Estimating natural interstage egg mortality ofAtlantic mackerel (Scomber scombrus) and horsemackerel (Trachurus trachurus) in the NortheastAtlantic using a stochastic model

Enrique Portilla, Eddie McKenzie, Doug Beare, and Dave Reid

Abstract: Egg mortality is a key parameter for understanding early life histories of fish. Small variations in estimatedmortality cause large differences on adult fish biomass estimates. Therefore, the assumption of a constant egg mortalityrate may be misleading. Here, we show how to estimate mortality rates for the individual egg stages of Atlantic mack-erel (Scomber scombrus) and horse mackerel (Trachurus trachurus) from triennial surveys conducted since 1977. Weuse a standard, continuous-time Markov process model that combines the numbers of eggs sampled in each stage withexperimental data on egg stage duration (dependent on water temperature). This is the first attempt to study mortalityamong egg stages in such detail and the first comprehensive effort to estimate horse mackerel egg mortality in theNortheast Atlantic. The results include detailed descriptions of spatial–temporal dependencies in mortality. The dailyegg mortality rates estimated are ~0.56·day–1 for Atlantic mackerel (far higher than suggested in the literature) and0.54·day–1 for horse mackerel. Although it was not possible to estimate stage 1 egg mortality directly, the results sug-gest high mortality in the first stage. This might lead to underestimation of fish biomass when assessed traditionally byegg survey data alone.

Résumé : La mortalité des oeufs est une variable essentielle pour comprendre les premières étapes du cycle biologiquedes poissons. De petites variations dans l’estimation de la mortalité produisent des différences considérables dans lesestimations de la biomasse des poissons adultes. Présupposer que le taux de mortalité des oeufs est constant peut doncinduire en erreur. Nous démontrons ici comment estimer les taux de mortalité des différents stades embryonnaires chezle maquereau bleu (Scomber scombrus) et le chinchard commun (Trachurus trachurus) à partir d’inventaires réaliséstous les trois ans depuis 1977. Nous utilisons un modèle standard de processus de Markov en temps continu qui com-bine le nombre d’oeufs récoltés à chaque stade et des données expérimentales sur la durée des stades embryonnaires(dépendante de la température de l’eau). C’est la première fois que l’on tente d’étudier la mortalité des différents sta-des embryonnaires dans un si grand détail et c’est aussi la première tentative d’envergure pour estimer la mortalité desoeuf du chinchard commun dans le nord-est de l’Atlantique. Nos résultats comprennent des descriptions détaillées desvariables spatio-temporelles explicatives de la mortalité. Le taux journalier de mortalité des oeufs du maquereau bleuest estimé à ~0,56·jour–1 (beaucoup plus élevé que ne l’indique la littérature) et celui du chinchard commun à0,54·jour–1. Bien qu’il soit impossible d’estimer directement la mortalité des oeufs de stade 1, nos résultats laissentcroire à une forte mortalité à cette étape. Cela peut mener à une sous-estimation de la biomasse des poissons, lorsquecelle-ci est évaluée de façon traditionnelle à partie des seules données d’inventaire des oeufs.

[Traduit par la Rédaction] Portilla et al. 1668

Introduction

The survival of fish during their early life history stages(eggs and larvae) is probably a key source of variability inrecruitment into the adult stocks (Hjort 1914). For any givenfish population, it is likely that most of the mortality will oc-cur in these early stages (Pitcher and Hart 1982). Thus, the

knowledge of the mechanisms and scale of early stage natu-ral mortality is likely to be critical in understanding the linkbetween generations, although clear cause–effect relation-ships in mortality variation are not clear.

Egg mortality in early life history stages is known tochange between populations and years (Pepin 1991). Mortal-ity could be considered as principally due to both exogenous

Can. J. Fish. Aquat. Sci. 64: 1656–1668 (2007) doi:10.1139/F07-128 © 2007 NRC Canada

1656

Received 5 September 2006. Accepted 21 June 2007. Published on the NRC Research Press Web site at cjfas.nrc.ca on22 November 2007.J19515

E. Portilla,1 D. Beare, and D. Reid. Fisheries Research Services (FRS) – Marine Laboratory, P.O. Box 101, 375 Victoria Road,Aberdeen AB11 9DB, Scotland.E. McKenzie. Department of Statistics and Modelling Science, University of Strathclyde, Livingstone Tower, Glasgow G1 1XT,Scotland.

1Corresponding author (e-mail: e.portilla@napier.ac.uk).

and endogenous factors (Heath 1992). Exogeneous mortalityis the result of predation (Bunn et al. 2000) and environmen-tal factors, e.g., temperature (Pepin 1991), and is believed tobe the more important (Bunn et al. 2000). Endogenous mor-tality also occurs and is due to inherent physiological factors(Lockwood et al. 1977; Pipe and Walker 1987).

In the marine–biological literature, mortalities are gener-ally calculated by fitting a mortality function to the data foreach egg stage, extrapolating this back to the number of eggsexpected at the time of spawning, and then calculating thearea under the curve. In general, exponential decay modelsare assumed in these calculations (Lasker 1985). Bunn et al.(2000), however, suggests that such models may not be opti-mal for estimating mortality by egg stage and uses a sigmoidcurve instead.

Here, we model the progress of a single egg through itsdifferent early-life stages as a Markov process in continuoustime. In any stage, an egg can either survive to mature intothe next stage or die before this happens. The occurrences ofthese two events are modelled as competing Poisson pro-cesses whose rates depend on the stage currently occupiedand temperature (Lockwood et al. 1977; Pipe and Walker1987). The value of this approach is that it allows us to com-bine large databases of egg catch data, which yield informa-tion on the probability of an egg being in a particular stage,with experimental data on survival rates in each stage and soderive new estimates of the rates of dying in each stage. Thisis the first attempt to estimate mortality in such fine detailfor fish eggs. The results are correspondingly more informa-tive about mortality in each egg stage, and therefore, the es-timate of overall mortality over the entire sequence of theseearly-life stages is more realistic.

Such Markov processes are at the centre of many stochas-tic models and are found in most areas of application (Coxand Miller 1965). They appear, for example, in differentfields of ecology (Pielou 1969; Renshaw 1993) and, in par-ticular, are used to model the effects of fisheries populationevolution over time under human exploitation (McGarvey1995). They have also been used to describe the probabilitythat fish schools will form to avoid predation (Swartzman1991) or simply to describe mechanisms of mortality in lar-vae during their early stages (Beyer and Laurence 1980).

The two species involved in our study of mortality, Atlan-tic mackerel (Scomber scombrus) and horse mackerel (Tra-churus trachurus), are migratory species widely distributedover the continental shelf region of the Northeast AtlanticOcean (Eaton 1989; Uriarte and Lucio 2001). Atlantic mack-erel in the Northeast Atlantic supports important fisherieswith landings estimated at 718 000 t in 2002 and a totalspawning stock biomass (SSB) estimated at 2.5 milliontonnes (ICES 2004). Horse mackerel is often found shoalingtogether with Atlantic mackerel and is also commercially ex-ploited.

The annual stock assessment of both species is carried outat an ICES (International Council for the Exploration of theSea) working group using both information from commer-cial landings (Patterson and Melvin 1996) and independentdata on biomass, based on egg surveys (Fig. 1) and the An-nual Egg Production Method (Lockwood et al. 1981). Exten-sive icthyoplankton surveys of the entire spawning area andseason have been carried out every 3 years since 1977 and

are currently used to estimate biomass via Total Annual EggProduction (TAEP) (Lockwood et al. 1981).

It is important to understand that the TAEP analysis forAtlantic mackerel and horse mackerel uses stage 1 eggsonly. This is done in an attempt to reduce the impact of eggmortality on the estimate. The full methodology is detailedon the annual reports of the ICES Working Group for Mack-erel and Horse Mackerel Egg Surveys (ICES 1996). Essen-tially the method assumes that no mortality occurs prior tosampling, although it is recognised that this is unlikely to betrue. Estimates of mortality are systematically included inegg production estimates for Atlantic cod (Gadus morhua),sole (Solea solea), and plaice (Pleuronectes platessa) in theIrish sea, which also use the TAEP method (Armstrong et al.2001), and also for anchovy (Engraulis encrasicolus) andsardine (Sardina pilchardus) in the Bay of Biscay and theIberian Peninsula (SGSBSA 2003). It is, therefore, arguablethat a better estimate of TAEP would be achieved if mortal-ity during every stage (and not simply the first) could beincluded in the Atlantic mackerel and horse mackerel analy-ses.

Very little information is currently available on naturalegg mortality in Atlantic mackerel. For the Atlantic mack-erel stock in the Gulf of St. Lawrence, Canada, natural eggmortality rates were estimated at 0.5·day–1 (Ware and Lam-bert 1985). Sea temperatures in this area are colder than inthe eastern Atlantic (Bez and Rivoirard 2000), where Atlan-tic mackerel egg mortality rates estimated using data fromthe 1977 and 1986 surveys are far lower. Mortality ratesrange between 0.045 and 0.162·day–1 (Thompson 1989),which is the main reason that no mortality correction ismade during the TAEP estimation. Thompson (1989) esti-mated egg mortality by fitting exponential decay curves toegg abundance as a function of time and temperature. In hisanalysis, the four last stages (2–5) were all added together.For horse mackerel, natural egg mortalities have been esti-mated in only the North Sea. The values given by Eltink(1991) are daily egg mortalities, which were estimated atbetween 30% and 38% using data collected in 1988 and1991 (equivalent to 0.36–0.48·day–1).

The aim of this paper is to combine a continuous-time sto-chastic model with experimental and catch data to estimatestage-specific natural egg mortality in Atlantic mackerel andhorse mackerel. We use an extensive data set from the trien-nial Atlantic mackerel egg surveys to examine causes ofstage-specific natural egg mortality in fine detail, which wethen relate to the calculation of biomass using the AnnualEgg Production Method (Lockwood et al. 1981).

Material and methods

Data collectionThe data used in this study were collected on the ICES tri-

ennial Atlantic mackerel and horse mackerel egg surveysbetween 1977 and 2001. The primary sampling tool waseither the Dutch Gulf III or Bongo nets, with a 200 µm cod-end mesh. The sampler was deployed at a constant speed ofaround 5 knots (1 knot = 1.852 km·h–1) in an oblique tow toa depth of 200 m (or 10 m off the bottom in depths less than200 m). All samplers were equipped with calibrated flowmeters to provide an estimate of volume of water filtered.

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Portilla et al. 1657

Plankton samples were then fixed with 5% formalin, eggswere removed, and identified to species. Eggs were classi-fied and scored into five developmental stages for Atlanticmackerel and four stages for horse mackerel (Lockwood etal. 1977; Pipe and Walker 1987). Samples were taken over aregular grid of 0.5 × 0.5 degrees from northwest Scotland tothe Bay of Biscay and, since 1995, the west coast of Portu-gal. In this study, we concentrated on the western spawningcomponent only, i.e., excluding Iberian waters, as this areahas the longest time series (Fig. 1). The sampling area is de-signed to cover the whole spawning area and season as faras possible (WGMEG 2003), and each station sampled in-cludes data on temperature, salinity, bottom depth, and timeof day.

The Markov process for egg stageThe estimation of natural interstage egg mortality is facili-

tated by modelling the progress of each egg through itsearly-life stages as a Markov process in continuous time.Thus, in stage k (for Atlantic mackerel, there are five devel-opmental stages, i.e., k = 1, 2, 3, 4, 5; for horse mackerel,four stages), an egg may die in a Poisson process of rate µ kper unit time or survive to enter stage (k + 1) in a Poissonprocess of rate λk per unit time. The properties of such pro-cesses are well known and well documented; see, for exam-ple, Cox and Miller (1965) and Renshaw (1993). Thus, thetime an egg spends in stage k follows a negative exponentialdistribution of mean 1/(λk + µ k). For an egg that matures

successfully to stage (k + 1), the time taken is negative ex-ponential of mean 1/λk, and for an egg that dies in stage k,the time to death is negative exponential of mean 1/µ k. Wealso envisage a stage 0, corresponding to spawning, whicheggs enter in a Poisson process of rate λ0. The entire processis shown schematically in Fig. 2.

Finally, we assume that the process is in equilibrium, andwe denote the corresponding stationary distribution of stagesfor an egg by (p0, p1, …, p5), i.e., the probability that an eggis in stage k is pk. It is perhaps worth noting that althoughwe are modelling the progress of a single egg through thesestages, we can also interpret this equilibrium distribution interms of the entire population of eggs. Thus, the probabilitythat an egg is in stage k, pk, may also be regarded as the pro-portion of the egg population at any given moment in stagek. Implicit in this interpretation is the assumption that theeggs behave independently, but each according to this samestochastic model. Further, since the process is in equilib-rium, we can employ the balance equations that match theimport and export flows for each stage. Thus:

(1) p p kk k k k kλ λ µ= + =+ + +1 1 1( ), ,0,1, 4�

We do not know the values of the pk, nor can we estimatethem easily because of the inaccessibility of informationabout stage 0. However, we can estimate the conditionalprobabilities, πk, given by

(2) πk P k k k= ≠ =( | ), ,egg in state 0 1, 2, 5�

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1658 Can. J. Fish. Aquat. Sci. Vol. 64, 2007

Fig. 1. Bubble plot showing numbers of stage 1 eggs per survey. Circle radii are proportional to the egg count per year.

directly from available data, since we observe the propor-tions of each sample in each of the five stages.

Further, we can rewrite eq. 1 in terms of πk, i.e.:

(3) π λ π λ µk k k k k k= + =+ + +1 1 1 1 2 4( ), , , ...,

As noted above, we can estimate πk directly for each sam-ple, and we denote these estimates by �πk. We also have esti-mates of λk, available from experimental studies in thestage-duration for Atlantic mackerel and horse mackerel(Lockwood et al. 1977; Pipe and Walker 1987). Stage-duration for each stage is modelled as a function of watertemperature (T, °C) and yields an estimate of λk in the form�λk = 1/exp[–(ak + bk logT)], where ak and bk are estimated inthe incubation experiments. Further, as noted, temperature isavailable in the survey data, and so an appropriate estimateof λk can be derived for every sample.

Hence, from eq. 3 we can derive estimates of µ k:

(4) � (� � � � )/ � ,µ π λ π λ πk k k k k k k+ + + += − =1 1 1 1 1, 2,3,4

Note, however, that we cannot derive �µ1 using this ap-proach, only �µ2, �µ3, �µ4, and �µ5.

Mortality: probability of dying and instantaneous rateIt is straightforward to show (Cox and Miller 1965) that

within our model, the probability of the death of an egg instage k is given by

(5) m kkk

k k

=+

=µλ µ

, ,1, 2, 5�

Again, we note the alternative population interpretation:mk is also the proportion of eggs in stage k at any given timethat will die before maturation to stage (k + 1). Therefore,from eq. 4 an egg that dies in stage k is given by

(6) ��

�

�

�, ,m kk

k

k

k

k

= − =− −

1 2, 3, 5π

πλ

λ1 1

�

Now, the proportion of eggs that survives through all fivestages to become larvae is given by s* = s1s2s3s4s5, where skis the proportion that survives from stage k to stage (k + 1).Note that sk = (1 – mk), and s* may be written as

(7) s s* = 15

1

5

1

ππ

λλ

Clearly, we cannot estimate s* directly for the same reasonwe could not estimate µ1 directly, i.e., there is no informa-tion in the samples about (p0, λ0). Nevertheless, we are ableto estimate the probability of dying, mk, and hence survivalfor stages 2–5. For practical convenience and to allow us toderive an estimate of s*, we will assign the same value to m1

as was estimated for m2, on the basis that younger eggs aremore vulnerable, and so the mortality in stage 1 should be atleast as high as that in stage 2. We would argue in favour ofthis assumption by considering three different types ofsurvivorship curves for natural populations of animals(Deevey 1947): Type I (negatively skew rectangularsurvivorship curves), with the mortality rate increasing withage; Type II (diagonal survivorship curves), with a constantmortality rate throughout the life span; and Type III (posi-tively skew rectangular curves), with a greater mortality ratein early life. Thus, here we are assuming a combination ofType III (typical of marine pelagic populations) and Type IIsurvivorships rather than a Type I (more typical of humanpopulations) survivorship (Deevey 1947).

It is common also to define an instantaneous mortality rate,z, based on a deterministic model of exponential decay for thepopulation size, so that the proportion of eggs present surviv-ing until a time t later is given by exp(–zt). We can derive anestimate of z for individual stages or combinations of them byequating this expression to the corresponding estimated sur-vival, for example, exp( � � ) �− =z t sk k k, or exp( � � ) �* * *− =z t s . Theappropriate time interval is the mean time spent in the stageor combination of stages, e.g., � /(� � )tk k k= +1 λ µ .

Thus, for example,

(8) �ln( � )

�*

*

*

zs

t= −

which, using � �m m1 2= and � ( � )s mk k= −1 in eq. 8, can be writ-ten as

(9) �ln[( � )( � )( � )( � )( � )]

�*

*

zm m m m m

t= − − − − − −1 1 1 1 12 2 3 4 5

To distinguish between these mortality rates, we refer tothe values zk as the interstage instantaneous mortality ratesand z* as the overall instantaneous mortality rate. It is clearthat mortality may be expressed either as the probability ofan egg dying before it matures to the next stage (mk) or as arate of exponential decay through the stage, i.e., (zk). Forease of comparison with the literature, however, we chose tosummarize the data using the latter, which, for convenience,we refer to in the rest of this paper simply as the (interstageor overall) mortality. It is also worth noting that the overallmortality is a weighted average of the interstage mortalities,where the weights are proportional to the mean times spentin each stage.

Problems with interstage mortality estimationThe estimation of µ k (the interstage Poisson mortality rate)

may be problematic because of the nature of the data, andthis also applies to the probability of dying, mk. The value µ kor mk cannot be estimated for cases where there is no samplefor any egg stage. The presence of zeros in sampling is verycommon, and indeed the majority of samples do not containan egg in some stages. Mortalities in such cases obviouslycannot be estimated, are assumed to be missing, and are nottaken into account during the analyses.

Another difficult situation occurs when some egg stagesare missing, i.e., a sample might contain only stage 2 andstage 4 eggs. This problem can be dealt with by aggregationof samples. This happens because sampling is separated by

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Portilla et al. 1659

Fig. 2. Schematic representation of the continuous-time Markovprocess for five egg stages.

large distances and times, and any single sample of eggsmight be derived from many different spawning females. Insuch a case, the sample yields no mortality estimate forsome stages. Finally, the number of eggs in a stage is some-times higher that the number of eggs in earlier stages. Thisseems to be a fairly common observation in studies on eggmortality (Bunn et al. 2000), including this study (Table 1),and often persists even after data aggregation. Following theassumptions of our conceptual model, i.e., eqs. 4 and 6,stage (k+1) would then have a negative value for �µ k+1,which is clearly nonsense. Consequently, such values wereset to 0 when estimating either z* through the stages z or zk.

It seems that some (but not all) of the problems might beovercome by aggregating samples before computation of anyof the forms of mortality. Nevertheless, it must be remem-bered that different temperatures apply for different samples,so separate mortalities expressed in any of the ways de-scribed above (µ k, mk, or zk) must be calculated each time.

Data aggregation and error estimatesThe observed distribution of eggs is very patchy in space

and time. Furthermore, natural egg mortality itself variesspatially and with time, both seasonally and long-term. Tomake useful summaries of natural mortality rates that weresimultaneously capable of exposing spatio-temporal patternswhile minimizing the effects of gaps, the data were aggre-gated into spatial and temporal blocks.

The egg surveys were grouped into nine zones distin-guished by three latitudinal blocks (North, Central, andSouth) and three blocks based on distance from the shelfedge (Deep Waters, 50 km around the contour, and ShelfWaters) (Fig. 3). These divisions are based on the assump-tion that each zone represents a degree of ecological homo-geneity, both spatially and temporally. North–southdifferences in stock distribution during the spawning season(migration) and differences perpendicular to the shelf break(a spatial spawning preference) have been reported in the lit-erature (Iversen and Skagen 1989; Uriarte and Lucio 2001).

So, for each month from February to July (note that dur-ing some years, February and March were not sampled) inevery third year from 1977 to 2001, interstage mortalityrates and overall daily mortality rates were estimated to pro-duce a data set grouped by 318 levels (spatial block + month +survey year). Mean estimates and associated standard errors(SE) were estimated by bootstrap generation of 200 repli-cates (Efron and Tibshirani 1993). (Although computingpower now allows bootstrap estimates for much larger num-bers of replicates, we found that 200 iterations producedsimilar confidence limits to that of larger numbers of repli-cates.)

Estimating and testing the overall mortalitiesFor both species, we modelled mortalities as a function of

month (March–July) and year (of survey) and location(zone). Our model involves these three covariates in an addi-tive form:

(10) mort month yearijk i j k ijkZ= + + + +µ ε

where mortijk is an estimated mortality in the ith zone, thejth month of the year (3–7), and the kth survey observed(from 1989 onwards for Atlantic mackerel and 1977 for

horse mackerel); the right hand side has the obvious inter-pretation as the sum of the corresponding effects due to thatcombination of the covariates and εijk being the unexplainederror. The advantage of modelling the mortalities in this wayis that we can estimate the forms of the vectors of the ef-fects, i.e., {Zi}, {monthj} and {yeark}, and so test formallyfor differences between zones, between months, or betweensurveys. This is a standard kind of decomposition in that{monthj} will yield the seasonal cycle and {yeark} the long-term trend. Another advantage in modelling the mortalitiesin this way is that we can use weights in the estimation tomirror the accuracy with which each mortality is known. Itis an important aspect of these data that we have measuresof accuracy associated with each estimate, and these shouldbe taken into consideration in any analysis. This is not cost-free, however, as there were several mortalities with zerovariance (based on single samples), and these were omitted.The reciprocals of the variances of the estimates (1/SE2) areused as the weights in the modelling process, as is commonpractice in such cases (Steel and Torrie 1980).

Results

Interstage mortalityMortalities were computed using eq. 9 and the distribu-

tions of the 318 pooled results for Atlantic mackerel andhorse mackerel are shown as boxplots for each stage(Fig. 4). When the information for all the samples availableis displayed together, a similar pattern is observed for bothspecies. Interstage mortalities tend to be higher in the earli-est stage (i.e stage 2) and, after dropping in stage 3, changelittle. There is some suggestion that Atlantic mackerel mor-talities may rise slightly from stage 3 onwards, whereasthose for horse mackerel seem to fall a little. These are con-sistent but very small changes, however, and there is consid-erable variation in the data.

The distributions of mortalities over months, years, andzones are displayed for Atlantic mackerel and horse mack-erel (Figs. 5 and 6, respectively). These boxplots of mortali-ties are not easy to interpret because the data are sparselydistributed through space and time and contain a number ofzero values in specific locations, but such plots can give auseful indication of trends, when they exist, and so we makea brief qualitative study of them here. For Atlantic mackerel(Fig. 5), mortalities in stages 2 to 4 increase during the sea-son, whereas those in stage 5 appear to decrease. As before,the changes are small but the trends are consistent. The Julyplots do not conform to this pattern, but this may be due inpart to an unusual number of zero estimates here, as evi-denced by the first quartile of zero. There is much less con-sistency in annual trends between the stages. In stage 2,

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1660 Can. J. Fish. Aquat. Sci. Vol. 64, 2007

Stage 2 Stage 3 Stage 4 Stage 5

Stage 1 1008 1434 990 625Stage 2 — 2643 1755 1088Stage 3 — — 1211 751Stage 4 — — — 975

Note: The number of samples analysed was 7564.

Table 1. Number of samples in which the number of eggs ateach stage is lower than that of the following stage.

mortalities were high in 1977, fell sharply in 1980, and thenrose steadily until the mid-1990s before levelling off andfalling again. Mortalities in stage 3 appear to fall generallyfrom 1977 to 1995 and then level out or perhaps even rise alittle. There are no apparent long-term trends in stages 4 and5. Spatial trends are much less evident, with possibly a smallrise moving north to south in all but stage 2, and possibly asmall increase along the shelf edge relative to oceanic andcoastal waters.

There are similar results for horse mackerel (Fig. 6). Eggsin all three stages exhibit increasing mortality throughout theseason, although in stages 3 and 4 mortalities fall in July.Annual trends show a single peak in 1992 for stage 2 and in1983 and 1986 in stage 3. Stage 4 values show no likelytrend. There are few mortality estimates available in thenorthern zones, and the corresponding boxplots yield no in-formation. However, mortalities appear to rise moving fromcentral to southern zones, and stages 2 and 3 show slightlyelevated mortalities along the shelf edge again.

Overall mortalitiesAs described earlier, we take a more formal approach

here, since these overall mortalities may be compared di-rectly with estimates from other sources and other species(the results are displayed in Tables 2, 3, and 4 and Fig. 7,and we explain and discuss their derivation and interpreta-tion here in detail).

There were no surveys in the three most northern zones(1–3) until 1986, and so we cannot draw any inferences forthat area before that year. In addition, we have mortality es-

timates for Atlantic mackerel for only one northern zone (ofthe three) in 1986, and so we model the Atlantic mackerelmortality estimates for all nine zones from 1989 onwards.Obviously, an alternative would be to model only zones 4–9for the entire period. However, there are almost no mortalityestimates for horse mackerel in the three northern zones inany of our survey years, and so this alternative is the modelwe used for that species. For Atlantic mackerel, we prefer tomodel all nine zones from 1989, since this is our onlyopportunity to draw some inference about these northernzones. The absence of horse mackerel mortality estimates inthe northern zones is due to that species’ southern prefer-ence for spawning grounds (Eaton 1989).

We describe the results of modelling in a little more detailfor Atlantic mackerel for clarity. Proceeding as describedabove and estimating and testing the components of the indi-vidual covariates yields the following results. The basic zonecomponent effect for the mean mortalities (with standarderrors) in the nine zones is displayed (Table 2). To obtain themean mortality in any zone, month, and year, we add to thezone effect, according to month and year, the seasonal cycle,and the annual trend effects listed in Table 2. By combiningthese estimated components appropriately, we can derive themean mortalities in any zone in any of the 5 months and inany of the 5 survey years. All tests in the model-fitting pro-cess were conducted at the 5% level of significance, al-though in many cases p values were very much smaller than0.05.

Although there are, as shown above, differences amongzones, months, and years, it is nevertheless useful to derive

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Portilla et al. 1661

Fig. 3. Division of survey area into nine different zones.

synoptic views by averaging over sets of covariates. Thus,by averaging over both years and months, we can generatemean mortalities for each zone (Table 3). Thus, for Atlanticmackerel, the highest mortalities occur in zones 5, 6, 8, and9, i.e., on the shelf and shelf edge in the central and southernzones. In the north, the mortalities in oceanic waters and onthe shelf are the same but much less than that found at theshelf edge. In the central and southern zones, mortalities arelower in oceanic waters than on the shelf or at its edge.

As noted, we have few valid horse mackerel mortalitiesavailable in the three most northern zones, and so we cannotadequately model mortalities in those zones over any timeperiod of interest. For this reason, we model horse mackerelmortalities in only the central and southern zones, i.e., zones4–9. Thus, the mean mortalities, averaged over all surveys,for horse mackerel are provided (Table 3). Mortalities are

shown to be lowest in oceanic waters increasing as we moveover the shelf edge and rising still further in the centralzone, but falling in the south, as we move on to the shelf it-self.

By averaging over years and zones, we can generate themean seasonal cycles for Atlantic mackerel and horse mack-erel (shown in Fig. 7a). They are very similar; both un-changing at the start of the season and then both risingrapidly, with horse mackerel starting first and Atlantic mack-erel suffering the apparently uniformly higher mortality.However, the much larger standard errors for horse mackerellater in the season make it clear that the two may be muchcloser by then. Testing equality would be very difficult,since the estimates are based on related samples, but a verycrude approximation suggests that equality is possible onlyin June. We may also note that because of the additivity ofthe model, the different cycles corresponding to differentzones for each species will be parallel to these.

Similarly, by averaging across months and zones, we canestimate annual trends for the mortalities of both species(Table 4). These annual trends for both species are shown(Fig. 7b). The annual trend in mortalities for horse mackerelrises until about 1992 and then falls a little. The Atlanticmackerel mortalities start only in 1989, but mirror the fall ofhorse mackerel mortalities. As before, the Atlantic mackerelmortalities are greater than those of horse mackerel in gen-eral, but once again the large standard errors in the lattercase make separation less certain.

Year-to-year variation of Atlantic mackerel and horsemackerel overall instantaneous mortality rates and the com-parison with historical data for the same periods and similarareas (Thompson 1989; Eltink 1991) are also available(Table 4). We estimate higher mortality than Thompson(1989) for Atlantic mackerel and, although lower than Eltink(1991), much closer estimates of mortality for horse mack-erel.

It is worthy of comment that temporal and spatial changesin the overall mortalities, i.e., those for the overall periodfrom fertilization to hatching, are closely related to trendsdisplayed in the interstage mortalities discussed above, espe-cially in stage 2, for both species.

Discussion

Comments on methodologyA secondary objective of this study was to introduce a

new tool for estimating natural fish egg mortalities with awide range of potential applications. Traditionally, egg mor-tality is assumed to be constant throughout all the stages(Lasker 1985). Consequently, an exponential decay model isoften fitted to the density of eggs as a function of age. Cur-rently, egg mortality rates are estimated for calculating eggproduction at age 0 (close to spawning) for anchovy, sardine,and sole (Lasker 1985; Armstrong et al. 2001).

The method used here differs in principal from the tradi-tional studies of mortality in that the mortality rate is esti-mated for each stage during the life span of the egg. Bunn etal. (2000) suggested fitting a sigmoidal model to lessen thesize of large residuals present on some egg stages that werenot well fitted. Previous workers (Beyer and Laurence 1980)have used stochastic models to estimate mortality, but this is

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Fig. 4. Boxplots showing distributions of interstage mortality ratesin each egg stage for Atlantic mackerel (Scomber scombrus)(a) and horse mackerel (Trachurus trachurus) (b) from all the dataavailable.

the first attempt to apply them to the especially difficultproblems presented by planktonic fish egg data. Difficultiesarise when comparing incubation studies to field data wherewe cannot observe eggs from fertilization until death or mat-uration (Mendiola et al. 2006). Instead, we observe eggs atage, and it is from this that we derive any mortality esti-mates, in contrast with what would be preferred (Dickey-Collas et al. 2003). The Markov process model can be ap-plied to any individual progressing through year classes(Renshaw 1993). It can also be used to summarize any sys-tem divided into more general classes or stages throughwhich individuals pass sequentially.

The methodology further allows us to estimate mortalityfor any sample. Difficulties in the application of the method-ology are invariably due to problems such as zero counts orwhen higher numbers are observed in the older stages, avery common feature of these data. It is also important tonote that although the Markov process model we use is richenough to allow us to combine different data sources (i.e.,field egg-at-age counts and incubation egg stage durationstudies) and hence estimate natural mortalities, this is notcost-free. The model is a detailed one, and we note here themajor underlying assumptions implicit in its structure andtheir consequences. The assumption that events (e.g., here,that death in a particular stage or maturation to the nextstage) are Poisson processes is a common one in very many

stochastic models and seems to be a reasonableapproximation in most cases (Dickey-Collas et al. 2003). Animmediate consequence, however, is that the process lacksmemory, i.e., the probability that an egg will die (or matureinto the next stage) at this moment does not depend on howlong the egg has already been in the current stage. For a sin-gle egg, this seems an unlikely property, but when viewed interms of the behaviour of the population of eggs and therates of change, i.e., death and maturation into the nextstage, the model is a reasonable first approximation.

Perhaps more problematic is another consequence of thememoryless property, viz that an egg’s progress in a particu-lar stage is independent of how long it spent in the previousone. The viability of an individual egg and its ability to com-plete developmental stages is a function of the environmen-tal conditions experienced throughout the developmentalperiod, e.g., salinity, temperature, or mechanical stress(Brooks et al. 1997; Bunn et al. 2000). Therefore, for bio-logical reasons, times spent in consecutive stages might beserially correlated. We have no simple way of estimating andmodelling such dependence, if it exists, and it is not clearhow we could connect it to the properties of interest, i.e.,embryo development and mortality, in any given stage.Again, however, it is a potential problem really only at thelevel of individual eggs, and much less so when consideringbehaviour of the egg population as a whole.

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Fig. 5. Boxplots showing distributions of interstage daily mortality rates for Atlantic mackerel (Scomber scombrus) by month, year,and zone for all the data.

The interpretation of the results in terms of the populationof eggs rather than a single egg is, as noted, dependent on anassumption that each egg independently undergoes thisMarkov process. Again, for biological reasons, we might an-

ticipate that this would be at best an approximation and thatthere might be correlation among individual eggs that arephysically close in the water. These dependences wouldhave little effect on mean values for the population of eggsas a whole, but would certainly affect all variances and stan-dard errors computed. Thus, estimates relating to meantimes, etc., might be expected to be robust against this kindof problem, but testing would be potentially unreliable with-out appropriate approximations. Tackling these problemswill be the topic of future work. This paper is a first effort atdeveloping this methodology and is not meant to provide thecomplete solution. We believe that when faced with suchcomplex processes, even a model that is only a first-orderapproximation can yield useful and important information.

Finally, the process is assumed to be in equilibrium sothat the balance equations can be used. This seems not anunreasonable assumption, at least during that part of thespawning season when egg production, which drives the sto-chastic model, is at a nearly constant level.

One of the most important assumptions we make in thiswork concerns the value of mortality in the earliest stage ofdevelopment. Nothing is known (in any spatio-temporal de-tail) about spawning rates that might be used as an input en-try rate into the system (λ0). The assumption made here isthat mortality between egg stages 1 to 2 is equal to that fromspawning to end of stage 1. Similar assumptions, therefore,

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Fig. 6. Boxplots showing distributions of interstage daily mortality rates for horse mackerel (Trachurus trachurus) by month, year, andzone for all the data.

Covariate

Zone (1–9)(µ + Zi)

Monthj

(March–July)Yeark

(1989–2001)*

0.257 (0.017) 0 00.403 (0.035) 0 0.147 (0.027)0.257 (0.017) 0 00.403 (0.035) 0.157 (0.029) 0.106 (0.029)0.491 (0.026) 0.464 (0.021) –0.084 (0.028)0.491 (0.026) — —0.257 (0.017) — —0.491 (0.026) — —0.491 (0.026) — —

Note: Mean mortalities are given by the appropriate sum,i.e., µ + Zi + monthj + yeark.

*Years used for the analysis were taken once every3 years.

Table 2. Estimates of components of instantaneousmortality rate model for Atlantic mackerel (Scomberscombrus).

would need to be made to estimate egg production at age 0(Dickey-Collas et al. 2003), where egg production ishindcast (or backcast) from the exponential decay model.Different assumptions could be made at this point, and arange of mortalities could be computed and considered. Wehave simply chosen one particular approach, which seemsplausible. Indeed, we have argued that our choice may beconservative, as mortality is often highest in the earlieststages of life in nature. On the other hand, we note that labo-ratory experiments suggest that mortality is generally con-stant throughout the developmental period; only in the laterstages does mortality tend to be higher. These experiments,however, do not take into account mortality due to preda-tion, which we might expect to be substantive (Bunn et al.2000). It is important to note that stage 1 mortality, howeverit is actually estimated, will have an important impact on thefinal value of the overall mortality and can dramatically af-fect the estimated numbers of eggs that survive at each stageof the embryo phase.

Comments on the resultsThe main aims of this work were to improve and update

the available information on natural egg mortality for Atlan-tic mackerel and horse mackerel. For Atlantic mackerel, thelast study available in the literature is based on the 1977–1989 surveys (Thompson 1989). The present work is farmore extensive in the range of its spatial and temporal ambi-tion than the one done in 1989. This study includes the egg

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Oceanic Shelf break Coast

Atlantic mackerel North 0.415 (0.013) 0.561 (0.034) 0.415 (0.013)Central 0.561 (0.034) 0.649 (0.017) 0.649 (0.017)South 0.415 (0.013) 0.649 (0.017) 0.649 (0.017)

Horse mackerel North — — —Central 0.226 (0.023) 0.441 (0.032) 0.619 (0.028)South 0.226 (0.023) 0.539 (0.063) 0.441 (0.032)

Table 3. Synoptic estimates (standard errors) of overall mortality rates for Atlantic mack-erel (Scomber scombrus) and horse mackerel (Trachurus trachurus) zones.

Atlantic mackerel Horse mackerel

Year Estimate SE Estimate SE Thompson (1989) Eltink (1991)

1977 0.655 — 0.293 0.024 0.094 —1980 0.469 — 0.293 0.024 0.162 —1983 0.423 — 0.459 0.026 0.128 —1986 0.404 — 0.293 0.024 0.045 —1988 — — — — — 0.3571989 0.519 0.018 0.595 0.048 — 0.4621990 — — — — — 0.4781991 — — — — — 0.3861992 0.665 0.021 0.595 0.048 — —1995 0.518 0.018 0.459 0.026 — —1998 0.623 0.025 0.293 0.024 — —2001 0.434 0.021 0.459 0.026 — —

Note: For years and species where modelling was possible, the estimates and their associated standard errors(SE) are displayed. For the rest (Atlantic mackerel pre-1989), the medians of estimated values are displayedinstead.

Table 4. Overall mortality rates for Atlantic mackerel (Scomber scombrus) and horse mackerel(Trachurus trachurus) compared with historical data (Eltink 1991; Thompson 1989).

Fig. 7. Averaged temporal changes in overall instantaneous mor-tality rates for Atlantic mackerel (Scomber scombrus) (continu-ous line) and horse mackerel (Trachurus trachurus) (brokenline). Vertical lines at the estimates represent 95% confidence in-tervals for the true average mortality rates. The plotted mortali-ties for one interval for one species are offset a little in time ineach plot to facilitate interspecies comparison.

survey data collected from 1977 to 2001 and also separatelyexamines any possible seasonal (from March to July) or spa-tial effect on natural egg mortality. For horse mackerel, theavailable bibliography is even briefer, and the only studyfound with which our results could be compared was doneby Eltink (1991) in the North Sea between 1988 and 1991.Information, however, does exist for a similar trachurid spe-cies (Trachurus symmetricus) (Pepin 1991).

This is the first exploratory study of this problem usingthis approach, and it yields estimates of mortality that aresurprisingly different from what has been obtained in thepast. On the other hand, the natural egg mortality estimatedin this study are comparable with those calculated for otherfish populations. Egg mortalities can be as high as 3.67·day–1,i.e., only 2.5% of the eggs survive per day, for eggs ofHarengula jaguana (Pepin 1991). Normally, it is consideredto be below 1·day–1, i.e., 37% of the eggs survive per day(Bunn et al. 2000). Relatively few studies have been madeon natural egg mortality in Atlantic mackerel. For the Gulfof St. Lawrence stock in Canada, egg mortalities wereestimated at 0.44·day–1 (Ware and Lambert 1985). Seatemperatures in this area, however, are colder than in theeastern Atlantic, where mortalities were estimated to belower, in the range 0.045–0.162·day–1 between 1977 and1986 (Thompson 1989). This value suggests (at the averagetemperature of 12 °C) that up to 76% of the numbers of eggsthat are spawned can develop into larvae. From our results,we believe that this value of survivorship is far too high andprobably incorrect.

For horse mackerel, we were able to compare our datawith only two other studies: (i) for the same species but on adifferent region (the North Sea), with values ranging from0.36 to 0.48·day–1 (Eltink 1991); (ii) on a similar species(Trachurus symmetricus, which is found along the Califor-nian coast), where mortalities were calculated at ~1.64·day–1

(Pepin 1991).We modelled the overall mortalities as a function of loca-

tion, seasonal cycle, and annual trend in a way that allowed usto draw inferences about the nature of these components forboth species. The model chosen was the simplest one thatwould allow us to model dependence of mortality on locationand time in a meaningful way, using as much of the informa-tion available as possible. Initially, we had considered model-ling dependence on month and year as linear or quadratictrends, but residual analyses showed that this was inadequate,and we treated these time covariates as factors instead. Ourmodel postulates that the long-term trends and seasonal cyclescombine in an additive way and that their forms depend onlyon location. Residual plots suggest that this is a reasonablerepresentation. It would be possible to attempt to model thesedata in more complex ways, e.g., allowing for any of the com-ponents to change in time (as well as location), but given thesparsity of the data over such a long period and wide area, itwould have been very difficult to have confidence in the real-ity of any such estimated interactions.

Factors regulating mortality

Temporal and spatial variation of mortalityOnce mortalities are estimated, their spatial and temporal

variation can be examined. The output from the models also

allows us to scrutinize both egg mortalities between individ-ual stages and the overall mortalities between stages 1 and 5.

Generally, interstage egg mortalities increase throughoutthe spawning season for all the stages, with the exception ofJuly for stages 3 onwards. Such seasonal variation might berelated to predation by zooplankters. Zooplankton abun-dance peaks at different times of the year, and different spe-cies have a range of feeding strategies and efficiences. Inspring, predatory zooplankton will be composed mainly ofzooplankton that feed visually (Beare et al. 2002) and hencemay preferentially target the older egg stages (e.g., stage 5in Atlantic mackerel). Later in the year, predatory jellyfishand fish larvae appear, and these have different feeding strat-egies from the crustaceans (Purcell 1985). Additionally,older egg stages tend to float to the surface. This means thatthe susceptibilities of eggs to predation will depend on acomplex mix of spatial, temporal, and environmental factors.

Similarly, the variation in time and space of mortalitiesmay be related to different aspects of spawning behaviour.Atlantic mackerel and horse mackerel present differentspawning preferences over the Northeast Atlantic, and dif-ferent year-class individuals arrive at the spawning groundsat different times during the spawning season (Macer 1977;Reid et al. 2003). Thus, for both Atlantic mackerel and horsemackerel, spatial differences might be related to adult pref-erences for spawning. Variation might also be influenced bythe time of arrival at the spawning grounds of individuals ofdifferent ages, e.g., for Atlantic mackerel (Uriarte and Lucio2001). Therefore, as is well known, recruitment success in-creases with year class (Nilssen et al. 1994), and so egg sur-vival could be higher for the older early-arrivers. In addition,since when and where adults spawn is also influenced tosome extent by local climatic conditions such as temperature(Bez and Rivoirard 2000), it follows that these may also befactors that regulate survival, as suggested by Pepin (1991).Finally, the co-occurence of predators on the spawninggrounds might also be responsible for the high mortality onthe shelf and shelf break south of the British Isles for Atlan-tic mackerel and horse mackerel (zones 5, 6, 8, and 9). Itshould be noted that there exists a southern preference forspawning grounds in horse mackerel (Eaton 1989), which isresponsible for the lack of overall mortality estimation in thenorth.

We have defined zones by creating blocks defined by loca-tion and time of year. Although this is a traditional approach,it is also somewhat arbitrary, and alternative approaches arepossible. One referee, for example, raised the possibility ofzonation using water temperature, which might well explainsome of the spatial and (or) temporal effects in a useful way.It would be more difficult to use than our version, however,and given the strong dependence of water temperature onlocation and time of year, it would be unlikely to contributesubstantially more information.

According to our data, mortality in the youngest egg stage(stage 2) is typically the highest; it falls off (stage 3) andthen may increase slowly for Atlantic mackerel (decrease forhorse mackerel) as the eggs mature. This general pattern forstages 2 and 3 is observed irrespective of how the data areaggregated prior to analysis. The high mortalities in theolder egg stages for Atlantic mackerel correspond to timeswhen embryos are clearly visible (Simpson 1959) and may

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be more susceptible to predation by fish (Ellis and Nash1997). Higher mortalities in later stages have also been ob-served in incubation studies for Atlantic mackerel (Mendiolaet al. 2006), suggesting endogenous factors controlling mor-tality. However, the aggregation of early stages in denserpatches is also likely to contribute to higher mortalities(Ellis and Nash 1997).

We routinely observed that egg mortalities for both Atlan-tic mackerel and horse mackerel eggs were highly variablebetween egg stages. Stage 2 eggs appeared the most vulnera-ble. There are a variety of possible explanations. Stage 2eggs are more abundant and their distribution is more patchythan the others. Predators encountering a patch, therefore,may be able graze on it. Similarly, older eggs will be fewerand more disperse (Bartsch and Coombs 2001) and hencemore difficult for predators to encounter en masse.

Another point to note is that the comparatively long dura-tion (Lockwood et al. 1977) of stages 1 and 2 (cf. 3, 4, and5) increases the length of the time they are exposed to inten-sifying predation. Traditionally, the egg mortalities shownherein are assumed not to be affected by stage class (Lasker1985), but they clearly are and, in fact, appear to be lower instages of shorter duration.

Finally, we recall that we have approximated the unknownmortality in stage 1, setting it equal to that in stage 2. Asdiscussed before, we believe that this approximation might,in reality, be conservative, and we have argued that mortalityin stage 1 could be higher. Indeed, a Type III survivorshipcurve can be intuitively assumed for pelagic species as op-posed to a Type I curve, more typical of human populations(Deevey 1947). In addition, we note that one possible causeof such higher mortality in stage 1 is the inability of unfertil-ized stage 1 eggs to develop to the next level. According toS. Milligan (Centre for Environment, Fisheries and Aqua-culture Science (CEFAS), Lowestoft Laboratory, PakefieldRoad, Lowestoft, Suffolk NR33 0HT, UK, personal commu-nication, 2006), it appears very unlikely that unfertilizedeggs would be recorded in the sampling. Failures in egg fer-tilization will cause an immediate entry of water into theembryo, which will then rapidly sink through the water col-umn, ultimately causing the cell to rupture. Consequently,there may be an even greater disparity between the estimatedfecundity of females and the numbers of eggs recorded inthe water, and this, of course, is crucial for the estimation ofthe SSB (Lockwood et al. 1981).

In this work, we have used the overall mortality (z) to fa-cilitate comparison of our results with those of other studies.It is, however, an oversimplification to assume a constantvalue of mortality over all of this period, not least becausewe actually observe different rates of mortality during thedifferent stages of the developmental process. Nevertheless,as we have noted, the overall rate is a weighted average ofthe interstage rates and displays similar behaviour so that itis reasonable to compare, at least tentatively, our z valueswith others from fish populations worldwide.

Implications for TAEP estimatesAn important aspect of this work is its potential impact on

stock estimation by the annual egg production method(WGMEG 2003). When the total annual egg production iscalculated, no estimate of mortality is made (Lockwood et

al. 1981). Since we do not have enough information to esti-mate stage 1 egg mortality from the data, we assumed thatstage 1 eggs suffered at least as great a mortality as the stage2 eggs. If this is approximately true, then around 70% ofstage 1 eggs will die before passing on to the next stage(note that this 70% value is an average over all the year,month, and zone combinations). Since Atlantic mackerelSSB is calculated (by TAEP) on the basis that a specificnumber of stage 1 eggs will be spawned by a specific num-ber of females, then a failure to account for mortality willbias the estimates of SSB upwards. However, sensitivitystudies on mortality show that varying values of mortalitycan change the final values of TAEP, but the variability onthe estimation of SSB comes from different sources(McGarvey and Kinloch 2001).

On the estimation of mortality using a stochastic modelWe have used a simple stochastic model of an egg’s prog-

ress through the early life stages of development to allow usto combine experimental data on egg stage duration timeswith egg catch data from nine triennial surveys to yield esti-mates of interstage egg mortalities for Atlantic mackerel andhorse mackerel. We believe that the use of our Markov pro-cess model for an egg’s progress through its early life stagesnot only greatly facilitates a methodology for estimatingmortality at high spatial and temporal resolution, but alsosensibly reflects the holistic dynamics of the spawning pro-cess. Further, the model allows us to estimate different mor-talities at different stages of the development process. Thisyields new insights into the mortality process of develop-ment to larvae.

On the other hand, a weakness of our approach is that wehave no estimate of mortality for stage 1, and so our overallconclusions are dependent upon the assumptions we makeabout this value. Variation in interstage mortality changesamong stages and does not always appear to vary in space ortime in a consistent way. In addition, Atlantic mackerel andhorse mackerel egg mortalities are very high compared withcurrent literature. Consequently, if the estimates of mortalityare correct, an unexpectedly high loss of eggs is sufferedbefore any sampling takes place. This will have seriousimplications for the current stock estimation process.

Acknowledgements

We thank all institutions that provided the the data used inthis study. Various European Union funds have also beeninstrumental in promoting the international collaborationamong various organisations necessary for the completion ofthis work. Finally, we are grateful to the anonymous refereeswho made important improvements to early versions of themanuscript. E. Portilla is funded by FRS Marine Laboratory,Aberdeen, and is supervised at the Department of Statisticsand Modelling Science, University of Strathclyde, Glasgow.

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