an expected value of 1. The levels ofσµ andσν were setto values observed in the original fit to the data, and 200 simulationswere conducted. Parameter standard deviations and the correlationmatrix were computed directly from the resulting 200 sets of esti-mates.

Monte Carlo simulations were conducted to assess the potentialbiases associated with the parameter estimates of the Steele and Hen-derson model. The same general procedure described above was fol-lowed, but now,σµ andσν were set to values representing four typesof error structure: equal process and measurement errors at each oftwo different levels (σµ = σν = 0.1 andσµ = σν = 0.2), predominatelymeasurement errors (σµ = 0.1, σν = 0.2), and predominately processerrors (σµ = 0.2,σν = 0.1).

Simulations assessing rebuilding under a variety of performanceand harvest policies were conducted with each production model.Each rebuilding simulation ran for 50 years, beginning in 1993 withthe estimated stock size of 13.8 kt. As before, catch was removed atthe beginning the year by multiplying the beginning year biomass bythe exploitation rateut, defined as (1 – e−qEt), whereq reflects theinstantaneous catching power of a unit of effort andEt is the level offishing effort for yeart; the productqEt is taken as the instantaneousrate of fishing mortalityFt. In contrast with other discrete-time mod-els of harvesting, this specification has the advantage that a givenlevel of effort cannot instantly remove the entire population. Twoperformance measures were considered: the sum of discounted yieldand sum of discounted revenue. The annual benefits obtained fromharvesting (i.e., yield or economic revenue) were discounted with thefactor e–δt, whereδ is the discount rate andt is time (i.e., year from1993); two discount rates (0.0 and 0.02) were considered.

The computation of revenue involved derivation of a price modelof the form

(5) P = a + bQ + cT

whereP is the observed price (dollars per kilogram),Q is the quantitylanded (kilotonnes) in the northeastern United States, andT is anindex for observation year (1964= 1,..., 1993= 30). The observedprices were adjusted to constant 1993 U.S. dollars. The temporal termis included to account for trends in factors not considered in the sim-ple formulation above (e.g., shifts in consumer preferences orimports). When projecting forward the revenues accrued under

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rebuilding strategies, any temporal trend in the price model was haltedby holdingT at its 1993 level.

The price model above can be used to project total revenue fromthe U.S. Georges Bank fishery after consideration of two factors. One,since enactment of theMagnuson Actin 1977, the Georges Bankfishery has been divided between Canadian and U.S. fishers. In recentyears, the catch has largely been taken by Canadian fishers (84% in1993; NEFSC 1995), but earlier years indicate a larger U.S. alloca-tion. As the simulated haddock stock was rebuilt, it was assumed thatthe U.S. allocation would increase by 5% annual increments from15% in 1993 until the U.S.–Canadian allocation was 50% apiece. Thistype of allocation was also applied by the New England FisheriesManagement Council (NEFMC) in simulating effects of the Amend-ment 7 regulations (NEFMC 1995). Two, the price model above isbased on all landings of haddock in the northeastern United States,which are only partially composed of Georges Bank haddock. From1964 to 1993, U.S. landings of Georges Bank haddock averaged 73%of the total northeastern U.S. landings. After adjusting for these twofactors, total U.S. revenues were estimated as the product of projectedprice and the U.S. share of the haddock catch.

For each combination of underlying production model,performance measures, and discount rates, the accumulated benefits

and likelihood of rebuilding the stock were assessed with three typesof harvest policies. First, a constant level ofF was applied for allyears, with theF levels evaluated ranging from 0.00, 0.03,..., 0.45.Second, the policy maximizing either the sum of discounted yield orsum of discounted revenue (referred to as maximum yield and maxi-mum revenue policies, respectively) was applied. These policies werederived from stochastic dynamic programming, an optimizationmethod that produces a control law indicating the level ofF, for agiven stock size, appropriate for maximizing a clearly defined objec-tion function (Spencer 1997). Separate maximum yield and maximumrevenue policies were derived for each combination of productionmodels and discount rates. These policies essentially attempt to keepthe stock at an “optimal” stock size and may involve rapid changes inF between years. Intermediate between these two harvesting policiesis a modified threshold policy in whichF is constant for stocks abovea critical threshold size and is proportional to stock size below thethreshold (Sigler and Fujioka 1993). In the notation below, theF levelfor the modified threshold policy refers to the value above thethreshold.

One hundred simulations were conducted for each case of thethree general types of policies, and the mean biomass and catch foreach year of the rebuilding simulation were computed. Rebuildingwas defined to occur when the stock grew to at least 80 kt, the criticalstock size for haddock in the Amendment 7 regulations; this stocksize also served as the threshold level of the modified thresholdpolicy.

Results

Parameter estimationReasonable parameter estimates were obtained with the Steeleand Henderson model (Fig. 2a; Table 1). Predicted surplusproduction did not vary substantially between the three as-sumptions regardingλ, particularily with stock sizes less than250 kt where most of the data occur; thus, only the fit withλ = 0.50 will be discussed. The fitted value ofr of 0.76 corre-sponds closely to the value of 0.83 reported by Edwards andMurawski (1993), although production at low stock sizes isreduced by the predation term. The predicted maximum

Fig. 2.Comparison of fitted production models with estimatedsurplus production. (a) Fitted Steele and Henderson model withλ =0.50 (solid line),λ = 0.25 (dotted line), andλ = 0.75 (broken line);(b) fitted Schaefer model withλ = 0.50 applied to data from 1931to 1993 (dotted line) andλ = 0.01 for data from 1976 to 1993(solid line).

Term λ = 0.25 λ = 0.50 λ = 0.75

r 0.91 (0.38) 0.76 (1.00) 0.74 (1.2)K 467.2 (1629.8) 520.2 (1438.1) 486.7 (330.84)D 41.2 (38.56) 28.2 (119.4) 22.1 (169.03)A 31.2 (10.0) 27.9 (23.9) 23.5 (36.16)σµ 0.15 0.09 0.05σν 0.07 0.11 0.15TSS 3.01 2.62 2.4

Parameter correlation matrix (λ = 0.50).

r K D A

r 1.00 –0.26 0.96 0.46K 1.00 –0.16 0.14D 1.00 0.61A 1.00

Table 1.Parameter estimates, total sum of squares (TSS),and the parameter correlation matrix of the Steele andHenderson model fit to the haddock data under differingassumptions ofλ (error variance ratio) (standard deviationsof parameters in parentheses).

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surplus production occurs at a biomass of approximately260 kt, near the upper end of the observed data.

The bootstrap standard deviations of all parameters werehigh, particularly with theK parameter. The rate at which pro-duction at high stock sizes is reduced by compensatory pro-cesses is represented by theK parameter, but the available datado not clearly show the level of compensation; thus, modelswith widely divergent levels ofK may fit the data similarly.The underlying cause of the parameter variability is the con-founding of parameters; the information available in the had-dock data is insufficient to provide unique estimates of all fourparameters. Because of the saturating type III functional re-sponse, the rate of predation is not substantially affected byhaddock biomass when biomass is large, a feature that, in ef-fect, divides the density-independant rate of growth into twoterms,r andD. The estimates ofr andD are positively corre-lated because they can be increased (or decreased) simultane-ously with little effect on the magnitude of populationproduction. As with the Schaefer model, the effect of a highlevel of r can be offset with a low level ofK, which explainstheir negative correlation.

Despite the uncertainty in the parameter estimates, the totalleast squares procedure withλ = 0.5 was able to provide rea-sonably unbiased parameter estimates under a variety of errorstructures. Boxplots in Fig. 3 summarize the distributions ofparameter estimates for various Monte Carlo simulations; theunderlying “true” levels of the parameters used to simulate thedata were standardized to 1 to facilitate comparisons. The lev-els ofσµ andσν used in the simulations were similar to thoseobserved in the model fits, which were 0.09 and 0.11, respec-tively, for λ = 0.50. For all parameters, the median value wasclosest to the underlying value whenσµ andσν were equal; thedistributions withσµ andσν = 0.1 were very similar to thoseof the bootstrap simulations. Observation errors causedr andD to be overestimated andK andA to be underestimated; pro-cess errors had the opposite effect. With higher levels of pro-cess errors, the variability in theK parameter was large. Whenthe error structure in the simulation differed fromλ = 0.5, themedian parameter estimates deviated farther from the under-lying values, but the interquartile ranges generally containedthe underlying values. In cases where the parameters were notwell estimated, the final estimates may be quite large and per-haps at the upper constraints; this explains why large standarddeviations of the parameters were observed despite fairly tightinterquartile distances.

An important feature of the fitted Steele and Hendersonmodel is that at low stock sizes (less than ~50 kt), surplusproduction does not substantially increase but remains at rela-tively constant, low levels. This pattern of reduced productionin the model fit is consistent with the low-biomass, low-production data of recent years and is made possible by thesigmoidal predator functional response. Due to the parametercorrelations, the choice ofD and A that fit the recent yearsinfluences the estimated value ofK. The lack of an explicitpredation term in the Schaefer model, in contrast, constrains itto a parabolic form in which the population is assumed to growat a rate approximate tor when at low stock sizes. Because thelevel, or even the presence, of compensation is difficult toascertain from the haddock data, it is difficult to obtain plau-sible parameter estimates for the Schaefer model. The fit withλ = 0.50 is nearly linear (Fig. 2b) and essentially assumes no

compensation asK is at its upper constraint of 5000. Plausibleparameter estimates (r = 0.40,K = 129 kt) were obtained byassuming essentially a process-error model (λ = 0.01) and us-ing only the later years of the data set (1976–1993) (Table 2);the correlation between estimates ofr andK was –0.25.

Rebuilding simulationsThe basic harvest policies evaluated provide contrast regard-ing howF may be adjusted in response to changes in stock size(Fig. 4). The maximum revenue policy showed gradualchanges inF as a function of stock size relative to the maxi-mum yield policy. The maximum revenue policy incorporatedthe fitted price model, with the estimated parameters ofa =1.82,b = –0.022, andc = 0.033; ther2 was 0.78. Some harvestwould be taken at low stock sizes (where marginal revenue ishigh), and some potential harvest would be spared at highstock sizes (where marginal revenue is low). In contrast, themaximum yield policy attempts to move the stock to the

Fig. 3.Boxplots of parameter distributions under simulations with avariety of error structures. Each column represents the distributionof 200 simulations, with the standard deviation of process andobservation errors, respectively, indicated in the column labels.Each boxplot shows the median (white line) and interquartiledistance (height of the box); the whiskers extend a maximum ofthrice the interquartile distance.

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optimal stock size as quickly as possible and involves rapidchanges ofF with stock size. The modified threshold policy isintermediate between policies that are sensitive (maximumyield and revenue) and invariant (constant harvest rate) tochanges in stock size.

The effect of various fishing intensities on population abun-dances can be seen from simulations incorporating the con-stant harvest rate policy (Fig. 5). The higher level ofproductivity in the Steele and Henderson model resulted in theunfished stock fluctuating at ~475 kt whereas the unfishedstock in the Schaefer model fluctuated at ~130 kt. In the Steeleand Henderson model, increases inF gradually reduced stockbiomass up toF = 0.12; further increases inF substantiallyreduced the stock size, and anF level of 0.24 tended to keepthe stock at a low equilibrium. In contrast, asF was increasedin the Schaefer model the biomass declined more gradually.

Interpretation of the various constant harvest rate policiesin the Steele and Henderson model is assisted by reference tothe equilibrium yield plot (Fig. 6). The triple-value region ex-ists forF values from 0.21 to 0.36 and has high and low equi-librium yields associated with the stable equilibria; theintermediate yields (broken line) correspond to unstableequilibria. Annual yields at the end of the 50-year simulationwere largest whenF = 0.18, as this was the largestF levelevaluated that produced a single, high-equilibrium point. Thesystem is at the edge of the triple-value region whenF = 0.21

and only a portion of the simulated stocks rebuilt to higherlevels, thus considerably lowering the average yield at the endof the simulation. Further increases inF precluded the systemfrom moving to the higher equilibrium level. Again, changesin yield varied more gradually in the Schaefer model, as thepotential for dramatically moving to alternate equilibria doesnot exist.

The effect of multiple equilibria is also reflected in the pro-portion of simulations in which the stock rebuilt to 80 kt andthe mean times for rebuilding (Fig. 7). In the Steele and Hen-derson model, all simulations with a constant harvest of 0.15or lower showed rebuilding whereas none rebuilt with a con-stant harvest rate of 0.24 or higher (Fig. 7a). In contrast, thetransition between all simulations either rebuilding or not re-building was more gradual in the Schaefer model, as 31% ofthe simulations rebuilt with anF level of 0.24. With the modi-fied threshold policy, more substantial differences between thetwo models occurred (Fig. 7c). The higher productivity of theSteele and Henderson model allowed all simulations with anF of 0.30 or less to rebuild whereas in the Schaefer model, allsimulations showed rebuilding whenF was 0.21 or lower.

In the Steele and Henderson model, the maximum yield and

λ Data used r K σµ σν TSS

0.25 1931–1993 0.31 5000 0.16 0.07 3.240.50 1931–1993 0.32 5000 0.10 0.11 2.910.75 1931–1993 0.33 5000 0.06 0.16 2.920.01 1931–1993 0.32 1792 0.26 0.005 4.410.01 1976–1993 0.40 129.0 0.31 0.005 1.55

(0.44) (1137.0)

Note: Parameter standard deviations (in parentheses) are reportedfor the fit to data from 1976 to 1993.

Table 2.Parameter estimates and total sum of squares (TSS)of the Schaefer model fit to the haddock data under differingassumptions ofλ (error variance ratio).

Fig. 4.General form of various harvest strategies, indicating howthe instantaneous rate of fishing,F, would be adjusted as a functionof stock size. The maximum yield and maximum revenue policieswere derived for the Steele and Henderson model withδ = 0.0.

Fig. 5.Simulated biomass and catch levels under various constantharvest rate policies for the Steele and Henderson (STH) andSchaefer (SCH) production models; the value for each year is themean of 100 simulations.

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revenue policies had the expected result of producing themaximum levels of yield and revenue, respectively (Table 3).However, the sum of discounted yield obtained with the maxi-mum revenue policy is close to that obtained with the maxi-mum yield policy. The sum of discounted revenue obtainedunder the maximum yield policy is considerably less than thatobtained with the maximum revenue policy, as the maximumyield policy would involve intensively harvesting at high stocksizes where marginal revenue is low. The modified thresholdpolicy provided substantially greater benefits than the constantharvest rate policy. In the Schaefer model, the modified thresh-old, maximum yield, and maximum revenue policies all pro-duced similar levels of yield and revenue. This feature isexplained by the tendency of all three policies to keep the stocksize at ~65 kt, thus resulting in similar patterns of biomassesand catches.

The large differences in production between the two modelsled to correspondingly large differences in benefits obtained.For example, the sum of yields obtained under the maximumyield policy was 2706.9 kt for the Steele and Henderson modelbut 563.1 kt for the Schaefer model; this corresponds to aver-age annual yields of 54.1 and 11.26 kt, respectively. Similarly,the sum of revenue obtained under the maximum revenue pol-icy was $2254.1 million for the Steele and Henderson modeland $703.5 million for the Schaefer model, corresponding toaverage annual revenues of $45.1 and $14.1 million,respectively.

Despite large differences in production, theF levels thatmaximize yield and revenue under the constant harvest rateand modified threshold policies were similar between the twomodels. Under the constant harvest rate policy, benefits weremaximized withF = 0.12–0.15 in the Steele and Hendersonmodel andF = 0.18 in the Schaefer model. For the modifiedthreshold policy, higher fishing mortalities are sustainable be-causeF is reduced at low stock sizes. Maximal benefits for theSteele and Henderson model were obtained withF = 0.18–0.21 compared with anF = 0.24 in the Schaefer

model. However, the fishing mortality rates that maximizebenefits under the Schaefer model are close to the values thatwould limit rebuilding in the Steele and Henderson model.

Discussion

Expectations about the rebuilding process for Georges Bankhaddock are dependent on our assumptions of stock productiv-ity. The fitted Schaefer model uses only the later years of thedata set, during which the stock was at very low levels. Thus,the productivity of the stock is estimated to be small, and areturn to pre-1965 levels of biomass and catch is inconsistentwith the model. The incorporation of the nonlinear predationterm in the Steele and Henderson model provides an explana-tion for rapid shifts in abundance and allows a reasonable fit tothe entire data set. Returns to pre-1965 levels of biomass andcatch are observed, thus providing a more optimistic view ofstock productivity. The two models presented here representcontrasting views of stock productivity a manager would wish

Fig. 6.Equilibrium yield curves of the fitted production models andactual yields and estimatedF levels (from NEFSC 1995; Clarket al. 1982); the high yields in 1965–1966 are not shown. TheSteele and Henderson curve has yields associated with the high andlow stable equilibria (solid lines), separated by a backward-bendingportion associated with unstable equilibria (dotted line); thetriple-value region exists for 0.21≤ F ≤ 0.36. The Schaefer curve(broken line) has the familiar parabolic shape.

Fig. 7.Number of simulations (of 100 total) that achieved therebuilding goal of 80 kt, and mean and standard deviation ofrebuilding year, for various levels of the constant harvest rate andmodified threshold policies applied to the Steele and Henderson(shaded bars) and Schaefer (open bars) production models.

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to consider, and both correspond to previously publishedestimates.

The fitted Schaefer model is similar to results obtained byEdwards and Murawski (1993) who fit a generalized produc-tion model of the form dX/dt = –HX + GXm– qEX to theGeorges Bank – Gulf of Maine haddock stocks and obtainedparameter estimates ofH = –0.83,G = –0.7× 10–5, andm =3.4. The unexploited equilibrium biomass for the Edwards andMurawski model is 130.0 kt, similar to that observed with theSchaefer model here. The optimal stock size for maximizingnet economic value in the Edwards and Murawski model was78 kt, similar to the stock size obtained here under the maxi-mum yield or maximum revenue policy (65 kt). The similarityof the two models stems from the observed low production inthe post-1976 data. Interestingly, the rebuilding target for theAmendment 7 regulations exceeds the optimal stock size ofboth the Edwards and Murawski model and the Schaefermodel here, suggesting a more optimistic view of stockproductivity.

The stock size corresponding to the surplus productionmaximum in the Steele and Henderson model is 260 kt, largerthan all but four of the haddock biomass estimates from 1931to 1993. While this may appear unduly optimistic, rebuildingprojections from the Steele and Henderson model are consis-tent with other assessments incorporating the higher produc-tivity (pre-1965) portion of the haddock data. Overholtz et al.(1986) found that a policy of initially reducing fishing

mortality on large year-classes for a period of 5 years couldproduce yields of 35–50 kt, similar to the yields obtained here(Fig. 5b). Application of the average recruitment from 1935 to1960 (71.4 million) to a yield-per-recruit model provided es-timates of stock sizes and yield of 405 and 37.8 kt, respec-tively, with an F of 0.10 (NEFSC 1995). It should be notedthat the difference in production between the fitted modelshere is related to the data sets used; a more productive Schaefermodel could be fit to the entire data set by further constrainingK, but this would require an additional a priori assumption.

The maximum yield and revenue policies provide the high-est level of their respective performance measures in eitherproduction model, but a manager may wish to consider otherharvesting policies. For the maximum yield policy, the drasticchanges inF required to move quickly to the “optimal” stocksizes may involve large costs that were not considered here.Additionally, there is considerable year-to-year variation incatch, similar to other escapement policies (Walters 1986).The maximum yield and revenue policies attempt to maintaina stock size that corresponds approximately to the peak of theproduction curve, which differs considerably between theSchaefer (65 kt) and Steele and Henderson (260 kt) models.Thus, the cost of applying such a policy to the inappropriateproduction model may be severe and may involve drastic re-ductions is biomass or highly conservative exploitation ratesand foregone yields.

The similarity between the models of theF levels that

Population model

Harvest policy

Performance measure(benefits)

Discountrate

Constantharvest rate

Modifiedthreshold

Maximumyield

Maximumrevenue

Sum of discounted yield 0.0 Steele and Henderson 1507.91 2095.98 2706.88 2649.55(256.71) (199.68) (180.22) (159.32)F = 0.15 F = 0.21

Sum of discounted yield 0.0 Schaefer 522.25 559.82 563.1 564.50(37.14) (40.96) (43.93) (43.29)F = 0.18 F = 0.24

Sum of discounted yield 0.02 Steele and Henderson 792.47 1135.27 1523.80 1485.64(89.06) (101.09) (110.51) (101.88)

F = 0.12 F = 0.18Sum of discounted yield 0.02 Schaefer 308.18 335.36 337.46 338.13

(22.00) (26.30) (26.60) (26.67)F = 0.18 F = 0.27

Sum of discounted revenue 0.00 Steele and Henderson 1542.55 1953.99 1755.09 2254.06(126.04) (111.10) (132.64) (84.65)F = 0.12 F = 0.18

Sum of discounted revenue 0.00 Schaefer 652.20 698.87 677.64 703.47(43.69) (47.52) (48.65) (50.16)

F = 0.18 F = 0.24Sum of discounted revenue 0.02 Steele and Henderson 827.55 1075.47 1003.35 1278.43

(85.42) (79.55) (78.45) (62.16)F = 0.12 F = 0.18

Sum of discounted revenue 0.02 Schaefer 381.58 415.99 406.92 420.81(25.93) (30.42) (29.40) (30.70)

F = 0.18 F = 0.27

Note: Values reported are means of 100 simulations, each conducted for 50 years; standard deviations are in parentheses. For the constantharvest rate and modified threshold policies, the maximum benefits (and correspondingF) are reported.

Table 3.Benefits (sum of discounted yield (kt) and sum of discounted revenue (millions of dollars)) with various harvest policiesunder two production models for the Georges Bank haddock.

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maximize benefits in the constant harvest rate or modifiedthreshold policy helps in choosing a policy robust to uncer-tainty in the stock productivity; this similarity stems from thesimilar rates of growth at small stock sizes. A modified thresh-old policy with 0.18≤ F ≤ 0.21 may provide such a strategy.The basic form of the modified threshold policy is similar tothe maximum revenue policy; at low stock sizes,F increasesrapidly with stock size, and at higher stock sizes,F changeslittle with stock size (Fig. 4). Rebuilding to the Amendment 7threshold would be expected on the order of 12 years (Fig. 7d),and the benefits obtained would be considerably higher thanin a constant harvest rate policy and comparable with the maxi-mum yield and revenue policies. A similar modified thresholdpolicy is used to set allowable catch levels for sablefish(Anoplopoma fimbria) in the Gulf of Alaska (Fujioka et al.1996). The threshold is 35% of the estimated unfished sable-fish biomass; below the threshold the fishing rate is propor-tional to biomass. Monte Carlo simulations, conducted bySigler and Fujioka (1993), showed that the overall perform-ance of the policy was superior to either a constant harvest rateor a threshold policy. Compared with the constant harvest ratepolicy, the modified threshold policy had a slightly reducedcatch and slightly higher catch variance, but much lower riskof overfishing. Sigler and Fujioka (1993) preferred the modi-fied threshold policy to a threshold policy because it does notrequire a complete closure of the fishery and is much lesssensitive to the choice of threshold or errors in measuringabundance relative to the threshold.

The policies that maximized yield or revenue in either pro-duction model are more conservative than the Amendment 7rebuilding strategy ofF0.1= 0.24. If the Steele and Hendersonmodel is correct, it would be necessary to reduceF to belowthe lower boundary of the triple-value region (0.21) to ensurerebuilding, even though the peak of the sustained yield curvecorresponds to anF level (0.273) similar to theF0.1 value. Ifthe Schaefer model is correct, the peak of its sustained yieldcurve corresponds to anF level of 0.20. In either model, thereis a low probability of rebuilding atF = 0.24 (Fig. 7a), and anyrebuilding that occurred in the Schaefer model would takeabout 30 years (Fig. 7b). Overholtz et al. (1986) concludedthat rebuilding would not occur unlessF was reduced to at least0.40, and the results here indicate thatF may have to be re-duced even further. Drastic reductions in fishing effort, suchas ~50% reduction from the 1990 level, can be expected tospeed the rebuilding process (Overholtz et al. 1995).

Marine fisheries stock assessment is typically conductedwith either age-aggregated surplus production models or age-structured models such as VPA; the assessment here mergesthese two approaches. Surplus production models are com-monly used with CPUE as an index of biomass, although,clearly, alternative measures of biomass can be used. Concernsthat effort data are not representative because they essentiallymeasure haddock bycatch have led to the use of research sur-vey data to tune VPA estimates (Hayes and Buxton 1992;NEFSC 1995); the problem of obtaining standardized effort isexacerbated when attempting to fit a production model to dataextending back to 1931. The use of VPA-derived biomass es-timates in conjunction with the haddock catch data provided adata set suitable for this purpose. To our knowledge, the fittedSteele and Henderson model represents the first time a surplusproduction model has been applied to the entire haddock data

set. The absence of effort data necessitates the discrete removalof catch and precludes estimation of a rate of removal typicallyfound in other estimation procedures (Pella and Tomlinson1969; Ludwig et al. 1988). The estimation procedure used herecan be easily modified to allow quarterly catches, although thisdid not substantially affect the parameter estimates. Addition-ally, with low effort levels the catchability coefficient in thediscrete-time harvest model closely approximates that in thecontinuous-time model dC/dt = qEX (Walters 1986), thus al-lowing comparisons between the two approaches.

An implicit assumption of the surplus production modelsused here is that the underlying ecological conditions remainunchanged, and variations in stock size reflect random vari-ability and the effects of fishing. This justifies the use of post-1976 data to fit the Schaefer model, as the dramatic changesin Georges Bank system structure are partially excluded. TheSteele and Henderson model suggests that high fishing mor-talities in the mid-1960s caused a dramatic decline of haddockbiomass, and the current production is constrained by the in-teraction of fishing and predation; the maximum rate of pre-dation D (which reflects predator population size) isunchanged. However, it is becoming increasingly recognizedthat low-frequency stock fluctuations, or “regime shifts,” inmany marine stocks are influenced by both fishing intensityand environmental variability (Souter and Issacs 1974; Lluch-Belda et al. 1989; Steele 1996). This is consistent with pre-vious work with single- and two-species versions of the Steeleand Henderson model in which environmentally driven vari-ation in the maximum rate of predationD or predator abun-dance induced rapid shifts in prey abundance (Spencer andCollie 1996; Spencer 1997). A statistical fit of the Steele andHenderson model necessitates that variation is represented interms for which there are estimates, such as stock size, leadingto the assumption here of a stationary production model. Thisdoes not obviate the important implications of environmentalvariability on system dynamics, and future rebuilding projec-tions may change substantially with nonstationary shifts inhaddock productivity.

The multiple stable equilibria in the Steele and Hendersonmodel result from depensation at low stock sizes. Depensatoryproduction models are often associated with stocks trapped atlow sizes and unresponsive to reductions in fishing, but theresults here suggest that rebuilding is possible in the Steele andHenderson model. This is due, in part, to the narrow range ofstock sizes over which depensation occurs and the positiveproduction levels over all stock sizes. The Steele and Hender-son model requires slightly more conservative fishing mortali-ties than the Schaefer model to ensure rebuilding. Myers et al.(1995) found some evidence of depensation in the haddockstock–recruitment data, but the depensation was not statisti-cally significant. Likewise in this study, the fit of the Steeleand Henderson model is not statistically better than that of theSchaefer model, even though the former model gave morebiologically feasible parameter estimates. The low surplus pro-duction observed in recent years (Fig. 1b) suggests that it isprudent to consider a range of plausible models when makingcritical management decisions.

Acknowledgments

We thank Mike Fogarty, John Steele, and Andrew Solow for

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their helpful comments on this work. Steve Edwards and SteveMurawski kindly provided bioeconomic and fishing effortdata, respectively. This publication is the result of researchsponsored by the Joshua MacMillan Fellowship and the RhodeIsland Sea Grant College Program, with funds from the Na-tional Oceanic and Atmospheric Administration, Office of SeaGrant, Department of Commerce, under grant No.NA36RG0503 (project No. R/F-9501).

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