Estimating stock status from relative abundance and resilience

Rainer Froese *, Henning Winker, Gianpaolo Coro , Nazli Demirel , Athanassios C. Tsikliras , Donna Dimarchopoulou, Giuseppe Scarcella, Maria Lourdes Deng Palomares, Manuel Dureuil, and Daniel Pauly 8 Department of Marine Ecology, GEOMAR Helmholtz Centre for Ocean Research, Düsternbrooker Weg 20, Kiel 24105, Germany DEFF—Department of Environment, Forestry and Fisheries, Private Bag X2, Vlaeberg 8018, South Africa Centre for Statistics in Ecology, Environment and Conservation, Department of Statistical Sciences, University of Cape Town, Private Bag X3, Rondebosch, South Africa Department of Engineering, ICT and Technologies for Energy and Transportation, Institute of Information Science and Technologies “A. Faedo”— National Research Council of Italy (ISTI-CNR), Via Moruzzi 1, Pisa 56124, Italy Department of Marine Biology, Institute of Marine Sciences and Management, Istanbul University, Istanbul 34134, Turkey Department of Zoology, Laboratory of Ichthyology, School of Biology, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Institute for Biological Resources and Marine Biotechnologies National Research Council of Italy (IRBIM-CNR), L.go Fiera della Pesca, 60125 Ancona, Italy Sea Around Us, Institute for the Ocean and Fisheries, University of British Columbia, 2202 Main Mall, Vancouver, BC V6T 1Z4, Canada Department of Biology, Dalhousie University, 1355 Oxford Street, PO BOX 15000, Halifax, NS B3H 4R2, Canada Sharks of the Atlantic Research and Conservation Centre, Trail, BC V1R 4B1, Canada *Corresponding author: tel: þ 49 431 600 4579; e-mail: rfroese@geomar.de.


Introduction
The Law of the Sea (UNCLOS, 1982) commits its signatories to manage the exploitation of fish and invertebrates so that these populations are large enough to generate maximum sustainable yields (MSY). National and regional implementations of this MSY framework have made it clear that for risk-avoidance and economic benefits, biomass (B) of stocks must be above the MSY level (Bmsy) or proxies thereof (CFP, 2013;HSP, 2018) and fishing pressure (F) must be below the MSY level (Fmsy) (UNFSA, 1995;MSA, 2007). However, exploitation level and stock status are unknown for most exploited populations or stocks because the data required for full stock assessments are missing. Methods that make best use of available data combined with general knowledge and Monte Carlo approaches have recently been developed on the basis of previous work in fish population dynamics (Graham, 1935(Graham, , 1939Schaefer, 1954Schaefer, , 1957Beverton and Holt, 1957;Ricker, 1975), such as CMSY (Froese et al., 2017) for catch data and LBB (Froese et al., 2018a to estimate relative biomass levels from length frequency data. This study applies a similar approach to observed trends in the relative abundance of exploited species and thus complements CMSY and LBB. Fisheries-independent surveys carried out year after year with standardized gear and in a random or stratified fashion produce time-series of indices of fish abundance also referred to as catch-per-unit-effort (cpue), conventionally in units of catch in numbers or weight per duration of gear deployment or per swept area. Cpue obtained from such standardized research surveys is a good indicator of abundance (see e.g. Silliman and Gutsell, 1958 for experimental confirmation). Abundance estimates (relative or absolute) can also be obtained using hydroacoustic methods, as practiced for half a century for the stock of Peruvian anchoveta (Engraulis ringens; see Pauly et al., 1987). Time-series of abundance are useful in that they allow trend analyses such as comparing current cpue to the average of previous years (e.g. ICES, 2017). However, it is often unclear whether such abundance trajectories refer to a stock fluctuating around unexploited stock size, around a stock size close to collapse, or somewhere in between. If reliable catch data are available, this ambiguity is best addressed by combining cpue trends with catch data, e.g. in surplus production models (Schaefer, 1954;Pella and Tomlinson, 1969;Fox, 1970;Froese et al., 2017;Pedersen and Berg, 2017;Winker et al., 2018).
If no reliable data are available for the total catch taken from a stock, as is often the case in migratory species, widely dispersed stocks, bycatch species, or species with high discard rates, independent assessments of relative stock size can be used as priors for modelling, such as derived from expert opinion or preferably from other data sources such as length frequency data (LBB, Froese et al., 2018a. One must be conscious, however, that such use of independent assessments of relative stock size to present the observed cpue in an MSY framework fully depends on the quality of the independent assessment and is not informed by the cpue data. This study aims to overcome this limitation by performing a joint analysis of abundance trends, independent stock size information, and readily available information on the resilience or productivity of the respective species.

Material and methods
Theoretical background For stocks that lack information on age, natural mortality, or recruitment, but have reliable time-series of catch and abundance, surplus production models are the method of choice for estimating stock status and exploitation. Based on Graham (1935), the Schaefer (1954) model estimates surplus production or equilibrium yield (Y) from biomass (B), maximum intrinsic rate of population increase (r, sometimes called rmax), and carrying capacity (k): The difference form of the Schaefer model predicts the biomass in the next year (Bt+1) from the current biomass (Bt) plus surplus production or yield (Yt) minus catch (Ct): Note that the expression (1 -Bt/k) describes the linear decline with relative biomass of the applicable fraction of r, resulting in a factor of 1 when Bt = 0 and zero when Bt = k.
In surveys that deploy a standard gear in a random or stratified fashion across an area, cpue is usually assumed to be directly proportional to the abundance or biomass of the target species in that area. The relationship between cpue and biomass is then determined by the catchability coefficient q (Arreguin-Sanchez, 1996;Maunder and Punt, 2004), which is here assumed to be constant over the considered time-period (see below for exceptions) such that: Multiplying both sides of equation (2) with q and replacing Bt q with cpuet gives: Solving equation (3) for Bt and inserting in equation (4) gives equation (5) (Froese et al., 2017): For the purpose of estimating relative exploitation and stock status, it is not necessary to know the absolute values of Ct, Bt, k, and q. One can instead treat Ct q as relative catch Cq t and k q as relative carrying capacity kq, or the cpue one would obtain if there were no commercial fishing: Equation (6) can be rearranged to predict relative catch Cq up to the second last year in the time-series: In the Schaefer model, the maximum sustainable catch (MSY) is obtained at half of k and half of r with MSY = r/2 × k/2. A similar expression is obtained in: where MSYq is the maximum sustainable value of relative catch Cq and, therefore, the ratio Cq/MSYq is the same as the ratio C/MSY. This logic, which is very similar to the matter covered in Ricker (1975, p. 316), also means that the relative catch predicted from equation (7) can be presented in an MSY framework. Similarly, since kq represents the expected value of cpue in the absence of fishing, the ratio cpuet/kq is the same as the ratio Bt/k; therefore, cpue can be presented as the relative biomass within an MSY framework.
Finally, in the Schaefer model, fishing mortality F is equal to the ratio of catch to biomass Ct/Bt, which is identical to the ratio Cq t/cpuet. Hence, the fishing mortality that corresponds to MSY is Fmsy = 0.5r; therefore, relative exploitation can be presented within an MSY framework: The abundance-MSY (AMSY) approach A time-series of cpue and prior ranges for r and relative stock size Bt/k in a given year are required input data for AMSY. A prior range for kq is derived from Bt/k and cpuet as described below. With this information, the time-series of cpue can be placed within a preliminary MSY framework where half of the endpoints of the kq range are used as ranges for Bmsy_q. A multivariate log-normal random sample or r-kq pairs is then created based on the variance-covariance matrix (VCM) shown in equation (10), where the prior log ranges of r and kq are assumed to represent four standard deviations, and variance is standard deviation squared. The r-k q covariance in log space is obtained from the empirical correlation between log r and log k q obtained as median = -0.607 across 140 stocks (EU_Stocks_BSM_Results.xls, Froese et al., 2018b) analyzed with a Bayesian Schaefer model (Froese et al., 2017), and from the prior standard deviations (s.d.) of log r and log kq, such that covariance log r-kq = -0.607 × s.d. log(r) × s.d. log(kq) (see cloud of grey dots in Figure 1).
As shown in the simulations in Section 1 of the Supplementary material and in Figure 1a, this procedure will result in a predicted central r-kq pair in the middle of the prior r-kq log space, with approximated confidence limits as wide as that space. In order to better accommodate "true" r-kq pairs that are off the centre and to reduce the amount of uncertainty, AMSY applies filters to exclude r-kq pairs that would give unreasonable results when combined with the priors and cpue data. For example, the relative catch predicted from equation (7) may not become negative or exceed cpue anywhere in the time-series because it is unlikely that a fishery catches all fish in a given year. Only r-kq pairs that fulfill these and additional conditions (see below) are considered "viable" and are retained and used by AMSY to determine the most likely r-kq pair, with approximate confidence limits (see examples in Section 2 of the Supplementary material and in Figure 1b).

Priors for r, kq, and F/Fmsy
Priors for r were derived from FishBase (www.fishbase.org, Froese and Pauly, 2019) for fish and from SeaLifeBase (www.sealifebase.org, Palomares and Pauly, 2019a) for invertebrates, from the section on the species summary page entitled "Estimates of some properties based on models", either as lognormal distributions based on previous assessments or as qualitative indications of resilience from very low to high (Table 1). Resilience was then translated into uniform prior ranges as described in Froese et al. (2017) and reproduced here for easy reference.
A prior for relative biomass B/k can be derived from experts who are asked how stock size was in a year of their choice compared to past stock size when there was little fishing of the species. For example, if the stock was only lightly fished in the beginning of the timeseries, it is reasonable to assume that stock size was more than half of the unexploited level in those years. Such qualitative assessment is then translated into B/k ranges, as indicated in Table 2. Alternatively, and preferably, a quantitative assessment of B/k or B/Bmsy is derived from a previous assessment or from independent data such as length frequencies analyzed with methods such as LBB (Froese et al., 2018a, see examples in Section 3 of the Supplementary material). The year for which the B/k prior is provided depends on the available data, i.e. a year with a good length frequency sample or unanimous expert opinion. For example, if fishing was very light at the beginning of the time-series, experts are likely to agree that stock size was close to unexploited, giving a B/k prior of e.g. 0.75-1.0.
A prior range for kq is derived from B/k as kq = cpuet/(Bt/k). If the lower bound of kq resulting from this exercise is less than the maximum cpue in the time-series, max(cpue) is used as the lower kq bound because abundance of an exploited species is unlikely to exceed carrying capacity. Also, in order to avoid unrealistically narrow or wide ranges, the upper bound of kq is set at least 30% larger than the lower bound, but not farther away than threefold the lower bound. In other words, the B/k prior together with population dynamics and scaling considerations is used to put the observed cpue into a preliminary MSY framework. This placement is then refined by the Monte Carlo filtering described below.

AMSY Monte Carlo filtering
Based on the prior knowledge of B/k, a time-series of cpue can be presented in a preliminary MSY framework, with the cpue range that is capable of producing MSYq given by cpuemsy = cpuet/(Bt/k), using the lower and upper limits of B/k. Pairs of r-kq are then randomly selected from their respective prior ranges, and the time-series of relative catch Cq corresponding to the time-series of cpue is calculated from equation (7).
To account for reduced recruitment and thus reduced productivity or surplus production at very small stock sizes, equation (7) is combined with a hockey stick recruitment function (Barrowman and Myers, 2000;Froese et al., 2016aFroese et al., , 2017. Thus, if relative stock size at the end of the time-series is smaller than half of Bmsy or 0.25 cpue/kq, a linear reduction of surplus production with declining biomass is assumed (similar to the MSY Btrigger rule in ICES, 2016): AMSY applies a state-space model formulation with an annual multiplicative lognormal random process error exp( ) and observation error exp( ) terms with ~(0, 2 ) and ~(0, 2 ), respectively. For cpue observation error, the default was set to 0.3 and for surplus production process error was set to 0.05, 0.07, 0.1, or 0.15, depending on the productivity of the stock from very low to high. These error terms are not shown in the equations for the sake of simplicity. The chosen values are preliminary, but worked well for the purpose of this proof-of-concept study.
For cases where cpue stems not from standardized surveys, but from commercial fisheries and where efficiency of commercial fishing may increase with time, an effort-creep correction can be applied by AMSY based on the average percentage of increase in catching efficiency, as provided by the user: where cpuecorr is the corrected cpue, t is a year in the time-series, p is the percentage of average increase of efficiency as a decimal (e.g. 0.02 for 2%, Palomares and Pauly, 2019a), and t0 is the first year in the time-series.

Filters used to find r-k pairs compatible with the provided cpue and priors
The following numerical settings or multipliers for the filters are derived from preliminary test runs against the simulated data. They were accepted as sufficient for the purpose of this presentation and proof-of-concept of AMSY and are expected to be further refined, also against real data, in subsequent research.
(1) Exclusion of r-kq pairs if predicted relative catch is negative. By definition, catch is an extraction of fish from the population and thus may become zero, but not negative. Therefore, combinations of productivity and carrying capacity that, in combination with the cpue time-series, predict negative relative catches in any given year can be excluded as unrealistic. However, periods of relative catches close to zero are realistic scenarios especially during recovery phases of species with low or very low resilience; during such periods, negative predictions of relative catch may result from the uncertainty and corresponding error terms used in the modelling. Therefore, during such periods, a negative relative catch of 2-6% of kq (for low or very low productivity) is allowed by AMSY.
(2) Exclusion of r-kq pairs if predicted catch in a given year exceeds biomass. It is unlikely that a fishery catches all fish of a stock in a given year. Thus, r-kq pairs that, in combination with the cpue time-series, predict relative catches above the available cpue appear unrealistic. However, looking at equation (7), the term r(1cpue/kq) determines the amount of surplus production, and it may exceed 1.0 if e.g. r > 1.2 and cpue/kq < 0.2, i.e. predicted annual catches may exceed biomass in species with high productivity and depleted stock size. AMSY accounts for this dependence on productivity by using empirical multipliers for the cpue value not to be exceeded by relative catch from 1.4 for high to 0.25 for very low productivity. This filter is skipped if, at any point in the time-series, cpue approaches zero [cpue < 0.1 max(cpue)] because catch may exceed mean biomass under those circumstances.
(3) Exclusion of r-kq pairs if predicted catch strongly exceeds MSY. While it is possible for fisheries to catch more than MSY for a few years, the degree of such overfishing is inversely correlated with the MSY/k = r/4 ratio; in species with very low productivity, MSY is only a small fraction of carrying capacity and can be easily exceeded several fold. In contrast, in species with high productivity, MSY is a quarter or more of carrying capacity and is unlikely to be overshot by more than MSY. Accordingly, a multiplier for maximum predicted relative catch was set from tenfold MSYq for very low productivity to twofold MSYq for high productivity. (4) Exclusion of r-kq pairs if F/Fmsy is negative or unrealistically high. If the timeseries of F/Fmsy ratios predicted by AMSY contains highly unrealistic values, such as less than -25 or more than 12, then that combination or r-kq with its specific error patterns is excluded from the analysis. Note that while negative F/Fmsy ratios require negative catches and thus are not possible in the real world, periods of very low or zero catches are realistic scenarios especially during recovery phases of species with low or very low resilience. Thus, during such periods, predictions of negative F/Fmsy ratios may result from the uncertainty and corresponding error terms used in the modelling. (5) Exclusion of r-kq pairs if modeled cpue/kq is outside the prior B/k range. If the relative cpue (cpue/kq) in the year specified for prior B/k falls outside of that prior range, then the r-k q pair is discarded. Note that all r-kq pairs are tested multiple times with different random error settings for surplus production and cpue and are only excluded from further processing if all of these runs fail to pass the filters. This processing leads to a modelled cpue time-series slightly different from the observed cpue, as peaks, troughs, and slopes that would lead to unrealistic catches or unrealistic productivity are smoothed.

Finding the most likely values for r, kq, F/Fmsy, and B/Bmsy
The r-kq pairs that passed the filters described above were considered as viable. Median values of viable r and kq were considered to be the most likely estimates, and 2.5 th and 97.5 th percentiles were taken as approximate confidence limits, respectively. The time-series of relative catch predicted by the viable r-kq pairs in combination with cpue were stored and a proxy for median Ft was obtained by dividing the median predicted catch by the median of cpue. An estimate of recent F/Fmsy was obtained by dividing Ft by the median estimate of r/2, with t set to the second to last year. Approximate 95% confidence limits were obtained similarly from the 2.5 th and 97.5 th percentiles of predicted catch. The time-series of modelled cpuet were stored, and a proxy for recent B/Bmsy was derived by dividing median cpuet by median kq/2 and setting t to the last year. Approximate 95% confidence limits were obtained similarly from the 2.5 th and 97.5 th percentiles of modelled cpue.

Simulated data
To assess the performance of AMSY, simulated catch and cpue data were created so that the "true" simulated parameter values and stock status estimates were known and could be used for comparisons. For convenience, kq was set to 1000 and r was set at 0.06, 0.25, 0.5, and 1.0 year -1 to represent species with very low (VL), low (L), medium (M), and high (H) resilience, respectively (c.f. Table 1). For time-series of 50 years, biomass patterns of continuously high (HH), continuously low (LL), high to low (HL), low to high (LH), low-high-low (LHL), and high-low-high (HLH) biomass were created. From an "above half" (0.5-0.85 k) or "small" (0.15-0.4 k) starting biomass in the first year, the desired pattern was produced by inserting high or low catches into equation (6) and calculating the biomass in subsequent years. These first year ranges of relative biomass were used as prior for AMSY to reduce the influence of the prior on the estimated relative biomass 50 years later. If relative biomass fell below 0.25 kq in any given year, surplus production was reduced, as described in equation (11) to account for potentially reduced recruitment. A catchability coefficient q = 0.001 was assumed to turn biomass, catch, and k into the desired values of cpue, Cq, and kq, respectively. The simulated data and the spreadsheet used to produce them are part of the Supplementary material.

Real data
For the evaluation of AMSY estimates against real data, 140 stocks from the Northeast Atlantic, the Mediterranean, and the Black Sea were used as a subset of the 397 stocks analyzed by Froese et al. (2018b). Criteria for stock selection were uninterrupted time-series of catch and abundance (cpue, indices, or predicted biomass) for at least 15 years. These data were then analyzed with a Bayesian implementation of the Schaefer model (BSM) which is part of the CMSY package (Froese et al., 2017). AMSY and BSM used the same cpue timeseries and the same priors for productivity and relative stock size in the first year of the timeseries, the only difference being that BSM in addition had time-series of catch as input. The BSM results for the 140 stocks were also used to derive the median correlation between r and kq in log space, as required for construction of a correlation matrix ( Table 1). The data and the results of the BSM analysis are part of the Supplementary material.

First assessments of data-limited stocks
To test the usefulness of AMSY for its intended purpose, 38 data-limited stocks were analyzed first with LBB (Froese et al., 2018a to obtain objective prior information on relative stock size from length frequencies, and then with AMSY to derive estimates of r, Fmsy, F/Fmsy, and B/Bmsy from cpue data. All of the sections in the Supplementary material, data, spreadsheets, and R-code used in this study are available as Supplementary material from http://oceanrep.geomar.de/47135/. The version of LBB (33a) used in this study is available from http://oceanrep.geomar.de/43182/.

Results
Verification against simulated data AMSY predictions of population dynamic parameters (r, kq, MSYq), fishing pressure (F/Fmsy), and stock status (B/Bmsy) at the end of the time-series were compared with the "true" values used to produce the simulated data. In order to better understand the influence of the priors and of the Monte Carlo filtering on the results, the simulated data were analyzed twice by AMSY, first without and then with the Monte Carlo filters described above.
Without the filters, all r-kq pairs of the multivariate distribution are "viable", the central values of predicted r and kq are in the centre of the prior log space, and the respective approximate 95% confidence limits are wide and equivalent to the respective prior ranges. By design, the "true" values of r and k q were within the prior ranges and thus fall within the approximate 95% confidence limits of the predictions. Similarly, all "true" values of MSY q , F/F msy , and B/B msy fall within the approximate 95% confidence limits of the respective estimates.
With Monte Carlo filtering, numerous r-kq pairs are excluded because of unrealistic predictions, and consequently the estimated central values of r and kq may move away from the centre of the prior log space and their approximate 95% confidence limits get narrower and may exceed the original prior bounds (see Figure 1 and more examples in Section 2 of the Supplementary material). With one exception (estimate of kq in LHL_VL, see Section 2 of the Supplementary material), "true" values of all parameters still fall within the narrower approximate 95% confidence limits. Table 3 shows a comparison of median absolute relative error (MARE=abs(trueestimate)/true) and of median relative lower confidence limits (MRLCL=(estimate-lcl)/estimate) for 24 AMSY runs without and with Monte Carlo filtering. In the runs with filters, MAREs are substantially reduced for r and F/Fmsy and only slightly increased for kq and B/Bmsy, and MRLCLs are substantially reduced for all estimates except B/Bmsy. Evaluation against real data AMSY predictions for 140 real stocks were compared with those of a Bayesian implementation of a regular Schaefer model (BSM). AMSY estimates of r were similar to those of BSM (Figure 2), with 128 (91.4%) BSM estimates included in the approximate 95% confidence limits of AMSY. AMSY predictions of relative biomass (B/Bmsy) in the last year included the BSM estimate in their approximate 95% confidence limits in 122 stocks (87.1%). AMSY predictions of exploitation (F/Fmsy) included the BSM estimate in their approximate 95% confidence limits in 123 stocks (87.9%). Note, however, that AMSY confidence limits for F/Fmsy estimates were often wide. The median ratios of AMSY vs. BSM predictions for r (0.92), final F/Fmsy (1.16), and final B/Bmsy (0.99) were used to summarize deviations and detect potential biases. Thus, AMSY predictions were, on average, 8% lower for r, 16% higher for F/Fmsy, and 1% lower for B/Bmsy. Note that these are not entirely fair comparisons because catchability q is not estimated by AMSY, and this may cause part of the observed deviations. A spreadsheet [EU_ StocksResults_2.xls] with the detailed results for every stock is part of the Supplementary material.

Application to data-limited stocks
Application of AMSY to data-limited stocks without reliable catch data produced the first MSY-level assessments for 38 stocks of mostly bycatch species (Table 4). This includes 23 species for which these are the first assessments globally (marked bold in Table 4). The details of these assessments are presented in Section 3 of the Supplementary material. We found overall very good agreement of predicted relative biomass trends between LBB (Froese et al., 2018a based on length frequencies and AMSY based on cpue. There is also general good agreement between current and retrospective analyses, i.e. AMSY runs where data from the last 1, 2, or 3 years were omitted from the analyses. In two North Sea stocks (syc.27.3a47d, rjc.27.3a47d), the retrospective analysis indicated a substantial deviation of predicted relative biomass estimates (B/Bmsy) if the respective last years were included because these years suggested a strong increase in biomass. These increases were accepted for the purpose of this study, but may turn out to be fluctuations once data for the subsequent years become available.
Predictions of exploitation (F/Fmsy) in the second-to-last year indicate that 24 stocks (63%) were subject to overfishing, but note the wide margins of uncertainty. Predictions of relative biomass (B/Bmsy) in the last year indicate that only 9 stocks (24%) were above the biomass level required by UNCLOS (1982) and 21 stocks (55%) were smaller than half of that level, suggesting that successful reproduction may be endangered. Margins of uncertainty for relative biomass are mostly less than 50% with regard to the relevant lower confidence limit (Figure 2c) and thus similar to assessments with more input data.

Choice of Schaefer vs. Fox or Pella-Tomlinson
Several types of surplus production models are used in fisheries, with Schaefer (1954), Fox (1970, and Pella-Tomlinson (1969) being the most common. Of these three, only the Schaefer model is derived from ecological principles, implementing the sigmoid population growth that has been observed in many animal populations (Hjort et al., 1933;Graham, 1939;Hairston et al., 1970;Smith, 1994;Yoshinaga et al., 2001). The Fox model is a logarithmic transformation of the Schaefer model, resulting in MSY being obtained at 37% of carrying capacity rather than at 50%, as in the Schaefer model. This results in the Fox model predicting higher equilibrium yields for a given biomass at small stock sizes, implying that the Schaefer model is more precautionary in the proposed biomass necessary for producing MSY and in the sustainable catch that a given biomass can support (Cadima, 2003; Figure A1 in Appendix 1 of Froese et al., 2011;Tsikliras and Froese, 2019). The Pella-Tomlinson model is a mathematical generalization introducing a shape parameter p for the sigmoid curve, corresponding to the Schaefer model if p = 1 and to the Fox model if p approaches zero (Pella-Tomlinson 1969, Cadima 2003, or with a shape parameter defined as m, such that m = 2 for the Schaefer model and m approaching 1 for the Fox model in later implementations (e.g. Pedersen and Berg 2017;Winker et al., 2018). Beverton and Holt (1957) have shown that the biomass that can produce MSY is actually a function of fishing pressure and selectivity, reaching about half of unexploited stock size if F is close to natural mortality M and length at first capture is close to optimum length (see Figure 2b in Froese et al. 2016b). For AMSY, the Schaefer (1954) model was chosen over the Fox model to err on the precautionary side and over the Pella-Tomlinson (1969) model to avoid estimation of a third parameter in a data-poor situation.

Performance of AMSY
The key point of this study is to explore whether a model that only has a time-series of cpue as input can produce similar results vs. a model that, in addition, has a time-series of catch data as input, everything else being equal. AMSY uses cpue data combined with independent prior knowledge about the resilience or productivity of the species and prior perceptions or estimates of stock status for the year with the best available estimate. It applies surplus production modelling with randomly selected parameters for r and kq to predict catches that are compatible with the cpue time-series and the priors. AMSY aims to improve the precision and plausibility of stock status estimates by applying a set of filters to exclude r-kq pairs that result in e.g. negative catches or unrealistic exploitation values.
To better understand the respective influence of the priors and the filters on the results, AMSY was run against simulated data with and without filters. If no filters were used, the priors determined the central r-kq values with 95% confidence limits about equal to the prior ranges and with already reasonable fits of predicted vs. "true" time-series of relative catch and stock size, albeit with wide margins of uncertainty (Figure 1a). The addition of the filters moved the estimates of r and kq closer to the "true" values and reduced the confidence limits for all estimates except B/Bmsy, which remained about unchanged (Table 3).
In other words, if the relationship between abundance and catch follows the logic of a surplus production model and if the priors for productivity and relative stock size include the "true" values, then AMSY predictions of r, F/Fmsy, and B/Bmsy are not significantly different from the "true" values in simulated data covering a wide range of productivity and relative stock size. The question then is how well these assumptions are met in real world data.
For this purpose, 140 European stocks from the Barents Sea to the Black Sea and including invertebrates from shrimp to octopus and fish from anchovy to halibut (see EU_Stocks_ID_8.csv in the Supplementary material) were analyzed with a Bayesian implementation of a full Schaefer model (BSM) with time-series of catch and cpue as input, and with AMSY with only cpue as input. Both models used the same priors for productivity and relative stock size at the beginning of the time-series.
Results from both models showed good agreement for r, F/Fmsy, and B/Bmsy, with more than 87% of the BSM central estimates included in the approximate 95% confidence limits of the AMSY estimates, thus being not significantly different. AMSY predictions for relative exploitation (F/Fmsy) in the penultimate year had, however, wide margins of uncertainty and thus deviations in predictions could be substantial. Note also that BSM estimates are not free of error, and some of the largest differences were found where the filters applied by AMSY prevented it from predicting extreme values of exploitation (compare Figure 2).
Application of AMSY to selected data-poor stocks from the North Atlantic, Mediterranean, and South Africa provided the first MSY-level assessments of exploitation and stock status for 38 stocks and 35 species (Table 4). The stocks were chosen because they had no previous MSY-level assessments and no reliable or no catch data, but length frequencies as well as cpue data available. The species range from bycatch, such as eelpout (Zoarces viviparus) in the Baltic, to commercially important species such as common octopus (Octopus vulgaris) in the Adriatic Sea. Note that for all these stocks, objective prior information on relative biomass depletion was provided from the analysis of length frequency data (LBB, Froese et al., 2018a. The wide margins of uncertainty for predictions of relative exploitation (F/Fmsy, Table 4) are not surprising, given that no information on catch was available for these stocks. Therefore, these predictions of exploitation should be used with caution. In contrast, the margins of uncertainty for predictions of relative stock size are within usual ranges and, therefore, are suitable for management advice. With few exceptions, the predicted relative biomass B/Bmsy was below the level that can produce maximum sustainable yields, and about half of the stocks were so small that successful reproduction may be endangered. While the selection of stocks was not random and was, therefore, not representative of non-assessed species in general, the results underline the need for MSYlevel assessments and management of data-poor stocks.

Properties and assumptions of AMSY
The Schaefer (1954) surplus production function used by AMSY captures with only two parameters the interplay among somatic growth, reproduction, and natural mortality. AMSY is implemented within a state-space modelling framework (Meyer and Millar, 1999;Froese et al., 2017;Winker et al., 2018) to account for process error due to the real-world variability in size structure, species interactions, natural mortality and recruitment, and observation error resulting from sampling error and variations in catchability. This allows the predicted biomass trajectories to deviate from the deterministic expectations resulting from equations (6) and (7), while keeping the trajectories within plausible biological limits through the use of the productivity prior, the associated process variance, and the filters imposed to identify viable r-k pairs. This means that the time-series pattern of the predicted relative abundance (B/Bmsy) may differ from the pattern of the cpue provided as input to the model, within the bounds determined by the error terms for process and observation, which can be set by the user.
AMSY assumes that there is a direct proportionality between cpue and exploited biomass. However, catch rates in commercial and survey fisheries may be influenced by factors such as fishing vessel type, where and when fishing occurred, gear used, depth of fishing, and whether fishing occurred during day or night. There are also cases of reduced catch rates because of depredation e.g. on longlines by various predators (Söffker et al., 2015) or because of predator avoidance behavior by fishers shifting into less optimal cpue areas (Haddon, 2018). Management regulations such as size and catch limits or closed areas and seasons may also impact cpue. These factors, and any changes therein over time, may obscure the inter-annual changes in cpue resulting from changes in stock size, which are the focus of AMSY (e.g. Sporic and Haddon, 2018).
As shown with the simulated data, predictions of AMSY come with high margins of uncertainty in stocks with very low resilience and periods of very low exploitation. Also, predictions of exploitation (F/Fmsy) come with wide margins of uncertainty and may be especially misleading during phases of low exploitation (Figure 2b).
When deriving management advice from cpue, it is important to consider situations where the catch per unit effort may be significantly biased, potentially resulting in biased advice. One such bias is the continuing increase in the efficiency of fishers to catch a certain species. This is often a combination of an increase in experience about when and where target species are likely to be found and an improvement in technology ranging from more efficient navigation (GPS, autopilots) to more efficient sonars to more efficient gear. Palomares and Pauly (2019b), based on a comprehensive review of published cases, found this "effort creep" to increase efficiency in commercial fisheries by 1-5% per year, with 2% per year being a reasonable assumption if no better information is available. AMSY provides a correction for commercial cpue depending on the percentage value provided by the user [equation (12)]. Cpue from standardized surveys should not be affected by this.
Another potential bias in commercial cpue data is known as "hyperstability", where cpue remains stable while abundance is declining, leading to overestimation of biomass and underestimation of fishing mortality (Quinn and Deriso, 1999;Harley et al., 2001). This may be caused by a fishery expanding into previously less-fished areas or depth zones (Morato et al., 2006;Kleisner et al., 2014) with the new catches masking the overall decline. It may also be caused by aggregating behavior of the target species, when the centre of the aggregation is fished primarily and the density there remains high even if overall density is declining, as occurred prior to the collapse of northern cod (Gadus morhua) in Canada (Hutchings, 1996), or when the spawning aggregations typical of some tropical species are exploited (see Sadovy de Mitcheson et al., 2008 and www.scrfa.org). In contrast, "hyperdepletion" (Quinn and Deriso, 1999) describes a situation where cpue declines faster than overall stock abundance. This occurred, for example, at the onset of some tuna fisheries, where fishers targeted rapidly declining accumulations of old tuna, but whose biomass was not representative of the entire, more resilient population (Ahrens and Walters, 2005). While "hyperdepletion" will lead to overly pessimistic assessment of stock status, the damage would be limited as fish not caught because of too conservative exploitation will increase the remaining biomass and future catches (Froese et al., 2016b). Cpue from standardized surveys should not be affected by either "hyperstability" or "hyperdepletion".

Conclusion
The purpose of this study was to explore whether a standard population dynamics model can approximate regular predictions if given only one instead of two time-series of input data, everything else (base model and priors) being equal. This is shown to be the case. The question then is the availability of independent reasonable priors. For productivity, this is solved through online databases which offer such priors for practically all commercially important species based either on previous stock assessments or on life history traits. The priors on relative stock size can be derived either from expert knowledge or better from typically available independent data such as length frequencies, as shown here for 38 datalimited stocks.
Summarizing the results of this proof-of-concept study, AMSY seems to be well suited for estimating productivity r and thus Fmsy = ½ r as well as relative stock size B/Bmsy. Estimates of relative exploitation F/Fmsy may come with wide margins of uncertainty and may be less suitable for management, especially at low levels of exploitation. As a first application of AMSY, the first MSY-level stock assessments are presented for 38 data poor stocks for which no reliable catch data are available.    Figure 1. AMSY analysis of simulated data for a stock with very low productivity and low biomass. The grey dots are a random sample of 50 000 points drawn from a multivariate distribution of r and kq in log space. The dotted rectangle indicates the prior ranges of r and kq and contains 95% of the random points. The black dots are "viable" r-kq pairs, with the red cross indicating the most probable value with approximate 95% confidence limits. The blue circle indicates the "true" r-kq pair used in the simulations. In (a), no logical filters are applied and the most probable r-kq pair falls in the centre, with confidence limits about equal to the prior ranges. In (b), logical filters are applied to the selection of "viable" r-kq pairs, with a central value much closer to the true one and much narrower confidence limits which slightly exceed the prior range [AMSY_68Fig1b.R].