International Couneil for the Exploration of the SeaStatisties Committee. ICES CM 1992/D:28ref. K.

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Phase-space diagrams, .a first apprmich to chaos modellingof North American landings of Homaros americanus.

by

Gerard Y. Conan, Franc;ois Söler, et Anne de KerrosDepartement de Mathematique,

Universite de Moncton, ~oncton N.B. EIC 9B6, Canada.

Summary

Over the past years the landings of North American lobster (Homarus americanus) havereached arecord high for the past century. Although strict enforcement of fishing regulationsmay be involved, the reasons for these high landings are not fully understood.

Positive eorrelations of lobster landings with environmerital parameters such as annualaverage water temperatures, or fresh water outflow from the St Lawrence have been invoked,or negative correlations with potential predator stocks such as eod. Such effeets tend to belocal and do not last for the whoie period of observätion. ,".

North American landings seem to follow global patterns, irrespeetive of loeal differences infisheries regulations and of local environmental disturbanees. A global system reacting nonlinearly to minute disturbances by positive and negative feedback ful(ils the prerequisites forchaos dynamics.

We have conducted a preliminary ehaos analysis of 100 years of lobster landings data usingtools borrowed from physics. We conclude that lobster landings may behave as a chaoticsystem gravitating around a iriple loop attractor in phase-space . The landings have presentlyreaehed a loop which is rarely visited and appears to be less stable 'than the other loops. .Slight disturbanees either positive or negative could have a major effect on the evolution ofthe landings.

A ehaotic behaviour does not imply that the fishery cannot be managed. However/management'approaehes may have to be reeonsigered if stabilizing beeomes"a more importantgoal than maximizing the landings. .

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The landirigs of North Americari lobster (Homarus ame1ricanus) have reached their highestlevel in a cen1ury over the past years, both iri Camida and the United Stites. Although strietenforcement of fishing regulations is prcbably involved; th6 reasons for these high landingsare not fully understood. ' i

The trends appear to be simiia~ in'both countries and nb partlcular stock management methodcan be ideniified 'as having determined these increases in landings. Stock management ,methods vary considerably between regulation areas or:States, and between countries. InCanada a' great einphasis is set on regulating effort (limitätion of number of fishing licences,of traps per licence, of fishing seasons) as weH as onminim3.l legal fishable size; seh~ctivityof the traps (escape gaps) arid prohibition to länd berried females. In United statt~s, emphasisis set cin regulation of legal fishable sizes (both minimal and maximal in certain locations)and on protection of the females iri reproduction (prohibition 10 land berried females and "Vnotched" fem~l1es which have carfied eggs). The enforced legal minimal fishable sizes .vati' 'greatly between regulation areas. I

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Authors have atiempted 10 identify environmental facto'rs which might have affeeted hirvalrecruitmerit. such as average water temperntures occurrence of storms beaching early lobsterstages, orfiver outflow (Drinkwateret al. 1991). Sometimes conviricing correlations, arideven predictioris have been identified between these factors and the laridings after a time lagof 6,to 10 years (Flowers and Saila. 1972; Sutcliffe, 1972; 1973; Sheldon er al., 1982;Driflkwater, 1987). However the.effects identified have always been limited to specific

. loeation, arid efficient oiter alimited perioo of years.I. •Since North American lobster landings seem to foHow global patterns, it is terripting 10 trea1the population(s) as a complex,. global system simihir to a meteorologicaI system.. Localeffects cari be propagated on the long terrri between fishirig locaticins. Geographie gerietiedifferences between lobster subgroups have so far beeri fotind to be limited. Large scalediffusion of benthie adults, as weIl as drift of larvae which may remain pelagie for severalweeks to months probably explain the lorig term uniformity in group dyriamics.

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The global system is evolving as a whole as a function of multiple minute lcical disturbanees.The. relati~nshi(b~tween the nu,m,ber ofIfldividuals or;t~e. biomass andpopulation erenvlronmental parameters is generally thought to be non hnear. The system reacts to ,disturbances hy positive and negative feed back rriechanisms such as density dependence ofgrowth recruitrrient, natural mortality, or fishing mortality through mariagement regulations.The system may' everi have a (genetic) memory. Such asystem seems to fulfil a11prerequisites for chaos dynamics (~fay er al. 1978; Wilson er al.; 1991).. , ,

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w~ have conducted a pretiminary chaos arialysis"of th~ avaiiable lobster landing data fromCanada and the United states. We used 'statistical iöols developed for the analysis of' physic3.lphenorriena, but which can also be apptfed to biological daia. We attempted to introduce

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supplementary variables such as phYsical environmental factars (average yearly temperatures;outflow of the 5t Lawrence) and an index of abundance of a reputed predator : records ofcod fish landings. 0

Generalizing from the chaos 0 analysis of lobster landings, we provide general thought on.fisheries management. We shall comment on the efficiency of long term predictions andmanagement regulation, as weIl as on the applicability of current fisheries modelling methodsto a chaotic situation.

l\laterial 3nd Methods

Biological dara

Cod and lobster have been actively fished for more than two centuries in Eastem Canada,and reliable ancient records of landings are available. A~cient effort information is notavailable.

Lobster landings per se may be a good measure of stock abundance, since it is generallybelieved that male lobsters are exhaustively caught soon after they reach legal minimal size..The 50uthem Gulf of 5t Lawrence the catch is believed to contain 'no more than two agegroups.

Many fishermen believe that cod is a major predator of lobster and. that the depletion of codstocks by fishing has boosted lobster stocks in recent years. We used cod catch as anapproximate index ofcod abundance in order to check for effects on lobster catch.'

Lobster laridings ,Canadian lobster landings were taken from Williamson (1989), laridings for the years 1990 to1992 were provided by the Industry Development and Program Directorate, Department ofFisheries and Oceans, Canäda. Additional information on lobster landings were provided bylohn Temblay and Mike Eagles, DFO, 5cotia Fundy Region. Precise information is availableby statistical areas or by county in ancient records.

Cod landings

Cod landings for Atlantic Canada were obtained from the 5econd Annual Report of theInternational Commission for the Northwest Atlantic Fisheries (ICNAF) for the years 1951­52, and from the 5tatistical Bulletins of ICNAF for the years 1953 to 1989.. .

Physical environment

The Gulf of 5t Lawrence to the Gulf of Maine inclusive may be considered as anoceariographic system (Sutcliffe er al. , 1976). The influence of the outflow of the StLawrence affects the 5cotian 5helf and the Gulf of Maine. Global effects of averagetemperatures and 5t Lawrence outflow on the catch of lobsters were investigäted.

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Linear correlatioris

Statistica/ analysis

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I 4ISurface water terriperatures I,I .

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Surface water temperature should weIl characterize the habitat of lobster in the Gulf of StLawrence. In this region,. lobsters are fourid only betw~n 0 and 80 m, the maximal depth ofthe mixed layer in summer. At ihe surface the salinity 'may be as low as 20%0 and thetemperature as high as 20 oe in summer. Lobsters apparently never penetrate the "intermediäte water layer tlowing from Labrador (stable' -1 to + l°e and circa 33 %° salinityall year round). Lobster larvae are found mostly withiri the most superficial centimetres cfwater. !

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Similar surface water temperature's should intluence all North Ainerican Lobster habitat. Theintluence of the outtlow of the" St Lawrence' affects the' Scotian Shelf and the Gulf of Maine

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. (Sutc1iffe er a!., 1976). Deep concentrations of lobsters are found off the Scotian Shelf andthe Gulf of Maine. in regions not affected by the Labrador intermediate water. However,Exchanges occur between shallow and deep concentrations at the larval and adult stages.

. IWeused iemperature records from Booth Bay Harbou~ (Maine), Halifax(Nova Scotia), andSt Andrews (New Bninswick) provided by lohn Tremblay and'Mike Eagles. In order tocompensate for gaps in data the temperatures from the' three locations were averagect..

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St Lawrence water run off. ;\

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The run offs were otitained by summirig detailed information available for the St Lawrenceriver. aftluents and the Saguenay river from 1923 to 1963. Anon. 1926 to 1966. WaterResource Papers 48 to 107. . ! . ..

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. iA common way of searching for causalityeffects from tiiological or envirorimerital factors onstock abundance consists in using simple scatter plots and calculate linear correlation . .coefficients. Setting a time lag between the measure on the factcir and the census on nie

. fishable stock allows to.take into account effects on early life history. We used higs of 0 Lo10 years for lobster landings vs average temperatures;1 St Lawrence run off and cod landings.

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!Chaos analysis j.

.' . I.May (1974) introduced the idea of chaos in population biology, and later in the theory ofharvesting natural populat~ons (May er a/. 1978). Intuitively, iri purely deterministic world, apopulation being displaced either negatively or posiiivdy by an external factor from itssimple equilibrium logistic growth curve should tend to return tö equilibrium conditi?ris bythe effect of negative density dependent feed back. However, May shows that the "return"to equilibrium may lead to dumped periodic oscillations, stable periodic oscillations; chaotic

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oscÜiations (pseudo rand~m), and eventually to extinetion. Iri a chaotic situation the e~olutionof a population becomes unpredictable, even in the simple 10giStic growth condiiions, becausethe slightest iriitial differences in. population aburidance or population parameters willgenerate divergirig behaviours. Global patterns or "attractois" can rievertheless be identified.The attractors can lead to adynamic stability within apparent chaos.

Natural populations are ccintinuously submitted to external <Üsturbances, ami their survivalmust have lead through evolutionary processes to the genesis of complex homeostaticsystems. These hoineostatic systems may be conceived as attractors. The attractors may besimple in the case of dampened periodic oscillations, or of stable oscillations. The attractors .rriay be of a very complex nature in the case of chaotic oscillations: when oscillations arerepeated a great number of limes, from apparent randomness organized patterns emerge.Although it is impossible to forecast where the population will be located from one step tothe next, it is possible to outline patterns for its trajectory.

Under such conditions ideritifying the effects of factofs on the population iJsing iinear .correlatioris becomes an fictitious task. The population may react lirieilrty to the effects of afactor over a number of generations; but its owns homeostatic feed back mechanisms 'will ..soon create oscillations which will eventually free the population from the em~ets of thefactor. Liriear correlations cannot be used for assessing the em~ets of a factor over longperiods of time in the presence of chaos. Specialized toolSare r~uired.

Chaos theory has been documented (Gleick, .1987; Briggs & David Peilt; 1990) and·successfully used for. some time iri many sciences such as physics, ariä meteorology, but hasbeen so far of limited application in fisheries scierice (Wilson et al. 1991). We haveborrowed tools from physical sciences (Sprott and Rowlands, 1992) to analyze the lobster

. landing data for chaotic behaviour.

Sprott and Rowland warn,that, in practice, with a limited amourit of data that may becontaffiinated by noise, success caririot be gua~nteed.Time series in fisheries biology ar~

available over much shorter sequerices tha~ iri physiCs. The smallest time unit for a lobsterpopulatiori is a year since reproduction is seasonal, but a more logical u~it inay be ageneration time (6 to 10 years). A hundred years of data on lobster landirigs (Williamson,1989) is an exceptionally good set of observations. In fisheries, records of landings can bequite precise; but population aburidance estimates are provided with no better than 30-50%confidence intervals. Three point running average smoothing was used at times to reducenoise iri the time sequences of lobster landings, no attempts were made to use populatioriabundance estimates.

As a guidance for the interpretation of the results of the·chaos data analyzer tools, we .processed reference data sets containing 100 data points. TypicaUy random, periödic andchaotic. data sets were assayed together with the' records on lobster landings: References for.the tools arid standard data sets are pro.vided in Sprott and Rowlarid. We avoided using trialSstich as return maps which showed to be urisuitable for small data se~s. .

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Graph of data: . . , '. . I ,The simplest graphie representatiori of data is landings ~s time: Simple visual inspectio~ mayreveal possible eycles or 'long term trends: Sprott and Rowland recommend to remove long

, term trends prior to ctiaos analysis. ,!.. i

Frequency distribution: . !.' .Purely random data will. give rise to a simple curVe. Pe'riodic data gives a simple histogramwith sharp edges. The distribution is likely to be fractal for chaotic data, but can also be asimple curve. ' , ' , . j , .

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Power spectrum: . .... .... 1.' ,'. .'This is a Fast Fourier transform of the data~ a usuaI procedure for periodie time seriesanalysis. The power (mean square amplitude) is displayed as a function of frequency; A •power spectrum with a few dominant freqlü~ncies shows that the data could oe weIl . .approximated oy a Fourier series with afew terms. Penodie and quasi periodic data. willproduce few dominant peaks in the spectrum. Random land chaotic give rise to oroad spectra~Power spectra which are straight lines on a log linear scale are thought to be good candidatesfor chaos~ I .Dominant frequencies: !The approach is similar to the Power Specirum; Instead of a fast Fourier transform, theMaximum Entropy method is used. The method can be used forliriear predictions of the riextvalues in the time series. iCorrelation dimension: ! ".Quantifies the underlying complexity of a chaotie time~series a dimension greater thari 5implies random data. As the embedding dimension inc~eases, the correlations 'diriH::nsionshould increase, but eventually saturate at the correct value (asymptote).. ,.Phase-space plots: .; •In a two dimensional plot the derivative is plotted vs the abundance of the population at eachtime of observation. the derivative is approximated byjhalf the difference of the two data .points adjacent to each point. periodic data should appear as a closed eurve. . .

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On a three dimensional plot the secorid derivative is plotted along with the first derivativeand the population aburidance on' three orthogonal axes. The second derivative is thedifference between the slopes of the lines conneciing each data point with iis two nearestneighbours. Same cases' which are not obviously periodie in 2 dimensions reveal theirpedadicity in three Idimensions. '••

. . .' . ILyapunovexponent: . . . " I' .Measure of the rate at which nearby trajectories in phase space diverge. Chaotic orbits haveat least one positive Lyapunov exponent. For periodie' orbits, all LyapuIlOv exponents arenegative. Only the most positive exponent is calculated here. The existence of a positiveLyapunov exponent implies a chaotic time series, although nöise will also produce a positiveexponent.· I .

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Correlation matrix: . .The approach is similar to a Pnncipal Com'porients Analysis of muItivariate data. The matrixanalyied, instead of containing correlations between multivariate observations, coritainsautocorrelations between observations for variable lags. Attempts are made to represent thedata as a finite sei of orthogonal functions, the number of significant eigen values of thecorrelation matrix., At least 3 eigen values are required to create chaos. The number ofsigriificarit eigenvalties is a measure of the complexity of the system.

One might try correlation matrrces with the goal of finding a set of model eqmitions whose 'solution as evidenced by the phase-space plot is astrange attractor that bears some .resemblance to the phase-space plot of the original data. Because of a change of coordimites;these'phase-space plots will be distorted images of one another, but because the chariges are .liriear the topological factors 'such as hole~ in phase space must be coriimo~ to both.

Prediciioris ,of next values can be made and tested using the principal componenis as modelvariables. With chaotic data predictioris will be realistic only on very short interVals.

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Results

En\.'ironmenuil parameters

Our data sets are still iricomplete, we did not succeed yet to assemhle informatioii over thefuH time iriterval over which:lobster landirig data is available, but we do have lang enoughtime series to visualize long term effects. An index of outflow froin the St Lawrence wascalculated from information in Anon. Surface Water Supply of Canada St Lawrence arid '.,Southerri Hudson Bay Drainage, Onta.rio and Quebec (1934 to 1966). The outflows from 32affluents of the St Lawrence were summed (fig. 1). Surface temperature records from BoothBay Harb0i..Ir (Maine), St A,ridrews (New Brunswick), arid Halifax (Nova scotia) ,were inturn processed separatelY, then averaged to obtain a general index from 1926 to 1989 (fig.2)..-' . .

Extensive assays .were in~de to attempt to identify trends in lobster landirigs within wide ..subareas of the Gulf of St Lawrence and the outflow index from the St Lawrence. No bettercorrelations co~ld be identified than for the l::indings over the whole AtIaritic Camida coast(figs. 3, 4). The correlatioris are weak and negative. Six years is usuaHy theaveiäge timeascribed to a lobster for growing from larVal to commercial stages, setting a lag of' 6 yws

. between catch and outflow index reduces ihe correIation in stead of improving it.

There is a cörrelatian between lobster landings in Canada arid' average temperatures thesame year as ,weH as 6 years earlier (figs. 5, 6). 'Ne did not ~etect ariy peculianiies in localeffects. The drasiic increas~s' in catch' ,for the last years cannot be ,explained by a temperatureeffect o'rily. '.'

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Predarion

Chaos dara analysis

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Cod Landings in Easter Canada were obtained from Arion. 1952 and Anon. 1953 to 1990ICNAF Stat. BuH.. Detailed information is available for geographie subregions. We tned torelate lobster landings to cod landings records' available from 1893 to 1989, we iricorPürated .lags of 0, 2 and 5 year lags to account for possible eod predation on early life stages..Thecorrelations are weak but positive instead of negative, arid best for a 2 year lag (fig. 7).

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Attempis were first made to segregate data into subregions into which sub populations oflobsters may be logically expected to exist. The local änalyses did not provide a hetterdefinition than the global analysis of the data, and are :not reported here. A. smooti1ing of thedata using repeatedly a three point runriing average did appear to erase random variabilitysuperimposed on the chaos process (fig. 8). This variability may. be simple white noise - ,generated by the poling of part of the data where and!when precise information is difficult to

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One hundred data points from typical periodic, white noise, and chaos data are provided for. comparison (figs 9, 10, 11). The same data is presented in frequeney distribution forITl in

figs 12 io 15, and as phase space diagrams in figures 16 to 19.. l

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,Using the criteria defined by Sprott and Rowlarid ; lobster time sequencesare best describedas chaos data. Particularly; the phase space plot (fig. 16), without having the regularity of'periodic data plots (fig. 17) c1early differs from random data plots (fig 18), and displays .irregular tracts as in the chaos data (fig. 19). ! .

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The power spectrum after fast fourier transforms provided a somehow log linear relationsliip. . •No 'clearly dominant frequencies were present. The predictions provided by the dominantfrequency techniques were poor as opposed to what w'ould be 'generated by periodic data;.' I ,"

The correlation dimension stabilized well and was of the order of 2, well below the value offive expected for white noise data. The Lyapunov exporients are all positive (max. 0.26+1-

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I .The correlation matrix analysis produced principle coinponents which allowed to generate anattractor in phase space topologically similar to the phase space diagram of the observed data(fig 20). The predictions [rom the pre~ent point were 'unstable beyond one single year. This

, . ., .is quite consistent with the two year recruitment catch compositi6n of the present landings;, . '-

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Discussion 1I .

The North American lobster stocks must react to highly v~riable environmental conditions,.and show a wide geographie variability of biologieal' characteristics. The environment in the

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lobster habitat widely varies from the eold oeeanie waters of the northem extent ofNewfoundland, to the highly seasonal Gulf of St Lawrence waters (-1 to 20°C), tlie deepeanyons off ~faine ami Nova Seotia (more seasonaIly stable iemperiliures around 6°(:), thewarmer seasonal waters of North Carolina. A strong geographie vanabiliiy exists in growth,age at first maturity, li fe eyc1es in alternating periodicities for moulting, mating andspawning; The fishing regulations also widely vary over the range of distribution' of lobster.

, Despite this high variability, the genetie composition of the stocks appear to be remarkablyhomogeneous. Published and unpublished 'results from modern techniques such asisoelectrophocusing of a wide variety of enzymes, on initoch'ondrial DNA analysis havefailed to identify disiinct genetie populations. The exchanges between geographie siies.canoecur through extensive movements at advaneed benthie stages as weIl as larv3J. drift duringseveral weeks to months at early stages. The distinet geographie life histoiy patterns revealedunder different life history conditions appear to be flexible adaptations rather than fixedgerietie differenees between individuals.

Under such eonditions; it can be logically expected that North Ainerican lobster stocks as ciwhole react cis a homeostatie system to minute loeal disturbances. Loc3J. disturbances mayhave locally drastie effects over a limited number of years, but these effects will dampenover a longer time. Through recruitment feed back the system has its own periodicity. Thelag time between oscillations of the effect and population oscillations may progressivelyextend until the effeets are dampened out. This is possibly what Drinkwater, and Sutcliffe .observed about loeal responses of lobster stoeks to fresh water outflow. The global effect offreshwater is not notieeable or may even be reversed.

Temperature may have global long term effects on the landings. Temperature changes areless locally variable than river outflow. The 0 lag effect may be a eatchability effeet. It hasbeen frequently reported that lobster are more active and enter more readily traps astemperatures seasonaIly raise in the Gulf of St Lawrence. Most of a recruitment group iscaught over two years; but it may be more advantageous in terms of yield per recruit toeatch most of the recruitment group over a single year. The 6 year lag can be explained byan enhancement of growth iri early stages, and subsequently a redtictiori of mortalitY.

The null to positive. relaiionship between cod and lobster landings was unexpected, and doesnot verify the intuitive perception that a depletion' of cod stocks may h~lVe a positive effeet onlobster landings. Predation has more subtle effects than what earl be detected I:>y ci linearcorrelation. The most effective lobster p'redators are not necessarily eommercial species.Conan (personal observations) has reported while divirig aetive predaiion of a small but veryeommon unpalatable fish, the blue perch (Tautogolabrus adspersus) on moulting lobsters.

Rather than tryirig to detect and predict the effects of loeal pheriomena on the iobster stocks,it may be more promising to understand their behaviour over time as the behaviour of aglobal system in w.hich order is the result of chabtic processes. The system reacts to .envirorimerital and human effeets aeeorqing to orbits that can be better vis,ualized in phase­space thari iri a simple plot of landings vs time. Although it may be urirealistie to exaetlyforeeast the next loeation in phase space from a given Ioeatiori; historical records allow tooutline the most usual possible trajeciories; their divergeriee and unstable vs stable loeatioris.

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. , i .In unstable loeätions, any everit as minute as it may be can have drastic effects, simplemanagement goals sueh as maXimum yield per recruit ~ilI not be re3.listic, while they mightbe realistie in stable loeatioris of the phase spaee orbits.~

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, Identifying a present fishery IoeatlOn 10 phase spaee may provlde IOformatIon on thC? Stablhtyof ci present stäte and ean be useful for risk analysis fOI- direeting investment on fishirig fleetdevelopment for instanee. Chaos analysis does not leadlto eontemplation of unmanageable. .' , '. ". ' . (. . . .

fishenes, but may provlde gUldanee for long term management approaehes more ,sophisticah~d and realistic than simple deierininistic intuitively attractive tools.

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Chaos analysis does provide serious eIues that the North American stoeks behave globally asa ehaotic system. Three interrelated orbits with different 'degrees of stability can be suspectedto exist in phase späee for Camidian data. The results are similar although less weil markedfor regiorial data. The set of data from USA that we proeessed' is shorter, 2 orbits 'eari berecognized and a third one woulrl probably be eompleted should we ineorporate start of theeentury reeords. Mixed CanadalUSA reeords behave similady to the Canadian reeords perse.' ., '. .. . '.,,'. I " , ,o, ' ,The present lOerease in landings over the past ten years appears to be areturn to an ancientinitial situation, whieh vanished at the onset of the disturbances eaused by fishirig. Based onhistorie reeords, it does not appear to be gravitating with stability round a high (laridings)point of attraeiion. It may be expeeted that this loeation be visited only very oecasiona.1ly... I

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This document was ,written in order to stimulate exchahge of ideas between scientists: Theopinions expressed in this doeumeni are personal arid do not imply any endorsement by theDepartment of Fisheries arid Oeeans Canada. The data1used is eIther, in 'the publie domain inthe form of published or about io be published information. No specifie managementreeommendations are implied. IAckriöwledgments 1

I\Ve wish to ihank iohn Tremblay and Mike Eägies frolm Scienee Br3.neh; Department ofFisheries and Oeeans, Seotia: Fundy Region, who pro~ided extensive data on diskettes forlandings and temperature reeords. This saved us eonsiderable time for enteririg data andbibliographical searehes. ' I·. '

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\Ve wish' to thank Paulette levesque and GiseIe Richa~d, Librarians at Gulf Fishei"ies 'Ceriter,Moneton, and Marilyn Ruddi, Librariari at the Biological Station St Aridrews fOI- having veryefficieriily helped us to reirieve putiliSherl historie ~am: on river discharge, arid eod landirigs.

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We ~lsh to thank laek Duong (Department of Frshen7s and ,Oeeans, Industry j)evelopmentand Programs Direetorate, Ottawa.) who provided detailed statisties on the 1990-1992lobsters landings ~or Eastern Canada. . ,i ' " .

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Bibliography

Arion. Surface \Vater Supply of Canada. St Lawn~rice and Souttlerri Hudson Bay Drairiage, and Quebec. Dep. of the Interior, Canada. Domiriion Water Power and Reclamatiori Service.

-1926. Water Resources Paper No.48. Climatie Years 1923-24 and 1924-25. '168p.-1929. \Vater Resources Paper N9.58. Climatie Years 1925-26 and 1926-27. 390p.

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-1950. Water Resources Paper No.99. CIimatic Years 1945-46 arid 1946:47. 409p.-1952. Water Resources Paper No. 103. Climatic Yeärs 1947-48 and 1948-49. 430p.-1954. Water Resources Paper No. 107. Climaiic Yws 1949-50 and i950-~1. 456p.

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-1957. Water Resources Paper No.ll1. Clirriatic Years 19~1-52 lind 1952-53. 459p.-1959. Water Resources Paper NO.115. Climatic Years 1953-54 arid 1954-55. 51Op.-1960. Water Resources Paper.No.119: Climatic Years 1955-56 and 1956-57. 544p.-1962. Water Resources Paper No. 126. CIimatic Years 1957-58. 326p.-1963. Water Resources Paper No. 129. Climatic Yeafs 1958-59. 328p.-1964. Water Resources Paper,No.133. Clirriatic Years 1959-60. 352p.-1964. Water Resources Paper 'No. 137. Climatic Years 1960-61. 376p.'-1964. Water Resources Paper No. 140. Climatic Years 1961-62. 362p.-1965. \Vater Resources Paper No.i43. Climatie Years 1962-63. 352p.-1966. Water Resources Paper ~0.147. Climatic Years 1963-64. 368p.

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i 'Anon. 1952. International Commission fcr the Northwest Atlantic Fisheries. Second annualreport for the year 1951-52. Dartmouth. N.S. 68 p. I '

, IIf ' .• I

Anon. 1953 to 1990. International Commission for the~Norihwest Atlantic Fisheries. .'Dartmouth, N.S. Stat. Bull. 3 to 40. ,I.

" '

Briggs 1. & F. David Peat. 1990. Turbuient Mirror. Aln illustrated guide to chaos th~ory andthe science of wholeness.Harper and Row. New York.~ 222 p.

i•, ,Drinkwater. K. F. 1987. "Sutcliffe revisited It: previously published correlations between fishstocks and environmental indices and their recent performance, p. 41-61. in,R.t Perry andK.T.Frank ed. Environmental effects on recruitment to Canadian AtlantiC fish stocks. Can.Teen. Rep. Fish. Aquat. Sei. 1556. i

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Drinkwater. K:F., G. C. Harrling, W.P. Vass, and DJ Gauthier. 1991. The relatirinship~fQuebee lobster landings to freshwater runoff and wind storms, p.179-187. In 1.~C. Theriault(ed.) The Gulf of St. Ulwrence: Small ocean or big estuary? Can. Publ. Fish.

'. IAquat.Sci.113. I

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Flowers, 1.M. and S.B. Saila, 1972. An analysis of temperature effeets on the inshorelobster fishery. 1. Fish.Res. Board Can. 29: 1221-1225.

1 ., .

Gleick J. 1987. Chaos. Sphere Books, Macdonald & Co. London. 352 p.,. , , I'

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Hall, C. A. S. 1988; An assessment of several of the historically most influencial theoreticalmodels used in ecology and the daci provided in their 'support. Ecological modelling. 43: 5-31 j'

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197019601950194019301920

400000 4- -....----r---.--........-~---~-----..,,..----r---.

1910

500000

600000

year

Fig. 1

2000198019601940

6-+-----------.;.......------or---.,..---.,..---.,....----,

1920

~

t 9 •=-~...~Q"

e~

~8~...=~

=~~:;

7

10

Years

Fig. 2

24000 y = 2 9760 - 0.0235x R=0.27 EI

0

"'" EIEI

EI~

EI~>. 22000

EI

~ EIEI

"0 EI EI~ 91Jc m~

U 20000EI

C.- m ElIIlI

.c(J-~(J

18000m

"'"m EI

~-r'l m,.Q0 m~

16000 EI EI mEI fJ

EI

EI EI

14000 m mEI

m

12000400000 500000 600000

RIVSUM, year 0

Fig. 3

24000y = 2 1190 - 0.OO53x R=0.06 EI

="'"

EI EI EI~~ 22000 EI>.• ~

m EI

"0 EI m.~

EI

CEI

~ EI EIU 20000c EI EI.-.c(J-~ EI(J 18000 EI

"'"m

~ mr'l

,.Q0~ 16000

EI EIm EIEI EI

EIEI

14000 EI EIliI

12000400000 500000 600000

RIVSUM, year -6Fig. 4

50000

Y= - 1366.0296 + 2762.1426x R=0.28

= EI

'"~~ 40000

EI>. EI

cf"0~ EI=~U EI

=....c 30000~

EI- EI.~~

'"~tIl~Q

....::l 20000

500000

III !!I!!I

!!Im m

400000

IIIm % III

III JPm m .

mm

200000 300000cod catch, year ·2

Fig. 7

100004----.-----T--....--.....,----r--r-----r-----.

100000

50000 y =11990 + 0.0312x R =0.24

20000

=a..=~~

:§~OOOO=tJa..~

ß.3 30000

1 time

Fig. 8Total Canadian lanqings of H.A.

188

2

x

1 t.ime

Fig. 9t 15-1 t

Data from sin (2) + cos <--z- . 2) Twosinus)

188

5

x

1 -time

Fig. 10Random data. White noise

188

--~------------------------

x

I time

Fig. 11Data from the Lorenz attractor

IBB

58888x

., ' .

................... , .. . . . . . . . . .. . . . . . . . . . .. ... .. . . . . . . . . . . . ..... . . .. . .. . .. ~ .

.................................................

................................................

III

..•.•........ _.. 0.... . ...........•. ,_ l

I\I

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

: I: I: I: I: I: I

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

: i, I

: I. I

\ I__ ~~~I·~·· ···················18 L- ��_.iiiiiii.üiii.iiü.iii.iiiiiii.iüii~iiülliii·iiiiiilliiülliii-iiiiiiilli-.-iiiliii-iiiiiilliiii-iiii...-iiii-i.iii-----.J1

8

1

ar

Fig. 12Probability distribution. Canadian data of H. A. landings

··.

. .· .................................................................................................................................................................· . .

· .

· ...... "" -. . .. . .. .ar

. 2

Fig. 13Probability distribution. Twosine.

. ..

1

........................................................................................................................................................

.... ....................................................................

...................................................................

••••••••••••••••••••••••••••••••••••••••••••••••••••••• 0'

81-------­-s x

. .

. .

5

Fig. 14Probability distribution. White noise.

e e

I,

• I

1 .

.............- .. ..

ar

~. . . I'............................................................................... . .

28x.

·····:·········~·:···~I~~~I.III II~'···I·;··················~I:·················8 L-_-iiiiiii-iiii-iiij-iiiiiiiUii-iji-Ü-iiij-iiiiiii-iii-..-iiiiiJ-iiiiiUiii-iiiiiiiü-iiiiüiiiiii-..-Ü-iiii.i·-iiiii.--iiiiüi__...iiiiiiüiiiiiiiiiiiiiiiii.:.-----l-28

, .

Fig. 15 .Probability distribution•. Lorenz;attractor•.

X'(t)

X" (t) X(t)

Fig. 16Phase-space plot. Canadian landing of H.A.

Fig. 17Phase-space plot: Twosine.

xet>

•

Fig. 18Phase-space plot. White noise.

X(t)

X'(t>

X" (t.)Fig. 19

Phase-space. Lorenz attractor

X(t.>

'."

"

SI

SI3Fig. 20

Phase-space. Canadian landing of H.A.Attractor generate from the three principal eigenfunctions.

SI1

.... I