A management oriented approach to stock recruitment analysis

A management oriented approach to stock

recruitment analysis

Jon T Schnute and AR Kronlund

Abstract This paper describes an approach to stock recruitment analysis in which model parameters relate directly to

management The proposed recruitment curve generalizes earlier models by Ricker Schaefer and Beverton and Holt where

a shape parameter γ determines the curve type A family including these classical models can be represented analytically in

terms of γ and two management parameters the maximum sustainable catch Clowast and harvest rate hlowast In particular the Ricker

curve can be expressed in terms of (Clowast hlowast) rather than the classical parameters (α β) We describe the biological and

management implications of this mathematical transformation and show that it is particularly amenable to a Bayesian

formulation We use simulationndashestimation experiments to demonstrate that the new parameter estimates reduce statistical

bias problems described in the literature for classical estimates Furthermore the new estimates are reasonably robust to the

choice of shape parameter whereas classical estimates are not By focusing on management parameters (Clowast hlowast) rather than

productivity parameters (α β) the approach here yields estimates with improved statistical properties and direct relevance to

policy

Reacutesumeacute Dans cet article nous expliquons une forme drsquoanalyse du recrutement des stocks dans laquelle les paramegravetres sont

directement lieacutes agrave la gestion La courbe de recrutement proposeacutee est le reacutesultat de la geacuteneacuteralisation des modegraveles de Ricker de

Schaefer et Berverton et Holt dans lesquels un paramegravetre de forme γ deacutetermine la nature de la courbe On peut repreacutesenter

analytiquement une famille regroupant ces modegraveles classiques au moyen du paramegravetre γ et de deux paramegravetres de gestion

soit les prises maximales en reacutegime de durabiliteacute Clowast et le taux de capture hlowast La courbe de Ricker en particulier peut ecirctre

formuleacutee avec les paramegravetres (Clowast hlowast) au lieu des paramegravetres classiques (α β) Nous deacutecrivons les effets de cette

transformation matheacutematique sur le plan biologique et au point de vue de la gestion et nous montrons qursquoelle srsquoexprime

particuliegraverement bien par la formule de Bayes Par des expeacuteriences de simulation agrave des fins drsquoestimation nous deacutemontrons

que lrsquoestimation du nouveau paramegravetre reacuteduit les problegravemes de biais statistique dont on fait eacutetat dans les publications traitant

des meacutethodes classiques En outre les nouvelles estimations possegravedent une robustesse raisonnablement eacuteleveacutee pour ce qui est

du choix du paramegravetre de forme ce que les meacutethodes classiques nrsquooffrent pas En privileacutegiant les paramegravetres de gestion (Clowast

hlowast) au lieu des paramegravetres de productiviteacute (α β) on obtient une estimation qui possegravede des proprieacuteteacutes statistiques ameacutelioreacutees

et qui est en rapport direct avec la politique de pecircche

[Traduit par la Reacutedaction]

Introduction

Attempts to quantify the relationship between stock and re-cruitment date back at least to the work of Ricker (1954 1958)Schaefer (1954 1957) and Beverton and Holt (1957) Each ofthese three approaches begins with a functional relationship

(01) R = f (S θ)

between stock S and recruitment R dependent on a vector θ oftwo parameters The approaches differ only in particularchoices for f and θ Deriso (1980) and Schnute (1985) showthat by including a third parameter in the vector θ a singlefunction f encompasses and generalizes all three approaches

Although juvenile production obviously requires a parentstock evidence for a stockndashrecruitment relationship is typi-cally noisy at best Serious statistical problems can occur when

trying to fit the relationship (01) to real data For exampleWalters (1985) and Kope (1988) discuss biased parameter es-timates for the Ricker relationship

(02) R = αSendashβS

with θ = (α β) Kopersquos (1988) conclusions differ somewhatfrom those of Walters (1985) however both authors agree thatanalytical methods and carefully fabricated simulations offertwo possible avenues for addressing the bias problem

In this paper we follow these suggestions and arrive at asimple approach that reduces both the bias and variance ofrecruitment parameter estimates as well as correlations amongthem Our approach also gives a management interpretation tothe relationship between stock and recruitment Specificallytwo parameters in our model relate directly to managementand a third distinguishes among curve types as in the DerisondashSchnute analysis We show that estimates of the two key man-agement parameters are often robust to the choice of the thirdThus the choice among Ricker Schaefer or BevertonndashHoltcurves may not greatly influence the management implica-tions

Our analysis reinforces conclusions reached by Shepherd(1982) who also proposed a three parameter relationship be-tween stock and recruitment His analysis and ours are bothmotivated by concern for the maximum sustainable catch and

Received October 11 1995 Accepted January 1 1996J13108

JT Schnute1 and AR Kronlund Department of Fisheriesand Oceans Science Branch Pacific Biological StationNanaimo BC V9R 5K6 Canada

1 Author to whom all correspondence should be addressede-mail schnutejpbsdfoca

Can J Fish Aquat Sci 53 1281ndash1293 (1996)

1281

copy 1996 NRC Canada

harvest rate both models include a parameter related to curveshape We incorporate management parameters directly intothe model itself and confirm by simulation that these are rela-tively well determined in comparison with classical parame-ters such as (α β) in (02) as anticipated by Shepherd

Section 1 presents the new model in deterministic form Insection 2 we devise a stochastic simulation model and likeli-hood estimation procedure for investigating statistical proper-ties of recruitment parameter estimates Our model specificallyaddresses Kopersquos (1988) concerns about model initializationSection 3 presents the results of our simulation experimentboth for a single test case and for 200 cases collectively Fi-nally we summarize conclusions limitations and future direc-tions of our research in section 4 We use tables to present mostof the paperrsquos mathematics where for example the notation(T25) refers to equation 5 in Table 2 Appendixes A and Bprovide mathematical proofs of the results described in sec-tions 1 and 2 respectively

1 Stock recruitment model

To motivate the recruitment function proposed here considera fish stock at equilibrium that produces a constant catch C Ifthe stock conforms to the deterministic recruitment function(01) then the equilibrium recruitment R and catch C are re-lated by

(11) R = f (R ndash C θ)

Furthermore the ratio

(12) h = CR

defines the equilibrium harvest rate h Combining (11) and(12) gives the relationship

(13) C = h f (1 minus h

hC θ)

between C and h Typically this equation can be solved toobtain a formula expressing C in terms of h For example theRicker function (02) and the equilibrium condition (13) implythat

(14) C = h

β(1 minus h)log [α(1 minus h)]

These arguments show that a stock recruitment function(01) generally implies another function

(15) C = g (h θ)

that expresses equilibrium catch C in terms of equilibrium har-vest rate h As in (14) C = 0 when h = 0 As h increases from0 C(h) increases until a critical value hlowast is achieved for whichC takes the maximum possible value Clowast = C(h) A sustainedharvest rate h greater than hlowast drives the stock down to a levelfor which the equilibrium catch C is less than Clowast

Mathematically hlowast is determined from (15) by the condi-tion

(16)

dg (h θ)dh

h = h

lowast= 0

Ricker (1975 p 285) discovered that no analytic solution hlowast

exists in the case defined by (02) and (14) consequently heproposed solving by trial Stated another way the transition

(17) (α β)r (Clowast hlowast)from Ricker parameters to optimal harvest parameters cannotbe achieved analytically To our knowledge no one has no-ticed that the opposite transition

(18) (Clowast hlowast)r (α β)actually can be expressed analytically Thus the Ricker func-tion can be rewritten explicitly in terms of (Clowast hlowast)

Table 1 summarizes an analysis (Appendix A) of the Rickerfunction from this point of view The classical function (T11)can equivalently be written in the new form (T12) The tran-sition (18) is given explicitly by (T13)ndash(T14) Furthermorealthough (17) is impossible analytically the iterative process(T16)ndash(T17) initiated by (T15) converges rapidly to hlowast as in(T18) Once hlowast is known then Clowast can be computed from(T19) Finally the Jacobian of the transformation (17) isgiven by (T110) and its reciprocal gives the Jacobian of (18)Because α gt 0 and 0 lt hlowast lt 1 it follows that the Jacobians areboth positive thus the transformations (17)ndash(18) are not sin-gular

The reader can see by inspection that the transformations(T13)ndash(T14) convert the classical Ricker function (T11) intoits new form (T12) In this paper we consider analyses basedentirely on (T12) rather than (T11) From this point of viewtwo key questions motivate the investigation of a stockrsquos char-acteristics First what is the maximum sustainable harvest ClowastSecond what proportion hlowast of a stable stock can safely beharvested in the long run To some extent these characteristics

Recruitment function

(T11) f (S α β) = αSeminusβS

(T12) f (S Clowast hlowast) = S

1 minus hlowast exphlowast minus hlowast2

1 minus hlowastS

Clowast

Transition (18) (C h)r (a b)

(T13) α = ehlowast

1 minus hlowast

(T14) β = hlowast2

(1 minus hlowast) Clowast

Transition (17) (a b)r (C h)

(T15) h0lowast = 05

(T16) αi = ehi

lowast

1 minus hilowast

(T17) hi+1lowast = hi

lowast +1 minus hi

lowast

2 minus hilowast log

ααi

(T18) hlowast = limirarrinfin

hilowast

(T19) Clowast = hlowast2

(1 minus hlowast) βJacobian

(T110)part (α β)

part (Clowast hlowast)=

2 minus hlowast

hlowast2αβ2

Table 1 Deterministic Ricker model in terms of parameters (α β)

or (Clowast hlowast) Transformations between (α β) and (Clowast hlowast) are also

shown along with a corresponding Jacobian

Can J Fish Aquat Sci Vol 53 19961282

copy 1996 NRC Canada

are independent For example consider two salmon popula-tions one originating from a large watershed with many tribu-taries and the other coming from a single small streamBecause Clowast relates to overall stock size the former populationmight be expected to sustain a larger optimal catch Clowast than thelatter By contrast the parameter hlowast measures stock productiv-ity the proportion available as surplus production This mightbe the same for both populations if individual spawning bedsin each watershed have similar characteristics

To gain a numerical sense of the distinction between (α β)and (Clowast hlowast) consider a stock with Clowast = 1 where the numberlsquo1rsquo denotes lsquoone unitrsquo of fish (One unit might for example be104 fish or 104 kg of fish) Table 2 illustrates how the classicalparameters (α β) vary with the corresponding choice for hlowastAs hlowast increases linearly both α and β increase nonlinearlyEach pair (Clowast hlowast) in Table 2 has the simple interpretation dis-cussed in the previous paragraph In contrast the two parame-ters α and β do not have obvious management interpretationsindependent of each other

From the perspective of dimensional analysis Clowast has unitsof fish and hlowast is dimensionless Similarly in the classical for-mulation 1β has units of fish and α is dimensionless Thedimensional relationship between new and old parameters isevident in (T13)ndash(T14) where α is a nonlinear rescaling ofhlowast Biologically α represents productivity at low stock levelsconsequently salmon biologists sometimes interpret α as ageneral productivity parameter For example α = 5 if each fishfrom one generation produces five in the next Similarly asustainable harvest rate hlowast = 08 is achieved if each spawnerproduces five adults from which four are caught in the nextgeneration This argument wrongly suggests that α = 5 whenhlowast = 08 or more generally that α = 1(1 ndash hlowast) In fact α = 111when hlowast = 08 (Table 2) as calculated from the correct non-linear relationship (T13) between hlowast and α

Table 3 extends the analysis of Table 1 to the DerisondashSchnute family of curves (Appendix A) A third parameter γ in(T31) determines curve type where the BevertonndashHoltRicker and Schaefer families correspond to fixing γ at ndash1 0or 1 respectively (Schnute 1985) In particular each equationin Table 1 can be obtained from its counterpart in Table 3 bytaking the limit as γ rarr 0 The parameters (Clowast hlowast) in Table 3have the same management interpretations as in Table 1 Thusin the extended model (T32) Clowast and hlowast still denote the optimalequilibrium catch and harvest rate respectively We regard(T32) as the paperrsquos central equation because it provides a

general recruitment function with direct relevance to manage-ment

An alternative version of our recruitment model (Table 4)can be expressed in terms of optimal stock parameters (Slowast Rlowast)which correspond to the following cycle of recruitment andcatch

(19) Rlowast = f (Slowast Clowast hlowast γ)

(110) Slowast = Rlowast minus Clowast

Thus the optimal escapement Slowast produces optimal recruitmentRlowast which reduces back to Slowast after the optimal catch Clowast isremoved The relationships (T43)ndash(T44) between (Clowast hlowast)and (Slowast Rlowast) are independent of γ (Appendix A) Substituting(T45)ndash(T46) into (T32) gives the alternative model (T41)which takes the form (T42) when γ = 0 The reader can verifyby inspection that Rlowast = f (Slowast) in (T41)ndash(T42)

Figure 1 illustrates geometrically the role of both parametervectors (Clowast hlowast γ) and (Slowast Rlowast γ) in this family of models Fourpanels correspond to four choices of γ Within each panelthree curves are determined by the fixed parameter Clowast = 1 andthree choices for hlowast Furthermore each panel shows two 45deglines R = S and R = S + Clowast The upper line is tangent to eachcurve at the point (Slowast Rlowast) which is emphasized by a verticalline of length Rlowast from the S-axis The dotted portion of eachvertical line represents maximum surplus production Clowast = 1which is also the vertical distance between 45deg lines The ratio

Clowast hlowast α β10 01 12 001

10 02 15 005

10 03 19 013

10 04 25 027

10 05 33 050

10 06 46 090

10 07 67 163

10 08 111 320

10 09 246 810

Table 2 Ricker parameters (α β) corresponding

to Clowast = 1 and hlowast = 01 09 Calculations are

based on the transformation (T13)ndash(T14)

Recruitment function

(T31) f (S α β γ) = αS (1 minus βγS)1γ

(T32) f (S Clowast hlowast γ) = S

1 minus hlowast1 + γhlowast minus

γhlowast2

1 minus hlowastS

Clowast

1γ

Transition (18) (C h)r (a b)

(T33) α =(1 + γhlowast)1γ

1 minus hlowast

(T34) β = hlowast2

(1 + γhlowast) (1 minus hlowast) Clowast

Transition (17) (a b)r (C h)

(T35) h0lowast =

05 γ ge minus1

minus1(2γ) γ lt minus1

(T36) αi =(1 + γhi

lowast) 1γ

1 minus hilowast

(T37) hi+1lowast = hi

lowast +(1 +γhlowast) (1 minus hi

lowast)2 + γhi

lowast minus hilowast log

ααi

(T38) hlowast = limirarrinfin

hilowast

(T39) Clowast = hlowast2

(1 + γhlowast) (1 minus hlowast)βJacobian

(T310)part (α β)

part (Clowast hlowast)=

2 + (γ minus 1)hlowast

hlowast2αβ2

Table 3 Counterpart of Table 1 for the DerisondashSchnute model

with parameters (α β γ) or (Clowast hlowast γ) Equations here reduce to

those of Table 1 when γ = 0

Schnute and Kronlund 1283

copy 1996 NRC Canada

of the dotted portion to the entire vertical line is the optimalharvest rate hlowast = ClowastRlowast

The Schaefer Ricker and BevertonndashHolt families corre-spond respectively to panels A B and C Notice that regard-less of the choice of γ in panels AndashC curves with the sameparameters (Clowast hlowast) exhibit similar behaviour for stock sizesS lt Slowast to the left of the corresponding vertical line The pa-rameter γ primarily influences curve shape at large stock sizeswhere as in the Ricker and Schaefer scenarios a stock increasemay actually lead to reduced recruitment This geometricanalysis helps clarify the prospects for estimating parametersin (T32) For example in highly exploited populations mostdata correspond to stock sizes S lt Slowast Consequently γ may bepoorly determined by the data and estimates of (Clowast hlowast) maybe robust to an arbitrary choice for γ

The geometry of the new family (T32) differs somewhatfrom its classical counterpart (T31) The difference stemspartly from the fact that α in (T33) is not real when γ lt ndash1hlowast

(ie 1 + γhlowast lt 0) This condition determines one of four possi-ble curve types portrayed in Fig 2 where the caption associ-ates each type with a specific range for γ Curves of type A orB have a global maximum at the stock size

(111) Sprime =(1 + γhlowast) (1 minus hlowast)

(1 + γ) hlowast2Clowast

Type A curves also indicate zero recruitment at the positivestock size

(112) Sprimeprime =(1 + γhlowast)(1 minus hlowast)

γhlowast2Clowast

Both Sprime and Sprimeprime are negative for type C curves which have noupper limit Type D curves for which α is not real have aminimum at Sprime and asymptotically approach +infin as S rarr SprimeprimeCurve c in Fig 1D also illustrates this anomalous behaviourWe regard type D curves as biologically meaningful only forS gt Sprime

2 Simulation model

We use the simulation model in Table 5 to investigate statisti-cal properties of parameter estimates obtained from (T32)The modelrsquos complete parameter vector F includes recruit-ment parameters (Clowast hlowast γ) a standard error (τ) integers (m n)associated with time periods for initialization and data collec-tion and initial and final harvest rates (h1 hn) One simulationfrom Table 5 generates data for years t = ndashm n

The population is initialized with a constant harvest rate h1

during years t = ndashm 0 We assume that the recorded fisherytakes place subsequently in years t = 1 n during which theharvest rate increases linearly to a final level hn Thus (T52)defines the simulated harvest rate ht for each year t The popu-lation Rndashm in (T53) corresponds to an equilibrium point for thedeterministic model (T32)

Rndashm = f (Rndashm Clowast hlowast γ)

Our strategy is to choose m reasonably large so that the popu-lation has adequate time to move from the unfished level Rndashm

to a level R1 commensurate with the initial harvest rate h1

Starting with the population Rndashm in year t = ndashm simulationproceeds recursively through equations (T54)ndash(T56) whichdetermine the catch Ct escapement St and subsequent recruit-ment Rt+1 The final equation (T56) introduces lognormal pro-cess error with variance τ2 where the variates εt are assumedindependent and normal with mean 0 and variance 1 By in-cluding process error during the initialization phase we obtaina random initial population R1 for the recorded fishery Thuswe address Kopersquos (1988) concern that recruitment simulationmodels be properly initialized The distribution of R1 in ourmodel depends on the pre-fishery harvest rate h1 and on sto-chastic properties of the recruitment process (T56) where theinitial point Rndashm is discounted by choosing m large

The dynamic equations in Table 5 primarily representsalmon populations to which the Ricker recruitment functionhas been widely applied We confine our attention to this rela-tively simple model where the recruitment function (T32)plays a dominant role Thus we assume implicitly that(1) adults are recruited just prior to the fishery (2) no naturalmortality occurs during the harvest period (3) spawning takesplace immediately after the fishery (4) all adults die afterspawning and (5) the recruitment function (T32) captures theentire population dynamics linking spawners from one genera-tion to adults of the next In this context even the notationlsquoyear trsquo needs special interpretation as a generation index Forexample pink salmon (Oncorhynchus gorbuscha) require twocalendar years to proceed from generation t to t + 1

The simulation process in Table 5 can be summarized bythe transition

(21) F r C S

from simulation parameters to the catch and escapement datavectors defined in (T63) Table 6 presents statistical inferencefunctions for the reverse transition

(22) C S r Q^

from data to parameter estimates where the parameter vector(T61) includes recruitment parameters (Clowast hlowast γ) and thestandard error τ As indicated in (T64) C and Sdetermine the

Recruitment function

(T41) f (S Slowast Rlowast γ) = Rlowast S

Slowast1 + γ

1 minus Slowast

Rlowast

1 minus S

Slowast

1γ

Ricker case (g = 0)

(T42) f (S Slowast Rlowast) = Rlowast S

Slowast exp

1 minus Slowast

Rlowast

1 minus S

Slowast

Transition ( C h)r (S R)

(T43) Slowast =1 minus hlowast

hlowast Clowast

(T44) Rlowast = Clowast

hlowast

Transition ( S R)r (C h)

(T45) Clowast = Rlowast minus Slowast

(T46) hlowast = 1 minus Slowast

Rlowast

Table 4 Version of the model in Table 3 based on the optimal

stock parameters (Slowast Rlowast)

Can J Fish Aquat Sci Vol 53 19961284

copy 1996 NRC Canada

recruitment vector R which can also be treated as data avail-able for estimating Q

The function G(Q) in (T67) represents twice the negativelog likelihood

(23) G(Clowast hlowast γ τ) = minus2 log P (R | C Clowast hlowast γ τ) + K(R)where K(R) is a constant independent of parameters and Pdenotes the probability of the recruitment series R given thecatch history C and the parameters Q (Appendix B) If τ isconsidered a nuisance parameter the vector Q reduces to Qprimein (T62) with the associated inference function

(24) H(Clowast hlowast γ) = minus2 log max

τP(R | C Clowast hlowast γ τ)

+ Kprime(R)

where again Kprime does not depend on the parameters Given thedata R and S both G(Q) and H(Qprime) can be computed from theresiduals (T65) and the sum of squares (T66)

Maximum likelihood estimates Q^

and Q^ prime can be obtained

by minimizing G and H respectively Furthermore because of

their derivation from twice the negative log likelihoodchanges in these functions can be used to construct approxi-mate confidence regions Table 7 illustrates four possibilitiesdepending on the parameters of interest

Recent fisheries literature (eg Walters and Ludwig 1994)has advocated the role of Bayesian statistics in fisheries re-search From this point of view the choice of parameters di-rectly influences the outcome of the analysis For examplewhen changing parameters from (α β) to (Clowast hlowast) in the Rickercurve (T11)ndash(T12) the prior distribution must be adjusted bythe Jacobian (T110) This implies for instance that a uniformprior for one pair of parameters transforms to a prior that is notuniform for the other pair Thus the Bayesian approach forcesa decision regarding which pair should be considered primary

As discussed earlier the pair (Clowast hlowast) has direct relevance tomanagement and these two parameters can be considered in-itially uncorrelated For this reason we propose the simplenoninformative prior (T610) for the four dimensional parame-ter vector Q in (T61) The standard error τ is a scale parameterwith natural prior 1τ (Box and Tiao 1973 p 31) in effect the

Fig 1 Examples of recruitment curves R = f (S Clowast hlowastγ) with f defined by (T32) Panels correspond to the four choices (A) γ = 10 (B) γ =00 (C) γ = ndash10 and (D) γ = ndash15 Within each panel three curves are determined by the fixed parameter Clowast = 1 and three choices (a) hlowast =04 (b) hlowast = 06 and (c) hlowast = 08 Thin lines in each panel represent two 45deg lines R = S (lower) and R = S + Clowast (upper) The optimal harvest

point for each curve is indicated by a vertical line whose dotted upper segment has length Clowast

Schnute and Kronlund 1285

copy 1996 NRC Canada

location parameter log τ has a uniform prior Similar argu-ments applied to the scale parameter Clowast give the prior 1ClowastBecause hlowast is a proportion parameter with 0 lt hlowast lt1 it can beassigned the natural prior 1[hlowast(1 minus hlowast)]12 (Box and Tiao 1973p 35) in effect sinndash1 [(hlowast)12] has a uniform prior Finallybecause γ can theoretically range from ndashinfin to infin we treat it asa location parameter with uniform prior If each of these fourpriors is considered independent of the others the overall priorbecomes (T610) To avoid mathematical singularities nearClowast = 0 hlowast = 0 and hlowast = 1 we assume that Clowast and hlowast are con-fined to intervals

(25) 0 lt Cminlowast le Clowast le Cmax

lowast 0 lt hminlowast le hlowast le hmax

lowast lt 1

where minimum and maximum values for each parameter arespecified by the analyst Alternatively (T610) might be re-placed by uniform or normal distributions of Clowast and hlowast con-fined to the intervals (25)

The posterior (T611) for Q can be expressed in terms ofG(Q) which differs from twice the negative log likelihood bya constant Bayes distributions for Q = (Clowast hlowast γ τ) can be

converted to distributions for (α β γ τ) via the Jacobian(T310) where hlowast is computed from α by (T35)ndash(T38)

3 Simulation results

In this section we use cycles of simulation (21) and estimation(22) to assess properties of recruitment parameter estimatesDuring the estimation phase we obtain maximum likelihoodestimates

(31) Q^ prime = (C^ lowast h

^ lowast γ^ )by minimizing H(Qprime) in (T68) based directly on our proposedrecruitment function (T32) Once these estimates are knownthe remaining estimates (α^ β^ S

^ lowast R^ lowast τ^ ) can be computed from

(T33)ndash(T34) (T43)ndash(T44) and (T69) respectively Fur-thermore from (T67)ndash(T69)

(32) G (Q^ ) = (n minus 1)(1 + 2 log τ^ )Parametric constraints such as (25) must be imposed

while minimizing H(Qprime) We enforce these constraints analyti-cally by expressing H(Qprime) in terms of surrogate parameters

Fig 2 Four recruitment curve types associated with the following four conditions on γ (A) γ gt 0 (B) ndash1 lt γ lt 0 (C) ndash1hlowast lt γ lt ndash1 (D) γ lt

ndash1hlowast Each type relates to stock sizes Sprime and Sprimeprime in (111)ndash(112) indicated here as points on the S-axis

Can J Fish Aquat Sci Vol 53 19961286

copy 1996 NRC Canada

Each actual parameter θ is computed from a correspondingunconstrained surrogate p by the formula

(33) θ = θmin + (θmax minus θmin ) sin 2

π p

2

Consequently for all values of the surrogate parameter p θautomatically lies in the constrained interval (θmin θmax) Inparticular the interval end points are achieved when p is aninteger Our computations employ the simplex search algo-rithm (Nelder and Mead 1965 Mittertreiner and Schnute1985) where the search is conducted in surrogate parameterspace

Figure 3 illustrates one simulation from the model inTable 5 based on the parameter vector F with components

(34)

Clowast = 10 hlowast = 06 γ = minus05 τ = 04

m = 100 n = 15 h1 = 03 hn = 08

Consistent with the dimensional analysis in section 1 thechoice Clowast = 1 merely sets the scale for the simulation Theoptimal harvest rate hlowast = 06 corresponds to a rather productivepopulation and the shape parameter γ = ndash05 strikes a compro-mise between the Ricker and BevertonndashHolt curves Whenτ = 04 the lognormal variate eτ ε

t in (T56) will fall in theinterval (046219) with probability 095 Thus process errorin our simulation causes the recruitment Rt+1 to vary fromabout half to double its deterministic prediction f (St Clowast hlowast γ)The initial population R1 is determined randomly by m = 100years of simulated fishing with the constant harvest rate h1 Wethen generate n = 15 years of data in the simulated fishery Inrelation to hlowast = 06 the simulation uses a relatively modesthistoric fishing rate h1 = 03 This is increased to an excessiveharvest rate hn = 08 in the final year n = 15

Data points (St Rt+1) in Fig 3 exhibit considerable scattercomparable to that often seen in real data The initially largepopulation (points lsquoarsquondashlsquodrsquo) is reduced to lower levels (pointslsquokrsquondashlsquonrsquo) by the increase in harvest rate from h1 = 03 to h15 =08 Table 8 summarizes parameter estimates obtained fromthese data depending on the choice of γ Within the data range

the true curve A ascends fairly rapidly and then flattens outnear the dome where S = Sprime = 156 as calculated from (34) and(111) Even though the data were simulated with γ = ndash05they show no evidence of declining recruitment at large stocksizes The best estimate (case E Table 8) ie the lowest valueG(Q) is obtained with γ^ = ndash102 Essentially then the datasuggest a BevertonndashHolt curve (case D γ^ = ndash1)

As anticipated from our discussion of Fig 1 estimatedcurves in Fig 3 for various values of γ are similar at low popu-lation levels and somewhat divergent at high population levelswhere the role of the shape parameter becomes more impor-tant Table 8 provides information about the robustness of pa-rameter estimates to the choice of γ For example as γdecreases from 0 to ndash1 (cases BndashD) Clowast declines about 6 from090 to 085 In summary percentage changes between casesB and D for various estimates are

(35)

C^ lowast minus6 h

^ lowast +15

a^

+115 β^ + 409

S^ lowast minus33 R

^ lowast minus16

Thus from the example in Fig 3 estimates (C^ lowast h

^ lowast) are reason-ably robust to the choice of γ (S

^ lowast R^ lowast) are somewhat less ro-

bust and (α^ β^ ) are quite unstable Similar results from otherexamples reinforce our preference for the recruitment parame-ters (Clowast hlowast γ)

Confidence contours (Fig 4) reveal another aspect of theparameter pairs examined in (35) The least robust parametersalso tend to be the most highly correlated Thus contours for(α β) in Fig 4B are particularly narrow elongated and tilted

Parameters

(T51) F = (Clowast hlowast γ τ m n h1 hn)Control policy

(T52) ht =

h1 minusm le t lt 1

h1 + (hn minus h1)t minus 1

n minus 1 1 le t le n

Initial condition

(T53) Rminusm =

1 minus (1 minus hlowast)γ + γhlowast

γhlowast2(1 minus hlowast) Clowast γ ne 0

hlowast minus log (1 minus hlowast)hlowast2

(1 minus hlowast) Clowast γ = 0

Simulation (t = ndashm n)

(T54) Ct = htRt

(T55) St = Rt minus Ct

(T56) Rt + 1 = f (St Clowast hlowast γ) eτ εt

Table 5 Stochastic simulation model based on the stock

recruitment function (T32) when γ ne 0 or (T12) when γ = 0

Parameters

(T61) Q= (Clowast hlowast γ τ)(T62) Qprime = (Clowast hlowast γ)

Data

(T63) C = Ct

t=1

n S =

St

t=1

n

(T64) R = Rt

t=1

n=

Ct + St

t=1

n

Residual sum of squares

(T65) ηt(Qprime) = log

Rt+1

f (St Qprime)

t = 1 n minus 1

(T66) T (Qprime) = sumt=1

nminus1

ηt2(Qprime)

Inference functions

(T67) G (Q) = (n minus 1) log τ2 + τ minus2 T (Qprime)(T68) H (Qprime) = (n minus 1) log T (Qprime)

(T69) τ^ 2 (Qprime) = 1

n minus 1T (Qprime)

Bayes prior and posterior

(T610) P (Q) ~1

[hlowast(1 minus hlowast)]12 Clowastτ

Cminlowast le Clowast le Cmax

lowast hminlowast le hlowast le hmax

lowast

(T611) P (Q | C S) ~ e minusG (Q)2 P (Q)

Table 6 Calculations leading to inference functions for the

parameter vector Q based on the data vectors C and S

Schnute and Kronlund 1287

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with respect to both axes Each panel in Fig 4 also indicatesthe true value (d) of the corresponding parameter pair whichlies between the 50 and 80 contours

Contours in Fig 4 have been obtained from condition(T74) in Table 7 with γ constrained at the actual value (γ =ndash05) used to generate the data in Fig 3 Similar contours areobtained with the constraints γ = 0 or γ = ndash1 However if γ istreated as a free parameter as in (T73) the resulting contoursbecome broader Our next analysis explores more fully theissue of appropriate constraints for γ

To extend our results beyond the single example in Fig 3we consider 200 simulations based on the parameters (34)Figures 5Andash5D represent maximum likelihood parameter esti-mates for these simulations with γ constrained at one of threefixed values (ndash10 ndash05 00) or with γ free We use the surro-gate technique (33) to impose the following bounds on pa-rameter estimates

(36)

01 le Clowast le 100

005 le hlowast le 095

minus10 le γ le 10

The bounds on Clowast bracket its true value (Clowast = 1) by a factor of10 The somewhat narrow bounds for γ define a restricted fam-ily of curves between the asymptotic BevertonndashHolt case(Fig 1C γ = ndash1) and the domed Schaefer case (Fig 1A γ = 1)Curves that rise indefinitely (Fig 1D γ lt ndash1) and left-skewedcurves (γ gt 1) are excluded

Each panel in Fig 5 pertains to the same set of 200 simula-tions where panels AndashD differ only in the condition imposedon γ during the estimation phase Boxplots represent the ratioof estimated to true parameter values on a log (base 2) scaleThus a value 0 indicates perfect fit and the illustrated range(ndash33) represents an eightfold deviation below or above thetrue parameter value This logarithmic representation is appro-priate only for positive parameters on the scale (0 infin) Because0 lt hlowast lt 1 we first apply the transformation

(37) hprime = hlowast

1 minus hlowast

to convert hlowast to a positive parameter hprimeFigure 5 confirms our conjecture that estimates (C

^ lowast h^ lowast) are

relatively unbiased and robust to the constraint on γ By con-trast the estimates (α^ β^ ) show bias that varies systematicallyfrom negative to positive as γ decreases from 0 (Fig 5A) to ndash1(Fig 5C) All estimates appear reasonably unbiased when γ isconstrained to its true value ndash05 (Fig 5B) Furthermore theestimates (C

^ lowast h^ lowast) appear to have somewhat lower variance

when γ is constrained at the incorrect value γ = 0 (Fig 5A) Thevariance of all estimates increases somewhat when γ is freewithin the constraints (36) (Fig 5D) Estimates γ^ (not shown)obtained for Fig 5D correspond to the three cases γ^ = ndash1 ndash1 ltγ^ lt 1 and γ^ = 1 in about 13 12 and 16 respectively of the200 simulations The distribution of γ^ toward the left end of theinterval [ndash11] and its median value γ^ = ndash053 reflect the truevalue γ = ndash05

Figure 6 shows pairs plots of estimates obtained in Fig 5DTo facilitate the display one outlier (C

^ lowast h^ lowast) = (254 095) has

been omitted Furthermore the ranges of α^ and β^ have beenrestricted to a maximum of 20 and 10 respectively for example

Fig 3 Results of one simulation from Table 4 with the parameters

(34) For t = 1 14 simulated data points (St Rt+1) are plotted

with solid circles (d) and labelled a n respectively True (A)

and estimated (BndashD) recruitment curves correspond to cases AndashD

in Table 8 The curve for case E (not shown) coincides almost

exactly with that for case D A diamond symbol (e) indicates the

point (Slowast Rlowast) on each curve

Parameters Constraints Region boundary condition

(Clowast hlowast γ τ) none (T71) G (Clowast hlowast γ τ) minus G (C^ lowast h^ lowast γ^ τ^ ) = X (p4)

(Clowast hlowast γ) τ free (T72) H (Clowast hlowast γ) minus H (C^ lowast h^ lowast γ^ ) = X (p3)

(Clowast hlowast) (γ τ) free (T73) minγ H(Clowast hlowast γ) minus H(C^ lowast h^ lowast γ^ ) = X (p2)

(Clowast hlowast) γ = γ0 τ free (T74) H (Clowast hlowast γ0) minus H(C^ lowast h^ lowast γ0) = X (p2)

Table 7 Conditions that determine the boundary of a p-level confidence region for various combinations of recruitment parameters (Clowast hlowast γ

τ) Other parameters may be free (ie treated as nuisance parameters) or constrained The function X(p k) denotes the 100 p th percentile for

the χ2 distribution with k degrees of freedom for example X(095 1) = 384

Can J Fish Aquat Sci Vol 53 19961288

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an estimate α^ gt 20 is portrayed in Fig 6 as α^ = 20 The pairs(C

^ lowast h^ lowast) exhibit a reasonably circular distribution around the

true value (Clowast hlowast) = (10 06) thus confirming our conjecturethat these two estimates are relatively uncorrelated By con-trast (α^ β^ ) exhibit a relatively high positive correlation alonga diagonal band in the scatter plot Similarly (Slowast Rlowast) show aweaker positive correlation The pairs (h

^ lowast α^ ) reveal thenonlinear relationship α^ = α^ (h^ lowast γ^ ) specified by (T33) Essen-tially the points (h

^ lowast α^ ) are confined to a band between two

curves determined by the extremes γ^ = ndash1 and γ^ = 1 whichcharacterize roughly 13 and 16 respectively of the pointsplotted

Each scatter plot in Fig 6 indicates a pattern like that of(h

^ lowast α^ ) which tends to become more diffuse as the constraints(36) on γ are widened We have investigated broader rangesfor γ as extreme as (ndash10 10) and found similar patternsamong the remaining parameter estimates Extreme values γ^

may however correspond to type D curves (Fig 2D) or othercurves with unrealistic biological properties For this reasonwe recommend confining γ initially to the rather narrow range(36) This range at least captures the curves types of greatesthistorical interest Parameter estimates for all 200 simulationsin Figs 5ndash6 were obtained without the benefit of residualanalyses An analyst dealing with data from a single stockwould of course exploit diagnostic procedures in the searchfor an appropriate model including a suitable range for γ

4 Discussion

We have shown that the DerisondashSchnute generalization ofthree classical recruitment models (Ricker 1954 1958 Schae-fer 1954 1957 Beverton and Holt 1957) can be expressedanalytically in terms of optimality parameters (Clowast hlowast) and a

Fig 5 Boxplots representing parameter estimates from 200 cycles

of simulation (21) and estimation (22) with simulation parameters

(34) During the estimation phase γ is constrained as follows (A)

γ = 00 (B) γ = ndash05 (C) γ = ndash10 (D) γ free Each boxplot

represents the ratio of estimated to true value for the specified

parameter on a log2 scale where hlowast is first transformed by (37)

Boxes represent the interquartile range (25 to 75 quantiles) of

the ratio where a central line indicates the median Whiskers are

drawn to the nearest value not beyond 15 times the interquartile

range Solid circles denote individual points beyond this range

Fig 4 Confidence contours for (A) (Clowast hlowast) (B) (α β) and (C)

(Slowast Rlowast) based on the simulation in Fig 3 Contours (A) are

calculated from condition (T74) in Table 7 with γ = ndash05 and τfree Transformation formulas in Tables 3ndash4 allow (B) and (C) to

be computed from (A) The symbols + and d indicate estimated

and true parameter values respectively Curve styles designate

confidence levels as follows dotted (50) broken (80) solid

(95)

Case Clowast hlowast γ τ α β Slowast Rlowast G (Q)

A 100 060 ndash050 040 51 13 067 167 ndash1334

B 090 061 000 036 47 11 058 147 ndash1466

C 087 065 ndash050 035 62 20 048 135 ndash1575

D 085 069 ndash100 034 101 56 039 123 ndash1616

E 085 069 ndash102 034 104 59 038 123 ndash1616

Table 8 Original parameters (case A) maximum likelihood

estimates (cases BndashE) and corresponding values of the inference

function G(Q) for the simulation portrayed in Fig 3 The shape

parameter γ is either constrained at a specified value (cases BndashD)

or estimated (case E) where the constraint γ = ndash05 in case C

matches the true value in case A

Schnute and Kronlund 1289

copy 1996 NRC Canada

shape parameter γ Computer code for parameter estimationalgorithms can use the new formulation (T32) directly ratherthan the old formulation (T31) combined with the transition(17) We have adopted the surrogate technique (33) to con-strain parameter estimates within reasonable bounds Esti-mates (C

^ lowast h^ lowast) show little bias and are relatively robust to the

choice of shape parameter γ in contrast with estimates (α^ β^ )of classical Ricker parameters When analyzing an actual data

set we recommend trying several fixed values of γ (eg γ =ndash1 0 1) before estimating γ as a free parameter Figures 1 and2 which illustrate the geometry of this family of recruitmentcurves provide an intuitive framework for assessing the roleof γ

Theoretically Clowast and hlowast represent the maximum sustain-able catch and corresponding harvest rate for the stock Thiswould literally be true if the stock deterministically followed

Fig 6 Pairs plots of the 200 estimates (C^ lowast h

^ lowast α^ β^ S^ lowast R

^ lowast ) obtained for Fig 5B

Can J Fish Aquat Sci Vol 53 19961290

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the recruitment law (T32) In this case the relationship (15)between equilibrium catch C and constant harvest rate h takesthe form

(41) C(h) =h(1 minus hlowast)hlowast(1 minus h)

1

γhlowast

1 + γhlowast minus

1 minus hlowast

1 minus h

γ

Clowast

The function C(h) defined by (41) rises from the origin (hC) = (0 0) to a maximum at (h C) = (hlowast Clowast) This determinis-tic picture is altered however by recruitment process errorassociated with τ in (T56)

To assess the influence of τ on the equilibrium catch (41)we performed simulations with four levels of error (τ = 0002 04 06) and the parameters

(42)

Clowast = 10 hlowast = 06 γ = minus05

m = 100 n = 10 000 h1 = hn = h

where h remains to be specified In other words we simulateda fishery with a fixed harvest rate h over a long time period (n =10 000) The average long term catch

(43) Cminus__ = 1

n sumt=1

n

Ct

provides a measure of overall returns from the fishery in rela-tion to h and τ Figure 7 shows the results of this analysis Thesolid curve (τ = 0) corresponds to the deterministic catch curve(41) which passes through the maximum point (hlowast Clowast) =

(0610) For increasing levels of process error τ the optimalharvest rate for the stochastic model remains close to hlowast = 06and the optimal catch increases slightly from Clowast = 10 Thisincrease in C

minus__when τ gt 0 results from the positive skewness

of lognormal population sizes generated from (T32) For ex-ample the stochastic recruitment Rt has equal probability ofbeing twice or half its prediction from (T32) Thus on aver-age gains from favourable years numerically dominate lossesfrom poor years in the calculation (43)

Figure 7 shows that the optimal long term average catch andassociated harvest rate for the stochastic model in Table 5 areclose to their deterministic counterparts Clowast and hlowast These re-sults are somewhat surprising in the light of earlier work (Bed-dington and May 1977 Thompson 1994) where differentmodels imply that the optimal harvest rate varies systemati-cally with the amount of process error Simulations analogousto that for Figure 7 could be used to assess model differencesas well as management strategies more complex than a con-stant harvest rate h We stress that even for the model in Table5 the parameters (Clowast hlowast) serve only as guidelines not literalmanagement recommendations

Although our analyses utilize likelihood methods we havealso described a Bayesian approach Because the parametershave simple management interpretations informative or non-informative priors can readily be formulated As a simple testwe examined the simulated data in Fig 3 using the Bayes prior(T610) with the bounds (36) The posterior mode computedfrom (T611) agrees closely with the maximum likelihood es-timate reported in Table 8 (case E)

Our model has features in common with that proposed byShepherd (1982) notably the presence of a shape parameterthat determines a variety of possible curve types The modelhere has the possible advantage that it includes the three clas-sical models cited at the start of this section as exact specialcases In addition we have presented simulation tests to exam-ine statistical properties of the parameter estimates in an at-tempt to resolve bias problems discussed by Walters (1985)and Kope (1988) We find that our formulation in terms ofparameters (Clowast hlowast) addresses these problems and also leads toestimates of quantities with direct management relevance

We should perhaps mention that the parameters (B Rlowast) inShepherd (1982) correspond to an ad hoc point on the recruit-ment curve chosen by the analyst to represent typical valuesof spawning and recruitment stock biomass These parametersshould not be confused with the optimality parameters (Slowast Rlowast)defined here We address the problem of simultaneously esti-mating all three parameters (Clowast hlowast γ) and we introduce thegeometric analysis in Figs 1ndash2 to assist this process

To maintain focus on the new recruitment model (T32) wehave considered only the relatively simple stochastic model inTable 5 We have not for example addressed the importantproblem of measurement error in stock recruitment data (Wal-ters and Ludwig 1981 Ludwig and Walters 1981) We regardthis as a subject for future study in the context of state spacemodels (Schnute 1994) In fact the formulation here resultedfrom our attempts to resolve technical problems in the analysisof recruitment state space models The robust statistical prop-erties of the estimates (C

^ lowast h^ lowast) may prove particularly helpful

when the data result from more complex stochastic processesthan the simple process error contemplated in Table 5

Fig 7 Mean catch Cminus__

in (43) as a function of the constant

harvest rate h for four levels of process error τ Curves with τ gt 0

have been slightly smoothed with the aid of a cubic spline

Schnute and Kronlund 1291

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Acknowledgements

We thank Grant Thompson and a second anonymous reviewerfor thoughtful comments leading to an improved final paperWe also thank Bill Ricker for helpful discussions and encour-agement

References

Beddington JR and May RM 1977 Harvesting natural popula-tions in a randomly fluctuating environment Science (Washing-ton DC) 197 463ndash465

Box GEP and Tiao GC 1973 Bayesian inference in statisticalanalysis Addison-Wesley Publishing Co Inc Menlo Park Calif

Beverton RJH and Holt SJ 1957 On the dynamics of exploitedfish populations Fish Invest Ser 2 No 19

Deriso RB 1980 Harvesting strategies and parameter estimation foran age-structured model Can J Fish Aquat Sci 37 268ndash282

Hilborn R 1985 Simplified calculation of optimum spawning stocksize from Rickerrsquos stock recruitment curve Can J Fish AquatSci 42 1833ndash1834

Kope RG 1988 Effects of initial conditions on Monte Carlo esti-mates of bias in estimating functional relationships Can J FishAquat Sci 45 185ndash187

Ludwig D and Walters CJ 1981 Measurement errors and uncer-tainty in parameter estimates for stock and recruitment Can JFish Aquat Sci 38 711ndash720

Mittertreiner A and Schnute J 1985 Simplex a manual and soft-ware package for easy nonlinear parameter estimation and inter-pretation in fishery research Can Tech Rep Fish Aquat SciNo 1384

Nelder JA and Mead R 1965 A simplex method for functionminimization Comput J 7 308ndash313

Ricker WE 1954 Stock and recruitment J Fish Res Board Can11 559ndash623

Ricker WE 1958 Handbook of computations for biological statis-tics of fish populations Bull Fish Res Board Can No 119

Ricker WE 1975 Computation and interpretation of biological sta-tistics of fish populations Bull Fish Res Board Can No 191

Schaefer MB 1954 Some aspects of the dynamics of populationsimportant to the management of the commercial marine fisheriesInt-Am Trop Tuna Comm Bull 1 26ndash56

Schaefer MB 1957 A study of the dynamics of the fishery foryellowfin tuna in the eastern tropical Pacific Ocean Int-AmTrop Tuna Comm Bull 1 245ndash285

Schnute J 1985 A general theory for the analysis of catch and effortdata Can J Fish Aquat Sci 42 414ndash429

Schnute JT 1994 A general framework for developing sequentialfisheries models Can J Fish Aquat Sci 51 1676ndash1688

Shepherd JG 1982 A versatile new stockndashrecruitment relationshipfor fisheries and the construction of sustainable yield curves JCons Cons Int Explor Mer 40 67ndash75

Thompson GG 1994 A general diffusion model of stock-recruit-ment systems with stochastic mortality ICES CM 1994T18International Council for the Exploration of the Sea Charlotten-lund Denmark

Walters CJ 1985 Bias in the estimation of functional relationshipsfrom time series data Can J Fish Aquat Sci 42 147ndash149

Walters CJ and Ludwig D 1981 Effects of measurement errorson the assessment of stock-recruitment relationships Can J FishAquat Sci 38 704ndash710

Walters CJ and Ludwig D 1994 Calculation of Bayes posteriorprobability distributions for key population parameters Can JFish Aquat Sci 51 713ndash722

Appendix A Proofs of Tables 1 3 and 4

The recruitment function (T31) serves as a starting point forthe entire analysis Schnute (1985 p 418) shows that (T31)reduces to the BevertonndashHolt Ricker and Schaefer functionswhen γ = ndash1 0 or 1 respectively Substituting (T31) into (13)and solving for C gives

(A1) C(h) =1 minus[α(1 minus h)]minusγ

βγ (1 minus h)h

The condition d log C(h)dh = 0 which is equivalent to (16)characterizes the point h = hlowast where C(h) takes its maximumvalue After algebraic reduction this leads to the condition

(A2)1

hlowast(1 minus hlowast)1 + γhlowast minus [α(1 minus hlowast)]γ

1 minus [α(1 minus hlowast)]γ = 0

relating hlowast and α Solving (A2) for α gives (T33)

By definition the maximum value of the function (A1) isClowast = C(hlowast) Thus (A1) can be used to express β in terms of(Clowast hlowast)

(A3) β =1 minus [α(1 minus hlowast)]minusγ

γ (1 minus hlowast)Clowast hlowast

Substituting (T33) into (A3) gives (T34) Furthermore sub-stituting (T33)ndash(T34) into (T31) gives the new recruitmentfunction (T32)

Equation (T33) expresses α as a function of hlowast To invertthis function numerically we apply Newtonrsquos method to finda root hlowast of the function

(A4) φ(hlowast) = log

(1 + γhlowast)1γ

1 minus hlowast

minus log α

This function is increasing convex upward and defined for0 lt hlowast lt 1 unless γ lt ndash1 in which case φ(hlowast) is defined only for0 lt hlowast lt ndash1γ The initial point h0

lowast defined by (T35) is thus atthe midpoint of the valid interval for hlowast The Newton iteration

(A5) hi+1lowast = hi

lowast minusφ(hi

lowast)φprime(hi

lowast)

where φprime = dφdhlowast reduces to (T36)ndash(T37) The proof usesthe fact that φ(hi

lowast) = log(αiα) where αi is defined by (T36)The limit (T38) of Newtonian approximations hi

lowast gives thevalue hlowast(α) that inverts the function α(hlowast) in (T33) Usuallyconvergence is so rapid that hlowastasymp h5

lowast is an adequate approxima-tion Hilborn (1985) proposes an approximation for the Rickercase (γ = 0) closely related to the choice hlowastasymp h1

lowast

After hlowast(α) is computed from (T38) Clowast(α β) can be com-puted from (T39) To prove this substitute (T33) into (A3)and solve for Clowast Finally the Jacobian

(A6)part(α β)

part(Clowast hlowast)=

partαpartClowast

partβparthlowast minus

partαparthlowast

partβpartClowast

can be computed directly from (T33)ndash(T34) where calcula-tion is simplified by the fact that partαpartClowast = 0 This proves(T310) and completes the proof of Table 3 Each equation inTable 1 is the limiting form of its counterpart in Table 3 asγ rarr 0 where proofs are simplified by the identity

Can J Fish Aquat Sci Vol 53 19961292

copy 1996 NRC Canada

(A7) limγrarr0

(1 + γ x)1γ = ex

The optimal recruitment Rlowast and escapement Slowast by definitionsatisfy the two equations

(A8) Slowast = Rlowast ndash Clowast Clowast = hlowast Rlowast

The pair (A8) is equivalent to both transitions (T43)ndash(T44)and (T45)ndash(T46) Furthermore substituting (T45)ndash(T46)into (T32) gives the recruitment function (T41) which re-duces to (T42) as γ rarr 0 by the identity (A7) This completesthe proof of Table 4

Appendix B Likelihood derivations

Likelihood for the model in Table 5 comes entirely from thestochastic equation (T56) From this point of view the catchCt escapement St and recruitment Rt = Ct + St are known fort = 1 n Because each random variable Rt+1 depends con-ditionally on the previous random variable Rt and the knowncatch Ct the likelihood can be expressed recursively as theproduct of conditional probabilities

(B1) P(R2 Rn|C1 Cnminus1 Q) = prodt=1

nminus1

P(Rt+1|Rt Ct Q)

where the lognormal probability distribution for each year t is

(B2) P(Rt+1 | Rt Ct Q) = 1

(2π)12τ Rt+1

times expminus 1

2τ2[log Rt+1 minus log f (Rt minus Ct Qprime)]2

Combining (23) with (B1)ndash(B2) gives the function G(Q)defined in (T65)ndash(T67) where

K = minus(n minus1) log(2π) minus 2sumt=2

n

log Rt

Minimizing G(Q) in (T67) analytically with respect to τ givesthe estimate (T69) Finally substituting (T69) into (T67)gives H(Qprime) in (24) and (T68) where

Kprime = K + (n ndash 1)[log(n ndash 1) ndash 1]

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