f,

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NOT TO BE CITEI? WITHOUT PRIOR REFERENCE TO THE AUTHORS

INTERNATIONAL COUNCIL FORTHE EXPLORATION OF THE SEA

STATISTICS COMMIITEEC.M. 1992/D:34

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STOCHASTIC MODELS FOR STATIONARY .GEAR FISHERIES

Michael J. Fogartyland

Julian T. Addison2

INational OCeanic and Atmospheric AdministrationNational Marine Fisheries SerVice

Woods Hole, MA 02543USA

2Ministry of Agriculture, Fisheries and FOOdDirectorate of Fisheries Research .

Lowestoft, Suffolk, NR33 ORTUK

Abstract

Stationary jishing gear is employed in a diverse array ofjisheries including those formany high unil-value species. The mode of operation of these gear types (longlines, gil/nets,traps etc.) requires special consideration in lhe developmenr ofabundance indices derivedfromcatch-per-uilil-ejJon. Specijically, factors ajJecting the probability ofcapture iuul ofescapemenror other sources of loss must be' addressed. Here, we describe two classes of modeli forslationary gear that includi existing models as special cases. Deterministic models are jirstdeveloped 10 illuslrate the basic principles and stochastic analogues of these inodels are thendescribed using /he /heory ofbinh-death stochastic processes. /t is shOwn for models wherec/osed fonn solurions are possible, /hat /he mean of /he slochastic mOdels is idenrical /0 /hedelenninistic case. The S/OChaslic models provide addilional diagnostic informarion in tenns oflhe variance and /he probabi/iry dislribution ofcatch levels. An applicaiion Oflhis approach isprovidedfor experimenral observations on ingress and egress rates in an American lobs/er lrapjishery.

Introduction

Stationary fishing gear (including traps, longlines, and gillnets), comprise a domiriantmode of capture in many fish arid irivertebrate fisheries. These gear types are Set at fixedlocations arid retrieved after vanable immersion (soaI<) intervals. Major fisheries for shellfish(notably crustaceans and gastropods; Kiouse 1989; ~lil1er 1990) and tropical (Munro 1983) andtemperate-boreaI tish species (von Brandt 1984) are prosecuted using traps of vanous designs.Longline arid gillnet catches accciurit for substantial portions of v,'orldwide fisheries includingthose for many high uriit-vaIue sPecies incIuding salmon, tunas, and halibut (von Bciridt 1984).These diverse fishenes share in common a passive mode of capture in which the behavior of thespecies sought plays a dominant role. Immersion limes can vary greatly in these fisheries andcatch is typically not a linear function of the socik intervaJ. These characteristics complicate thedevelopment of scindardized measures of abundance based on ccitch-per-tinit-effort.

Methods of effort staridaidiiation for longÜne ami gillnet fisheries were proPOsed byGul~and (1955; pp 34-36) and Beverton arid Holt,(1957; pp 94-95) based on a saturatingcapturemodel with no loss or escapement. This model provided a descripiion of the relationshipbetweencatch and soal time. Murphy (1960) subsequentlyrefined this approach with explicitcorisidemtiori of the individtiaI prOcesses of capture, escapement; and other sources of 10ss forloriglines (see also Sinoda 1981). Munro (1974) described the process of entry and escapementto unbaited Antillean fish traps and propösed a model basecl on ccinstäni entrY rates arid consiantproportional escapement. ,This model cari be expressed in a form ideniical to the Gulland­Beverton-Holt model but the interPretation of the pärameters is entirely different. Munro'smodel has been applied io many crustacean fisheries (see Miller 1990 for a comprehensiveoverview).

Several additional models for tcip fishenes have been propo5ed; Ausiin (1977) describedci Power function mOdel for thespiny iobster (Panuiiius argUs) fishery. Somerton arid Merrit(1986) independently derived Murj>hy's (1960) model in an application to a king crab fisherybut did not consider multiple sources of loss. Smith and Jamieson (1989) developed a trappingmodel incorJ>orating behavioräl iriteraetions arid changes in bait effectlveriess in an applicationto a Dungeness crab fishery.

• -" ,,' ., .... ,. C', ...... '1 .. '" ,. '.' .. ' I.', " '.' .', '

Here, we descnbe a genenuized approach to modellirig ihe relatoriship between catch andimmersion time for sciiionary gear fisheries with consideration of capture processes (enuy totraps, hooking in longlines, cl' enciitglenierit in gillriets); and escapement or other sourees ofioss. Mariy.of the modelsdeScribed above are sPecial eases urider this approach. DeterrninisticmOdels in diserete time are. first deScribed to illuStr3.ie the baSic prlnciples. It is shown that theasymptotic (saturating) modelemployoo for longline and gÜlnet fisheries and that applied io triipfisheries have veiy different implieatioris for the development of abundance .indices baSed onCPUE dat.ä: Time-dependent prOcesSes are also considered. We then illustrate the development.öf stochastic anatoguesöf the deterministic models using the theory of birth-death stochasticprocesses (see FeIler 1957; Bailey 1964; and Karlin arid Taylor i975 for an overview).

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Stochastic models have previously .been d~veloped for hock and line fisheries using thetheory of Markov. pr6cesses (Rothschild 1960) and renew3.l theory (Deriso and Parma 1987).Reed (1986) illust.rated the development of a stochastic catch model arid SampSon (1988)described a catch model based on apur~ death stcichastic process; These modelsdirectly addressthe issue of variability in catch prOcesses arid are therefore. more reaIistic than determinisiic

.approaches. These models also provide additional diagnosÜc information on the probabilitydistribution of the catch, its meari, arid variance. . '. ". .. .

Background

The catch rate of stationai-y gear can be expressed as the balance between räies of capture .arid escapement. The catch cf a unit of gear can, iri geriefaI, be expressect:

>0 where Cis the number in the catch, h(C) describes the capture rate as a fune'tien of the number• in the catch arid};(C) represents the escaPement oe, rate of loss.. Iri this paper, we consider two

general forms for the capture furiction. TC> simplify the following analyses; only the case of aconstant per capita eScape or loss rate will be examined.

•

. \ ~. , . . ... . .' ,

Below,. we describe deterministic models embodying alternatlve. mechanisms for thecapture process to set the stage for the later developmertt of stochastic mOdels. .The first modeltype to be considerect. is more appropriate for longline arid gillnet fisheries as develoPed byGul1<üid (1955) and Beverton arid Holt (1957) while the second is more appropriate for triip...ba5ed fishenes (Munro 1974; Miller 1990). .

LOnglinelGiiInet F1sheries

We first consider the Case of a Sahifatlng cäpriire rate arid a propOrtional eScape rate äsan extension of the Gullarid-Bevertöri-Holt mOdel. This mOdel assumes thät the eapture ratedeclines linearlywith wiili eatch (resiJ1tlng in, an aSymptotlc relationship l>etween cumulatlvecatch and wak time) arid thai there is aconstant per capiia eseaj>e 01' loss rilte; The numtJer ofindividuals caught after t+~t time uriitS is given by:' . , .

.'

c.+At.;:;: C. + r(C· ..: CJ~t - bC.~t

where c~,is the,maximurn eatch level, r is therate at ~hich the maXimum ~tch is äpproached;arid bisthe 10s5 rate.The maximum cateti (C)is definoo for lorigline fisheries by, the illimberof hooks i~ the set. For gillnd fisheri~s,.C· could, inpnnCiple, t>e. defined as the number ofnlesh openirigs; however, this undoubtedly represents än unieaIistiCally· high level for themaximumcatch (Beverion änd. Holt 1957) becaUSe riet ävoidance presumably iricrl:ases as thecatch increäses. Gillnet fisheries cc>uld alsO,be mOdded using the approach deScribed t>elow forti'ap fisheries but here we will follow histoncil prece<Jent. T3.king the limit as ~t - 0 and

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solving gives:

Cl = {[,c"/(r+b)]}Ü - exp[-(;+b)t]}

for the initial conditions: Co=O, 'to=O. Note that the asymptotic catch is less that C" when b >O. Gulland (1955) and Beverton and Holt (1957) proposed a special case of this function as a

, model for longline/gillnet gear with no escapement or lass (Le. b=O). The rate at which theasymptotic catch is reached (r) has been proposed as' an index of abundance (Gulland 1955;Bevertonand Holt 1957). '

Trap Fisheries

We next consider a simple model incorporating a power'function for the capture processand a constant per capita escapement rate. The catch after an immersion time of t+.1t units canbe expressed:

~here Cl is the number of individuais caught after a so~ time of t units, a and mare parametersdescribing the rate of capture and b is the escapeinent or loss coefficient. Taking the limit as.1t - °and solving gives: ' .

Cl = {(alb)[l ... exp(-(l-m)bt)]}II(1-m~

. . ..• . I

for the initial conditions Co =0, 10 =0 (note that m~ 1 is undefined). This model again describesan asymptotic 'catch with increasing soak time; the limiting catch is given by the ratio of thecapture arid escapement rates (alb). Examples of the relationship between catch and soak timefor several levels of the shape parameter m,~e 'pr?vided in Figure 1.

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, . .For the Case where m=O (representing a constant capture rate and constant per capita

eScapement), we, have Munro's model for unbaited Antillean fish traps [where in Munro'soriginal notation Coo=alb arid R=b]. Inthe present notation, Mlmro's model is expressed: •

Ct = {(alb)[l - exp(-bt)]}

, . .which is indistinguishable in general form from the Gullaild-Beverton-Holt model but embodiescompletely different mechanisms. Here, , the entry rate (a) could be used as the abundance

'.' index: Note that this difft:rs considerably from the approach under' the saturating capture modeldescribect above. . ..

For the case, m > 0, the model is approprlate for ceItain gregarious species harvested in fish

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ärid crustacean tiäps where consPecifics in the trap attraet others (Murirci 1974; Miller 1990). 'This effect has been routinely observed and 'exploited to increase capture rates in spiny lohsterfisheiies (Miller 1990). Note that in this case, an' irifleciion in the catch' at 10w soale times ispredicted (Figure 1); For the case, In < 0, the capture rate declines with increasirig catch.This .effect has been. reported in niany crustacean trap fishenes (~Iiller 1990) and refleets-

,agonistic encounters and displays ~d/or chemical ~igrials which reduce entry rates.

FinaIly, for the case of apower function describing the capture process and noescapement or loss (b = 0), we can obtairi aversion of Austin's (1977) model.

_Tune.-Dependent Processes·-

Iri the developmerit ahove, we have impÜcitlY assumed that the capture and eSc3.Pementparameters do not varY with time. If these factors do change over time as a consequerice of lossof attractiveness of bait or other factors, an 3.ltemative approach is required. If entry to baitedtraps decreases. and/or escapement rates increase with time, it is necessary to replace the .'constint coefficients with tirrie-dependent parameters. For example, the representation of a thnedependent eritrY rate ais:

~'= f a(r) dr

An iiIustration of the implicatlons of a liriearly decreasing capttire rate with ,time (due~ forexample, to deereasing bait effectiveness oi consumption of ihe hciit over time) is provided inFigure 2 where it is shown that the catch reaches a maximum arid subsequently deciines withincreasing saale time. -This is in markerl coritciSt to the mOdels with time invariant cOefficients(which all predict a rion-deereasing catch level). In triip fisherit:~s where eseape is reliüiveiyeasy, it is not tinconimori to ohserve catches reaching a maximum and subSequently decliningwith increasing soak time (Bennett 1974; Sktid 1979). The mOdels 6f Murphy (1960),Somerton arid Merritt (1986) do allow for deereäsing catch with long immersion times but donot explicitlyconsider time-dependent processes~ Smith arid Iamieson (1989) do treat the caseof time dependent eapture. In the following;, we will consider the time inväriarit case. hut notethat _time deperident p3.icirneters ciin he readily accom~Odated in both the deteiministic andstochastic models described here. .

StOchastic l\fodelS

. . In thissection we describe the developmerit oi stochastic arialogues of the dete~inisticinOdels outlined aoove baSed on the thoory of biith-death stOchastic proeesses (See FeIler 1957; .Cox and Miller (1964), Bailey (1964) for overviews).

. The probabÜity of an individuaI being eaptured (e.g. entering airap Cl' heing hooked)during a short interval .of time t to t+h can be desigriated ach + oh,' ihe probability ofeScaPement during this Interval is ßch + o(h) [where o(h) is a qUantity such that lim~ o(h)/h

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= 0 (implying tl1at o(h) - 0 more rapidly than h - 0)]. The probability of two or more events(capture ana/or escapement) in the iriterval is taken to be o(h). The probability of obtairüngexactly C individuals in the catch at time t+h is:

Pc<t+h) .= Pc(t)(l - Cich - ßch + o(h)] + PC.1(t)(Cic.1h + o(h)]

+ PC+1(t)[ßc+/2 + o(h)] + o(h)

The first term on the right hand ~ide gives the probability of no change given that the catch attime t is exactly C individuals. The second term represents the probability of an increase fromC-l individuals and the third component is the probability of ci decrease from a catch of C + 1during the time intervaL The probabilities are taken to be independent and therefore additive.Rearranging and taking the limit as h-o of (Pc(t+h) -.Pdt)]/h gives:

Pc' =~(Cic + ßdPC<t) + Cic.IPC.1(t) + ßC+IPC+1(t)

where Pc' = dPddt and.' .

. .This is the basic system of equations for a linear birth-death stochastic process. Here, the birthcomponent is equated with capture processes and the death component relates to escapement orother sources of 1055. The terms Cic and ßc can, in general, be specified as functions of catch.Whether the above system of .eq~ations can be solved in closed form depends on the' exactftinctional forms arid the initial conditions specified. Below, we illustrate the derivation ofstochastic versions of severiiI of the deterministic models described earlier. .

, LongUne/GilIIiet Fisheries

For a saturating capture model with proportional eseapement, let Cic' = r(C~ - C) and ßc= bC; the catch model is then: .

PC'(t) = -[T(C·-C) - bC]Pc(t) + [r(C·-(C-l»]PC_1(t) + b(C+ 1)PC+1

and

. These equations can be solved in closed form by the method of probability generating functions.The initialconditions are taken to be Po(O) = 1 and PC<0) = 0 for al1 C ~ 0 (i.e the catch attime 0 is 0). The probability density function for this model is found to be binomiaI with mean:

. c; ={rC·/(r+b)}{l - exp[-(r+b)t]}

arid varianCe: .

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V(CJ={rC°'(r+b)}{l - exp[-(r+b)tJ}{exp[-(r+b)t]}

Notice that the stochastic mean is identical to the deterministic solution. For the case whereb=O, we recover the Gulland-Beverton-Holt model.

Trap Fisheries .

\ye first consider the simple case of. a constant capture rate and constant per capitaescapement or loss rate (Le. ac = a and ße = bC) corresponding to Munro's model (m=O).We then have:

Pe' = -ra -+- bC)Pc<t) + a(C-l)Pc-1(t) +' b(C+ l)Pe+\(t)

and

This system of equations is again amenable to solution using the method of probabilitygenerating functions. We again take the initial conditions to be Po = 1 and Pc<O) = 0 for allC ~ O. The probability generating funciion for· this model found to Poisson with mean:

Ct = (a/b){l - exp(-bt)}.

ReCall further that for the Poisson distribution, the mean and variance are identiCaI. - Note thatthe stochastic mean is again the same as the deterministic level,;

We next consider the full power function model for the capture process.. Letting ae =acm and ße ==: bC, we have for the basic system of equations:

Pe'(t) == -(acm + bC)PC<t) + a(C-l)mPCo1 (t) + b(C+ 1)Pc +1

andPo' ~ bP1(t).

• unfoitimately, application of the method of probability generating functiöns here leads to anonlinear partial differential equation that cannot be solved in closed form. Although alternativesolution methods are possible, it is likely that they will not easily yieId expressions for the meanand variance. Äccordingly, werecommend a numerica1 solution to the above equations.

ApplicatioD

An illustration of these issues is provided below for catch of American lobster (HomarusameriCanus) in baited traps (Auster .1985). Auster (1985) set strlngs of sixteen traps in LongIsland Sound (USA) and monitored ingress arid escapement from these traps over immersiontimes of up to. seven days. Observations by SCuBA were made daily and each individual in the

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trap was tagged for subsequent ideritificatiori. For the pUrp05eS of the present analysis, .,i,.eexamined data collected during two sets (10-17 September and 28 September-5 October 1982).Data from these two sets of observations were consisterit; accordirigly, we pooled data over boihsets. We determiried the daily ingress and egress from each tcip for these sets for. applicitionof the models. It should be noted that these data appear to be representative of relativeIy lowabundance coriditions where riegative interaciions affecting entry' rates were relativelyunimportai1t. Auster (1985) did provide information for other sets at higher density sites inwhich entry rates declined with iricreasing catch levels. Our objective here is simply to illustrate,the stoehastic mOdels using the simple case when:: m=0. '

The enii"y' rates for these strings were relaiively constant and the cumulative entry wasapproximately linear over time (Figure 3). Iri coritrast, the cumulative catch was clearlynonlinear with increasing soak time (Figure 3); This suggests that ihe Munro mOdel (constanteritry and constarlt per capiul escapement) is appropnate forthese observations. We estimatedthe parameters of this mOdel by non-linear least squares. The final model was:

Cl = [(0.138/0:215)][1 - exp(-0.215 t)]

The asymptoiic, staridard,errors for the entry arid escäPeinerit rates were 0.0218' and 0.0735resPectively. A comparisori of the observed andexpeeted catch with increasing soak time isprovided in Figure 4. The obsei-ved and predicted probability distribution of number ofiridividuals per tcip for this model is provided iri Figure 5. We tested the hypothesis that theobserved probability ,distributions wen~ drawn froma Poisson distribution with the predicted

'mean catch ,for. each soak time using 'Komogorov-Smimov testS. ,The null hYi'othesis of aPoisson distribution could not be rejected for any of the seven distributions.

Discussion

Stationarygear flsheries preserit unique challenges for the deveiopmeni of standardizedmeasures of aburidance baSed on catch- per-unit-of-effori. Catch rates in these fisheries aredetermined by the interplay of factors at"fectirig capture and eScapement or loss from the gear.The exact nature of the eapture and loss processes determiries the most appropriaie stiaiegy fofdevelopmerit of an abundance index. ,For scitionary geai fisheries;. the behavior of the targetspCcies plays a vital fole in the capture process.. Physiologic3l and erivironmentäI factors havea dominant erfect on the behavioi- of the species Sciught and therefore,iriust be eonsidered in thedevelopment of approaches to staßdardising catch data. This may require consideririg restiictedtempöräIlspatial units \Vhere erivirorimental conditions and physiologica1 state of the targetspecies can be considered relatively homogeneous.

. " There is considerable variation in the. interval betWeeri setllng arid hauÜng siaiioriary gear.. " in fisheiy operations. Fisherrnen may vary sOO.k times in anattempt io opumize C3.tch raies arid

aiidchänges in soak schedules may be cau5ed by adverse weather conditions. This variation insocik times creates difficulties the interpretation of catch;.effort data aS an index of abundance

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...."'.,•

~•

t··<-'·•

beeause of the characterlstic nariWiear relationship between catch arid so3.k time. The processis further complicated beeause entry rates may decline due reduction in the effectiveness of thebait and because of gear saturntion. For longlines; the numberof avaiiable h06kS declines asthe cateh incre3.Ses, arid for traps the presence ofanimaIs in the trap may inhibit the ftirthereritry of corispeeifics~ After lang soak periods, the escaperilte may be greater thein the eritryrate eind hence the total catch may decline with tinie... These factars must be considered in anyatiempt to standardize effort. Direct experimental observation of cäpture and lass processes willbe necessary to discern the relative iinpciitarice of these mechariisms. The· functional forms ofthese processes must be determined and the issue of time invariance of the proeesses must alsobe directly addressed to accOliilt for these diverse factofs.

There has been corifusion in the literature conceming the impIications of the nonIinearrelationship between catch arid soak time for siaiionary gea.r fisheries because mOdels based ondifferent undertying assumptions provide seemingly analogous solutions. For exarnple, Muiuo'smOdel is often equated with ihe GuiIarid-Bevertori';'Holt model; when in fact the uridertyingassumpiions and interPretationsof the parameters differ markedly. The Gullarid-Beverton-HoltmOdel (tioih with arid without 105s terms), Munro's constant eniry mOdel, and the power functianmodel described in this· paper all predict ein asymptoiic relationship of catch with time. Thisconfusion may lead to erröneotis interpretation of parameter va.1ues estimated from· fitting thesemodels to catch rlatil with imponani consequerices for abundance estiination.

. . Stocha.stic m(xif~ls are preferable to their determiriistic analogues hecause they provideadditional diagnostic clues to the applicability Of alternative mOdels. This approach providespiedictiC?ns of the mean, variance and probabilitY distribution ofthe riumber of iridividUa1scapturoo with time in coniparison. to their deterministic arialogues (which .provide only einestimate of ihe mean). The application corisidered here for an Amencan lobster Populationdemonstrated the utility of comparlng not only the observed and predicted mean catch but thedensiiy function of the predicted catch.

The mOdels preSented in this paper consider oniy asingle species in the capture prOcess.However in most stationary gear fisheries, sevefal speeies may be caught by tbe gear, arid there

may be competitive interactions thatinfluence the. capture process. In longline fisheries morethein one speeiesmaybe caught on the lines (e.g. Shomuraand Murphy, 1955) arid a high catchrate of ein undesirable species will reduce catChes of the .target species. .Trap fisheries are oftencirgeted at two or InOre sPecies ärid there is often a dominant (ar predatory) speeieswhosepresence,in and around the trap inhibits the entrY. to the trap of other sPecies (e.g. Richards etal;, 1983; AddiSon, in press). Our objective is to consider extensions of the approachesdeScobed aoove to the multispecies case in future research.

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References

Addison, J.T. Influence of behavioral interactions on lobster distribution and abundance asinferred from pot-caught sampIes. ICES Mar. Sci. Symp. in press.

Auster, P.J. 1985. Aspects of American lobster, Homarus americanus, catch in baitedtraps.M.S. Thesis, University of Connecticut.' 79 pp. .

Austin, C.B. 1977. Incorporating soak time into measurement of fishing effort in trap fisheries.Fish. BuB. U.S. 75:213-218.

Bailey, N.J. 1964. The elements ofstochastic processes with applications to the natural sciences.J. Wiley and Sons. New York. ' '

Bennett, D.B. 1974. The effect of pot immersion time on catches of crabs, Cancer pa~urus L.and lobsters, Homarus ~ammarus (L). J. Cons. int. Explor. Mer 35:332-336. •

Beverton, R.J.H. and S.J. Holt. 1957. On thedynamics cf exploited fish pOpulations. Fish.Invest. Min. Agric. Fish Food (Lond.) (Ser.2) 19: 533pp.

Deriso, R.B. and A.M. Parma. 1987. On the odds of catching fish with angling gear. Trans.Am. fish. Soc. 116:244-256. '

FeIler, W. 1957. An introduction to probability theory and its applications. Vol. 1: J. Wileyand Sons. New York. '

GuIland, J.A. 1955. Estimation of growth and mortality in commercially exploited fishpopulationS. Fish. Invest. Lond. Ser. 2, 18: 1-46.

Karlin, S. and H.M. Taylor. 1975. A first course in stochastic processes. 2nd Ed. AcademicPress. 557 pp.

Krouse, J.S. 1989. Performance and selectivity oftrap fisheries for crustaceans, In J.F. Caddy, •. (&L) Marine invertebrate fisheries: their assessment and management. J..WiIey and

Sons. New York. . ' . .

. .', .

Miller, R.I. 1990. Effectiveness ofcrab and lobster traps. Can.J. Fish. Aquat. Sei. 47:1228-1251.

Munro, J.L. 1974. The mode of operation of AntiIleari fish traps and the relationships betweeningress, escapement, catch, and soak. J. Cons. int. explor. Mer. 35:337-350.

Munro, J.L. (Ed.) 1983. Caribbean Cora! Reef Fishery Resources: Int. Center Living Aquat.

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Res. Manilla, Philippines.

Murphy, G.I. 1960. Estimating abundance from longline catches. I. Fish. Res. Board Can.17:33-40: .

Reed, W.I. 1986. Analyzing eatch and effort data allowing for randomness in the eatehingprocess. Can. J. Fish. Aquat. Sei. 43:174-186.

Richards, R.A., I.S. 'Cobb, and M.I. Fogarty. 1982. Effects ofbehavioral interactions on theeatehability of Ameriean lobster (Homarus amerieanus) and two species of Cancer erabs.Fish. Bull. U.S. 81:51-60.

Rothsehild, B.I. 1967. Competition for gear in a multiple species fishery. 1. Cons. int Explor. _'_Mer.31:102-110.

Sampson, D.B. 1988. Fish eapture as a stochastic process. 1. Cons. int. Explor. Mer. 45:39­60.

Shomura, R.S. and G.I. Murphy. 1955. Longline fishing for deep swimming tunas in theCentral Pacific, 1953. U.S. Fish and Wildlife Servo Spec. Sei. Rept. Fisheries No 157.

, pp. 1-70.

Sinoda, M. 1981. Competition for baited-hook in a multiple species fishery. 'BulI. lap. Soc.Sei. Fish. 47:843-848.

Skud, B.E. 1979. Soak time and the eateh per pot in an offshore fishery for lobsters (Homarusamerieanus) Rapp. P-v. Reun. Cons. int. Explor. Mer 175 180-189.

5rnith, B.D. and G.5. Jamieson. 1989. A model for standardizing dungeness erab, (Cancermagister) catch rates among traps which experienced different soale times. Can. J. Fish.Aquat. Sei. 46: 1600-1608.

Somerton, D.A. and M.F. Merritt. 1986. Method of adjusting crab catch per pot for differencesin soale time and its application to Alaskan Tanner crab (Chionoecetes Q.air® catches.North Am. J. Fish. Mgmt. 6:586-591

von Brandt, A. 1984. Fish catching methods of the world. Fishing News Books, 418 pp.

11

SOAK TIME

Figure I. Relationship between catch and soak (immersion) time for the power function capture •model for three values of the shape parameter (m).

--....._- TIME-DEPENOENT

~-- TIME INVARIANT

SOAl( TIME

Figure 2. Illustration of the effects of time-invariant and time-dependent capture processes forthe power function model. The time dependent model is based on a linearly decreasing capturerate with time.

----- -----1

w·

I

CL.<a::I-a::~a::wCO~

~ 05

Z<UJ~ CUMULATlVE

CATCH

o 2 3 • 5

SOAI( TIME (DAYS)e 7 8

Figure 3. Cumulative entry and catch (mean number per trap) for American lobster in LangIsland Sound (USA) in experimental fishing operations during September-October 1982 (Auster1985).

0.6,...--------- ---,

-a~

t=~

~0.4

•i~5Z 02 •~

870.0 .....-~--_.,...--..,......-......,..--..,...--!"'",__~-~

o 234 5 eSOAK TIME (DAYS)

Figure 4. Observed (closed circles) and predicted mean cateh per trap as a function of soak timebased on a constant entry and constant per capita escapement model for Long Island Soundlobster data.

:

ffiOBABIUTYo ,0 0

k1 .' m

(J

~N

Figure 5. Observed (shaded bars) and predicted (solid bars); probability distribution or catch.levels for· immersion times of up to seven days for Long < Island Sound lobster. data..'.,f

'.