ICES Journal of Marine Science: Journal du Conseil Advance Access originally published online on November 3, 2006
ICES Journal of Marine Science: Journal du Conseil 2007 64(1):3-17; doi:10.1093/icesjms/fsl017
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Short-term stock assessment of Loligo gahi at the Falkland Islands: sequential use of stochastic biomass projection and stock depletion models
Department of Fisheries, PO Box 598, Stanley, Falkland Islands
Correspondence to R. Roa-Ureta: present address: Departamento de Oceanografia, PO Box 160-C, Universidad de Concepción, Concepción, Chile; tel: +56 41 203765; fax: +56 41 256571; e-mail: rroa{at}udec.cl
Roa-Ureta, R. and Arkhipkin, A. I. 2007. Short-term stock assessment of Loligo gahi at the Falkland Islands: sequential use of stochastic biomass projection and stock depletion models – ICES Journal of Marine Science, 64, 3–17.Two short-term stock assessment models are combined to examine the pre-season, in-season, and post-season dynamics of the Loligo gahi fishery off the Falkland Islands over four consecutive fishing seasons. A stochastic biomass projection model (SBPM) projects a pre-season survey-based biomass estimate from the date of the survey to the start of the season. A stock depletion model (SDM) assesses in-season biomass from commercial daily catch-and-effort data. The SBPM projects the SDM biomass estimate at the end of the season to a post-season date of spawning. Combining the SBPM and the SDM helps to clarify the spatio-temporal functioning of the stock and to assess the comparability of survey- and fishery-based estimates of biomass. For the first 2005 season, projected length frequencies indicate two pulses of recruitment onto the fishing grounds. Survey-based projections of biomass were lower than equivalent fishery-based estimates. Over two surveys, the sex ratio was balanced, suggesting full recruitment of both sexes onto the fishing grounds, and the ratio of survey-projected to fishing-estimated biomass was constant. This constant is interpreted as a scaling factor between survey biomass and absolute biomass.
Keywords: biomass projection model, Falkland Islands, Loligo gahi, stock assessment, stock depletion model
Received 2 June 2006; accepted 14 September 2006; advance access publication 3 November 2006.
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Introduction |
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 Top Introduction Material and methodsResultsDiscussionReferences |
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The development and use of stock assessment models for squids present special challenges because of the rapid dynamics of their life history. Their short life (generally 1 y), positive acceleration of growth during early ontogeny, semelparity, and recruitment of the whole stock every year, together with both time- and labour-intensive ageing techniques, make it difficult to use age-structured models (Caddy, 1983). Moreover, squid have a highly plastic life cycle, including inter alia high fecundity, year-round spawning, and the existence of interbreeding seasonal cohorts. Such plasticity makes it difficult to trace cohorts over time through length frequency analysis and then to employ length-based models (Pierce and Guerra, 1994). The emphasis in squid stock assessment has been on rapid, short-term approaches (but see Roel and Butterworth, 2000) that permit assessment of the magnitude of recruiting cohorts entering the fishing grounds immediately before or after the start of the fishing season.
For the purposes of short-term stock assessment, the life cycle of a squid cohort subjected to exploitation can be described as having three sequential phases: (i) from hatching to recruitment onto the fishing grounds, (ii) a period of fishing pressure, and (iii) from the end of the fishing season until spawning. Boyle and Rodhouse (2005) summarized existing assessment methods for squid into three main categories: pre-season, in-season, and post-season assessment. The main objective of this paper is to demonstrate a combined modelling and statistical approach that involves assessment of the cohort in all three phases. It is important for squid, because the dynamics of individual growth and cohort decay are too fast to rely on global abundance estimates valid for the fishing season only.
The Patagonian longfin squid, Loligo gahi, is among the best studied and managed commercial squid resources in the world (Hatfield and des Clers, 1998). It is a relatively small squid (in the commercial catch, typically 13–17 cm mantle length) that lives over the continental shelf (20–350 m deep) around the Falkland Islands and is a target of the local trawl fishery (currently 16 vessels), yielding an average annual catch of 
50 000 t. Recent investigations of population structure (Shaw et al., 2004), environmental variables affecting stock distribution and abundance (Agnew et al., 2000; Arkhipkin et al., 2004), age and growth rates (Hatfield, 2000; Arkhipkin and Roa-Ureta, 2005) shed light on the life cycle of its two main cohorts, the autumn-spawning cohorts (ASC) and spring-spawning cohorts (SSC) (Arkhipkin et al., 2004). Moreover, the existing management scheme of almost instantaneous collection of biological and fisheries data from the fishing fleet has allowed the application of stock assessment models of the stock-depletion type to estimate the biomass at the start of each season (Agnew et al., 1998, 2002; McAllister et al., 2004). Pre-recruitment surveys have recently been introduced, and their results are used here to provide a pre-season assessment and fisheries-independent projection of the biomass at the start of each season, to be compared with the in-season estimate.
In the Falkland Islands, management aims to allow an absolute escapement biomass of >10 000 t at the spawning season that follows each of the two fishing seasons every year. However, the first season finishes more than 1 month before the assumed time of spawning, more than 10% of the cohort's lifespan. Therefore, the biomass estimated during the fishing season can be quite different from the spawning biomass, especially if the length frequency distribution of the cohort at the time of assessment is dominated by squid still growing fast, because in such a case, the length frequency distribution in the population changes more rapidly, and so does biomass. Another problem with the pre-recruitment survey is that it is carried out too early (2.5 months) before the second fishing season of the year. Therefore, depending on the length frequency distribution of the stock during the survey, the index of abundance at the start of the second season can be different from the index evaluated at the time of the survey. In addition, the possibility that several waves of recruits enter the fishing grounds at different times means that the stock assessed during a pre-season survey can be different from the stock fished during the season. In this paper, we address these issues with a combination of a stochastic biomass projection model (SBPM) for pre-season and post-season assessment and a stock depletion model (SDM) for in-season assessment.
Stock assessment in general relies on two main sources of information: fishery-generated data and scientific data from surveys. The former is a much larger source of information, but its analysis requires complex demographic models that may be slow to detect the early signs of stock decline not apparent in a stock's demographics (Hutchings and Myers, 1994). Therefore, there is interest in developing purely survey-based assessment techniques, as witnessed by the FISBOAT project of the EU. Here we provide an example of how survey- and fisheries-based estimates of abundance (pre-season and in-season estimates, respectively) can be compared directly, giving clues about the value of the scaling factor between the survey index and absolute biomass, and how the biological condition of the stock influences determination of this scaling factor.
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Material and methods |
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Overview of sequential use of models
Figure 1 is a schematic representation of our approach to short-term stock assessment of L. gahi in the Falkland Islands, by sequential use of two different models: SBPM and SDM. For the pre-season assessment, a 14-day survey provides an advanced estimate of squid biomass, and proportions by sex and size structure in the stock at a point in time, taken as the day at mid-survey, and this biomass is then projected forward to the starting day of the season with the SBPM. A separate and independent estimate of biomass on the starting day of the fishing season is produced in the in-season phase using the SDM, based on commercial daily fishing data. These independent biomass estimates from pre-season and in-season phases are used for comparative purposes. In the final post-season assessment, the biomass estimate at the end of the season from the SDM is projected forward to the assumed date of spawning by new application of the SBPM, this time directly connecting the two models. Statistical uncertainty is propagated throughout the three phases by Taylor series approximation to estimation variances.
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Survey procedures
Four trawl surveys were made, each on a different vessel of the commercial fleet and following a pre-designed non-random plan. The main objectives of the surveys were to estimate an index of abundance in biomass units on the fishing grounds, an area of ca. 7027 km2 (Figure 2), as well as the biological structure of the stock in terms of proportion by sex and length frequency. To provide an incentive to fishing companies, surveys were planned to fulfil both the scientific purpose of obtaining a large number of local observations of squid density and the commercial purpose of taking a reasonable catch. Under this scheme, three or four long trawls (2–4 h) for commercial purposes were carried out daily, at locations selected by scientists. This scheme produced too few observations of local squid density to allow a statistically meaningful estimate of biomass index to be produced. Long duration trawls pool several capture events of discrete aggregations of squid encountered by the fishing gear as the trawl progresses. Therefore, a method was devised to allocate the total, catch of a trawl into discrete capture events, using acoustic information from the net's echosounder. The method consisted of recording two variables related to the passage of squid "marks" on the net's echosounder screen during every
15 min of each trawl. The variables were the frequency with which marks were seen during each 15-min period (ratings of 0 for no mark, and 10 for continuous marks) and the average quality of the mark (1 for small, light-coloured marks, and 10 for large, dark-coloured marks). The records were taken by the two most experienced fishers only, the skipper and the first officer, after some calibration of the scales used. The latitude and longitude and the initial and final time of each 15-min period were also recorded. A convex linear combination of mark size, mark frequency, and precise duration was applied to the total trawl catch to allocate it into each 15-min period. Positional and net opening information was used to compute the distance travelled and the area swept during each 15-min period. Thus, the number of local density points was augmented between fivefold and eightfold in relation to the number of trawls (Table 1).
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Estimation of total biomass was carried out using a parametric spatial model on the geoR package of the GNU statistical programming system R (Ribeiro and Diggle, 2001). An important advantage of the parametric geostatistical approach is that it produces a maximum likelihood estimate (MLE) of total biomass, whereas estimates from conventional geostatistics are not based on maximizing a likelihood function. Strictly, the geostatistical model should account for the extra dependence in the data generated by the catch-allocation scheme. This is because local density observations in each
15-min period that belong to the same trawl might be correlated not only through spatial proximity (a factor accounted for in the geostatistical model), but also through specific conditions of that trawl, such as weather, currents, and seabed features. This potential source of dependence in the data is ignored in the present applications.
The catch during the surveys of February 2005 and 2006 was composed exclusively of ASC squid, whereas the catch of the May surveys in both years contained both ASC and SSC squid. Therefore, the total catch of every May survey trawl was further split into ASC and SSC components using maturity stages. Females of maturity stages 3, 4, and 5 and males of maturity stages 4 and 5 of Lipi
ski's (1979) maturity scale were considered to belong to the ASC. Furthermore, because of differential growth rate by sex, the catch of each trawl was also split into a female and a male fraction. The spatial model was then fitted to SSC squid of each sex separately. Total numbers were estimated by dividing total biomass by mean body weight. The latter was estimated for each sex using length frequency data and a model of the relationship between length and body mass, fitted by maximum likelihood (discussed later).
Stochastic biomass projection model
The main output of the SBPM is a maximum likelihood curve of the biomass of a single cohort from a given date when the size in numbers of the cohort was estimated, into the future, and a measure of uncertainty along the curve by Taylor series expansion. The SBPM can be used when the numbers have been estimated from a survey or from a fisheries-based stock assessment model. Although the SBPM accounts for almost all sources of statistical uncertainty in the original data, it does not, per se, involve any process of optimization (i.e. the model is not fitted). Rather, it is an analytical projection of previous optimization exercises that carries over the uncertainty from the estimation processes to the final result.
The SBPM considers five pieces of information. First, a MLE of total numbers in the stock and estimation variance before the date to which the projection applies. The pre-season applications consider the results of surveys carried out before the second seasons of 2004 and 2005 and the first seasons of 2005 and 2006. The post-season applications consider the numbers left at the end of the seasons, estimated by the SDM. Second, it takes into consideration concurrent biological information on structure in the form of a length frequency sample. For a pre-season application, the SBPM uses the length frequency from random samples of the catch of the surveys, and for a post-season application, it uses the same type of information from the last day of the fishing season. Third, it requires MLE for parameters of a length/body mass model and the estimated covariance matrix of those estimates. Fourth, individual growth-rate model parameters are needed. In L. gahi, the growth rates of males and females differ (Arkhipkin and Roa-Ureta, 2005), so for each fishing season the model was run as if it was composed of two separate cohorts, a female and a male cohort, with their given total number, length frequencies, length–body mass model, and growth-rate model. The fifth item of information required is the rate of natural mortality. Currently, the model uses a fixed value.
By definition, the biomass B of a cohort of identical individuals at a point in time t0 is
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The age-structured formulation in Equation (3) is not convenient because it requires age reading of statoliths, a time-consuming procedure that does not easily provide the rapid information needed in the present approach to stock assessment. On the other hand, a length-structured formulation utilizes data readily available from biological sampling of the catch, at the cost of making the cohort dynamics mathematically more complicated [Equations (6)–(9)]. In terms of length structure, the biomass at time zero is
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(body mass/length) and ß (dimensionless) are the parameters in the power function of the mantle length/body weight relationship. All parameter estimates in Equation (5) are MLEs.
Daily evolution of the cohort from time t=t0=0 to t0+T, where T is the number of days in the projection, is determined by dynamic models for the total number of individuals Ns and the length frequency distribution fl,s. The biomass at any time step after t0 is
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Deterministically, the proportion growing from k to l depends on the rate of growth at mk, the width of length categories (constant and equal to 0.5 cm), and the length of the time step (constant at 1 d). Arkhipkin and Roa-Ureta (2005) show that Schnute and Gompertz models are equally good growth models for L. gahi. For generality's sake, we here adopt the former because the Gompertz model is a particular case of the Schnute (1981) model. A reparameterized version of the Schnute model, where the inflection point of the curve replaces one of its length-dimensional parameters
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A stochastic version of the growth model described by Equations (4)–(10) can be constructed by filling the entries of the growth transition matrix with the values from a probability density function (p.d.f.). We use here the Gamma p.d.f. because it is a flexible probabilistic model for a positive random variable (r.v.). The r.v. is the mantle length Ml that individual squid of mantle length mk will reach after one time step. In our model, it has expected value mk+
k t, meaning that it is expected that after a time step of 1 d, individual squid of mantle length mk will be of mantle length mk plus the daily growth rate determined by Schnute growth model. As in the Gamma p.d.f., the expected value equals the product of its two parameters, the shape 
k and the scale 
, where the shape parameter k=(mk+k)/, the scale parameter is a new free parameter that arises as a result of the stochastic growth formulation. Under the stochastic formulation, some squid will have a positive probability of growing many centimetres in 1 d. This is deemed impossible on biological grounds, so we introduced a further parameter representing the maximum increase in length categories that any individual can perform in 1 d. We set this at 1 cm or two 0.5 cm length categories, so in effect our stochastic model utilizes a truncated Gamma model for stochastic growth, where only the main diagonal and the two neighbouring lower diagonals of the growth transition matrix can have positive values and all other cells are zero.
To fix the value of the parameter for each sex in the pre-season applications, we projected the population relative length frequencies from the pre-season surveys to the first day of the depletion episode, using the SBPM. Then, we visually compared this prediction with the relative length frequencies observed by sampling from the catch by observers at sea. To reduce sampling variation in the latter, we averaged the relative length frequencies observed during the first 5 d of the depletion episode. We visually fixed the parameter so that the modal length of both the predicted relative length frequency and the observed relative length frequencies coincided. For the post-season application, there were no data at the end of the projection (the assumed spawning date) to fix the parameter, so we used the same value from the pre-season application.
Under these specifications, the entries of the growth transition matrix are
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With this information, the starting biomass is projected to a desired time, and through the functional invariance property of MLEs, the projected biomass estimates are also MLEs. The propagation of statistical uncertainty through time was carried out by Taylor series expansion of the estimation variances and covariances of the parameters of the model for B(t0) [Equation (6)], with the exception of the natural mortality rate, which we took as fixed in the current version of the model and with the same value as in the SDM. At time t0, the estimated parameters are the single total in numbers for a given sex, the two parameters of the length/body mass model, and the proportion of the abundance in every length category of the relative length frequency distribution. For the first three parameter estimates (
s(t0), 
s, and 
s), we used the approximate estimation variances and covariances produced by numerical maximization of corresponding log-likelihood functions. For the proportion estimates in the relative length frequencies, we computed approximate estimation variances analytically using 
l(t)(1–l(t))/n. Thus, at any time step, the full approximate variance-covariance matrix 
t was a symmetrical matrix (83x83 in females, 103x103 in males), with the variance in total numbers at t0 in cell [1, 1], the 2x2 variance–covariance matrix of the length/body mass model in cells [2,2; 2,3; 3,3; 3, 2], the time-updated diagonal of relative frequencies variances in the remaining diagonal cells and zeroes in all other cells. The Taylor series approximation is 
(
(t))
D'D, where D is the column vector of derivatives 
B/i with i
, the set of estimated parameters in the model. The SBPM was coded in the GNU programming system Octave v. 2.1.5 (www.octave.org) and in a Calc spreadsheet of the GNU suite OpenOffice.org (www.openoffice.org). Mortality, growth, and length–weight input parameters are shown in Table 2 and abundance parameters in Tables 1 (pre-season) and 3 (post-season). All code written for this work is available from the authors.
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Stock depletion model
The SDM produces an estimate of initial abundance in numbers of a single cohort given data on catch in numbers and effort during a period of depletion attributable to fishing, assuming that no new individuals recruit into or emigrate from the fishing grounds once the process of depletion has started. This type of model started as linear regressions in Leslie and Davis (1939) and DeLury (1947), and was generalized by Chapman (1974), to account for natural mortality. Rosenberg et al. (1990) introduced Chapman's generalized model as a stock assessment tool for squid stocks, noting that squid populations suffer high rates of natural mortality.
Unlike Rosenberg et al. (1990) and later researchers (Agnew et al., 1998; McAllister et al., 2004), we did not assume the existence of a single depletion episode on the fishing grounds. Instead, the fishing grounds were divided into three major areas (Figure 2), based on biological and fisheries data (Arkhipkin and Middleton, 2002), then the existence of separate depletion episodes in these three areas was assumed.
The deterministic model considers catch and population dynamics equations:
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t, obeys
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2 is the variance in the distribution of 
and a nuisance parameter. The likelihood function is
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2 nuisance parameter was replaced by its MLE for every value of the parameters of interest, leading to the profile likelihood function |
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The SDM (and the length/body mass model, discussed later) was implemented in Auto Differentiation Model Builder (ADModel Builder version 6.0.2, Otter Research Ltd., otter-rsch.com/admodel.htm), with GNU GCC C++ compiler version 2.95.5 (gcc.gnu.org).
Natural mortality
In the implementation of both models (SBPM and SDM), the M parameter was assumed known. The value 0.009 d–1 has been used previously (Rosenberg et al., 1990; McAllister et al., 2004). When M=0.009 d–1, a cohort is reduced to 1% of its original size after 500 d. Intuitively, it is expected that across the many years of biological sampling and age reading, some squid would have appeared that were 500 d old or older. However, the maximum age observation obtained by counting daily growth increments in statoliths from squid collected on the fishing grounds has been about 350 d, for both sexes (Arkhipkin and Roa-Ureta, 2005). Therefore, the assumed value of M was recalculated using Hoenig's empirical equation based on longevity, as argued in Hewitt and Hoenig (2005), to arrive at M0.0133, with longevity as the maximum observed age.
Length/body mass model
The SBPM deals directly with biomass, but requires sex-specific estimates of the parameters of the length/body mass relationship. On the other hand, the SDM requires as input the daily catch in number for both sexes pooled, and the global daily catch is reported in kg. Therefore, for the SBPM, sex-specific power length/body mass models were estimated, and for the SDM the same model was estimated, but with data pooled across sexes (2795 females, 3310 males). In the latter, the daily catch in numbers was calculated as the daily catch in kg divided by the daily mean body mass. This in turn was estimated by transforming the daily length frequency into a daily body mass frequency using the sexes-pooled length/body mass model, and then computing the daily mean body mass (
t).
In each of the three models (females, males, and pooled), the stochastic model considered a multiplicative random process,
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is a lognormal random variable whose logarithm has mean 1 and variance w2, a nuisance parameter. MLEs for and ß were obtained by maximizing the profile likelihood |
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In calculating the daily catch in numbers, we ignored the uncertainty attributable to using the sample mean body mass and the MLEs of and ß, and . However, we considered this uncertainty when estimating the initial biomass from the SDM estimate of initial numbers. Mean body mass of the sample of length frequency data at day t was
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(..) is the estimated covariance. In Equations (18) and (19), the sums are over the individuals making up the daily sample of length frequency, of size nt. Continuing with the expansion, the approximate estimated variance of initial biomass in the SDM, 
, is
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Spawning-stock biomass
It is assumed that the time of spawning for the ASC is 15 May, 1 month after the end of the first season, and that the time of spawning of the SSC is 15 October, 15 d after the end of the second season (Arkhipkin et al., 2004). The numbers at the last day were estimated from the results of the SDM. The total number in the stock at any time step except the initial step was calculated as a function of the MLEs of N0 and q. They are MLEs by virtue of the functional invariance property, but their measure of precision is not directly available. Taylor series expansion was used to calculate a measure of precision for the numbers at the last time step of the depletion episode, NT , as follows:
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and 0 was ignored, because in our applications it was almost zero. Octave code was written to carry out this computation.
The SBPM was run up to the assumed spawning date using the length frequency of the last day of each depletion episode, with the parameter fixed at the same value obtained from the pre-season application in the corresponding season (Table 1). For all episodes except Period 1 in the first season of 2005, the SBPM started to run the day after the end of the season. The assumption of no migration in or out of the fishing grounds during the depletion episode is relevant here again, because migration into the spawning grounds is assumed to occur after the last day of the season.
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Results |
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Survey-based stochastic biomass projection
On any given trawl there were low, high, and intermediate density observations (Figure 3). It, therefore, appears that any dependence in the data created by the catch allocation scheme based on net-sounder information was weak. The SBPM-based predictions of relative length frequencies from the survey to the sample taken by observers on the first day of each depletion episode show that the modal length value can be predicted reasonably well, but that the dispersion is always higher in the prediction (Figure 4). For the first season of 2005, the SBPM could not predict the relative length frequencies observed at the start of the first depletion episode (discussed later), but it does predict the relative length frequencies at the start of the second depletion episode (Figure 4). The SBPM predicted decreases in the biomass index from the survey to the first day of the depletion episode for both second seasons and predicted relatively unchanged biomass indices for both first seasons (Table 1). Projected biomass index estimates for the first seasons are more precise than those for the second seasons (Table 1). In the second season of 2005, there was a strongly biased sex ratio (towards males) during the survey (Table 1).
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Fisheries-based stock depletion and spawning biomass
During the second season of 2004, the vessels showed a spatially dynamic behaviour, switching grounds often after the first two weeks of activity. All 16 vessels started in the south (Beauchene), then moved to North and Central. Greatest abundance of squid was estimated to have been in the Beauchene region (Table 3). The model followed the ups and downs of total daily catch by the fleet, although it underestimated the largest catches (Figure 5). A gradual decline in daily catch, consistent with the assumptions of the model, was observed in the three regions (Figure 5). The estimate in the Central area is more imprecise than in the other two areas because much less effort was expended there. The biomass projected with the SBPM to a date averaged over the starting date of the three depletion episodes (22 July; Table 1) was only 35% of the biomass estimated by the SDM over the three areas. The post-season spawning biomass was well above the minimum escapement level (10 000 t; Table 4).
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During the first season of 2005, the fleet fished exclusively in the Beauchene region, although this did not mean that there was a single depletion episode. Closer inspection of the spatial distribution of the fleet (using INMARSAT vessel positioning data, not shown) revealed two episodes. During the first 21 d (1–21 March, or the 60th to the 81st day of the year), the fleet fished mostly inshore of Beauchene Island, but also offshore, and thereafter they fished exclusively offshore of Beauchene. Examination of the length frequency of the squid caught revealed a break between the 81st and the 82nd day of the year, with larger squid in the second period (Figure 6). We interpret these results as showing that in the austral summer of 2005, there were two pulses of recruits into the Beauchene region, and that during the subsequent fishing season there were two separate depletion episodes, one for each pulse (Figure 7). The evolution of the length frequencies as predicted by the SBPM shows that during the survey, only the offshore stock was observed (Figure 4). The abundance of the offshore stock was very high, and initial numbers and catchability coefficients were estimated with great precision for both depletion episodes (Table 3). The biomass projected with the SBPM to the exact date of the depletion episode over the offshore stock (22 March; Table 1) was just 36% of the biomass estimated by the SDM. The spawning biomass was far above the escapement level (Table 4).
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During the second season of 2005, the fleet operated almost exclusively in the Beauchene and North regions, and as a consequence, the biomass estimate for the Central region is unreliable and is not reported here. The estimated abundance was lowest during that season (Table 3), and a sudden drop in daily catches was observed by mid-season in Beauchene and early in the season in the North region (Figure 8). The biomass projected with the SBPM to the average starting date of the depletion episodes in the Beauchene and North regions (31 March; Table 1) was just 20% of the biomass estimated by the SDM. The spawning biomass was slightly below the escapement level (Table 4).
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During the first season of 2006, the fleet operated almost exclusively in the Beauchene region. The estimated abundance by number was high, but the squid were smaller, leading to only moderate estimates of biomass (Table 3). The model predicted a gradual decline in catch, but it could not accommodate the lowest observed catches (Figure 9). The biomass predicted with the SBPM to the exact date of the start of the depletion episode was just 31% of the biomass estimated by the SDM. The spawning biomass was well above the escapement level (Table 4).
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Synoptic assessment
Biomass trends from the pre-season application of the SBPM the in-season application of the SDM, and the post-season application of the SBPM show that the biomass decreased when projected from the survey to the start of the depletion process and from the end of the depletion episode to the assumed spawning date in the second seasons, but increased slightly in both projections in the case of the first seasons (Figure 10). Clearly, the biomass decreases overall during the in-season phase, but owing to variations in the sample mean weight used to calculate catch in numbers, it may increase during specific time steps (Figure 10).
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Discussion |
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The schematic representation by Boyle and Rodhouse (2005) of the three phases in the rapid dynamics of fishing on squid stocks can be applied to real squid fisheries. Here it is applied to the L. gahi stocks of the Falkland Islands, by combining a pre-season survey-based SBPM, an in-season fisheries-based SDM, and a post-season fisheries-based SBPM. In this combined approach, the pre-season survey-based SBPM potentially gives an alternative estimate of absolute biomass to the fisheries-based SDM estimate with in-season data. In our applications, the pre-season biomass projected to the start of the depletion process is much smaller than the absolute biomass estimate from the SDM. We believe that the latter is more solid as an estimate of absolute biomass because it uses more information from the stock, so we interpret the difference as an underestimate on the part of the pre-season, survey-based SBPM. Two factors can be identified as leading to the SBPM's underestimate of absolute biomass at the start of the depletion episode. First, survey-based estimates of biomass are connected to true stock biomass by a "scaling factor" parameter [Equation (3) of Hilborn, 2000]. This scaling-factor parameter is generally related to avoidance of the fishing gear by part of the stock, which would explain the lower estimate from the SBPM in relation to the SDM. Note that in the latter model a parameter comparable to the survey-abundance scaling factor, namely the fraction of the total stock taken by one unit of effort or catchability coefficient, is included in the model so is taken into account in the estimate of abundance. Second, the pre-season application of the SBPM may be based on a survey carried out before the stock has fully migrated onto the fishing grounds where the survey is carried out. A clear indication that this timing factor related to the biological condition of the stock has also affected the pre-season results lies in the fact that when the survey-projected biomass has been biased towards one sex (male-biased in the second season of 2005, female-biased in the first season of 2006; Table 1, Figure 10), the underestimate has been stronger (20% in the second season of 2005, 31% in the first season of 2006). A corollary of this argument is that when the sex proportion is balanced, the underestimate of biomass at the start of the depletion process only reflects the value of the scaling factor between the survey index and the absolute biomass. Coincidentally, in the second season of 2004 and the first season of 2005, the sex ratio was unity, and the underestimate of biomass was constant at 35%. We interpret this value as the scaling factor between the survey biomass estimate and the absolute biomass of the L. gahi stock around the Falkland Islands.
Currently, there is interest in more complete use of fisheries-independent data to assess stocks (the FISBOAT project of the EU is an example), given some notorious failures of complex populations models based on commercial catch data. Our results indicate that, at least in estimating absolute biomass, survey data and models must account for the scaling factor parameter or "survey catchability". Also, our results show that determination of the scaling factor parameter can be influenced by the timings of surveys and the biological processes of the stock. In the case of the L. gahi stock around the Falklands, migration from spawning grounds and spatio-temporal segregation of the sexes (Arkhipkin and Middleton, 2002) means that the survey must be carried out when both sexes have completed their migration onto the fishing grounds for the scaling factor to be determined and applied correctly.
In addition to its use as an implementation of the Boyle and Rodhouse (2005) conceptual model, the combination of the off-season SBPM and the in-season SDM leads to relevant insights into the functioning of the stock and the fishery. First, the SBPM predicts decreases in squid biomass from survey to season only for the second seasons, whereas for the first season it predicts stability (2005) or an increase (2006). This is explained by the fact that the second seasons are closer to the spawning date and mortality is decreasing biomass faster than it is created by individual growth. Therefore, the two seasons are timed differently in relation to the evolution of cohort biomass. Second, the projection of length frequencies from survey to season in the first season of 2005 implies that there were two pulses of recruits into the fishing grounds in the Beauchene region. In fact, two depletion episodes were observed and fitted. The pre-season survey biomass was observed mostly inshore, and the projection of the length frequencies from survey to season with the SBPM showed that only offshore length frequencies could be predicted by the SBPM. This means that the stock observed inshore during the survey was offshore during the season, and that the season inshore stock recruited onto the fishing grounds after the survey. Within a month, the whole squid stock in the Beauchene region was replaced. This again demonstrates very fast spatial dynamics in squid stocks and highlights a need to disaggregate information into the smallest possible spatio-temporal units.
In the light of the above, an examination of the spatio-temporal dynamics of the fleet becomes a necessary component of the stock assessment of the L. gahi stock by the SDM. More than one depletion episode, some quite short-lived, may occur in a given season, either in distant regions within the fishing grounds (as in the second seasons of 2004 and 2005) or at short distances within the same region (the first season of 2005). Previous studies (Rosenberg et al., 1990; Agnew et al., 1998; McAllister et al., 2004) have successfully developed several aspects of the depletion modelling approach but have ignored the spatial dimension of the fishing process, and have pooled data into coarse time steps (weeks), attempting to fit a single depletion episode. The pooling of data may obscure the occurrence of several depletion episodes at time scales shorter than the duration of the season itself. First, when fishers observe depletion in a given area, they switch to an alternative area to maintain the greatest yield. Second, the stock itself may be structured in waves of recruits of the same cohort, and the fleet can fish on each wave separately. In such cases, the SDM on the pooled data may at best estimate average processes or may fail entirely, such as in 1998 and 2000 (McAllister et al., 2004).
From a statistical perspective, implementation of the SDM here differs from previous work. In Rosenberg et al. (1990, p. 346), implementation of the model depended on a single parameter, N0, whereas the parameter representing fishing catchability was in practice treated as nuisance and calculated after estimation of N0. The likelihood function was only maximized in the N0 dimension. This need not be so. The catchability coefficient is a parameter of interest in itself, so here the likelihood function was maximized with respect to both N0 and q. The observed correlations between these two parameters for all fits were less than 0.01. Additionally, in Rosenberg et al. (1990), the dependence of the value of the only fitted parameter, N0, on the nuisance parameters q and 2 was treated in such a way that the likelihood function was approximated by an estimated likelihood function for N0. In this work, elimination of the single nuisance parameter 2 leads to a profile likelihood function. Estimated likelihood functions are too narrow, giving a false impression of precision (Royall, 1997). Moreover, in the SDM implemented here, the observable variables are catch and effort, whereas in other implementations (e.g. McAllister et al., 2004) the observable variables are the catch rate (i.e. catch divided by effort) and the catch. Here the magnitude of the catch depends stochastically on the magnitude of the effort, whereas in the alternative catch-rate approach, the magnitude of the catch rate depends stochastically on the magnitude of the catch. When the catch vs. effort formulation is combined with the vessel-day unit of effort, the statistical aspects of the model are greatly simplified. This is because the model takes the traditional form of a random variable (catch) being dependent on a variable known exactly (effort in vessel-days), whereas in the catch-rate approach of McAllister et al. (2004) and others, with hours of trawling as the unit of effort, the model portrays the dependence relation between the ratio of two random variables (catch divided by the number of hours of trawling) and the numerator of the ratio.
The SDM assumes that the stock in numbers on the fishing grounds only decreases, and that it does so only through natural and fishing mortality. This assumption may hold approximately for the case of the shorter first season, but it can be problematic for the longer second season. One indication that emigration could have been occurring in the L. gahi stock is the fact that the catch at the end of both second seasons, and consequently the spawning biomass estimated by the SBPM, was strongly biased towards males. A further biological signal of emigration that applied in particular to the second season of 2005 was that maturation was earlier compared with the second seasons from previous years (data not shown). Also, in the second season of 2005 the sudden drop in daily catches observed early in the North region and at mid-season in Beauchene may be due to early emigration inshore to spawn. If there was an unaccounted process of emigration during the second season of 2005, then the catchability parameter was overestimated and the initial abundance parameter underestimated. Consequently, the SDM applied to the L. gahi stock needs to be generalized to include migration. The model that Brodziak and Rosenberg (1993) developed for L. pealei included a linear term for population flow that depended on data on catch rates inside and outside the fishing grounds, data that are not available for the L. gahi stock. We believe that the SDM applied to the L. gahi stock can be generalized by including a migration rate term dependent on the readily available biological information on maturity stages.
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John Barton suggested significant improvements to the approach to stock assessment shown here, and Panayiota Apostolaki produced a thorough review that contributed hugely to improving the original manuscript. An anonymous reviewer also helped clarify the submitted text. Part of this work was presented at the 2004 ICES Annual Science Conference in Vigo, Spain.
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