Age-structured Production Model Research Articles

Investigating trends in process error as a diagnostic for integrated fisheries stock assessments

Integrated stock assessments consist of fitting several sources of catch, abundance, and auxiliary biological information to estimate parameters of equations that describe the population dynamics of fish stocks. Stock assessments are subject to uncertainty, and it is a common practice to characterize uncertainty using alternative hypotheses and assumptions within an ensemble of models to develop scientific advice for fisheries management. In this context, there is the need to assign levels of plausibility to each of the combinations of factors that ultimately reflect the uncertainty on different biological and fishery processes. In this study, we describe and apply a model diagnostic to identify trends in process error in recruitment deviation estimates within ensembles of integrated assessment models of tropical tunas. We demonstrate that assessment model ensembles for tropical tunas contain distinct scenarios with significant trends in process error that are overlooked, with the associated implications for fisheries management. Using the Indian Ocean yellowfin as a case study, we found that trends in recruitment deviates are linked to extreme productivity scenarios which strongly diverged in scale from deterministic models fitted without recruitment deviates. This indicates that when recruitment deviates show an increasing trend, these can compensate for the loss of biomass in periods of high catch beyond the surplus production. In these cases, variation in recruitment is not a random process, but rather takes the function of a compensatory, systematic driver in productivity. Significant trends in recruitment were positively correlated with increased standard deviations and auto-correlation coefficient, non-random residual pattern in fits to abundance indices, and particularly poor performance of the Age-Structured Production Model (ASPM) diagnostic. We suggest that trends in recruitment deviates can be caused by misspecification of the biological parameters used as fixed values in integrated assessment models. The process error diagnostic described here can provide a statistical criterion in support for hypotheses and assumptions when using ensembles of models to develop fisheries management advice.

Auxiliary diagnostic analyses used to detect model misspecification and highlight potential solutions in stock assessments: application to yellowfin tuna in the eastern Pacific Ocean

Abstract Integrated models (IMs) for stock assessment are simultaneously fit to diverse data sets to estimate parameters related to biological and fishery processes. Model misspecification may appear as contradictory signals in the data about these processes and may bias the estimate of quantities of interest. Auxiliary diagnostic analyses may be used to detect model misspecification and highlight potential solutions, but no set of good practices on what to use exist yet. In this study, we illustrate how to use auxiliary diagnostic analyses not only to identify model misspecification, but also to understand what data components provided information about abundance. The diagnostic tools included likelihood component profiles on the scaling parameter, age-structured production models, catch-curve analyses, and two novel analyses: empirical selectivity and monthly depletion models. While the likelihood profile indicated model misspecification, subsequent analyses were required to indicate the causes as unmodelled changes in selectivity and spatial structure of the population. The consistency between the catch-curve models, the monthly depletion models and the IM information on abundance comes from a strong signal shared by several purse-seine fisheries data sets: the length composition data informs absolute abundance while the indices of abundance constrain the trend in relative abundance.

JABBA-Select: Incorporating life history and fisheries’ selectivity into surplus production models

Age-structured production models (ASPMs) are often preferred over biomass aggregated surplus production models (SPMs) as the former can track the propagation of cohorts and explicitly account for the effects of selective fishing, even in the absence of reliable size- or age data. Here, we introduce ‘JABBA-Select’, an extension of the JABBA software (Just Another Bayesian Biomass Assessment; Winker et al., 2018), that is able to overcome some of the shortcomings of conventional SPMs and allows a direct comparison to ASPMs. JABBA-Select incorporates life history parameters and fishing selectivity and distinguishes between exploitable biomass (used to fit indices given fishery selectivity) and spawning biomass (used to predict surplus production). Applying JABBA-Select involves using an age-structured equilibrium model to convert the input parameters into multivariate normal priors for surplus-production productivity parameters. We illustrate the main elements of JABBA-Select using the stock parameters of South African silver kob (Argyrosomus inodorus, Scienidae) as a case study. This species is exploited by multiple fisheries and was selected as an example of a data moderate fishery that features strong contrast in selectivity over time and across fleets. For proof-of-concept, we use an age-structured simulation framework to compare the performance of JABBA-Select to: 1) a conventional Bayesian state-space model using a Pella-Tomlinson (PT) production function, 2) an ASPM with deterministic recruitment; and 3) an ASPM with stochastic recruitment. The PT model produced biased estimates of relative and absolute spawning biomass trajectories and associated reference points, which by contrast could be fairly accurately estimated by JABBA-Select. JABBA-Select performed at least as well as the ASPMs in accuracy for most of the performance metrics and best characterized the stock status uncertainty. The results indicate that JABBA-Select is able to accurately account for moderate changes in selectivity and fleet dynamics over time and to provide a robust tool for data-moderate stock assessments.

The case for estimating recruitment variation in data-moderate and data-poor age-structured models

The birth and survival of juveniles (termed recruitment) is highly variable in marine populations, and this variation is not precisely predicted by spawning output or environmental effects in isolation. Variation in recruitment after accounting for differences in spawning output (termed “recruitment deviations”) is commonly estimated in age-structured populations models that are fitted to data including samples of length or age-composition (“age-structured integrated models”, ASIM). However, recruitment deviations often are not estimated when fitting only to catch and abundance-index data (“age-structured production models”, ASPM), or catch data given an assumed value for final stock status (“age-structured stock-reduction analysis”, ASSRA). We use a simulation and real-world example to illustrate the potential benefits of estimating recruitment deviations in data-rich (ASIM), data-moderate (ASPM), and data-poor (ASSRA) models, while using either marginal likelihood or tuning algorithms with penalized likelihood estimation. The simulation experiment shows that marginal and penalized likelihood methods are both unbiased and precise for ASIM when estimating recruitment deviations, but biased and imprecise when not. The ASPM is somewhat biased both with or without estimating recruitment deviations, but is more precise when estimating recruitment deviations than not. The ASPM with recruitment deviations estimated using marginal likelihood has somewhat better performance than using penalized likelihood, although maximum likelihood also sometimes does not converge. The ASSRA has similar precision with or without estimating recruitment deviations, but estimating recruitment deviations is important to have reasonable estimates of confidence intervals. Both marginal and penalized likelihood can accurately estimate the variance of recruitment deviations for ASIM and ASPM. Estimating recruitment deviations is necessary to accurately estimate uncertainty in spawning output, and results are robust to mis-specifying the recruitment model (e.g., when simulating data with regime-shift dynamics). A sensitivity analysis shows mis-specifying maturity-at-age causes bias in all models, but bias is somewhat mitigated when estimating recruitment deviations. Finally, a case-study involving Pacific hake off the US West Coast emphasizes that it is necessary to estimate recruitment deviations to closely match results from the original assessment.

Get the biology right, or use size-composition data at your own risk

Weighting of size-composition data (length or weight composition of the catches) can have a large influence on the results of contemporary integrated stock assessment models in the presence of model misspecification. Model misspecification leads to conflicting information among data sets, and the choice of data weighting will determine the results. Information content on absolute abundance and abundance trends contained in size-composition data is particularly susceptible to misspecification of the biological processes. Biological processes are often misspecified in assessment models for exploited fish stocks due to lack of information. The misspecification can be in a functional form (e.g., the growth curve) or in the values assumed for pre-specified parameters. Our application to bigeye tuna in the eastern Pacific Ocean shows how one needs to “get the biology right”, i.e. minimize model misspecification, to reduce the dependency of stock assessment results on the weighting of the various data components. The stock assessment results are sensitive to the conversion from processed weight to total weight, a common, but often overlooked, component of model specification, and to the asymptotic length of the growth curve. The results are also sensitive to the weighting of the composition data. Application of the Age-Structured Production Model diagnostic shows that recruitment variation must be taken into account to interpret the absolute abundance and trend information contained in a CPUE-based index of relative abundance. Unfortunately, recruitment cannot typically be estimated from the relative index of abundance alone, so composition data are needed. The abundance estimates from an age-structured production model with estimated recruitment deviates are too uncertain (i.e., have wide confidence intervals) to be of use for management advice. Therefore, there is a trade-off between using composition data to estimate recruitment and its influence on estimates of absolute abundance through a catch-curve type process. We conclude that (i) integrated analysis, the current approach for assessing fish stocks, is supported by our results; (ii) composition data are needed to estimate recruitment; and (iii) addressing key model misspecifications should be a major component of integrated analysis.

Searching for M: Is there more information about natural mortality in stock assessments than we realize?

Natural mortality (M) is both a highly consequential and difficult process to estimate in stock assessments. Because of its correlation with other influential parameters, and the inadequacy of data to understand how it varies with time and ontogeny, estimates of M from integrated assessment models are often considered implausible. This has lead to the assumption that estimates of M are highly uncertain within integrated assessments and it is commonly fixed at values consistent with understanding about the stock’s life history or derived from external analyses. Researchers recently used simulation analysis to challenge this assumption and provided evidence that M might be estimable under certain conditions. Our research builds upon those results by using the recently proposed age-structured production model diagnostic to help identify the conditions under which M might be estimable. This diagnostic aims to determine if changes in the scale and trend of stock abundance can be explained by catch alone, which is a key indicator of the presence of a production function. We apply the production model diagnostic to the same suit of assessments used in the aforementioned simulations to determine if a relationship between estimability of M and the presence of a production function can be identified. Statistical and subjective approaches to interpreting the production model diagnostic were developed with the aim of providing guidance on when M might be estimable. Statistical approaches to identifying the presence/absence of a production function did not outperform the subjective measure, but meaningful guidance about estimating M is still apparent. Our results provide more weight to the notion of M being estimable under certain conditions, and we provide guidance on identifying those conditions.

Can diagnostic tests help identify model misspecification in integrated stock assessments?

A variety of data types can be included in contemporary integrated stock assessments to simultaneously provide information on all estimated parameters. Conflicts between data, which are often a symptom of model misspecification and evident as model misfit, can affect the estimates of important parameters and derived quantities. Unfortunately, there are few standard diagnostic tools available for integrated stock assessment models that can provide the analyst with all the information needed to determine if there is substantial model misspecification. In this study, we use simulation methods to evaluate the ability of commonly-used and recently-proposed diagnostic tests to detect model misspecification in the observation model process (i.e., the incorrect form for survey selectivity), systems dynamics (i.e., incorrect assumed values for steepness of the stock-recruitment relationship and natural mortality), and incorrect data weighting. The diagnostic tests evaluated here were: i) residuals analysis (SDNR and runs test); ii) retrospective analysis; iii) the R0 likelihood component profile; iv) the age-structured production model (ASPM); and v) catch-curve analysis (CCA). The efficacy of the diagnostic tests depended on whether the misspecification was in the observation or systems dynamics model. Residual analyses were easily the best detector of misspecification of the observation model while the ASPM test was the only good diagnostic for detecting misspecification of system dynamics model. Retrospective analysis and the R0 likelihood component profile infrequently detected misspecified models, and CCA had a high probability of rejecting correctly-specified models. Finally, applying multiple carefully selected diagnostics can increase the power to detect misspecification without substantially increasing the probability of falsely concluding there is misspecification when the model is correctly specified.

A continuous time delay-difference type model (CTDDM) applied to stock assessment of the southern Atlantic albacore Thunnus alalunga

A continuous time delay-diff erence model (CTDDM) has been established that considers continuous time delays of biological processes. The southern Atlantic albacore (Thunnus alalunga) stock is the one of the commercially important tuna population in the marine world. The age structured production model (ASPM) and the surplus production model (SPM) have already been used to assess the albacore stock. However, the ASPM requires detailed biological information and the SPM lacks the biological realism. In this study, we focus on applying a CTDDM to the southern Atlantic albacore (T. alalunga) species, which provides an alternative method to assess this fishery. It is the first time that CTDDM has been provided for assessing the Atlantic albacore (T. alalunga) fishery. CTDDM obtained the 80% confidence interval of MSY (maximum sustainable yield) of (21 510 t, 23 118t). The catch in 2011 (24 100 t) is higher than the MSY values and the relative fishing mortality ratio (F2011/FMSY) is higher than 1.0. The results of CTDDM were analyzed to verify the proposed methodology and provide reference information for the sustainable management of the southern Atlantic albacore stock. The CTDDM treats the recruitment, the growth, and the mortality rates as all varying continuously over time and fills gaps between ASPM and SPM in this stock assessment.

Evaluation of virgin recruitment profiling as a diagnostic for selectivity curve structure in integrated stock assessment models

Virgin recruitment (R0), the equilibrium recruitment in the absence of fishing, is an often used parameter in fisheries stock assessment for scaling population size. We describe and evaluate the use of the R0 likelihood component profile to diagnose selectivity misspecification, using simulation analysis for bigeye tuna in the eastern Pacific Ocean. The profile is evaluated under two types of selectivity misspecification: (1) misspecified shape and (2) misspecified temporal variation. The results indicate that length-composition data can provide substantial information on R0 estimation when the model is correctly specified, but can substantially bias estimates of absolute abundance when selectivity is misspecified. Although contradictory profiles for length-composition and abundance index data result from selectivity misspecification, they may not be useful in determining which survey or fishery selectivity is misspecified. The R0 profile selectivity diagnostic is based on the influence of composition data on absolute abundance. However, perhaps a more problematic and difficult to detect issue is the impact of length-composition data on biomass trends. The age-structured production model diagnostic could be applied to identify bias in both absolute biomass and biomass trend caused by age- or length-composition data in the presence of model misspecification.

Evaluating alternative management strategies for bigeye tuna, <i>Thunnus obesus</i>, in the Indian Ocean

Bigeye tuna ( Thunnus obesus ) support a large commercial fishery in the Indian Ocean. However, explicit management strategies and harvest control rules are yet to be developed for the management of this fishery. We used a stochastic age-structured production model as an operating model to evaluate several potential management strategies under different assumptions of stock productivity. Five management strategies—constant fishing mortality, constant catch, quasi-constant catch, constant escapement, and status-dependent strategies—were evaluated and compared using the performance indicators including average catch, average spawning stock biomass, variation in catch, average fishing mortality and lowest biomass during the time period considered in the simulation. This study shows that (1) for the constant catch strategy, an annual catch of 90000 t would result in a low risk of stock being overfished while obtaining a stable catch; (2) for the constant fishing mortality strategy fishing mortality of 0.3 per year could yield a higher catch, but might have a high probability (64%) of stock dropping below the spawning stock biomass (SSB) that could achieve maximum sustainable yield ( SSB msy ); and (3) for the quasi-constant catch strategy an annual catch of 110000 t was sustainable if the current SSB was higher than SSB msy . Constant escapement and status-dependent strategies were robust with respect to different levels of virgin recruitment and steepness. This study suggests that it is important to incorporate uncertainties associated with key life history, fisheries and management processes in evaluating management strategies.

Stock synthesis: A biological and statistical framework for fish stock assessment and fishery management

Stock synthesis (SS) is a statistical age-structured population modeling framework that has been applied in a wide variety of fish assessments globally. The framework is highly scalable from data-weak situations where it operates as an age-structured production model, to complex situations where it can flexibly incorporate multiple data sources and account for biological and environmental processes. SS implements compensatory population dynamics through use of a function relating mean recruitment to spawner reproductive output. This function enhances the ability of SS to operate in data-weak situations and enables it to estimate fishery management quantities such as fishing rates that would provide for maximum sustainable yield and to employ these rates in forecasts of potential yield and future stock status. Complex model configurations such as multiple areas and multiple growth morphs are possible, tag-recapture data can be used to aid estimation of movement rates among areas, and most parameters can change over time in response to environmental and ecosystem factors. SS is coded using Auto-Differentiation Model Builder, so inherits its powerful capability to efficiently estimate hundreds of parameters using either maximum likelihood or Bayesian inference. Output processing, principally through a package developed in R, enables rapid model diagnosis. Details of the underlying population dynamics and the statistical framework used within SS are provided.

Assessment and management of Cape horse mackerel Trachurus capensis off Namibia based on a fleet-disaggregated age-structured production model

The Cape horse mackerel stock off Namibia is assessed using a fleet-disaggregated age-structured production model including historical International Commission for the Southeast Atlantic Fisheries (ICSEAF) and current data. Data from acoustic biomass surveys, on commercial catch per unit effort and on commercial catch-at-age, were used to estimate current stock status over a range of model specifications. A management tool, the so-called management monitor graph, was introduced in order to track past management relative to stock status. Target, limit and precautionary reference points, decision rules and harvesting strategies, according to which the stock should be managed, are described and tested. The target reference point was set at maximum sustainable yield (MSY), the precautionary reference point at 80% of MSY and the limit reference point at 30% of MSY. Owing to some inconsistencies in the dataseries, evaluation of the current state of the resource is challenging, but estimates are that the resource is around the MSY level, somewhere between 250 000 and 300 000 t.

Model averaging and probable status of North Pacific striped marlin, Tetrapturus audax

We show how model averaging can be applied to estimate the probable status of a fishery resource under assessment scenario uncertainty. This approach is applied to North Pacific striped marlin ( Tetrapturus audax ), an apex predator that may be vulnerable to recruitment overfishing in pelagic longline fisheries targeting tunas. In the current striped marlin assessment, two assessment scenarios were used to account for different hypotheses about the steepness of the stock–recruitment dynamics. Estimates of spawning stock and recruitment from these scenarios are used post hoc to fit age-structured production models that represent alternative hypotheses about the degree of compensation in stock–recruitment dynamics and the degree of serial correlation of environmental forcing. Model-averaged estimates of target spawning biomass to produce maximum sustainable yield (SMSY) and the associated limit fishing mortality (FMSY) characterize relative stock status (S/SMSY and F/FMSY) under each scenario. Scenario-weighted averages of relative status determine probable stock status, with weightings reflecting the credibility of each scenario. Estimates of the variance of probable status account for both model selection and assessment scenario uncertainty in risk analyses. Using model averaging to estimate probable stock status from multiple assessment scenarios is analogous to using ensemble averages from multiple predictive models to make weather forecasts.

Application of age-structured production models to assess oyster Striostrea margaritacea populations managed by rotational harvesting in KwaZulu-Natal, South Africa

Abstract de Bruyn, P. A., Moloney, C. L., and Schleyer, M. H. 2009. Application of age-structured production models to assess oyster Striostrea margaritacea populations managed by rotational harvesting in KwaZulu-Natal, South Africa. – ICES Journal of Masrine Science, 66: 408–419. Oysters have been harvested in KwaZulu-Natal for more than a century, by both commercial and recreational users. However, management of the resource has been based on ad hoc measures rather than quantitative, defensible management plans. Early in the history of the fishery, a rotational harvesting strategy was employed, but it has changed over time. At present, there are 10 management zones, each fished 2 years out of five, and recreational harvesting is allowed in the year preceding commercial harvesting. To assess the status of the oyster resource as well as the historical levels of exploitation, age-structured production models were applied to the oyster populations in the five North Coast management zones. The model fits to the observed data were good. Oyster abundance differed among zones, with the lowest abundance in the two most northern zones. As these zones had only recently been established, however, the paucity of data renders their assessments uncertain. The southernmost zone had the greatest oyster abundance. The benefits of rotational harvesting were demonstrated by the model outputs, which showed rapid population recovery during fallow years, suggesting sustainability of the resource. It is recommended that rotational harvesting continues to be used to manage oysters in KwaZulu-Natal.

A Bayesian analysis on the anchovy stock ( Engraulis japonicus) in the Yellow Sea

As the most abundant fish species in the Yellow Sea, anchovy ( Engraulis japonicus) is decreasing rapidly. Therefore, it is urgent to use a more conservative management approach to protect this declining fish resource. Based on the stock-recruitment (SR) data from 1987 to 2002, the Bayesian method in which uncertainties are treated explicitly is used to assess the SR relationship, the age-structured production model is applied to calculate the biological reference points (BRPs). The Monte Carlo simulation method is performed to predict the expected outcomes in the future for a range of harvest rate. Then, a conservative management approach is recommended.

Preliminary assessment of the Falklands Patagonian toothfish ( Dissostichus eleginoides) population: Use of recruitment indices and the estimation of unreported catches

Longline fishing for Patagonian toothfish ( Dissostichus eleginoides) has taken place in Falkland Island waters for over 10 years. This species was previously only caught as bycatch in the Loligo gahi and finfish trawl fishery. This paper presents a preliminary assessment of the Falkland Islands population of Dissostichus eleginoides using an age-structured production model (ASPM). Two models were investigated; one using a Beverton–Holt stock recruitment relationship and another using trawler CPUE based estimates of toothfish abundance to estimate yearly recruitment. A stock recruitment relationship was not included within the model that used abundance estimates, as recruitment to the population was estimated directly from the index of relative recruit abundance calculated from trawler CPUE. The model was fitted to longline standardised CPUE and to the catch-at-length data. The models produced estimates that provided similar declines in the toothfish population although the start and end biomass estimates varied slightly. The models provided estimates of between 13,000 and 26,000 tonnes of current spawning stock biomass and showed current biomass to be between 38 and 46% of virgin biomass. The fit to CPUE was poor between 1994 and 1996, which we hypothesised could have been due to unreported catches or changes in q or M. This was a time when there was considerable IUU fishing in the southwest Atlantic. When the model was allowed to estimate a level of extra catch the fit was improved and 5000 tonnes of extra catch was estimated. Two further models were briefly examined to investigate the possibility of changes in M and q but both required large changes in those parameters in order to fit. Estimates of MSY varied widely, from 912 to almost 3000 tonnes.

Model averaging to estimate rebuilding targets for overfished stocks

Reducing overfishing and recovering overfished fish stocks is a challenging and important global problem. Rebuilding targets are essential ingredients for guiding overfished stocks to recovery. Having robust estimates of rebuilding targets is likely a necessary condition for effective long-term management of fishery resources. In this paper, we show how Bayesian model averaging can be applied to estimate rebuilding targets under alternative hypotheses about stock–recruitment dynamics. Using alternative hypotheses about stock–recruitment dynamics and environmental variation is important because using only a single hypothesis can lead to substantially different reference points and policy implications. The alternative hypotheses are augmented with prior information collected from meta-analyses of stock–recruitment data to construct a set of age-structured production models. We illustrate our approach using three overfished New England groundfish stocks. We fit alternative model likelihoods to observed data using Bayesian inference techniques. The Schwarz goodness-of-fit criterion was used to calculate model probabilities. Bayesian model averaging was used to estimate rebuilding targets that were robust to model selection uncertainty. Model-averaged estimates suggested that rebuilding targets for overfished stocks can be reasonably well determined if adequate prior information on stock productivity is available. Nonetheless, results had wide confidence intervals that reflected the underlying uncertainty in rebuilding targets.

Reconstructing ecosystem dynamics in the central Pacific Ocean, 1952–1998. I. Estimating population biomass and recruitment of tunas and billfishes

Commercial yield of tunas in the central Pacific increased severalfold between 1952 and 1998. We developed age-structured production models that incorporate information from multiple fisheries to estimate population biomass and recruitment trends of tunas (Thunnus alalunga, Thunnus albacares, Thunnus obesus, and Katsuwonus pelamis) and billfish in the central north Pacific (0°N to 40°N and 130°E to 150°W). Our results suggest that all tuna stocks remain above 40% of 1950s levels, whereas blue marlin (Makaira nigricans) declined to 21% and swordfish (Xiphias gladius) to 56%. Estimated biomasses of juvenile bigeye (T. obsesus) and yellowfin (T. albacares) tuna increased to 112 and 129%, respectively, of 1950s levels. Juvenile albacore (T. alalunga) decreased during the 1970s and 1980s but recovered to historical highs (121%) in recent years. Skipjack (K. pelamis) remained relatively stable between 1952 and 1980, declined by 35% between 1981 and 1990, and then increased to 68% of 1950s levels. These changes generally represent decreases in top predators and increases in small tunas, which make up their prey. Application of stock assessment methods set in a food web context provides an important step toward developing a method that recognizes fishery exploitation as a component of ecosystem dynamics.

Evaluation of management tools for Australia's South East Fishery.2. How well can management quantities be estimated?

The ability to estimate some of the quantities relevant to the management of South East Fishery species is examined using Monte Carlo simulation. The analyses are based on scenarios for four of the species of this fishery (spotted warehou, tiger flathead, jackass morwong and pink ling). Integrated Analysis was found to perform best out of six potential methods of fisheries stock assessment (Schaefer and Fox production models, age-structured production model, Integrated Analysis, ad hoc tuned VPA and ADAPT) that are, or have been, applied to data for South East Fishery species. However, its performance, particularly for spotted warehou, is nevertheless relatively poor. The sensitivity of estimation ability to many factors including model uncertainty and the extent of observation and process error is examined. Factors that influence estimation performance markedly include: violation of the assumption of no population spatial structure; time-trends in catchability; the depletion of the resource at present; and uncertainty about the value of the instantaneous rate of natural mortality. Little improvement in estimation performance can be expected without resolution of issues related to model structure. This highlights the importance of considering model uncertainty in future studies of the estimation performance of stock assessment methods.

Equilibrium yield-curve analysis through an analytic age-structured production model: A sensitivity study for the Chilean jack mackerel fishery

An analytical age-structured production model is used to explore the equilibrium yield-curve properties, and management-related quantities, for the Chilean jack mackerel ( Trachurus symmetricus murphyi) fishery. The objective is to summarize management-related quantities for the jack mackerel fishery according to uncertainties in both the unexploited spawning stock biomass ( S 0) and the functional shape of the stock–recruitment relationship. The age-structured model includes age-specific weights, selectivity, maturity, and natural mortality rate, which permit yield-per-recruit (YPR) and spawning biomass-per-recruit (SPR) to be determined as a function of the fishing mortality rate ( F). Equilibrium yield ( Y e) curves were obtained by combining YPR, SPR and the stock–recruitment (S–R) relationships of Ricker, and Beverton and Holt. The parameters of the S–R models are reparameterized in terms of the unexploited spawning stock biomass ( S 0) and a factor that permits the slope of the S–R models to be guessed as a function of the unexploited spawning biomass-per-recruit, i.e. SPR at F=0 ( SPR F=0) . The Y e curves derived using the Ricker S–R model were slightly symmetric as functions of F and S, while asymmetric equilibrium yield curves were derived using the S–R model of Beverton and Holt. The maximum sustainable yield (MSY), the spawning biomass at MSY ( S MSY), the fishing mortality at MSY ( F MSY), and the fishing mortality at collapse ( F crash) were the management-related quantities, which were analyzed in relative terms (e.g. MSY /S 0). In the case of the Ricker S–R model, it was found that MSY fluctuated between 10.9 and 24.4% of S 0, while F MSY ranged between 0.8 and 1.6 times the natural mortality rate ( M=0.3 yr −1). In the Beverton and Holt S–R model, MSY ranged between 6.2 and 11.5% of S 0, and F MSY fluctuated between 0.44 and 1.74 times M. The quantity F crash was about 2.3–2.4 times F MSY by considering the S–R model of Ricker, but in the case of Beverton and Holt recruitment model F crash ranged between 2.7 and 9.4 times F MSY due to the asymmetric yield-curve. The values for F MSY and F crash were independent of S 0 for each S–R model, but these mortalities were dependent of the slope at the origin of the S–R relationship. It is now necessary to clarify the shape of the S–R relationship for Chilean jack mackerel because the results were very sensitive to this uncertainty.