Science, Faculty of
Oceans and Fisheries, Institute for the
DSpace
UBCV
Licandeo, Roberto
Doctor of Philosophy - PhD
University of British Columbia
Fish stocks often show persistent changes in fish production with hypothesized causes including time-varying recruit parameters, natural mortality of mature fish, and somatic growth. Changes in fish productivity can affect biological reference points and the expected robustness of harvest strategies in meeting long-term conservation and economic objectives. Current assessment approaches commonly assume âstationarityâ in stock-recruit parameters and implemented harvest strategies commonly lack consideration of changes in fish productivity. This dissertation covers three studies based on statistical analysis and simulation methods that aim to advance understanding of time-varying stock-recruit parameters and harvest control rules (HCRs) to cope with changes in fish production. Chapter 2 examined the performance of a Kalman filter algorithm that estimates dynamic state variables (e.g., changing stock-recruit parameters) from inaccurate observations obtained from stock assessment outputs. In Chapter 3, stochastic optimization in policy space (SOPS) was used to find an optimal HCR that is robust to changes in productivity under closed-loop simulation or management strategy evaluation (MSE). By using SOPS and a simple population dynamics model, this chapter examined optimal HCRs under different types of regime shifts (e.g., changes in carrying capacity and intrinsic growth rates). Chapter 4 examined empirical HCRs in an extreme situation of ânon-stationarityâ in stock-recruit parameters (i.e., a spasmodic recruitment pattern), using Atlantic redfish stocks as a case-study. The Kalman filter detected the overall trend in the stock-recruit parameters over time. Also, it was found that optimal HCRs depend on the timing of biomass estimates relative to harvest, observation error, intrinsic population growth, and regime shifts types. In the case of spasmodic stocks, it was found that simple HCRs that used data from well-designed fishery-independent resource surveys were capable of handling changes in productivity. The present dissertation shows the importance of well-designed fishery-independent surveys and simple HCRs for tackling changes in productivity and guidelines are provided for developing HCRs that are robust to the occurrence of such changes. The models and findings presented here can be applied to improve the management of other stocks that show persistent changes in fish productivity.
Fisheries harvest management procedures under time-varying productivity by Roberto Licandeo M.Sc., Fisheries. University of Concepcion, Concepcion, Chile B.Sc., Marine Biology, Austral University, Valdivia, Chile. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Oceans and Fisheries) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2020 © Roberto Licandeo, 2020 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Fisheries harvest management procedures under time-varying productivity_______________ submitted by Roberto Licandeo in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Institute for the Oceans and Fisheries Examining Committee: Dr. Murdoch McAllister, Institute for the Oceans and Fisheries, UBC Supervisor Dr. Carl Walters, Institute for the Oceans and Fisheries, UBC Supervisory Committee Member Dr. Villy Christensen, Institute for the Oceans and Fisheries, UBC Supervisory Committee Member Dr. Marie Auger-Méthé, Department of Statistics and Institute for the Oceans & Fisheries , UBC University Examiner Dr. Peter Arcese, Department of Forest and Conservation Sciences, UBC University Examiner iii Abstract Fish stocks often show persistent changes in fish production with hypothesized causes including time-varying recruit parameters, natural mortality of mature fish, and somatic growth. Changes in fish productivity can affect biological reference points and the expected robustness of harvest strategies in meeting long-term conservation and economic objectives. Current assessment approaches commonly assume âstationarityâ in stock-recruit parameters and implemented harvest strategies commonly lack consideration of changes in fish productivity. This dissertation covers three studies based on statistical analysis and simulation methods that aim to advance understanding of time-varying stock-recruit parameters and harvest control rules (HCRs) to cope with changes in fish production. Chapter 2 examined the performance of a Kalman filter algorithm that estimates dynamic state variables (e.g., changing stock-recruit parameters) from inaccurate observations obtained from stock assessment outputs. In Chapter 3, stochastic optimization in policy space (SOPS) was used to find an optimal HCR that is robust to changes in productivity under closed-loop simulation or management strategy evaluation (MSE). By using SOPS and a simple population dynamics model, this chapter examined optimal HCRs under different types of regime shifts (e.g., changes in carrying capacity and intrinsic growth rates). Chapter 4 examined empirical HCRs in an extreme situation of ânon-stationarityâ in stock-recruit parameters (i.e., a spasmodic recruitment pattern), using Atlantic redfish stocks as a case-study. The Kalman filter detected the overall trend in the stock-recruit parameters over time. Also, it was found that optimal HCRs depend on the timing of biomass estimates relative to harvest, observation error, intrinsic population growth, and regime shifts types. In the case of spasmodic stocks, it was found that simple HCRs that used data from well-designed fishery-independent resource surveys were capable of handling changes in productivity. The present iv dissertation shows the importance of well-designed fishery-independent surveys and simple HCRs for tackling changes in productivity and guidelines are provided for developing HCRs that are robust to the occurrence of such changes. The models and findings presented here can be applied to improve the management of other stocks that show persistent changes in fish productivity. v Lay Summary Large-scale atmospheric and oceanic conditions can restructure marine ecosystems for decades. In response to the prevalent long-term environmental conditions, fish populations can become more or less productive because the restructured ecosystems can generate persist changes in vital population parameters. This situation is problematic for fishery managers because it is challenging to discriminate if the resulting changes in population size are due to natural or fishing causes. Moreover, detecting time-varying population parameters is difficult. The information for detecting time-varying population parameters is obtained from population model outputs. However, population models usually use noisy data and limited time coverage. To deal with such changes in fish productivity it is necessary to develop harvest strategies that are robust to fluctuation. This dissertation advances the understanding of estimation of time-varying stock-recruit parameters in fish populations and possible harvest strategies to cope with variations in fish productivity. The developed models and findings obtained can help to improve management in situations where stocks show changes in fish productivity. vi Preface This dissertation constitutes to my original and independent work. I conceptualized the studies, performed statistical analyses and data simulations, interpreted the findings and wrote the manuscripts. Dr. Walters and Dr. McAllister played a key role supervising and guiding my work and provided insightful input. I am the lead author for all chapters. Each chapter was prepared with the intention of being submitted to international peer-reviewed scientific journals. Chapter 4 as already been published. Licandeo, R., Duplisea, D.E., Senay, C., Marentette, J.R., and McAllister, M.K. (2020). Management strategies for spasmodic stocks: a Canadian Atlantic redfish fishery case study. Can. J. Fish. Aquat. 77: 684â702. Chapter 2 and 3 are in final preparation for submission. Chapters 2 and 3 were completely based on data simulation. In the case of Chapter 4 there was data simulation, but also data analyses based on already collected data (real case study). The data were provided by the Department of Oceans and Fisheries (DFO) after approval of a data request (personal communication Dr. Daniel Duplisea). vii Table of Contents Abstractâ¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦.iii Lay Summary...â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦.v Preface...â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦...â¦.vi Table of Contents...â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦.........vii List of Tables..â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦......xi List Figures..â¦...â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦xiii Acknowledgementsâ¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦xvii Dedicationâ¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦â¦.xviii 1 Chapter 1: Introduction ........................................................................................................ 1 1.1 Addressing non-stationarity in stock productivity parameters ......................................... 1 1.2 Harvest strategies to approach uncertainty in productivity population parameters ......... 4 1.3 Simpler versus complex harvest strategies rules to deal with changes in fish productivity .............................................................................................................................. 6 1.4 Thesis structure ................................................................................................................ 9 2 Chapter 2: Evaluation of the Kalman filter for detecting changes in fish stock-recruit parameters using outputs from virtual population analysis and statistical catch-age models ................................................................................................................................... 10 2.1 Abstract .......................................................................................................................... 10 2.2 Introduction .................................................................................................................... 11 2.2.1 Overview ................................................................................................................. 14 2.2.2 Operating Model ...................................................................................................... 14 2.2.3 Data generation ........................................................................................................ 19 2.2.4 Estimation models ................................................................................................... 20 2.2.5 Stock-recruitment analysis ...................................................................................... 23 2.2.6 Simulation scenarios for changes in productivity and density-dependence ............ 25 2.2.7 Reference points analysis ........................................................................................ 26 2.2.8 Performance statistics .............................................................................................. 26 2.3 Results ............................................................................................................................ 28 2.3.1 VPA and SCA assessment model performance ....................................................... 28 2.3.2 Stock-recruitment analysis ...................................................................................... 28 2.3.3 MSY reference points .............................................................................................. 32 viii 2.4 Discussion ...................................................................................................................... 37 2.4.1 VPA and SCA assessment performance .................................................................. 37 2.4.2 Stock-recruitment analysis ...................................................................................... 38 2.4.3 Conclusions and recommendations ......................................................................... 42 3 Chapter 3: Stochastic optimization within closed-loop simulation for the identification of optimal harvest control rules for fisheries that undergo regime shifts ...................... 44 3.1 Abstract .......................................................................................................................... 44 3.2 Introduction .................................................................................................................... 45 3.3 Materials and methods ................................................................................................... 48 3.3.1 General framework .................................................................................................. 48 3.3.2 Operating model ...................................................................................................... 50 3.3.3 Biomass estimates .................................................................................................... 53 3.3.4 Harvest strategies based on stochastic optimization ................................................ 54 3.3.5 Maximum sustainable yield (MSY) HCRs .............................................................. 56 3.3.6 Performance metrics ................................................................................................ 57 3.3.7 Operating models scenarios ..................................................................................... 57 3.4 Results ............................................................................................................................ 58 3.4.1 Risk-neutral utility ................................................................................................... 58 3.4.2 Risk-averse utility .................................................................................................... 59 3.4.3 Performance metrics ................................................................................................ 62 3.4.4 AR regime shift ....................................................................................................... 62 3.4.5 r regime shift ........................................................................................................... 62 3.4.6 K regime shift .......................................................................................................... 62 3.4.7 Closures ................................................................................................................... 63 3.4.8 Overfished ............................................................................................................... 63 3.4.9 Overfishing .............................................................................................................. 63 3.4.10 Variability in catch biomass yield ........................................................................... 64 3.4.11 Optimal HCRs vs MSY HCRs ................................................................................ 64 3.5 Discussion ...................................................................................................................... 66 3.5.1 Optimal harvest strategies under observation error ................................................. 66 3.5.2 Optimal harvest strategies under SPMs ................................................................... 68 3.5.3 Optimal harvest strategies for different biomass estimates ..................................... 70 3.5.4 Optimal harvest strategies under risk-neutral and risk-averse utility functions ...... 71 3.5.5 Observation error in economic performance for risk-neutral and risk-averse utility functions .................................................................................................................. 72 ix 3.5.6 Optimal vs. MSY harvest strategies under regime shifts ........................................ 73 3.5.7 Caveats ..................................................................................................................... 73 4 Chapter 4: Management strategies for spasmodic stocks: a Canadian Atlantic redfish fishery case study ................................................................................................................. 76 4.1 Abstract .......................................................................................................................... 76 4.2 Introduction .................................................................................................................... 77 4.2.1 Management approaches for spasmodically recruiting stocks (SRS) ..................... 78 4.2.2 Atlantic redfish fishery as a case study ................................................................... 81 4.3 Materials and methods ................................................................................................... 84 4.3.1 Framework ............................................................................................................... 84 4.3.2 Conservation and fishery objectives and performance metrics ............................... 84 4.3.3 Management procedures .......................................................................................... 87 4.3.4 Conditioning the operating model ........................................................................... 92 4.3.5 Operating model description (base case) ................................................................. 92 4.3.6 Selection of uncertainties ....................................................................................... 100 4.3.7 Present and future recruitment ............................................................................... 104 4.3.8 Species identification ............................................................................................. 104 4.3.9 Discarding .............................................................................................................. 105 4.3.10 Change in vulnerability ......................................................................................... 106 4.3.11 Life history parameters .......................................................................................... 106 4.4 Results .......................................................................................................................... 107 4.4.1 Operating model fit and diagnostics ...................................................................... 107 4.4.2 Operating model outputs ....................................................................................... 108 4.4.3 Management procedure performance .................................................................... 111 4.5 Discussion .................................................................................................................... 116 4.5.1 Harvest control rules responsive to the biomass of large-sized fish ...................... 117 4.5.2 Use an empirical index to formulate harvest control rules .................................... 118 4.5.3 Capped harvest control rules ................................................................................. 119 4.5.4 Test harvest strategies against a variety of recruitment assumptions .................... 122 4.5.5 Other considerations .............................................................................................. 124 4.5.6 Conclusion: recommendations for managing spasmodically recruiting stocks ..... 127 5 Chapter 5: Conclusion ....................................................................................................... 130 5.1 Conclusion .................................................................................................................... 130 5.2 Future research ............................................................................................................. 139 6 References ........................................................................................................................... 141 x 7 Appendices .......................................................................................................................... 153 7.1 Appendix A Chapter 2: Tables and Figures ................................................................. 153 7.2 Appendix B Chapter 3: Tables and Figures ................................................................. 169 7.3 Appendix C Chapter 4: Tables and Figures ................................................................. 191 xi List of Tables Table 2.1. Notation and parameter values used in the operating model for jack mackerel, anchovy, and hake. .................................................................................................. 15 Table 2.2. Age-structured population dynamics for the operating model for jack mackerel, anchovy, and hake. .................................................................................................. 17 Table 2.3. Parameter estimated, residuals, and likelihood function for the SCA model. ............ 20 Table 2.4. Parameter estimated, residuals, and likelihood function the VPA. ............................. 21 Table 3.1. Notation and parameters values used in the operating model. .................................... 49 Table 3.2. Population dynamics model and harvest control rules (HCRs) used in the closed-loop simulation. ............................................................................................................... 52 Table 3.3. Deterministic MSY reference points for the continuous surplus production model. .. 56 Table 3.4. MSY harvest control rules used under the base case operating model. ...................... 58 Table 3.5. Operating models under the optimal and MSY harvest control rules. ........................ 58 Table 4.1. Conservation and fishery objectives and their respective performance metrics (PMs) agreed by the redfish working group ....................................................................... 83 Table 4.2. Harvest control rule (HCR) parameters for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). ........................................................................................................... 89 Table 4.3. Description for the empirical (target-based) harvest control rule (HCR) for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). .......................................................... 90 Table 4.4. Base case operating model (OM1) for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). .......................................................................................................................... 94 Table 4.5. Equations for the base case operating model (OM1) for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). ........................................................................................ 99 Table 4.6. Core, stress, and sensitivity test operating models formulated for the redfish management strategy evaluation (MSE).. .............................................................. 102 xii List of Figures Figure 2.1. Outputs from the SCA and VPA models for hake.. ................................................... 27 Figure 2.2. Stock-recruit parameter estimates from the Kalman filter obtained from the SCA and VPA model outputs for jack mackerel .................................................................... 29 Figure 2.3. Stock-recruit parameter estimates from the Kalman filter obtained from the SCA and VPA model outputs for anchovy.. ........................................................................... 30 Figure 2.4. Stock-recruit parameter estimates from the Kalman filter obtained from the SCA and VPA model outputs for hake. .................................................................................. 31 Figure 2.5. MSY reference points calculated from the Kalman filter outputs for jack mackerel.. ................................................................................................................................. 34 Figure 2.6. MSY reference points calculated from the Kalman filter outputs for anchovy.. ....... 35 Figure 2.7. MSY reference points calculated from the Kalman filter outputs for hake. .............. 36 Figure 3.1. Optimal harvest parameters, B* (minimum based-stock) and Q*, under a risk-neutral and risk-averse utility functions for anchovy.. ........................................................ 60 Figure 3.2. Performance in average yield for anchovy from optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models including regime shifts, surplus production models, and biomass estimates .......................... 61 Figure 3.3. Performance in average yield for optimal harvest strategies (risk-neutral and risk-averse utility functions) and MSY harvest control rules for three stocks under different regime shifts and biomass estimates for the continuous surplus production model (cPHM) under the base case. ........................................................................ 65 Figure 4.1. Units 1 and 2 of the Canadian Atlantic redfish fishery. The overlapping region (Unit 1+2) constitutes. PEI (Prince Edward Island), NS (Nova Scotia). .......................... 80 Figure 4.2. Historical landings and catch limits (CL) for Sebastes spp. for the management Units 1 and 2, off eastern Canada. .................................................................................... 81 Figure 4.3. Example of the harvest control rule (HCR) applied in the management procedures (MPs) for Sebastes mentella (and S. fasciatus). ...................................................... 86 xiii Figure 4.4. Example for determining the catch limits (CL in kt) for each species. The lines are the raw CL for S. mentella and S. fasciatus, CL caps, and the realized CL for MP1 under the base case operating model.. ..................................................................... 87 Figure 4.5. Flow chart illustrating steps to calculate the realized catch limit (CL) for Sebastes mentella (S.m) and S. fasciatus (S.f.) under management procedure #1 (MP). ...... 88 Figure 4.6. Plots for the base case operating model (OM1) for S. mentella of the 1) catch biomass killed and retained, 2) harvest rates, 3) observed and predicted fits for the Units 1 and 2 survey indices, and 4) age 1 recruitment and spawning stock biomass (SSB).. ................................................................................................................... 109 Figure 4.7. Plots for the base case operating model (OM1) for S. fasciatus of the 1) catch biomass killed and retained, 2) harvest rates, 3) observed and predicted fits for the Units 1 and 2 survey indices, and 4) age 1 recruitment and spawning stock biomass (SSB) ..................................................................................................................... 110 Figure 4.8. Average catch retained (in kt) for the 10-20 and 10-40 year horizon for the 21 management procedures (MPs) under the core operating models. ........................ 113 Figure 4.9. Average number of years with catch limits (CL) â¥40kt in the 2028-2057 year horizon for the 21 management procedures (MPs) under the core operating models. ............................................................................................................................... 115 xiv Acknowledgements I would like to express my gratitude to Professors Carl Walters and Murdoch McAllister for their constant intellectual and economic support during all these years as a PhD student. I am also thankful for the opportunity of being part of the Institute for the Oceans and Fisheries at UBC where I have grown significantly through the years. I would also like to thank Dr. Yuying Zhang (Department of Biological Sciences, Florida International University) for her comments and corrections on my final dissertation. I would also like to acknowledge my family, my partner, and close friends for being around me during all this time as well as the warm company of my Lab group at UBC. xv Dedication To my parents 1 1 Chapter 1: Introduction 1.1 Addressing non-stationarity in stock productivity parameters There have been numerous instances where marine ecosystems have changed rapidly from one state to another (deYoung et al. 2004; Polovina 2005). Such abrupt changes can restructure marine ecosystems with the restructured state sometimes persisting for decades (Steele 2004; Cahuin et al. 2009). The restructuring of marine ecosystems can be so rapid that impacts on fish stocks are evident within their reproductive life-scale (deYoung et al. 2004; Steele 2004). In pelagic stocks (anchovies and sardines), such abrupt fluctuations can change the abundance generating âboomâ and âbustâ cycles over time (Schwartzlose et al. 1999; Barange et al. 2009a, 2009b; de Moor et al. 2011). These persistent fluctuations in marine ecosystems are known as âregime shiftsâ (deYoung et al. 2004; Polovina 2005). Marine ecosystem functioning can respond to changes in large-scale atmospheric and oceanic conditions. From a fishery perspective, regime shifts can affect the productivity of fish stocks (e.g., fish recruitment). Depending on the direction of the regime shifts, stocks can become more or less productive (Cahuin et al. 2009). Further, such abrupt changes are especially problematic when the stocks go to an unproductive regime and the fishing capacity is high. In this situation, it is difficult to discriminate if the decrease in stock biomass is due to fishing mortality or because of fluctuations in fish productivity (Walters and Parma 1996). In contrast, when the stocks transition into a productive regime, if management regulations fail to control for fishing overcapacity, this can lead to hardship when the stocks return to an unproductive system. This is particularly so when the fleet is too specialized to target other stocks or species. Therefore, fishing management needs to be aware of variations in productivity to provide adequate alternative fishing opportunities to avoid unnecessary hardship. 2 When regime shifts occur, they can affect the estimation of population parameters that determine population growth and sustainable harvest rates (Walters and Martell 2004). Regime shifts can change the juvenile survival rate and the available habitat for spawning or nursery, or both, and therefore the âcarrying capacityâ of the stocks. When regime shifts affect juvenile survival rates, the resilience of the stock to fishing mortality is also affected. Thus, stock assessments should account for changes in the productivity caused by the regime shift. Detecting regime shifts in fish stocks presents many research challenges. In fisheries, the relationship between spawning stock and the production of juvenile offspring (i.e., the stock-recruitment relationship) is essential for the understanding of fish population dynamics. This relationship is determined by two key population parameters; a density-independent parameter, which measures the maximum reproductive rate of the stock at low stock size (α) and a density-dependent parameter (β) that determines the carrying capacity of the stock (Hilborn and Walters 1992). It is expected that regime shifts generate non-stationarity in stock-recruit parameters (Beamish and Bouillon 1993; Walters and Parma 1996; Rose 2004; Rademeyer et al. 2008); however, detecting which parameter is changing is a difficult task. This situation leads to a problem in fishery management because management strategies based on the maximum sustainable yield (MSY) concept require a clear definition of these parameters. At present, most of stock assessment agencies worldwide assume that key fish population parameters such as juvenile survival rate and carrying capacity are stationary (Collie et al. 2012; Deroba et al. 2015; Edwards and Dankel 2016). This is due to at least three main factors. First, in fishery modelling, testing for non-stationary in productivity parameters is not a recurrent practice in stock assessment (Walters 1987). Second, non-stationarity in stock-recruit parameters is difficult to assess due to noisy data (Walters 1987). Third, the use of advanced statistical tools 3 (e.g., software) and specialized training for evaluation of non-stationarity are usually not considered in the ecological (and stock assessment) curriculum. Testing for non-stationarity in productivity parameters has been performed for salmon fisheries, and there is evidence indicating that non-stationarity in salmon populations parameters is quite common (Peterman et al. 2000, 2003; Dorner et al. 2008; Peterman and Dorner 2012; Malick and Cox 2016). Testing for non-stationary productivity in salmonids has been often because stock-recruitment data are more direct to collect than for marine fisheries. In marine fisheries, the time-series records of spawners and recruits are usually obtained indirectly (e.g., via stock assessment) and difficult to obtain (Ricard et al. 2012). The time-series for most marine stocks are obtained from model outputs (Walters and Martell 2004) instead of direct counts as occurs for some Atlantic and Pacific salmon stocks. As a result, marine stock-recruit time series are susceptible to many sources of uncertainty (Brooks and Deroba 2015; Dickey-Collas et al. 2015); e.g., model uncertainty (misspecifications), observation uncertainty (e.g., abundance indices), and process errors (variability not explained in the model structure such as recruitment variation but it could include other processes such as time-varying selectivity and catchability)(Maunder and Piner 2014). All these factors contribute to error in the reconstruction of historical spawning stock and recruitment trends. Recent advances in sophisticated statistical methods and software have given us insight into some worrisome trends in fish productivity (Minto et al. 2014; Britten et al. 2016, 2017; Szuwalski et al. 2019). Recently, a global meta-analysis evaluating 262 marine stocks examined non-stationarity in stock-recruit parameters (Britten et al. 2016). This study showed that changes in carrying capacity were more frequent than changes in productivity. Similarly, Vert-pre et al. (2013), systematically analyzed 230 fish stocks and found that 69% of them showed regime 4 shifts in productivity. Taken together, this evidence suggests that testing for non-stationarity in productivity parameters should be a routine part of fisheries stock assessment (Walters 1987; Walters et al. 2008). 1.2 Harvest strategies to approach uncertainty in productivity population parameters There are two main approaches that have been used to find robust management strategies under conditions of environmental change; stochastic dynamic programming (SDP) (Walters, 1975; Palma, 1990; Walters and Palma 1996; Hawkshaw and Walters, 2015) and closed-loop simulations (Walters 1986) or management strategy evaluations (MSEs) (Aâmar et al. 2009a, 2009b; Ianelli et al. 2011; Collie et al. 2012; Szuwalski and Punt 2013a, 2013b; Siple et al. 2019). A fundamental difference is that in SDP, the HCR parameters are found by optimizing a utility function while the MSE approach (Edwards and Dankel 2016) uses the âa prioriâ or âad hocâ HCR parameters. These HCRs are evaluated using simulation. The use of ad hoc HCR parameters allows examining a limited range of potential harvest options. As a result, the relative performance of candidate harvest strategies are only valid over the pre-defined harvest control rules; hence, this heuristic approach does not provide general solutions for regime shifts. On the other hand, SDP takes into account all possible harvest strategies under all possible harvest sequences for all possible realizations of the population dynamics (Memarzadeh and Boettiger 2019). Moreover, SDP allows for a harvest strategy to be completely flexible (e.g., it can take any âshapeâ), while also being optimal (i.e., it maximizes a utility function). However, incorporating observation and process errors in the state dynamics simultaneously is challenging in SDP. Recently, Memarzadeh and Boettiger (2019) applied a new developed algorithm in robotics to explicitly include (as opposed to SDP) observation uncertainty. This algorithm is 5 called partially observed Markov decision process, POMDP. Unfortunately SDP and POMDP still rely on simple population dynamics models that have a single dynamics state (e.g., surplus production and delay difference models); this is due to the âcurse of dimensionalityâ (Hilborn and Walters 1992; Marescot et al. 2013). The MSE approach, although more heuristic, allows for the accommodation of greater model complexity and uncertainty than SDP and POMDP and is able to find the âbestâ management under conflicting management objectives. Another approach that has not been widely used to find management strategies is the stochastic optimization in policy space (SOPS) (Moxnes 2003, 2005; Walters and Martell 2004). This method shares features with SDP and can incorporate MSE components as well. The SOPS, instead of using the sophisticated mathematical programming required in SDP, uses stochastic nonlinear optimization to find HCR parameters that maximize a utility function. In this case the HCR is defined by a deterministic function with unknown HCR parameters. As in SDP, the SOPS also uses a utility function, can accommodate entirely flexible harvest strategies (Moxnes 2003, 2005), and can replicate the optimal HCR found in SDP (e.g., Reed 1979; Sethi et al. 2005). The most appealing features compared with SDP (and POMDP), are that SOPS allows multiple dynamic states (e.g., age-structured dynamics models), allows different sources of uncertainty simultaneously (e.g., process, observation, and model errors), and the HCR can be found using feedback control (i.e., as in MSE). All these features make SOPS a prominent approach to finding optimal harvest strategies in non-stationary populations without imposing any a priori HCR, in contrast to what has been commonly used (e.g., Edwards and Dankel 2016). 6 1.3 Simpler versus complex harvest strategies rules to deal with changes in fish productivity Fishery managers are commonly tasked with applying a harvesting approach to the current state of the marine ecosystem (de Moor and Butterworth 2009; Ianelli et al. 2011; Punt et al. 2013; Szuwalski and Punt 2013a; King et al. 2015). Regime shifts in fish productivity create two difficult problems in this context. Firstly, the change in productivity needs to be detected within an appropriate time frame to avoid suboptimal harvest or to prevent undesirable overfishing. Secondly, it may be important to determine which productivity parameters are being affected to define appropriate harvest strategies. For example, in many fishery agencies, biological reference points need to be well-defined to provide management advice (e.g., to prescribe a quota or allowable fishing effort) (King et al. 2015). Therefore, it is of interest for fishery managers to have well-defined harvest control rules (HCRs) that balance conservation and management objectives while remaining robust to changes in fish productivity independent of the cause of these fluctuations (Walters and Parma 1996). Several studies have evaluated the robustness of HCRs to non-stationary relationships in productivity population parameters (Parma 1990; Walters and Parma 1996; Collie et al. 2012; Hawkshaw and Walters 2015). In general, there are two HCRs to cope with regime shifts; time-invariant harvest strategies and time-varying harvest strategies (or regime-specific strategies). Time-invariant harvest strategies do not update the HCRs parameters to adjust to the current regime, while the time-varying harvest strategies do. A time-invariant harvest strategy can perform well depending on how the regime shift affects the population parameters. For example, Walters and Parma (1996) found that a constant harvest rate strategy performs well in terms of long-term yield when only the carrying capacity of the stock-recruitment function changes over 7 time. This finding emphasizes the importance of determining which population parameters are affected by a regime change. In the case of time-varying harvest strategies, several approaches have been developed to respond to changes in fish productivity. One method is to use models that detect sudden changes in productivity using time-series of recruit-per-spawner or recruitment estimates obtained from data analysis (e.g., stock assessments). These methods include the âdynamic B0â, âmoving windowsâ, and âSTARS approachâ (see Punt et al. 2013 for details). Such approaches use recruitment estimates over the more recent years (e.g., 20 years) to calculate biological reference points (e.g., Bmsy and B0) that reflect the current productivity regime. Thus, biological reference points change over time as new data are collected, which consequently update the HCR. However, simulation studies have shown that these methods only perform well if the current regime system is well-known and they can be risky when the regime shift is incorrectly identified (i.e., when there is no regime shift) (Punt et al. 2013; Szuwalski and Punt 2013b). Moreover, these methods can only detect the overall change in productivity and are thus unable to determine which productivity parameter is changing. Another approach is to model the changes in productivity in the actual population dynamics. Recently, the Kalman filter method (Kalman 1960) has been used to estimate non-stationary parameters in ecological applications. The Kalman filter is a recursive algorithm that estimates unobserved states (e.g., stock-recruit parameters) based on inaccurate observations. In this approach, the time-varying stock-recruit parameters are treated as system state variables while the time-series of spawning abundance are treated as arbitrary, fixed values (i.e., the dynamic states are not for spawning stock dynamics; they are instead for the stock-recruit parameters). As the Kalman filter estimates time-varying stock-recruit parameters, it is possible 8 to calculate time-varying reference points. Thus, the Kalman filter allows linking the changes in the productivity parameters with the HCR. Only a few studies have examined Kalman filter methods to deal with time-varying harvest strategies for changes in productivity (Peterman et al. 2000; Collie et al. 2012). Collie et al. (2012) used the Kalman filter to examine time-varying harvest strategies for changes in productivity in Pacific salmon. They found that compared with time-invariant harvest strategies, time-varying harvest strategies generated higher escapements (spawner abundances), higher yields, and reduced the risk that run size would fall below the escapement target under a range of harvest rates for salmon populations in Alaska. Although it has not been reported, HCRs that make direct use of biomass information derived from survey indices (e.g., trawl surveys) can be used to track changes in fish productivity. In MSE literature these models are referred to as data-based HCRs (also called empirical or model-free HCR) (Rademeyer et al. 2007). Data-based HCRs ignore population dynamics models, and the prescribed quota is adjusted up or down as a function of perceived trends in the survey index. This type of HCRs has a key advantage compared with the more sophisticated methods described above (e.g., Kalman filter or models that detect sudden changes in time-series of productivity). The data-based HCRs do not depend on MSY reference points (or any biological reference points). It is well-know that MSY reference points are challenging to estimate from historical reconstructions and they are affected by different sources of uncertainty (e.g., model structure uncertainty, non-stationary population parameters, process error, and observation error), which are problematic to control (Ludwig and Walters 1981; Walters 1985; Martell and Stewart 2014; Brooks and Deroba 2015; Dickey-Collas et al. 2015). In short-lived species (e.g., sardines and anchovies) that show evident boom and bust recruitment dynamics 9 due to changes in productivity parameters, data-based HCRs have been successfully applied to manage highly variable stocks (e.g., de Moor et al., 2011; de Moor and Butterworth 2016). 1.4 Thesis structure The research carried out in the present thesis contributes to the current knowledge on uncertainty in the estimation of productivity population parameters, advancing management decisions, in particular, for fish stocks that exhibit abrupt changes in stock productivity as a result of non-stationary processes in population parameters due to regime shifts. In Chapter 2, entitled: âEvaluation of the Kalman filter for detecting changes in fish stock-recruit parameters using outputs from virtual population analysis and statistical catch-age modelsâ, a simulation-estimation framework was developed to test the ability of the Kalman filter to track regime shifts in time-varying stock-recruit parameters derived from stock assessment model outputs. In Chapter 3, entitled: âStochastic optimization within closed-loop simulation for the identification of optimal harvest control rules for fisheries that undergo regime shiftsâ, a closed-loop simulation was developed to evaluate harvest control rules robust to regime shifts using different model configurations, with an emphasis on harvest control rule parameters obtained from optimization methods (i.e., SOPS). Lastly, Chapter 4, entitled: âManagement strategies for spasmodic stocks: a Canadian Atlantic redfish fishery case studyâ, provides a case study of a stock with extreme non-stationarity in recruitment. This was used to test the ability of simple harvest control rules that do not rely on model-based methods, but instead use biomass information derived from survey indices, to track changes in fish productivity. 10 2 Chapter 2: Evaluation of the Kalman filter for detecting changes in fish stock-recruit parameters using outputs from virtual population analysis and statistical catch-age models 2.1 Abstract Fish stock-recruit parameters, such as the productivity at low stock size and the density-dependence parameter, e.g., âcarrying capacityâ, have often shown persistent changes (i.e., are non-stationary). Many stock assessments still assume stationarity in those parameters. Non-stationarity has implications for biological reference points. Simulations were applied to test the ability of the Kalman filter to estimate time-varying stock-recruit parameters from outputs from virtual population analysis (VPA) and statistical catch-at-age (SCA) models. Scenarios investigated regime shifts (RS) that varied productivity, density-dependence, or both parameters simultaneously. When the stock-recruit parameters were time-varying, the Kalman filter detected that changes had occurred. Overall, it was difficult to discriminate whether productivity or density-dependence (or both) had changed. The detection of productivity or density-dependence RS depended on the attributes of the stock and the catch history. The VPA had better performance in detecting recent RS than the SCA model. The Kalman filter was better at tracking productivity RS in hake than in anchovy and mackerel. The Kalman filter was better at detecting non-stationarity with extended RS. It is recommended that analysts test for non-stationarity in results from both VPA and SCA models. Stock assessment configurations that were robust to non-stationarity in the SRR were then identified. 11 2.2 Introduction The relationship between spawning stock and the production of juvenile offspring (i.e., the stock-recruit relationship, SRR), is essential to understanding fish population dynamics, and to establishing sustainable harvest rates (Beverton and Holt 1957; Ricker 1975). Various models are used to describe the SRR (e.g., Ricker, Beverton and Holt, and hockey stick models; see Hilborn and Walters 1992 and Quinn and Deriso 1999). The parameters in the SRR are the maximum reproductive rate at low stock size (α) and the density-dependence parameter (β). The maximum reproductive rate parameter is critical in fisheries management because it determines the sustainable limits of fishing mortality. Thus, it is called a âresilienceâ or productivity parameter. The density-dependence parameter reflects the effects of the spawning stock biomass (SSB) on juvenile offspring and typically is related to the âcarrying capacityâ of the stock (Myers 2001). It affects the survival of juveniles due to competition among juveniles or cannibalism by the adults. Here, the stock-recruit parameters are referred to as productivity and density-dependence parameters. A common assumption in fishery stock assessment models is that population dynamics follow a long-term steady-state (Walters 1987). This means that although the stock-recruit parameters may change over time, the stock-recruit dynamics remain stationary. There is considerable evidence that these parameters are non-stationary for many stocks (e.g., Peterman et al. 2003; Dorner et al. 2008; Minto et al. 2014; Britten et al. 2016; Ichinokawa et al. 2017). Oceanographic regime shifts, for example, can generate abrupt and persistent physical and biological changes in an ecosystem (Polovina 2005). In response to these changes, stocks produce unexpected collapses or sudden so-called âboomsâ in biomass (Oliveros-Ramos et al. 2010; de Moor et al. 2011; de Moor and Butterworth 2016). These persistent changes in 12 productivity are referred as regime shifts (RS) (Mohn and Chouinard 2007). Under RS, it is expected that the stock-recruit parameters are non-stationary. Recently, there has been much interest in using Kalman filter (Kalman 1960) to detect non-stationarity in stock-recruit parameters for marine stocks (Dorner et al. 2008; Minto et al. 2014; Britten et al. 2016; Holt and Michielsens 2019; Szuwalski et al. 2019). The Kalman filter is a method that recursively estimates unobserved states (in this case, changing stock-recruit parameters) based on inaccurate observations. Under the Kalman filter, the stock-recruit parameters are treated as dynamic state variables. In contrast, the time patterns of spawning abundance are arbitrary fixed values. Peterman et al. (2000) applied the Kalman filter to estimate productivity parameter trends using a salmon simulation model. Dorner et al. (2008) and Minto et al. (2014) estimated trends in the productivity parameter for multiple populations of Pacific salmon and Atlantic cod. Previous studies (e.g., Minto et al. 2014) showed that a Kalman filter with time-varying productivity and constant density-dependence fitted empirical data better (as measured by Akaike Information Criterion, AIC; Burnham and Anderson 2002) for cod stocks. However, in a global meta-analysis of 262 fish stocks, the AIC supported density-dependence (i.e., carrying capacity) variation in half of the stocks (Britten et al. 2016). Szuwalski et al. (2019) found that half of 52 forage fish stocks supported productivity variation or density-dependence variation. SSMs that examine non-stationary processes are becoming more common in population dynamics and ecology (e.g., Lamon et al. 1998; Scheuerell et al. 2002). Unfortunately, even simple linear Gaussian SSMs can generate biased parameter estimates and thus bias in the unobserved states (Auger-Méthé et al. 2016). Another layer of uncertainty exists in fisheries. The data used to describe the non-stationary processes (such as time-varying fish stock productivity) 13 are derived from stock assessment outputs rather than direct measurements (e.g., SSB and recruitment). Thus, the estimated trends in stock-recruit parameters may be an artifact of model misspecification, bias, parameter correlations, and structural assumptions of the original assessment model (Brooks and Deroba 2015). Two age-structured models are commonly used to generate time-series of stock-recruit data: virtual population analysis and statistical catch-at-age models (VPA and SCA, respectively). These methods make different model assumptions (Hilborn and Walters 1992; Quinn and Deriso 1999; Walters and Martell 2004), but both use catch-at-age data. The reconstruction of historical recruitment (and abundance-at-age) needs to be solved backward in time for the VPA and forward in time for the SCA. Also, VPA assumes time-varying fishing selectivity and negligible error in observed catch-at-age data. By comparison, SCA assumes that the differences between the observed and predicted catch-at-age data are results of errors associated with age reading and sampling (and other sources) given an assumed fishing selectivity pattern (which can be time-varying or not). Therefore, implicit assumptions in age-structured models and data may affect the Kalman filter estimates of stock-recruit parameters. The combination of structural bias and non-stationarity can bias biological reference points (e.g., maximum sustainable yield, MSY) and sustainable harvest rates if they are not correctly accounted for. In contrast to simulation studies of salmonids (Peterman et al. 2000; Collie et al. 2012; Holt and Michielsens 2019), no studies have evaluated the ability of the Kalman filter to detect non-stationarity in stock-recruit parameters from VPA and SCA outputs; moreover, this situation has challenges different from those for salmonids. Simulation-evaluation experiments may help provide a better understanding of the estimability of stock-recruit parameters in natural 14 populations. Estimability refers to when, given the data, the model produces a unique estimate (Auger-Méthé et al. 2016). We used three stocks (mackerel, anchovy, and hake) under hypothetical RS that vary the productivity, or the density-dependence, or both parameters simultaneously. Estimation performance (i.e., precision and bias) was examined for key population parameters and management quantities (e.g., MSY reference points). Stock assessment configurations that were robust to non-stationarity in the SRR were then identified. 2.2.1 Overview 2.2.2 Operating Model The OMs had age-structured population dynamics with discrete-time survival equations to project abundance-at-age based on harvest rates (ut) and fishery selectivity (Eq. 2.2.13). The OM was parameterized using three stocks: jack mackerel (Trachurus murphyi), anchovy (Engraulis ringens), and hake (Merluccius gayi)(Table 2.1). These stocks were analyzed using SCA models (Canales 2014; Tascheri et al. 2014; Zuniga et al. 2014). These stocks were reanalyzed using the Walters and Punt (1994) VPA to obtain estimates of recruitment compensation (steepness) and fishing selectivity for the simulations. The jack mackerel and hake had similar maximum age and recruitment compensation, but still differed in terms of, for example, natural mortality and fishery selectivity. The Kalman filter showed distinct patterns for the time-varying stock-recruit parameters. It was considered appropriate to present the results of these two stocks. 15 Table 2.1. Notation and parameter values used in the operating model for jack mackerel, anchovy, and hake. Symbol Value (jack mackerel; anchovy; hake) Description Indices T 40 Maximum year A 12; 5; 13 Plus-group t 1,â¦,T Annual time step a 1,...,A Age Model parameters ð 0 1.1; 41.6; 1.1 Mean unfished recruitment (age-1) (millions) ððâ² ; ðð¡â²â² 1,â¦,A-1; 2,â¦,T Deviates for the initial abundance and recruitment deviates in year t κ 13.7; 8.3; 14.2 Compensation ratio M 0.23; 0.7; 0.33 Instantaneous natural mortality rate (year-1) ÏR 0.2 Standard deviation of the recruitment variation ut Uninformative, Fig. 2.1; Informative, Fig.7.3A Harvest rate in year t Ω1 2.4; 2.1; 3.8 Age-at-50% maturity Ω2 0.6; 0.2; 0.4 Age maturity slope-1 Ω3 3.6; 1.7; 2.6 Age-at-50% selectivity Ω4 0.5; 0.2; 0.4 Age selectivity slope-1 Ï1 0.3 Standard deviation for survey biomass index n1 T Number of years with survey index q 1.0 Catchability coefficient Ï2 0.3 Standard deviation for the proportion and catch-at-age for the fishery n2 T Number of years with fishery age composition ðð¼; ðð½ Multiplier factor for âstep functionâ in the regime shift scenarios c 0.001 Constant in plus-group for the VPA p 5 Number of year to calculate terminal selectivity in VPA afull 8; 3; 6 Age for fully vulnerable fish in VPA wa Canales 2014; Zuniga et al. 2014;Tascheri et al. 2014 Weight-at-age (kg) 16 In the OM, many parameters (except life-history parameters such as natural mortality, recruitment compensation, and maturity) were set to specific values to facilitate comparison across stocks and assessment models (Table 2.1). For example, the standard deviation of the observation error in the biomass survey index (Ï1) and age-composition data (Ï2) were set to a relatively high value (i.e., 0.3) (Martell and Stewart 2014; Wiedenmann et al. 2015; Siple et al. 2019) to indicate a typical situation where data collection is noisy. For simplicity, the catchability coefficient was fixed at q=1, and the natural mortality and the fishing selectivity were time-invariant (Schnute and Richards 1995; Martell and Stewart 2014). The standard deviation of the process error (ÏR) in the RS scenarios was set at 0.2. This value generated plausible variability in the RS scenarios (see Results). This level of variability was similar to that applied in other simulation studies that estimate process and observation error variances in SSMs (Linton and Bence 2008; Auger-Méthé et al. 2016) and SCA models (Forrest et al. 2013). Previous studies have overlooked that the time-series of SSB (and the recruits) are fixed and known quantities in the Kalman filter. Two catch histories diagnosed the estimability of the stock-recruit parameters under different contrast in the time-series of SSB. The uninformative catch history was an increasing harvest rate to a maximum equal to ðððð and then constant (Fig. 2.1). The informative catch history was an increasing harvest rate to ðððð and then decreasing (Fig. 7.3A). 17 Table 2.2. Age-structured population dynamics for the operating model for jack mackerel, anchovy, and hake. Description Equations (1) Survivorship ð ð = expâ¡(âð) (2) Survivorship-at-age per recruit ðð = {1â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð = 1ððâ1ð ðâ1â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡1 < ð < ð´ððâ1ð ðâ1 (1 â ð ð)â â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð = ð´ (3) Proportion mature-at-age ðð =11 + ðâ(ðâÎ©1)/Î©2 (4) Weight-at-age ð¤1, â¦ , ð¤ð´ (5) Fishery selectivity-at-age ð£ð =11 + ðâ(ðâÎ©3)/Î©4 Stock-recruit relationship (6) Unfished spawning biomass per-recruit (7) Unfished spawning biomass (8) Stock-recruit parameters Productivity parameter ð¼ =ð ð Density-dependence parameter ð½ =lnâ¡(ð )ðððµ0 Regime shifts (9) Time-varying productivity ð¼ð¡ = ð¼exp[ðð¡â²â²ðð â 0.5(ðð )2]ðð¡ð¼â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡1 < ð¡ â¤ ð (10) Time-varying density-dependence ð½ð¡ = ð½exp[ðð¡â²â²ðð â 0.5(ðð )2]ðð¡ð½â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡1 < ð¡ â¤ ð (11) Recruitment (t>1; a=1) Population dynamics (12) Initial abundance (t=1) ðð,1 = ð 0ðððððâ² ðð â0.5(ðð )2â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡1 < ð < ð´ (13) Abundance-at-age (t>1) ðð,ð¡ = {ððâ1,ð¡â1ð ð(1 â ð£ðâ1ð¢ð¡â1)â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡1 < ð < ð´ðð´â1,ð¡â1ð ð(1 â ð£ð´â1ð¢ð¡â1) +â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ðð´,ð¡â1ð ð´(1 â ð£ð´ð¢ð¡â1)â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð = ð´ (14) Spawning stock biomass (15) Vulnerable biomass (16) Catch biomass Fishery age compositions (17) True catch-at-age 1a Aa a aal m wï¦ï½ï½ï½ï¥0 0SSB R ï¦ï½~ (0,1)tNï·ï¢ï¢1, 1 1exp( )t t t t tN SSB SSBï¡ ï¢ï­ ï­ï½ ï­~ (0,1)aNï·ï¢,1 0 A AN R l a Aï½ ï½,1At a t a aaSSB N m wï½ï½ï¥,1At a t a aaVB N v wï½ï½ï¥t t tC u VBï½, ,a t a t a tC N v uï½18 (18) Proportion-at-age (19) Sampling error (20) Observed proportion-at-age (21) Observed catch-at-age (numbers) (22) Expansion factor (23) Observed survey biomass index ð¼ð¡â² = ðððµð¡ðð¿ð¡ð1â0.5(ð1)2 The initial (unfished) population was set by assuming lognormal variation around an average unfished recruitment (Eq. 2.2.12). Annual recruitment of age-1 fish followed a Ricker SRR (Ricker 1975) parameterized in terms of compensation ratio (κ) and unfished mean recruitment (ð 0) for the three stocks (Eqs. 2.2.6â2.2.8) (Walters and Martell 2004). While pelagic stocks as anchovy, sardine, and mackerel can follow a Beverton-Holt SRR, this is not always the case. For example, Ichinokawa et al. (2017) examined 26 stocks, including anchovy, sardine, herring, and mackerel and in many cases (20 stocks), a bias-corrected AIC could not support a Beverton-Holt, Ricker or a hockey stick (Clark and Kirkwood 1986) SRR. The SSR is usually poorly defined and has high uncertainty (Myers and Pepin 1994) even in data-rich stocks. The Ricker model is mathematically convenient in the Kalman filter estimation procedure. However, the Kalman filter could be extended in a future study to include a Beverton-Holt SRR. ,,,1a ta t Aa taCpCï½ï½ï¥,~ (0,1)a tNï¥, , , 2 , , 211ln (ln ) Aa t a t a t a t a tax p pAï¥ ï´ ï¥ ï´ï½ï½ ï« ï­ ï«ï¥,,,1exp( )exp( )a ta t Aa taxpxï½ï¢ ï½ï¥, ,ta t a taCC pwï¢ ï¢ï½,1Aa a t aaw p wï½ï½ï¥~ (0,1)tNï¤19 The regime shifts were simulated using a âstep functionâ. First, a lognormal error, with standard deviation ÏR, multiplied the âtrueâ productivity or density-dependence parameters. Then, a multiplier factor, ðð¡, simulated the abrupt increases and decreases depending on the RS scenario (Eqs. 2.2.9â2.2.10). The simulations included five RS (see simulation scenarios section). A RS consisted of a decrease in productivity or density-dependence over a period (e.g., ten years), followed by a return to the initial state. Parameter definitions and equations for the OMs are in Tables 2.1â2.2. 2.2.3 Data generation The OM generated data for the SCA and VPA models by adding observation error to the âtrueâ survey biomass index and fishery age-composition data. The survey biomass index was proportional to the vulnerable biomass with a bias-corrected lognormal error term with a standard deviation Ï1 (Eq. 2.2.23). The age-composition data were simulated using a multivariate-logistic distribution (Schnute and Richards 1995; Martell and Stewart 2014) with standard deviation Ï1 from true catch-at-age values (Eqs. 2.2.19â2.2.22). The true catch-at-age was the product of the abundance-at-age, selectivity-at-age, and the harvest rate. The survey biomass index and fishery age-composition data were generated for every year. To facilitate comparison between assessment models, the Ï1 and Ï2 had the same value. The OM simulated âobservedâ proportions-at-age for the SCA. For the VPA, the âobservedâ catch-at-age in numbers was obtained by scaling the simulated proportions-at-age to the total catch biomass using a ratio estimator (Eqs. 2.2.21â2.2.22). 20 Table 2.3. Parameter estimated, residuals, and likelihood function for the SCA model. The parameter with a â^â symbol indicates that the value was estimated. Description Equation (1) Estimated parameters ?Ì?ð ðð = ?Ì?0, {?Ì?ðâ² }ð=2ð=ð´, {?Ì?ð¡â²â²}ð¡=1ð=ð , Î©Ì1, Î©Ì2, ?Ì?12, ?Ì?22, ?Ì? (2) Initial abundance ðð,1 = ?Ì?0ððð?Ì?ðâ² (3) Recruitment ð1,ð¡ = ?Ì?0ð?Ì?ð¡â²â² (4) Residual error for the survey biomass ln( ) ln( ) t t td I VB zï¢ï½ ï­ ï­ (5) Residual error for fishery age-composition , , , , ,11ln( ) ln( ) [ln( ) ln( )]Aa t a t a t a t a tap p p pAï¸ï½ï¢ ï¢ï½ ï­ ï­ ï­ï¥ (6) Catchability coefficient ?Ì? = ðð¥ð(ð§Ì ); â¡â¡â¡â¡â¡â¡â¡ð§Ì =1ð1â[ðð(ð¼ð¡â²) â ðð(ððµð¡)]ð1ð¡=1 (7) Variance for survey biomass index ?Ì?12 =1(ð1 â 1)â ðð¡2ð1ð¡=1 (8) Variance for fishery age-composition ?Ì?22 =1(ð´ â 1)ð2ââðð,ð¡2ð´ð=1ð2ð¡=1 (9) Harvest rates ð¢ð¡ =ð¶ð¡ððµð¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡0 < ð¢ð¡ < 1 (10) Priors for initial abundance and recruitment ðð =12(ðð )2â ?Ì?ðâ²2 +12(ðð )2ð´ð=2â ?Ì?ð¡â²â²2ðð¡=1 (10) Negative log-likelihood ð¿ð ðð = ð1ðð?Ì?12 + (ð´ â 1)ð2ðð?Ì?22â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ (11) Objective function ððð(Î) = ð¿ð ðð + ðð 2.2.4 Estimation models 2.2.4.1 SCA The SCA followed the same population dynamics equations as the OM (Table 2.3) except that the abundance-at-age in the first year, the annual recruitment, and population and fishery parameters were estimated in the model fitting process (Eq. 2.3.1). The SCA reconstructs the abundance-at-age forward in time. The annual recruitment was estimated as ?Ì?0 multiplied by a lognormal error with a bias correction factor, assuming a ÏR for the logarithm of the recruitment residuals (Eqs. 2.3.2â2.3.3). Thus, the model did not include a SRR in the population dynamics. 21 The abundance-at-age for the next year decreased given the abundance-at-age, natural and fishing mortality, and fishing selectivity from the previous year (Eq. 2.2.13). The selectivity-at-age was estimated and was asymptotic and time-invariant based on a logistic model. The ut were conditioned on the âobservedâ catch biomass (Eq. 2.3.9); i.e., it followed the stock reduction analysis (SRA) approach. Table 2.4. Parameter estimated, residuals, and likelihood function the VPA. The parameter with a â^â symbol indicates that the value was estimated. Description Equation (1) Estimated parameters Îð£ðð = (?Ì?ð , ?Ì?12, ?Ì?) Population dynamics (2) ðð,ð¡ =ðð+1,ð¡+1ð ð+ ð¶ð,ð¡â² â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð¡ < ð; ð < ð´ â 2 (3) ðð´â1,ð¡ = (ðð´,ð¡+1ð ð+ ð¶ð´â1,ð¡â² + ð¶ð´,ð¡â² ) â â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡(ð¶ð´â1,ð¡â²ð¶ð´â1,ð¡â² + ð¶ð´,ð¡â² + ð) â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð¡ < ð; ð < ð´ â 1 (4) ðð´,ð¡ = (ðð´,ð¡+1ð ð+ ð¶ð´â1,ð¡â² + ð¶ð´,ð¡â² ) â â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡(ð¶ð´,ð¡â²ð¶ð´,ð¡â² + ð¶ð´â1,ð¡â² + ð) â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð¡ < ð; ð = ð´ (5) ðð,ð = ð¶ð,ðâ² ?Ì?ðâ ?Ì ?ð,ðâ¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð¡ = ð; ð < ððð¢ðð (6) ðð,ð = ð¶ð,ðâ² ?Ì?ðâ â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð¡ = ð; ð â¥ ððð¢ðð Relative selectivity (7) , , ,( / ) / ;1a t a t a t tv C N u t T a Aï¢ï½ ï¼ ï£ ï£ (8) ð£ð,ð = (ð¶ð,ðâ² ðð,ðâ ) ?Ì?ðâ â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð¡ = ð; 1 â¤ ð â¤ ð´ (9) Terminal selectivity 1, ,1,1 1 t T pa T a t fullt Ta T fullv v a apv a aï½ ï­ ï­ï½ ï­ï½ ï¼ï½ ï³ï¥ (10) Vulnerable biomass , ,1At a t a T aaVB N v wï½ï½ï¥ (11) Harvest rate , ,, fullfullAa ta at Aa ta aCu t TNï³ï³ï¢ï½ ï¼ï¥ï¥ (12) Negative log-likelihood ð¿ð£ðð = ð1ðð?Ì?12 22 2.2.4.2 VPA Contrary to the SCA, the VPA runs backward in time along cohorts. First, the VPA calculates the abundance-at-age in the terminal year (most recent) as: ðð,ð = ð¶ð,ðâ² /ð¢ð?Ì ?ð,ð (1) where ð¶ð,ðâ² is the âobservedâ catch-at-age in numbers, uT is the terminal harvest rate, and ð£ð,ð is the terminal fishery selectivity-at-age. The VPA estimated uT by minimizing the differences between the backward predicted and âobservedâ survey biomass index (i.e., a process called âtuningâ). As in the SCA, the predicted survey biomass index depends on the predicted vulnerable biomass and q. At year T, for ages not fully vulnerable, the ð£ð,ð were calculated using the relative selectivity-at-age averaged over the last five years (Eq. 2.4.7â2.4.8), without including year T. For ages fully vulnerable, ð£ð,ð = 1. Then, for the years <T, the VPA propagated the population dynamics backward in time by solving the equation ðð+1,ð¡+1 =ð ð(ðð,ð¡ + ð¶ð,ð¡â² ) for ðð,ð¡ (see Eq. 2.4 and Table 2.4 for notation). The VPA included a plus group split into the age ð´ â 1 and ð´, based on their catch ratios (where ð´ is the plus-group) (Eqs. 2.4.3â2.4.4). For the years <T, the annual harvest rates were the average of the âobservedâ catch-at-age (in number) over the predicted abundance-at-age for ages fully vulnerable (Eq. 2.4.11). Note that VPA estimated far fewer parameters than the SCA: uT and two conditional parameters (q and Ï1) used in the likelihood function for the survey biomass index (Eq. 2.4.1). For the VPA and SCA models, individual simulations were discarded when the maximum gradient from the optimization procedure was greater than 0.001. Details of the VPA are in Table 2.4. 23 2.2.5 Stock-recruitment analysis The traditional time-invariant Ricker SRR is given by: ð ð¡ = ð¼ðððµð¡ðâð½ðððµð¡+ðð¡ (2) where ð ð¡ is the age-1 recruitment, SSBt is the spawning stock biomass in year t, and α and β are the productivity and density-dependence parameters, respectively, and ððð¡ are log-normally distributed errors ðð¡ â¼ ð(0, ðð2). The Ricker model was linearized as: ð¦ð¡ = ðâ² â ð½ðððµð¡ + ðð¡ (3) where ðâ² = ðð(ð¼) is the logarithm of the productivity parameter. Equation (3) is a linear regression model with ð¦ð¡ = ðð (ð ð¡ðððµð¡). This model allows productivity and density-dependence to vary over time (either independently or simultaneously). However, the underlying pattern for RS in nature is unknown (Peterman et al. 2000). In a simulation study, Peterman et al. (2000) showed that a random walk tracked different temporal trends, including a âstep functionâ, a sine wave, and an autoregressive at lag 1 (AR(1)) model, when the productivity parameter was time-varying. Here, the simulations showed that the âstep functionâ can be tracked using a random walk, but depended on the scenarios (see Results). Thus, a random walk was also used for the changes in the stock-recruit parameters: ð¦ð¡ = ðð¡â² â ð½ð¡ðððµð¡ + ðð¡ (4) ðð¡â² = ðð¡â1â² + ð¤ðâ²,ð¡ (5) ð½ð¡ = ð½ð¡â1 + ð¤ð½,ð¡ (6) where ð¤ðâ²,ð¡ and ð¤ð½,ð¡ are the âprocess errorsâ distributed as ð¤ðâ²,ð¡ â¼ ð(0, ðð¤ðâ²2 ) and ð¤ð½,ð¡ â¼ð(0, ðð¤ð½2 ). The time-varying stock-recruit parameters can be estimated using the Kalman filter, 24 which has two main equations: the state and observation equations. The state equation describes the temporal changes in the stock-recruit parameters (Eqs. 5â6). However, we do not observe the state variables directly, but through the observation equation, the Ricker SRR (Eq. 4). This equation links the observations yt with the unobservable states at the time t. Note that yt and SSBt were outputs from the VPA and SCA models. The Kalman filter is a recursive algorithm that first starts with a set of the âprediction equationsâ to provide estimates for both the state variables and their variances (or a variance-covariance matrix if the state variables are correlated). Then, the algorithm uses a set of âupdate equationsâ to update the state variable estimates and their variances conditioned on a new observation as it becomes available, outputs from the prediction equations, and a variance ratio called âKalman gainâ. The prediction equations predict the state variables at time t given data up to time tâ1 while the updated equations using data up to time t. The state variable estimates (and their variances) from the update equations are called posterior or filter estimates. The posterior state estimates (and their variances) are obtained applying the âKalman gainâ which is a ratio of the state variances to the total variance (i.e., it includes variance components for the state and observation equations). The Kalman filter algorithm needs an initial condition (t=0) for the state variables (ð0â² ,β0) and their variances (ðð0â²2 , ðð½02 ) to iterate through the predicted and updated equations for a time-series of observation from t=1, 2, 3, â¦, T. Note that the Kalman filter algorithm provides predicted and posterior state variable estimates at any time t. The Kalman filter uses the difference between the observation t and the predicted value to calculate a prediction error at time t (or also called innovation). Then, the prediction error and the innovation variances are used to calculate a log-likelihood function for the Kalman filter parameters ð. Here, the parameters estimated were the initial condition for the state variables and 25 variance components for the state and observation equations, ð={ð0â² , ð½0, ðð2, ðð¤ð0â²2 , ðð¤ð2 }. The ðð0â²2 and ðð½02 were set at 1. Note that the state variables at time t are analytically calculated conditional on ð. The negative log-likelihood was minimized by applying the Kalman filter and the smoother provided in Holmes et al. (2012). Because we were interested in reconstructing the time-series of stock-recruit parameters given the data already available (i.e., the data are not arriving sequentially in time), we applied the retrospective analysis (smoothing). Full details of the Kalman filter are found in Harvey (1989) and Petris et al. (2009). It is not clear how the stock-recruit parameters change over time, and in particular, when both change simultaneously. In this situation, it is possible that estimating the covariance for the stock-recruit parameters could improve the estimates. This hypothesis was tested by estimating the covariance parameter (ððð½) in the state equation. Two Kalman filter estimators were applied to RS scenarios; 1) a filter that did not estimate the covariance parameter (KFÏ0,0) and 2) one that estimated this parameter (KFÏaÎ²). 2.2.6 Simulation scenarios for changes in productivity and density-dependence The âstep functionâ simulated five RS: 1) both stock-recruit parameters were stationary; 2) the productivity was non-stationary and the density-dependence was stationary; 3) the density-dependence was non-stationary and productivity was stationary; 4) both stock-recruit parameters were non-stationary and negatively correlated, and 5) stock-recruit parameters were non-stationary and positively correlated. Scenarios 1-5 were denoted as stationary-SRR, productivity -RS, density-dependence-RS, non-stationary-SRRâ, and non-stationary-SRR+, respectively. The 26 âââ and â+â symbols indicate that productivity and density-dependence changes were negatively or positively correlated, respectively. 2.2.7 Reference points analysis To examine how the stock-recruit parameters derived from the Kalman filter affected key reference points, the MSY, the harvest rate that produces the MSY (ðððð), and the SSB at MSY (ðµððð) were recalculated for each time t. The MSY-reference points were derived using the âincidence functionsâ (Walters and Martell 2004). These depend on survivorship-at-age, selectivity-at-age, and fecundity-at-age integrated with an equilibrium stock-recruitment analysis to calculate equilibrium yield and ðððð (Table 7.1A). The MSY-reference points used the estimated selectivity-at-age in the terminal year in the SCA or VPA models. 2.2.8 Performance statistics Performance statistics included time-series of recruitment, SSB, and time-varying MSY reference points. The simulations assessed both bias (i.e., the closeness of the average parameter estimate to the true value) and precision (i.e., the degree of reproducibility between repeated estimates) of the estimates by calculating the relative error (RE) and the absolute relative error (ARE) for each trial, relative to the OM values. The median of 100 trials was calculated over the years and then over simulations (MRE and MARE, respectively) (Table 7.2A). 27 Figure 2.1. Outputs from the SCA and VPA models for hake. Black lines indicate true trends. The median and 95% confident intervals for the estimates are the thick line and shaded regions, respectively. The scenario had an uninformative catch history. Columns show the regime shift (RS) scenarios (from left): stationary-SRR, productivity-RS, density-dependence-RS, non-stationary-SRRâ, and non-stationary-SRR+. The âââ and â+â indicate that productivity and density-dependence changes were negatively or positively correlated, respectively. 28 2.3 Results 2.3.1 VPA and SCA assessment model performance Under the simulated RS, both assessment models (AMs) fit the time-series of recruitment, SSB, and harvest rates at close to the true values for the three stocks (Tables 7.3Aâ7.5A). The time-series for hake are in Fig. 2.1. For jack mackerel and anchovy, and the scenarios with the informative catch history, are in Figs. 1.1A and 7.2A. The SCA showed higher precision than the VPA for the three stocks. However, the VPA showed more precise estimates across RS, stocks, and catch histories. For recent years, the SCA overestimated recruitment and SSB, and underestimated harvest rates, in particular, under the productivity-RS and non-stationary-SRRâ. The informative catch history improved the time-series estimates under the productivity-RS, but still were positively biased for the density-dependence-RS and non-stationary-SRRâ (Figs. 7.3A, 7.4A, and 7.5A). 2.3.2 Stock-recruitment analysis Under the stationary-SRR, the two Kalman filters (KFÏ0,0 and KFÏaÎ²) produced unbiased stock-recruit parameter estimates across stocks, AMs, and catch histories (Figs. 2.2â2.4). However, the SCA had higher precision than the VPA. The KFÏaÎ² had lower precision, in particular, under the VPA (Tables 7.3Aâ7.5A). Under the productivity-RS, in jack mackerel and anchovy, and under both AMs, the filters tracked the changes in productivity. However, both stock-recruit parameter trends were overestimated under the uninformative catch history, in particular, productivity (Figs. 2.2â2.3; Tables 7.3Aâ7.5A). The KFÏaÎ² produced better estimates under the VPA; it tracked the productivity in magnitude, but it overestimated the true trend over time. The filters tracked better 29 the changes in productivity when the catch history was informative (Figs. 7.6Aâ7.8A), but the density-dependence still showed an upward bias for jack mackerel and anchovy. Assessments applied to hake had the best performance in estimating both stock-recruit parameter trends; the KFÏ0,0 had better performance (Fig. 2.4) (Tables 7.3Aâ7.5A). Figure 2.2. Stock-recruit parameter estimates from the Kalman filter obtained from the SCA and VPA model outputs for jack mackerel; subscripts indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The median and 95% confident intervals are the thick line and shaded regions, respectively. Black lines indicate true trends. The scenario had an uninformative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 30 Figure 2.3. Stock-recruit parameter estimates from the Kalman filter obtained from the SCA and VPA model outputs for anchovy; subscripts indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The median and 95% confident intervals are the thick line and shaded regions, respectively. Black lines indicate true trends. The scenario had an uninformative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 31 Figure 2.4. Stock-recruit parameter estimates from the Kalman filter obtained from the SCA and VPA model outputs for hake; subscripts indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The median and 95% confident intervals are the thick line and shaded regions, respectively. Black lines indicate true trends. The scenario had an uninformative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 32 Under the density-dependence-RS, the KFÏ0,0 reliably detected the trends for both stock-recruit parameters across AMs and stocks under the uninformative catch history (Figs. 2.2â2.4). The SCA, overall, had higher precision in parameter estimates than the VPA (Tables 7.3Aâ7.5A). However, when the catch history was informative, the performance decreased for both stock-recruit parameters; the productivity trend was overestimated and the density-dependence trend was overestimated when the density-dependence has a low RS (Figs. 7.6A, 7.7A and 7.8A). For the non-stationary-SRRâ, the VPA performed better than the SCA for both stock-recruit parameter trends across stocks (Figs. 2.2â2.4). Both filters produced a positive bias for the density-dependence at the end of the time-series (Figs. 2.2â2.4). The performance was better under the KFÏaÎ² for both AMs (Tables 7.3Aâ7.5A).When the catch history was informative, the performance was worse for the productivity and the density-dependence trends across stocks and AMs, particularly for the SCA (Figs. 7.6Aâ7.8A). The KFÏÎ±Î² reduced a positive bias for productivity, but not for density-dependence. For the non-stationary-SRR+, the filters could not track the stock-recruit parameter trends across AMs, stocks, and catch histories (Figs. 2.2â2.4). 2.3.3 MSY reference points For the stationary-SRR, the MSY-reference point trends were close to the true values, and the VPA had less precise estimates (Figs. 2.5â2.7) (Tables 7.3Aâ7.5A). For the productivity-RS in jack mackerel and anchovy, the ðððð¡ trends were tracked. However, the filters did not capture the magnitude of the changes, while for hake the filters 33 performed well (Figs. 2.5â2.7). In jack mackerel, both filters underestimated the ðµðððð¡ under both AMs. For anchovy, both filters tracked the ðµðððð¡ but underestimated the trend when there was a high ðµðððð¡ , particularly under the SCA model. For jack mackerel, only the KFÏ0,0 under the VPA tracked the ððððð¡ but overestimated the last low ððððð¡ regime. For anchovy, the best ððððð¡ estimates were obtained under the VPA. When the catch history was informative, the ððððð¡ estimates improved for the SCA model (Figs. 7.9Aâ7.11A). Across AMs, filters, and catch histories, the last low ððððð¡ regime was overestimated. For hake, both AMs showed good performance in estimating the ððððð¡ . For the density-dependence-RS across AMs and stocks, the filters tracked the ðððð¡, ðµðððð¡ , and ððððð¡ trends, and the KFÏ0,0 had more precise values (Figs. 2.5â2.7) (Tables 7.3Aâ7.5A). The informative catch history decreased performance, particularly for the SCA, where both filters overestimated the last low ðððð¡ and ðµðððð¡ regimes (Figs. 7.9Aâ7.11A). Also, both AMs underestimated the last high ðððð¡ and ðµðððð¡ regime. For the non-stationary-SRRâ across stocks, the VPA tracked the MSY-reference point trends and the KFÏ0,0 had more precise estimates (Figs. 2.5â2.7) (Tables 7.3Aâ7.5A). The SCA model overestimated the last low ððððð¡ regime across stocks. The informative catch history decreased performance, particularly for the SCA model (Figs. 7.9Aâ7.11A). For the non-stationary-SRR+, the MSY reference points were biased (Figs. 2.5â2.7). The ðµðððð¡ were underestimated when the ðµðððð¡ regime was high, while the estimated ððððð¡ ranged between the high and low ððððð¡ estimates. 34 Figure 2.5. MSY reference points calculated from the Kalman filter outputs for jack mackerel. Gray and blue-red lines indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The median and 95% confident intervals are the thick line and shaded regions, respectively. Black lines indicate true trends. The scenario had an uninformative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 35 Figure 2.6. MSY reference points calculated from the Kalman filter outputs for anchovy. Gray and blue-red lines indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The medians and 95% confident intervals are thick line and shaded regions, respectively. Black lines indicate true trends. The scenario had an uninformative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 36 Figure 2.7. MSY reference points calculated from the Kalman filter outputs for hake. Gray and blue-red lines indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The medians and 95% confident intervals are thick line and shaded regions, respectively. Black lines indicate true trends. The scenario had an uninformative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 37 2.4 Discussion 2.4.1 VPA and SCA assessment performance Both assessment models estimated recruitment and SSB trends very close to the true values (i.e., MRE ± 2%). However, under the productivity regime shift and when both stock-recruit parameters were non-stationary and negatively correlated, the SCA overestimated recruitment and SSB for the most recent period while the VPA did not. This pattern remained even when the catch history was more informative. This result was surprising because forward-projection models (i.e., SCA) typically âseeâ final recruitment trends better compared with backward projection models (i.e., VPA). Therefore, forward-projection models under non-stationary stock-recruit parameters could be less likely to detect âterminal regime shiftsâ as suggested via the simulations. Unless that there is clear non-stationarity in the SRR, stock assessments typically do not examine this issue. In salmonids, however, considerable attention has focused on examining changes in productivity (e.g., Adkison et al. 1996; Peterman et al. 2003; Dorner et al. 2008; Malick and Cox 2016). Detecting non-stationarity in the SRR is challenging. This is because many stock assessments parameterize the SRR in a form that makes it difficult to detect non-stationarity. For example, the SRR can generate artificial patterns because the population dynamics model will be forced to balance estimation of the recruitment deviates with other sources of variation in the model and data (e.g., typically fishery selectivity in SCA models). A more suitable alternative is to analyze the stock-recruit data after model fitting (e.g., as we did here). If the stock is believed to have stationary stock-recruit parameters, the Kalman filter or other simpler methods (Wada and Jacobson 1998; Walters and Martell 2004 p.159) could help to determine whether this belief is justified. 38 Error propagation (Hilborn and Walters 1992; Quinn and Deriso 1999) and retrospective patterns (Parma 1993) have demonstrated bias in backward projection models. Consequently, VPA has received little attention for reconstructing the time-series of stock-recruit trends. The results showed that the VPA was more consistent in tracking regime shifts than the SCA, possibly because the VPA has fewer assumptions and parameters (Walters and Martell 2004). However, the VPA had lower precision than the SCA, particularly at the end of the time-series (Parma 1993; N.R. Council 1998). The fishery selectivity was time-invariant in the simulations. As a result, the model outputs were similar between the SCA and VPA. Nevertheless, the VPA out-performed the SCA under some regime shifts, as mentioned above. The VPA has some key advantages for regime shift detection. The VPA does not assume any SRR or recruitment variance, and more importantly, does not require any assumptions about selectivity-at-age. Assumptions about these are standard features of SCA models. The time-varying selectivity is problematic in SCA models, requiring much work to control for (or estimate) changes over time (Sampson and Scott 2012; Martell and Stewart 2014; Monnahan et al. 2019). Under regime shifts, for example, misspecification of selectivity may generate artificial patterns in recruitment estimates (Bence et al. 1993; Linton and Bence 2011). 2.4.2 Stock-recruitment analysis No studies have compared the estimability of time-varying stock-recruit parameters in age-structured models under regime shifts. Previous studies have examined stock-recruit parameters but under time-invariant scenarios (e.g., Ludwig and Walters 1981; Magnusson and Hilborn 2007; Conn et al. 2010). Under the stationary SRR scenario, the Kalman filter identified correctly the stationarity of stock-recruit parameters. Scenarios with non-stationarity were more 39 challenging to estimate. In particular, when the productivity was changing, or when both parameters were changing for most recent regime shifts. Under the productivity regime shift and the uninformative catch history pattern, positive bias occurred in the time-series of productivity for jack mackerel and anchovy but not for hake. For anchovy, the Kalman filter detected periods of high productivity but overestimated the low productivity regimes, and the density-dependence trend was overall overestimated. As a result, for productivity regimes, ððððð¡ could be overestimated during low productivity while ðµðððð¡ underestimated. For hake, the Kalman filter tracked the productivity regimes. However, the longer duration of the regimes could have resulted in improved performance. When process errors become less correlated (e.g., a shorter time between regime shifts), the state equation will have âmoreâ random stochasticity, making it difficult to discriminate process error from observation error. In an extreme case, for example, the process equation for the productivity parameter (Eq. 5) could be just random stochasticity. As a result, the model will not be identified for the productivity parameter. Jack mackerel had productivity regimes similar to those for hake. However, the time-series of stock-recruit parameters were overestimated for jack mackerel. The Kalman filter detected periods of low and high productivity regimes, but both stock-recruit parameters were positively biased. Time-series bias (e.g., Walters 1985) could have generated this effect (i.e., both α and β would be overestimated). Here, the simulated time-series was long enough (n=40) to expect relatively little time-series bias despite the stable harvest rates (e.g., uninformative catch history). Moreover, the collection of information started from unfished conditions and covered a wide 40 range of spawning biomass, and the recruitment compensation was relatively high for the stocks. In short time-series (n=20) for low productivity stocks with fixed harvest rates, the stock-recruit parameters can be seriously biased because the time-series bias effect (e.g., Walters 1985; Korman et al. 1995). The time-series bias effect is because of non-representative sampling of the spawner abundance caused by correlations between recruitment anomalies, subsequent recruitment and spawner abundance. Regression errors-in-variables effects are less likely to have generated the positive bias in the productivity regime. Under the simulated catch history, where the stock biomass was fished from unfished condition and then not too heavily (e.g., at Umsy level), the bias in the productivity parameter would be expected to be negative (Kehler et al., 2002). The informative catch history pattern improved the time-series of productivity estimates, especially for jack mackerel under the SCA (Fig. 7.6A). Apparently, for jack mackerel, to obtain unbiased productivity estimates higher harvest rates are needed to inform the slope at the origin (Hilborn and Walters 1992). The observation equation (Eq. 4) suggests the time-series of productivity might be easier to estimate than the density-dependence changes since productivity parameter estimation tends to be less dependent on the spawning biomass time series; it is more challenging to separate the process error for density-dependence from observation errors. Under the density-dependence regime shift, both VPA and SCA models performed well, contrary to expectation. The duration of the regimes, as mentioned above, and the catch history probably influenced stock-recruit parameter estimation. The simulation had more extended regime shifts for density-dependence because changes in carrying capacity likely have extended environmental periods (e.g., decadal oscillations). For instance, Wilderbuer et al. (2002, 2013) found that for three Eastern Bering Sea flatfish stocks, the changes in productivity were consistent with decadal-scale variability. 41 However, the density-dependence parameter is confounded with observation error, in particular, when the observation error is larger than the process error (De Valpine and Hastings 2002; Knape 2008; Auger-Méthé et al. 2016). In stock assessment, the process error is likely larger than observation error, making the estimability less critical for the density-dependence parameter, but likely biased for other factors. Some catch time-series patterns generate more contrast in SSB, which produces better estimates for stationary stock-recruit parameters (Hilborn and Walters 1992; Conn et al. 2010). Here, a more informative catch history pattern did not necessarily lead to better time-series estimates of density-dependence. In particular, for the density-dependence regime shift and when both stock-recruit parameters were non-stationary and negatively correlated. In fact, the performance was poorer (see Figs. 7.6Aâ7.8A). The confounding effects of the observation and the process equations could explain this finding; increasing or decreasing spawning biomass trends (because the more informative catch history) will be confounded with periods of low and high density-dependence. For example, the Kalman filter will not discriminate between increasing spawning biomass under low fishing mortality and low density-dependence. At the same time, the productivity changes will be more reliably estimated because productivity estimates are less dependent on the spawning biomass time-series and because of better contrast on the data. Therefore, more âstableâ catch histories may improve detection of density-dependence regime shifts, but not necessarily productivity regime shifts. Even when the operating model simulated time-invariant trends in productivity or density-dependence, the Kalman filter still produced persistent positive bias that continued over time. The âupdate equationsâ did not correct the bias, although new data became available with 42 each new time step. This showed the importance of simulations to examine potential issues of estimability that cannot be achieved with empirical studies. The Kalman filter assumed that the temporal changes in the stock-recruit parameters followed a random walk process while the regime shifts were simulated using a âstep functionâ. This probably produced a departure from the Gaussian process assumed by the Kalman filter. This situation was expected. However, as Peterman et al. (2000) indicated, we showed that a random walk could track both stock-recruit parameters (either independently or simultaneously) generated by a âstep functionâ under some conditions. We do not know for sure the underlying pattern stock-recruit parameters in nature due to the numerous uncertainties associated with fitting models to field data (Francis 2011). More importantly, in stock assessment and management strategy evaluation (MSE) (Punt et al. 2016), it is not appropriate for the operating model that simulates data to be identical to the estimation model. Ideally, the operating model should challenge the estimation model or harvest strategy to detect robustness, even when the statistical assumptions of the estimation model do not conform to those of the simulation model. The analyses presented here are only the first step to assess non-stationarity in stock-recruit parameters. Future studies could consider a long-lived life-history type, like rockfish and a Kalman filter including a Beverton-Holt SRR for pelagic stocks. 2.4.3 Conclusions and recommendations The simulations showed that if stock-recruit parameters are non-stationary, the Kalman filter can likely detect at least broad patterns of change. The detection of productivity or density-dependence regime shifts depended on the attributes of the stock and catch history. Surprisingly, the VPA had better performance in detecting recent regime shifts than the SCA model. 43 The simulations suggest that the Kalman filter has a better chance of detecting non-stationarity in stock-recruit parameters with more extended regime shifts. For example, for stocks with longer periods of low and high productivity regimes, the filter will probably perform well detecting these shifts. Non-stationarity should be tested under both the VPA and SCA models. If the stock-recruit parameters are non-stationary, the SCA will likely be more prone to model misspecification (e.g., selectivity) and spurious trends. VPA will avoid imposing any selectivity pattern (or SRR) on the recruitment estimates but will do so at the cost of lower precision. To make it easier to examine non-stationarity in stock-recruit parameters, analysts should avoid including a SRR in SCA models (or at least include a very high ÏR in the SRR). If the analyst insists on using a SCA model, the time-varying selectivity calculated in the VPA could be used as a fixed input in the SCA. This could reduce the effects of model misspecification. Then, the stock-recruit data could be analyzed outside the stock assessment model. The analyses could be complex (e.g., a Kalman filter) or use simpler approaches. The latter could include, for example, examining residuals in the SRR and looking for persistent residual patterns and examining changes in the stock-recruit parameters using temporal frames (e.g., Wada and Jacobson 1998; Peterman et al. 2000; Ichinokawa et al. 2017). 44 3 Chapter 3: Stochastic optimization within closed-loop simulation for the identification of optimal harvest control rules for fisheries that undergo regime shifts 3.1 Abstract A novel approach based on stochastic optimization within closed-loop simulation was used to identify robust optimal harvest control rules (HCRs). This study evaluated optimal HCRs under persistent changes (i.e., regime shifts, RS) in intrinsic population growth r, carrying capacity K, and an autoregressive RS. Three surplus production models (SPM) using three stocks (anchovy, hake, and redfish) were examined. The optimal HCRs were compared to maximum sustainable yield (MSY) HCRs. The optimal HCR parameters depended on the level of observation error, on how the population dynamics were modelled (discrete and continuous fisheries), the utility function, and stock productivity (r). The risk-averse utility function and MSY HCRs produced less or similar average biomass yield (ð¶Ì ) compared to the optimal fixed escapement rule (FER). The fixed harvest rate (FHR) strategy was not always supported under a risk-averse utility function. When the stocks had low productivity (hake and redfish), the optimal FHR was supported more often. For productive stocks (i.e., anchovy), the continuous SPM was more robust to observation error and the risk-averse utility function had the best ð¶Ì under the carrying capacity and productivity RS. For hake and redfish, the risk-neutral and risk-averse utility functions had similar a ð¶Ì when the RS was in carrying capacity and productivity. For the autoregressive regime shift, the performance depended on the SPM and the utility function. For the base case scenario and across stocks, the time-invariant and time-varying optimal HCRs (both utility functions) had a similar ð¶Ì â¡ to that for the MSY HCRs under the carrying capacity and productivity RS. Overall, the performance (e.g., ð¶Ì ) of including time-varying HCR parameters was marginal. For the 45 autoregressive RS and across stocks, the time-invariant risk-neutral HCR and the time-varying MSY escapement HCR had the highest ð¶Ì . Simple HCRs (e.g., survey biomass estimates) performed similarly to complex HCRs. 3.2 Introduction A common assumption in fishery stock assessment is that populations follow a long-term steady state (Walters 1987). Under this assumption, although the stock productivity may change over time, the population parameters that govern the stock dynamics remain stationary. Recent research has shown that this assumption is not always correct. Several fish stocks have shown non-stationarity in the biological parameters controlling population dynamics (Peterman et al. 1998; Dorner et al. 2008; de Moor et al. 2011; Peterman and Dorner 2012; Vert-pre et al. 2013; Minto et al. 2014; Britten et al. 2016). For example, some stocks that undergo persistent changes in fish productivity can generate unexpected collapses or sudden so-called âboomsâ in biomass (e.g., Oliveros-Ramos et al., 2010; de Moor et al. 2011; de Moor and Butterworth 2016). These persistent changes in productivity are referred as regime shifts (RS) (e.g., Mohn and Chouinard 2007). It is not clear which harvest strategies can be applied under RS to maximize the average yield when stocks are characterized by periods of low or high productivity. When the parameters that control the intrinsic population growth (r) change over time, the annual exploitation rate needs to be adjusted. However, when the carrying capacity (K) changes alone, the annual exploitation rate is unaffected (Walters and Parma 1996). The estimates of the r and K parameters are often confounded in stock assessment models, making it difficult to identify the proper harvest strategy. Moreover, it is not clear if harvest strategies need to be regime-based 46 (e.g., Mohn and Chouinard 2007; Szuwalski and Punt 2013a, 2013b) or time-invariant (e.g., that do not depend on the RS) to cope with changes in fish productivity. It is possible that a simple harvest strategy would perform well regardless of the RS (e.g., changes in r or K). Some harvest strategies, for example, require a full description of the population dynamics to provide the current biomass estimate to prescribe a quota, while other harvest strategies use direct information from the biomass estimates derived from trawl research surveys. These approaches are referred as model-and data-based harvest strategies, respectively (see Rademeyer et al. 2007; Edwards and Dankel 2016). At present, the most common approach to evaluate harvest strategies is by closed-loop simulation or feedback control (also called management strategy evaluation, MSE) (Kell et al. 2005; Rademeyer et al. 2007; Punt et al. 2016). Usually, MSE uses âa prioriâ or âad hocâ harvest control rule (HCR) parameters to prescribe a quota (e.g., Edwards and Dankel 2016). MSE can accommodate complex models and uncertainty and are useful to find the âbest harvest strategiesâ under conflicting management objectives. Under this approach, however, it is only possible to examine a limited range of potential harvest options. Consequently, the relative performance of candidate harvest strategies is only valid over the pre-defined HCRs; hence, MSE cannot be used to identify broadly optimal solutions. A second method corresponds to finding harvest strategies using stochastic dynamic programming (SDP). SDP is a powerful method to find optimal harvest strategies. In contrast to MSE, SDP takes into account all possible harvest strategies under all possible harvest sequences for all possible realizations of the population dynamics (Memarzadeh and Boettiger 2019). SDP allows for a harvest strategy to be completely flexible (e.g., it can take any âshapeâ), while also being optimal (i.e., it maximizes a utility function). However, SDP has some significant 47 limitations. Due to the âcurse of dimensionalityâ, SDP relies on population dynamics models that have a single state dynamic (e.g., stock-recruit, surplus production, and delay difference models). Even more critical, the incorporation of simultaneous observation and process errors in SDP is extremely challenging. Observation error can be quite significant in stock assessment models and produces a considerable impact on optimal harvest strategies (Moxnes 2003; Sethi et al. 2005; Memarzadeh and Boettiger 2019). Few studies have examined optimum harvest strategies under RS or persistent productivity changes (Walters 1987; Walters and Parma 1996; Hawkshaw and Walters 2015). The present study was based on using a novel approach to find harvest strategies robust to persistent changes in stock productivity. In particular, the stochastic optimization in policy space (SOPS) has been proposed and used to fill the gap of the âcurse of dimensionalityâ (Moxnes 2003, 2005; Walters and Martell 2004). Importantly, this approach is can replicate the optimal HCR found in SDP (e.g., Reed 1979; Sethi et al. 2005). As in SDP, the SOPS also uses a utility function and the harvest strategy can be entirely flexible (Moxnes 2003, 2005). SOPS allows multiple dynamic states and different sources of uncertainty at the same time (e.g., process and observation error) and it can include feedback control (i.e., as an MSE). Therefore, SOPS is a promising approach to find optimal harvest strategies in non-stationary populations without imposing any a priori HCR, and thereby offers an advantage to more common approaches. This research extended previous studies in optimal harvest strategies (e.g., Reed 1979; Walters and Parma 1996; Moxnes 2003, 2005; Sethi et al. 2005; Memarzadeh and Boettiger 2019), including closed-loop simulation under multiple sources of uncertainty (e.g., observation and process error) and explicit incorporation of RS in the population dynamics. The optimal 48 harvest strategies were examined under risk-neutral (e.g., maximize total yield) and risk-averse (e.g., impose a high penalty to low catches at any time) utility functions for three stocks (i.e., Peruvian anchovy (Engraulis ringens), Namibian hake (Merluccius capensis), and Atlantic redfish (Sebastes mentella) that represent short-lived, âin-betweenâ, and long-lived life history types. The analyses included discrete (occurrence of harvest before or after growth production) and continuous fisheries with different approximations to estimate current biomass and test for optimum strategies. 3.3 Materials and methods 3.3.1 General framework Evaluation of HCRs using closed-loop simulation involved the following steps (see Edwards and Dankel 2016): a) Simulate the stock population dynamics under multiple hypotheses (e.g., RS in K or r) and in each year simulate abundance data (e.g., from a trawl biomass survey); b) Estimate the stock biomass from the simulated abundance data; c) Prescribe a quota as a function of the estimated biomass and the HCR; d) Apply the fishing mortality obtained from the HCR and update the population dynamics; and, e) Repeat steps a-d for the preset number of future years until the simulation is completed. 49 Table 3.1. Notation and parameters values used in the operating model. Symbol Value Description Indices T 1,â¦,T Annual time step Operating model parameters T 500 Last year of simulation r 1, 0.4, 0.1 Intrinsic population growth rate for anchovy, hake, and redfish K 1 Population carrying capacity p 1 Non-linear production growth parameter for continuous surplus production model (SPM) ðð 0, 0.6 Standard deviation for process error ðð 0, 0.7 Autocorrelation in process error ðð¡â² Annual deviates in year t for process error ðð¼ 0, 0.2, 0.4 Standard deviation for observation error in the trawl survey index ðð¡â² Observation errors in year t ðµðððð¡ Initial biomass at time t=1 Parameters assessment model ?Ì? 1, 0.4, 0.1 Intrinsic population growth rate for anchovy, hake, and redfish ?Ì? 1 Population carrying capacity ?Ì?ð 1 Standard deviation for process error ?Ì?ð¼ 0.4 Standard deviation for observation error ðð¼ 1 Coefficient of proportionality for the trawl survey Harvest control rule b 1 Non-linear parameter B* ðµâ â¥ 0 Minimum base-stock biomass Q* 0 â¤ ðâ â¤ 1 Slope (harvest rate) c 0.000001 Constant Derived variables ðð¡ , ðð¡â² , ðð¡â²â² Quota at time t from the time-invariant, time-varying, and nonlinear HCR ð¶ð¡ Catch biomass at time t ð¹ð¡,ðð¡ Instantaneous fishing mortality and harvest rate at time t Observations ðµð¡â² , ðµð¡â²â², ðµð¡â²â²â² Survey biomass estimates at time t derived from trawl survey, SPM (EKF), and local model (LM) 50 3.3.2 Operating model Fishing mortality can be modelled as a discrete event (e.g., seasonal pulse) or continuous on time. Regarding biomass estimates, these can be obtained before the harvest occurs (e.g., in salmon and some small pelagic fisheries) using the information obtained from multiple research surveys (de Moor and Butterworth 2016). This allows the quota to be set before the fishing season. Three surplus production models (SPM) were used to represent the population dynamics and harvest of fish stocks: a) Discrete SPM (Eq. 3.2.2), where a biomass estimate is known exactly before the harvest decision and the harvest occurs before biomass production (dHPM); b) Discrete SPM (Eq. 3.2.3), where the harvest occurs after biomass production (dPHM); and, c) Continuous SPM (Eq. 3.2.4), where the harvest occurs after and during biomass production (cPHM). The dHP and dPH models typically appear in SDP and fishery literature, respectively (e.g., Hilborn and Walters 1992; Memarzadeh and Boettiger 2019). The cPHM follows a Ricker model (Ricker 1954). A similar cPHM was recently evaluated through simulation (Pedersen and Berg 2017), but has not been tested under closed-loop simulation. Note that the above SPMs models are comparable because they derived from a more general SPM (Pella and Tomlinson 1969). Typically, the function that describes stock dynamics is multiplied by a process error term with autocorrelation to simulate persistent changes in fish productivity (Hawkshaw and Walters 2015; Siple et al. 2019). However, recent studies using state-space models showed that the parameters that describe the stock dynamics can change over time (e.g., Dorner et al. 2008; Peterman and Dorner 2012; Minto et al. 2014; Britten et al. 2016). To be consistent with these 51 studies, the population dynamics or the r and K parameters of the SPM was multiplied by an autocorrelated process error term that generated an autoregressive RS (AR-RS), a RS in K (K-RS) and RS in r (r-RS), respectively (Eqs. 3.2.2 to 3.2.4). Note that when the autocorrelation parameter was zero, the simulations only included white noise in the population dynamics or the r and K parameters, respectively. This scenario ignores RS, but assumes stochastic r and K parameters over time. The closed-loop simulation included three stocks with different life histories: Peruvian anchovy (Oliveros-Ramos et al. 2010), Namibian hake (Hilborn and Mangel 1997), and Atlantic redfish (Licandeo et al. 2020). Life history information (e.g., individual growth rates, instantaneous natural mortality, and recruitment compensation) were used to derive r using demographic methods (McAllister et al. 2001). The r values were approximated at 1.0, 0.4, and 0.1 for anchovy, hake, and redfish, respectively, and these values were consistent with the literature for similar stocks and life-histories (e.g., Rose 2004; Memarzadeh et al. 2019) (Table 3.1). The results for hake and redfish are shown in the Appendix B. 52 Table 3.2. Population dynamics model and harvest control rules (HCRs) used in the closed-loop simulation. Description Equation (1) Optimal harvest parameters ï¨ ï©,B Qïª ïªï ï½ (2) Discrete SPM with harvest before production (dHP) ï¨ ï© ï¨ ï©20.51 1( ) 1 ( ) / AR-RS( ) 1 ( ) /t Rtinitt t t t t ttt t t t t tB tB Q r B Q B Q K e ifBB Q r B Q B Q Keï· ï³ï·ï¢ï­ï¢ï«ï½ï­ ï« ï­ ï­ ï­ï© ï¹ï« ï»ï½ï­ ï« ï­ ï­ ï­ ï¨ ï©ï¨ ï©ï¨ ï© ï¨ ï©220.50.5 -RS( ) 1 ( ) / -RSRt Rt t t t t tif KB Q re B Q B Q K if rï³ï· ï³ï­ï¢ï­ï¬ï¯ï¯ï¯ï­ï© ï¹ïª ïºï¯ï« ï»ï¯ï© ï¹ï¯ ï­ ï« ï­ ï­ ï­ïª ïºï« ï»ï® (3) Discrete SPM with harvest after production (dPH) ï¨ ï© ï¨ ï©20.501 11 / AR-RS1 /t Rtinitt t t ttt t tB tB rB B K e Q ifBB rB B Keï· ï³ï·ï¢ï­ï¢ï­ï«ï½ï« ï­ ï­ï© ï¹ï« ï»ï½ï« ï­ ï¨ ï©ï¨ ï©ï¨ ï©ï¨ ï© ï¨ ï©22.50.5 -RS1 / -RSRt Rtt t t ttQ if KB re B B K Q if rï³ï· ï³ï¢ï­ï¬ï¯ï¯ï¯ï© ï¹ï­ ï­ïª ïºï¯ï« ï»ï¯ï© ï¹ï¯ ï« ï­ ï­ïª ïºï« ï»ï® (4) Continuous SPM with harvest after production (cPH) ï¨ ï©ï¨ ï© ï¨ ï©ï¨ ï©ï¨ ï©220.51 1exp 1 / 0.5 AR-RSexp 1 / t Rinitpt t t t Rptt t tB tB r B K F ifBB r B Ke Fï· ï³ï· ï³ï¢ï­ï«ï½ï© ï¹ï¢ï­ ï­ ï« ï­ï« ï»ï½ ï¦ ï¶ï­ ï­ï§ ï·ï¨ ï¸ï¨ ï© ï¨ ï©ï¨ ï©20.5 -RSexp 1 / -RSt Rpt t tif KB re B K F if rï· ï³ï¢ï­ï¬ï¯ï¯ï¯ï¯ï© ï¹ï­ïª ïºï¯ï« ï»ï¯ï¯ ï© ï¹ï­ ï­ïª ïºï¯ ï« ï»ï® (5) Process error 2 1/ 2-1~ (0,1) 1 (1 )tR ttR t R t RNtï·ï³ ï·ï·ï² ï· ï² ï·ï³ï½ï¬ï¯ï¢ ï½ ï­ï¢ ï« ï­ï¯ï® (6) Trawl survey biomass ï¨ ï©ï¨ ï©2exp 0.5t t t IB B ï¥ ï³ï¢ ï¢ï½ ï­ (7) Observation error 2 1/ 21~ (0,1) 1 (1 )tI ttI t I t INtï¥ï³ ï¥ï¥ï² ï¥ ï² ï¥ ï³ï­ï½ï¬ï¯ï¢ ï½ ï­ï¢ ï« ï­ï¯ï® (8) Quota from HCR with time-invariant parameters ï¨ ï©max 0, , ,bt t t tQ Q B B B Bïª ïªï© ï¹ ï¢ ï¢ï¢ ï¢ï¢ï¢ï½ ïï­ ïïïª ïºï« ï» 53 (9) Quota from HCR with time-varying HCR parameters ï¨ ï©ï¨ ï©2* ***0.5*max 0, , ,Ë -RSË t Rbt t t t t ttQ Q B B B BQ if KQQ reï· ï³ï¢ï¢ï­ï© ï¹ï¢ ï¢ ï¢ï¢ ï¢ï¢ï¢ï½ ïï­ ïïïª ïºï« ï»ï½ï« ï­ï¨ ï©ï¨ ï©ï¨ ï©20.5*** -RSË -RSË t Rtr if rB Ke K if KBBï· ï³ï¡ï¢ï­ï¬ï¯ï­ï¯ï®ï« ï­ï½ -RS,if rï¡ ï¢ï¬ï¯ï­ï¯ï®ïï (10) Quota from nonlinear HCR ï¨ ï© ï¨ ï©2max 0, , ,t t t tQ Q B B B B Bïªïªïª ïª ïªï© ï¹ï¢ï¢ ï¢ ï¢ï¢ ï¢ï¢ï¢ï½ ï ï­ ï« ïï­ ïïïª ïºï« ï»ïï (11) Harvest rate and catch biomass for dPH and dPH models ï ïmin .99,t t tt t tU Q BC U Bï½ï½ (12) Approximation for the fishing mortality and catch biomass the for continuous SPM /t t tF Q Bï½ ï¨ ï©ï¨ ï©ï¨ ï© ï¨ ï©2220.50.5( / ) 0.5 AR-RS( / ) -RS1 ( / ) t Rt Rpt t t Rrs pt t tpt tr r B K F ifg r r B Ke F if Kre B K Fï· ï³ï· ï³ï· ï³ï¢ï­ï¢ï­ï¢ï­ ï­ ï« ï­ï½ ï­ ï­ï­ ï­ -RSif rï¬ï¯ï¯ï­ï¯ï¯ï® ï¨ ï©exp 1 RS , ,rs rstt trstBB g AR K rgï¦ ï¶ï© ï¹ï½ ï­ ïï§ ï· ï« ï»ï¨ ï¸rst t tC F Bï½ (13) Utility function ï¨ ï© risk-neutralln risk-averseTtTtC ifC c ifï¬ï¯ï¯ï ï½ ï­ï¯ ï«ï¯ï®ï¥ï¥ 3.3.3 Biomass estimates Three approximations to estimate current biomass were used to evaluate how model complexity affects the optimal HCR parameters: 54 1) A data-based model (DBM) where the biomass estimate was the true biomass plus a multiplicative error (Eq. 3.2.6); 2) A model-based approach where the biomass estimate is derived from an extended Kalman filter (EKF) (Kalman 1960; Petris et al. 2009) (see Table 7.1B for a full description of the KF); and, 3) A model-based approach where the biomass estimate is derived from a local model (LM; Petris et al. 2009) using a KF (Table 7.2B). For the EKF and LM, the KF generated a posterior biomass estimate (Eqs. 7.1B.5 and 7.2B.5) for each time step based on a weighted average from the previous dynamics state and a new biomass observation (e.g., trawl survey index) (Eq. 3.2.6). The weighted average is the Kalman gain (Eqs. 7.1B.8 and 7.2B.8). When the process error is large, less weight is given to the state dynamics equation, so that the filter corrects more with the measurement (trawl survey) information (and vice versa). Note that EKF uses a population dynamics model for obtaining the posterior biomass estimate while the LM ignores the population dynamics (as the DBM); i.e., in the LM, the biomass is the state dynamics. In the EKF, the state equation had the same model structure as the operating model (Eq. 7.1B.3), but did not model RS in K or r parameters. 3.3.4 Harvest strategies based on stochastic optimization The evaluation maximized two utility functions: a risk-neutral and risk-averse utility function over a time horizon of 500 years (Eq. 3.2.13). A long simulation eliminates the initial and final conditions and avoids running multiple simulations. 55 The HCR was not fully flexible as Moxnes (2003), but still reproduced the fixed escapement rule (FER) and fixed harvest rate (FHR), under a risk-neutral and risk-averse utility function, respectively (Eq. 3.2.8): ðð¡ = max(0, ðâ(Î© â ðµâ)ð) (1) where Qt is the quota at time t, Q*, B*, and b are the optimal HCR parameters, and Ω is the biomass estimate for year t based on DBM, EKF, or LM. For the two utility functions, b was fixed at 1. In Eq. 1, under a risk-neutral utility function, a FER with slope of 1 is supported when Q*=1 and B*>0, hence B* determines the minimum based-stock (or escapement) for the HCR. Under a risk-averse utility function, a FHR is supported when B*=0 and thus Q* defines the harvest rate. The HCR parameters were optimized using a grid search. Eq. 1 assumes that optimal harvest parameters are stationary. Optimal HCRs with time-varying parameters were also evaluated (Eq. 3.2.9) to test if changes over time of K and r could result in improving performance (e.g., in average yield). This HCR had an additional parameter (α or β) depending on the RS. The first step was to find B* and Q* using Eq. 1 for each utility function and then to find the optimal α or β. If α=0 or β=0, the HCR produced the time-invariant HCR (Eq. 1), indicating no improvement in performance. Note that under a K-RS, the parameter search is only for α and for a r-RS the parameter search is for β. The optimal time-varying HCR parameters were evaluated under the base case operating model (see below) and the cPHM. For the cPHM the fishing mortality was calculated using the equation 3.12. The time-varying HCR parameters were evaluated under the K and r RS. For anchovy, a non-linear HCR was also evaluated under a risk-neutral utility function (Eq. 3.2.10). 56 3.3.5 Maximum sustainable yield (MSY) HCRs The previous harvest strategies allowed optimal HCR parameters to be found. To test if âa prioriâ HCRs results in improving performance (e.g., in average yield), six additional HCRs were evaluated. These âa prioriâ HCRs were based on MSY reference points and the base case operating model (see below). This analysis was performed using the cPHM. The parameters Q* and B* were replaced with time-invariant or time-varying Fmsy and Bmsy (i.e., the fishing mortality rate and the biomass that produces the biomass at MSY, respectively). For example, for a time-varying Fmsy HCR: ð¹ðð ð¦ð¡ = ðððð¡â²â0.5(ðð )2[1 â (ðµðð ð¦/ð¾)ð] (2) ðð¡ = max(0, ð¹ðð ð¦ð¡ (Ω â 0)ð) (3) where r is the true value from the operating model and the exponential term is the process error with autocorrelation for the continuous SPM. The MSY HCRs used true values from the operating model which describes a scenario where a stock assessment provides MSY reference points according to the current RS. ð¹ðð ð¦ð¡ and ðµðð ð¦ð¡ were derived using the equations in Table 3.3. A full list of the MSY HCR parameters are presented in Table 3.4. Note that, in some scenarios, the HCR was mis-specified with respect to the true RS. For example, under a K-RS, the simulated HCR could be a time-varying Umsy HCR, instead of an escapement HCR with time-varying Bmsy. Table 3.3. Deterministic MSY reference points for the continuous surplus production model. MSY reference points Equation Bmsy ð¾/(1 + ð)1/ð Fmsy ð â ð(ðµðð ð¦/ð¾)ð MSY FmsyBmsy 57 3.3.6 Performance metrics Five performance metrics were monitored over the simulation period: 1) Average biomass yield per year, ð¶Ì ; 2) Average number of years with closures; 3) Average overfished status, 1ðâ ðµð¡ â¤ ðµðð ð¦ð¡ðð¡=1 ; 4) Average overfishing status, 1ðâ ðð¡ â¥ ððð ð¦ð¡ðð¡=1 ; and, 5) Standard deviation (SD) in catch yield over the simulation period. When the K and r changed over time, Bmsy and Umsy were calculated for each time t, respectively. Metrics 3) and 4) were calculated based on the true biomass. 3.3.7 Operating models scenarios In total, six operating models were evaluated. A summary of the scenarios is presented in Table 3.5. These included a model with only white noise (process error ÏR=0.6 and no autocorrelation ÏR=0) in the population dynamics. This operating model compared results of the SOPS to those of the SDP. The base case (BC) included high process error (ÏR=0.6), with autocorrelation (ÏR=0.7) and high observation error for the survey biomass index (ÏI=0.4). The BC was modified for the remaining operating models: no observation error; low observation error; and white noise with high observation error. The range of values covered the levels of uncertainty in observation and process error (and autocorrelation) typically found in stock assessments and MSE simulations (e.g., Punt et al. 2008; Thorson et al. 2014; Geromont and Butterworth 2015b; Wiedenmann et al. 2015; Siple et al. 2018). 58 Table 3.4. MSY harvest control rules used under the base case operating model. HCR parameters Description (Notation) ðâ = 1;ðµâ = ðµðð ð¦ð¡ Time-varying escapement at Bmsy (tv_Escp) ðâ = ð¹ðð ð¦ð¡; ðµâ = 0 Time-varying Fmsy (tv_Fmsy) ðâ = 1;ðµâ = ðµðð ð¦Ì Ì Ì Ì Ì Ì Ì Ì ð¡ Average escapement at Bmsy (avg_Esp) ðâ = ð¹ðð ð¦Ì Ì Ì Ì Ì Ì Ì Ì ð¡; ðµâ = 0 Average Fmsy (avg_Fmsy) ðâ = ð¹ðð ð¦ð¡; ðµâ = ðµðð ð¦ð¡ Time-varying Umsy and escapement at Bmsy (tv_Fmsy_Bmsy) Table 3.5. Operating models (OMs) under the optimal and MSY harvest control rules. The numbers in the OMs indicate: process error (ÏR), autocorrelation in process error (ÏR), and observation error (ÏI), respectively. Base case (BC). Operating models Surplus production models Regime shift (RS) Biomass estimates Optimal harvest strategies MSY HCRs 0.6_0_0 dHP, dPH, cPH AR-RS, K-RS, r-RS Data-base model Model-based (EKF) Model-based (LM) ,,tttQQQï¢ï¢ï¢ see Table 3.4 0.6_0.7_0 0.6_0.7_0.2 0.6_0.7_0.4 (BC) 0.6_0._0.4 3.4 Results 3.4.1 Risk-neutral utility In the AR and r scenarios, the FER harvest strategy (i.e., Q*=1) was supported when the stocks included process error or had autocorrelation in the process error (Figs. 3.1, 7.5B, 7.11B). When the r was high (i.e., anchovy), the FER harvest strategy was supported at Bmsy (i.e., Q*=1, B*=Bmsy), but only under the dHPM. The cPHM and the dPHM also supported the FER, but the B* was lower than Bmsy. As the observation error increased, both B* and Q* values decreased, but the cPHM was more robust to observation error. In the case of redfish, both cPH and dPH models generated âpulseâ fishing, leading to an extremely high number of closures in the AR-RS. Also, in this stock, overall the FER at Q*=1 was not supported. In the K-RS, overall, the FER harvest strategy (i.e., Q*=1) was only supported under the model with process error for anchovy and hake (Figs. 3.1, 7.5B). For stocks with higher r, the B* 59 is lower, but the Q* values are higher; however, the Q* values decreases more as r become lower (i.e., redfish). Also, the values for B* are lower than for the r-RS and AR-RS scenarios. As r decreases, the observation error has less impact in the optimal HCR parameters. 3.4.2 Risk-averse utility The risk-averse utility function did not always support a FHR (i.e., B*=0), but this harvest strategy was more common in the stock with low r (Figs. 3.1, 7.5B, 7.11B). For anchovy, for example, when the model included autocorrelation in process error and had low or no observation error, the FHR was not supported (e.g., B*>0). In contrast, for the base case, the FHR was supported. Overall, when FHR was supported, K-RS had higher Q* values than the AR and r RS. However, as r decreased the differences in Q* values decreased across the stocks. For the risk-neutral utility function, as r decreased, the optimal HCR parameters were less impacted when observation error increased. 60 Figure 3.1. Optimal harvest parameters, B* (minimum based-stock, black) and Q* (slope, red), under a risk-neutral and risk-averse utility functions for anchovy. The operating models including different regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). 61 Figure 3.2. Performance in average yield for anchovy from optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). 62 3.4.3 Performance metrics 3.4.4 AR regime shift The best ð¶Ì varied across stocks. For anchovy, it was under the dHPM for both utility functions. The cPHM also performed well under the risk-neutral utility function. For hake, it was under the cPHM and the risk-neutral utility function and under the dPHM and the risk-averse utility function. For redfish, the best ð¶Ì was under dPHM and a risk-neutral utility function generated through a âpulseâ fishery. The cPHM also performed well under the risk-neutral utility function (Figs. 3.2, 7.6B, 7.12B). 3.4.5 r regime shift For anchovy, under the BC, the cPHM performed better than the dHPM under both models and under both utility functions. For hake and anchovy, the performance was similar across utility functions and biomass estimates (slightly higher for the risk-neutral utility function) (Figs. 3.2, 7.6B, 7.12B). 3.4.6 K regime shift The cPHM had the best performance under both utility functions with similar average biomass yield. For hake and anchovy, the performance was similar across utility functions and biomass estimates (slightly higher for the risk-neutral utility function) (Figs. 3.2, 7.6B, 7.12B). 63 3.4.7 Closures Closures should be avoided. Thus an arbitrary 10% (i.e., 50 closures/500years) of closures in simulations was considered as undesirable threshold. For the AR-RS, the cPH and dPH model generated high levels of closures for both utility functions. For redfish, in some scenarios the closures were >80%. Here, the optimal HCR parameters generated a âpulseâ fishery (only a few years of fishing followed for closures). In contrast, the dHPM avoided closures under the risk-averse utility function. In the r and K RS, the SPMs avoided closures under the risk-averse utility function but not under K-RS in redfish. In this scenario, only the dPHM with DBM did not passed the 10% threshold when the observation error was low (Figs. 7.1B, 7.7B, 7.13B). 3.4.8 Overfished A fishery was considered overfished when the threshold of Bt/Bmsyt was below 0.8 (i.e., a buffer of 20% due to time-varying Bmsy). Overall, across stocks and across the RS types, most SPMs provided results above the threshold, except for the dPHM under the AR-RS in both utility functions (Figs. 7.2B, 7.8B, 7.14B). 3.4.9 Overfishing Overfishing was defined for instances when Ut/Umsyt was above the threshold of 1.2 (i.e., a buffer of 20% due to time-varying Umsy). Overall, overfishing was avoided under both utility functions for the SPMs. For redfish, however, overfishing was avoided under the r-RS with the risk-neutral utility function, in particular under the dHPM. Note that under the AR-RS and the 64 dPHM, overfishing did not occur because of the large number of closures (Figs. 7.3B, 7.9B, 7.15B). 3.4.10 Variability in catch biomass yield As the r increased, the catch variability increased as well. In anchovy, the catch variability (i.e., SD) was high, but below 0.2 in most cases. The risk-neutral utility function resulted in more catch variability than the risk-averse utility function. The AR-RS produced more catch variability then the r and K RS (Figs. 7.4B, 7.10B, 7.16B). 3.4.11 Optimal HCRs vs MSY HCRs For the AR-RS, the time-invariant risk-neutral HCR had a ð¶Ì higher or similar to those for MSY HCRs with time-varying escapement. For redfish, the MSY escapement HCRs had a higher ð¶Ì than those for the optimal HCRs. The non-linear HCR had performance similar to that for the risk-neutral HCR across regime shifts (Fig. 3.3). For the r and K-RS, across stocks, optimal HCRs with time-varying and invariant parameters (with neutral and risk-averse utility function) did not show much difference in particular for hake and redfish. Time-varying MSY HCRs also had similar ð¶Ì âs to those for optimal HCRs. For anchovy, time-varying Umsy HCRs performed slightly better than optimal HCRs under the r-RS (Fig. 3.3). 65 Figure 3.3. Performance in average yield for optimal harvest strategies (risk-neutral and risk-averse utility functions) and MSY harvest control rules for three stocks under different regime shifts and biomass estimates for the continuous surplus production model (cPHM) under the base case. Across stocks, all the HCRs generated frequent closures in the AR-RS; for the stocks with lower r, the closures were more frequent. Risk-averse HCRs with time-varying and time-invariant parameters and Umsy HCRs avoided closures under the K and r RS. However, all HCRs avoided closures for redfish under a K-RS (Fig. 7.17B). 66 For the AR-RS, for the stocks with lower r, more HCRs overfished these stocks. For redfish, a time-invariant risk-averse HCR and Umsy HCR avoided the overfished level. For the r and K-RS, most of the HCRs avoided the overfished level. However, in terms of overfishing, some HCRs were riskier, in particular under an r-RS, and when the stock had low r (Figs. 7.18-7.19B). The catch biomass variability was higher in the AR-RS and when the stock had a high r (i.e., anchovy). In this situation, a time-invariant HCR with a risk-averse utility function avoided high variability. This HCR also performed well for redfish together with Umsy HCRs. Overall, for the r and K RSs, the optimal and MSY HCRs avoided high levels of catch variability. Catch variability was high for stocks with high r irrespectively of the RS type (Fig. 7.20B). 3.5 Discussion 3.5.1 Optimal harvest strategies under observation error Reed (1979), using dynamic programming, found that for a population under stochastic growth (i.e., only process error), the optimal strategy is the same as when the model is deterministic (i.e., no uncertainty) (Clark 1973, the FER at Bmsy). These findings have led to a paradox in fishery management because according to optimal control theory, reducing harvest should not matter under increasing uncertainty. In contrast, under the precautionary principle, the higher the uncertainty (e.g., stochastic growth and observation error), the more cautious the management strategy should be. Clark and Kirkwood (1986) attempted to resolve Reedâs paradox, but for reasons of tractability, they adopted a simplified assumption to include observation uncertainty. Their results just deepened the paradox (Memarzadeh and Boettiger 2019), but they showed that when the stock cannot be measured perfectly, the optimum strategy is no longer the FER with 67 slope 1. Later, Sethi et al. (2005, p312), found counter-intuitive results: when observation error increased, the expected optimal escapement was lower compared to a deterministic model. In addition, they found that observation error had the highest impact on the optimal harvest strategy, especially when combined with process and implementation error. Moxnes (2003; 2005) used a different approach to include observation error. In this research, SOPS was used to find optimal harvest strategies, hence avoiding the restrictive assumptions in SDP. However, it was found that the FER became increasingly cautious as observation error increased (Fig. 4 in Moxnes 2003). For example, the minimum base-stock decreased as the observation error increases (i.e., closures occur at lower stock sizes), but the slopes âbentâ (slopes were <1) as observation uncertainty increases; hence reflecting caution. Moxnes (2003) demonstrated that under uncertainty (observation and process error), the optimal strategy follows the precautionary principle and that under increasing observation error the economic performance decreases. He also showed that an optimal strategy that makes direct use of the current biomass estimate (e.g., trawl survey index), or a few of the most recent stock measurements, have better performance than ones that rely on inputs from a KF state estimate (i.e., filtering current and past data). Memarzadeh and Boettiger (2019) used a recently developed algorithm in robotics, which explicitly included (as opposed to SDP) observation uncertainty (called a partially observed Markov decision process, POMDP), which corroborated Moxnesâ (2003) findings that the paradox in optimal control theory follows the precautionary principle. This study expanded the current knowledge of optimum harvest strategies under closed-loop simulation with different sources of biological, model, and observation uncertainty. In particular, through SOPS, this study evaluated RS (i.e., persistent changes in r, K, and an 68 autocorrelation in the surplus production) under discrete and continuous fisheries (where the harvest can occur before, during or after growth production) with different approximations to estimate current biomass to adjust the optimum strategies using three stocks (anchovy, hake, and redfish). Moreover, a suite of HCRs was evaluated under different scenarios. These included: i) optimal strategies under risk-neutral and averse utility functions; ii) as i), but with time-varying HCR parameters to accommodate RS; iii) a non-linear HCR with a risk-neutral utility function; and iv) six MSY HCRs with true parameters based on the current RS (no optimization involved). 3.5.2 Optimal harvest strategies under SPMs Previous studies have followed conventional assumptions in the optimal control literature (e.g., Moxnes 2003; Sethi et al. 2005; Memarzadeh and Boettiger 2019); a discrete population model that assumes that the biomass measurement occurs just before harvest, followed by growth production (i.e., dHPM). These assumptions form the basis for much of bioeconomic theory (Clark 1990; Memarzadeh and Boettiger 2019). However, the results showed that both the discrete-time formulation and the timing of the harvest relative to the growth production were important to the optimal harvest strategies. Many fisheries follow a continuous harvest and in general, the measurement occurs after harvest. Mainly in salmon fisheries, it is possible to assume âsafelyâ that the populations can be measured (precisely) just before harvest. Across stocks, the results showed for a discrete or continuous fishery with stochastic growth (or deterministic) when the harvest occurs after growth production that optimal harvest strategies do not follow the FER at Bmsy (i.e., B*=Bmsy and slope of 1, Q*=1). Moreover, for both the discrete or continuous model, the optimal strategies differed significantly from the conventional assumption of harvest followed by growth production (e.g., Reed 1979; Memarzadeh and Boettiger 2019). 69 For example, for anchovy under the dPHM, the FER at Bmsy was not supported (e.g., B*âªBmsy and Q*<1), but it was supported under the dHPM model when the population had stochastic growth (only white noise). For this scenario and stock, the cPHM supported a FER with Q*=1, but with B*âªBmsy (e.g., 0.25). For redfish, the dHPM did not support a FER at Bmsy and only a few cases had a FER with Q*=1.Therefore, it is relevant to define how the fishery harvest occurs (e.g., discrete or continuous fishery), when the stock is measured (i.e., after or before growth production), and under what level of observation error to establish optimal harvest strategies. The SPM assumption (i.e., model uncertainty) also affected performance (e.g., average yield). For example, for anchovy, under the AR-RS, the dPHM did not perform well in terms of average yield compared with the cPH and dHP model. The poor performance relates to when the biomass is measured relative to harvest. In the dPHM, the optimal harvest parameters were adjusted after the production growth, thus the HCR do not âseeâ future fishing opportunities under high productivity events (or vice versa for period of low productivity). However, when stocks were less productive (e.g., redfish), the performance of the optimal HCRs between SPMs, utility functions, and RS were more similar and robust to e.g., increasing observation error. Thus, the optimal harvest strategies were stock and model-dependent. The different SPM configurations show the importance of the biomass estimates relative to harvest, in particular, under an AR-RS. Across stocks, both the dPH and cPH models generated frequent closures, regardless of the utility function. The dHPM model avoided closures and still had high average yields, indicating that biomass estimates before harvest are more important than the utility function and the level of observation error. Biomass estimates before harvest do not occur only in salmon fisheries, but also in stocks that show rapid changes in biomass as occurs in the South African sardine and anchovy fisheries (de Moor and Butterworth 70 2016), and the Peruvian anchovy fishery, where multiple surveys have been conducted before the harvest decision was made. 3.5.3 Optimal harvest strategies for different biomass estimates The optimization showed that for an SPM and RS combination, the optimal parameters were not affected much with the different approximations to estimate current biomass. Under most of the scenarios, similar optimal parameters were obtained under data-and model-based approaches. The posterior biomass estimates did not improve the performance in terms of average yield or robustness in the optimal harvest parameters. This could be due to two issues; i) the KF did not include time-varying r or K in the state equation and ii) both the standard deviations for the state and observation equations were set too high (ÏI=1 and 0.4, respectively) compared to the true population that had lower process (ÏR=0.6) and observation error (ÏIâ¤0.4); i.e., the model-based assessment had too much uncertainty, as usually occurs. Including non-stationarity in the state equation in part was discarded because the KF without population dynamics incorporated (i.e., local model) had similar optimal harvest parameters. Lower variance for the state and observation equations will unlikely improve performance for more complex models; e.g., models with increasing complexity have increasing variance in parameter estimates (Walters and Martell 2004). Moxnes (2003) also found that optimal harvest strategies make direct use of the measurements having good performance. Our results are similar to previous simulation studies in MSE where more complex management procedures led to more variability in catch limits without reducing conservation risk (Punt 1993; Geromont and Butterworth 2015b). 71 3.5.4 Optimal harvest strategies under risk-neutral and risk-averse utility functions A risk-averse utility function did not necessarily lead to a FHR (e.g., B*=0). For example, for anchovy, when the observation error was low (or zero) most of the optimal harvest parameters supported a FER (e.g., B*>0 and Q*<1). An FHR mostly occurred when the observation error was high (e.g., ÏI=0.4). However, when r was lower (hake and redfish), the FHR was usually supported. In anchovy, under an AR or r RS, the harvest rate Q* was lower compared to a model without RS and high observation error, thus indicating precaution. In stocks with high r (e.g., anchovy) with r and AR-RS, Q* was always way below Umsy (especially under the dHPM); in contrast while under a K-RS, Q* was close or below Umsy. For unproductive stocks (e.g., redfish), Q* was always close or at Umsy across RS, but the risk of overfishing was high across harvest strategies at the cost of infrequent closures. FHRs avoided overfishing, but had frequent closures. The degree of precaution in the optimal FHR depended on the SPM as well; e.g., the cPHM required less precautionary action (e.g., high Q*) than the dHPM. For redfish, however, the optimal FHR showed little difference compared to results under a model with no RS. The FER with slope=1 at B*=Bmsy occurred under restrictive scenarios, for example, under a dHPM, when the stock was productive, the observation error was low, and when the RS was in r or AR. A FER with slope=1 (Q*=1), but at B*<Bmsy also was supported, but the most common optimal HCR was a FER with slope <1 and B*<Bmsy. Across stocks, under high observation error, the B* was higher under the AR and r-RS, indicating that the minimum based-stock should be higher when stocks have K-RS. Also, with increases in observation error, both B* and the slope decreased. This indicates that under increases in observation error, HCRs with a minimum based-stock parameter would be decreased (closures can occur at lower stock sizes), but the harvest rate has to be decreased accordingly to maximize average yield. This tradeoff was 72 more evident for productive stocks. For productive stocks, data-based models generated lower B* (and lower harvest rates) than the model-based approaches, but in terms of average yield and closures they were similar. 3.5.5 Observation error in economic performance for risk-neutral and risk-averse utility functions A recent study that focused on stochastic dynamic optimization implicitly taking into account observation error (i.e., POMDP) found that the FER attained a consistently higher economic yield despite increasing observation error (Fig. 2 in Memarzadeh and Boettiger 2019). This could mean that reducing survey uncertainty could not improve economic performance under a POMDP FER. Our results do not agree with Memarzadeh and Boettiger (2019). In our study, for all SPMs, approximations to estimate biomass, and RS combinations, indicates that when observation errors increased, the economic performance decreased (similar to Walters and Parma 1996; Moxnes 2003, 2010; Sethi et al. 2005). This also occurred for the scenario with stochastic growth (only white noise) and observation error. The discrepancy could be due to the far higher uncertainties used in my simulations compared to those in Memarzadeh and Boettiger (2019); e.g., 0-0.4 vs. 0-0.15 and 0-0.6 vs. 0.15 (standard deviations), for observation and process error, respectively. However, for some scenarios (e.g., continuous vs. discrete SPM), the average yield was very similar (but not increasing as Memarzadeh and Boettiger 2019) when observation error increased (from 0-0.4). This result indicates that, for example, the model structure could be effective to control for high observation error. Moxnes (2003), used a far more flexible HCR than used in our study, but still 73 the economic performance decreased with observation uncertainty. Future simulation studies, comparing SOPS and POMDP could investigate this issue further. 3.5.6 Optimal vs. MSY harvest strategies under regime shifts Overall, including optimal time-varying parameters and time-varying MSY parameters in the HCRs did not substantially improve the performance in average yield. Across stocks, the differences were mostly for the AR-RS. The best performance was under the risk-neutral utility function with time-invariant and non-linear parameters. For MSY HCRs, the best performance was under a time-varying Bmsy escapement. However, the higher performance in average yield was at cost of high catch variability. For the r and K RS, the optimal and MSY HCRs performed similarly and overall avoided high catch variability. Also, under r and K RS, for hake and anchovy, the risk-averse utility function and Umsy HCRs were more effective at reducing closures. However, for the AR-RS and across stocks, all HCRs produced frequent closures. A similar situation was observed for redfish under the r and K RS. The closures were thought to be due to an overfished status in particular when the stock had low productivity. 3.5.7 Caveats Early studies demonstrated that FER produces the best economic performance (average yield), while FHR produced more stable yield over time (Walters and Martell 2004). Memarzadeh and Boettiger (2019) and Memarzadeh et al. (2019) suggested that POMDP FER harvest strategies are better for avoiding overexploitation, and for rebuilding stocks, while still producing higher economic performance compared to MSY HCRs and FE, derived from SDP. This does not 74 contradict previous findings. The FER performs well in conservation objectives because it will close the fishery at the minimum based-stock as many times as needed to bring the biomass to the optimal long-term level (e.g., Bmsy) (Fig. 5 in Memarzadeh and Boettiger (2019) and Figs. 3 and S3 in Memarzadeh et al. 2019). While FER met conservation and (optimal) economic objectives, it generates severe hardship for fishers and the community (Walters et al. 2019). FER (with both optimal and MSY parameters) are potentially too harsh to be implemented as a tool to rebuild fisheries. Management strategies need to be examined in a broader sense than just limiting attention to economic and conservation performance. Heuristic approaches such as MSE can accommodate a greater model complexity and uncertainty than SDP and POMDP helping to find the âbestâ management procedure under conflicting management objectives. The present study used a simple model to examine optimal harvest strategies. The main reasons were; i) to compare SDP (and POMDP) and SOPS studies that have traditionally used SPMs (e.g., Moxnes 2003; Sethi et al. 2005), ii) to make the presentation easier because the optimal harvest strategies (e.g., under observation uncertainty) are not well understood in the literature, and iii) to avoid nuisance parameters (e.g., fishing selectivity) that could obscure the results. In contrast to SDP (and POMDP), the SOPS allows multiple dynamic states to be included in a straightforward way, and thus can be applied to a wide range of models. More work is needed in age-structured models because multiple cohorts are not equally vulnerable to fishing mortality. Under multiple states, different sources of uncertainty (e.g., RS, observation errors), and utility functions, the optimal HCRs are still an open question. Finally, SOPS could be a more straightforward alternative to current MSE which explores a restricted set of âad hocâ HCRs. As in SDP (and POMDP), the SOPS HCR can be fully flexible (e.g., Moxnes 2003). 75 Finally, this study shows additional directions that can be applied to data-limit stocks that can only use SPMs. For example, Dowling et al. (2019), recommended obtaining better data (e.g., biomass survey information) to guide better management decisions. However, it is also possible and less costly to adjust instead the HCR parameters to the level of the observation error (Moxnes 2003, 2010). This would be achieved by knowing the âdirectionâ of the HCR parameters with respect to the uncertainty or by using a SPM robust to observation error, as we showed here. Dowling et al. (2019) recommended precautionary HCRs, but optimal HCR can also be used. Optimal HCRs are less time-consuming to evaluate that using MSE and can also expose the tradeoff of conflicting conservation and economic goals. 76 4 Chapter 4: Management strategies for spasmodic stocks: a Canadian Atlantic redfish fishery case study 4.1 Abstract There exist few recommendations for managing stocks with spasmodic recruitment, despite such stocks being not uncommon. Management procedures (MPs), developed for two rockfishes (Sebastes mentella and S. fasciatus) in Eastern Canada, are recommended for setting catch limits during high and low periods of abundance. A well-designed fishery-independent trawl survey is essential to provide advance warning of strong recruitment events and project future recruitment. Under an âinventory managementâ strategy, a more appropriate aim in spasmodic stocks may be to maximize the number of years with "good catches", instead of maximizing total catches as is traditionally considered in management strategy evaluation (MSE). Following a spasmodic recruitment event, an empirical harvest control rule based on larger fish delays the harvest of large cohorts by a few years, targets more commercially valuable fish sizes and reduces the risk of growth overfishing. Capped MPs produced longer periods of large catches than uncapped MPs. MPs allowed for low harvests rather than complete harvest curtailment during periods of low abundance, thus avoiding unnecessary hardship in the industry. MPs evaluated here could be good candidates for other stocks with similar or less extreme recruitment variability. 77 4.2 Introduction Developing an understanding of population dynamics and identifying management approaches consistent with this understanding have been among the chief aims of fisheries management science. To facilitate the achievement of these aims, fish stocks have been categorized as steady-state, cyclical, irregular, and spasmodic (Caddy and Gulland 1983; Hilborn and Walters 1992). Spasmodically recruiting stocks (SRS) are typically the most variable of fish stocks; these stocks have been commonly characterized by infrequent and irregularly occurring episodes of strong recruitment, followed by long periods of weak recruitment. This recruitment pattern is known as âspasmodic recruitmentâ (Caddy and Gulland 1983; Spencer and Collie 1997; Caddy and Agnew 2005); spasmodic recruitment has been observed in a wide range of species groups including rockfishes, hakes, sardines, and mackerels (Templeman 1976; Spencer and Collie 1997; Roel and De Oliveira 2007; Needle 2016; He and Field 2017). Spasmodically recruiting stocks can be quite productive once the strong cohort(s) enter the fishery because these cohorts are often much larger than the long-term average year-class strength. For example, for the Western horse mackerel a strong cohort was about 18 times higher than the long-term average (Roel and De Oliveira 2007). Especially in long-lived species, once a large recruitment occurs, the larger harvests can be sustained for several years, as has occurred for the Atlantic redfish fishery (Templeman 1976; CAFSAC 1984a). To maintain a viable fishery for as many years as possible before the next strong cohort appears, careful management planning is required (Caddy and Gulland 1983). By imposing high harvest rates on strong cohorts, the long-term biomass potential of these strong cohorts can be negatively impacted. Moreover, shortly following a strong year class, there is a predominance of small pre-recruit fish with little or no market value which are sometimes unavoidably captured by commercial fishing 78 gear. Without appropriate regulation of fishing activities, strong cohorts can thus be at risk of growth overfishing, but management approaches also need to be designed to avoid high discarding rates. A challenging aspect of SRS is that the steady-state assumptions of classical fishery models cannot be applied to these stocks (Spencer and Collie 1997). In particular, key recruit parameters that can influence the potential long-term yield, such as recruitment compensation (Hilborn and Walters 1992), are difficult to quantify even when long time series of stock-recruit data are available. For example, providing that recruitment can be estimated, management strategies that trade off short-term yields of smaller fish for delayed larger yields of larger fish could be reliably quantified for the short-term when a strong cohort appeared. However, due to the high year-class strength variability and their irregularity, the potential long-term benefits and trade-offs of different management approaches for such stocks will inevitably remain highly uncertain. 4.2.1 Management approaches for spasmodically recruiting stocks (SRS) Despite long-standing awareness of the behaviours of SRS, there exist few guidelines for their management. Early studies suggested limiting the number of permanent participants to endure periods of low recruitment, and enabling flexible access to the fishery for periods of high productivity (e.g., additional licenses; Caddy and Gulland 1983). Two recent studies have focused on stocks with spasmodic recruitment and tested alternative management strategies using simulation (Roel and De Oliveira 2007; Needle 2016). However, the scope of these studies did not include formulating more general guidelines for managing SRS. For example, Needle (2016) used a rectilinear harvest control rule broadly used in the management of European fish 79 stocks. This approach by itself however may be inadequate due to the lack of any additional measures to facilitate fleet adaptability to periods of building, high and low abundance characteristic of SRS. Roel and De Oliveira (2007) tested a series of harvest control rules (HCRs); an empirical vs. model-based approach. These authors concluded that either a fixed harvest rate or a constant catch strategy would maximize catch without increasing biological risks for Western horse mackerel. However, due to the paucity of fishery-independent data, a constant catch strategy was preferred. While an empirical approach (i.e., a "slope strategy" in Roel and De Oliveira (2007)) is probably flexible enough for SRS, the traditional approach of maximizing the average annual catch throughout the "outburst" may be insensitive to the limits of realizable industrial capacity. Typically, during the âoutburstâ, HCRs are designed to maximize annual yields. Without considering the potential for growth overfishing and also the rapid liquidation of biomass during periods of abundance, this approach may also generate hardship for the fishery in the long term. A better approach may be to develop harvest strategies and rules that set harvest caps at agreed levels so that the annual yields produced by the strong cohorts can be within the realizable capacities for industrial processing and marketing and spread over an extended number of years. This management strategy represents an âinventory managementâ strategy (Carl Walters com. pers.). Under this approach, fishing opportunities would be increased in a pre-agreed-upon way as stock biomass builds with the recruitment of a strong cohort. Management strategies would also need to avoid high discarding rates during the âoutburstâ. 80 Figure 4.1. Units 1 and 2 of the Canadian Atlantic redfish fishery. The overlapping region (Unit 1+2) constitutes. PEI (Prince Edward Island), NS (Nova Scotia). It is widely agreed that the management strategy evaluation (MSE) framework is the most appropriate way to evaluate alternative strategies for achieving fishery and conservation objectives under multiple sources of uncertainty (Punt et al. 2016). To evaluate and identify management strategies that could be generally appropriate for stocks with highly spasmodic recruitment, we applied the MSE framework to the redfish fishery in the Gulf of Saint Lawrence and Laurentian Channel (GSL-LCH), i.e., Management Units 1 and 2, on the East coast of Canada (Fig. 4.1). 81 Figure 4.2. Historical landings (bars) and catch limits (CL) (lines) for Sebastes spp. for the management Units 1 and 2, off eastern Canada. 4.2.2 Atlantic redfish fishery as a case study The redfish fishery is composed of two species, Deepwater Redfish (Sebastes mentella) and Acadian Redfish (S. fasciatus). These species have exhibited spasmodic recruitment dynamics (Valentin et al. 2015). Similarly to other pairs of Sebastes species that are caught together, these two species are indistinguishable based on external morphologic features (Ni 1981). Redfish are also slow-growing and long-lived species (>65 years) (Campana et al. 1990). These species have sustained the Canadian Atlantic redfish fishery for many decades (Fig. 4.2). The 1956, 1958, and 1970 cohorts were the first strong ones recorded (Sévigny et al. 2007). After that, the last strong 82 recruitment occurred in the early 1980s which consisted mostly of S. mentella, but with a significant amount of S. fasciatus (Valentin et al. 2015). These strong cohorts were subjected to high harvest rates. As time progressed, no further strong cohorts appeared and the redfish stocks became less abundant in the fishing grounds, reaching very low levels for the last two decades (i.e., 1995-2017). As a result, in 2010, the Committee on the Status of Endangered Wildlife in Canada (COSEWIC) designated S. mentella as endangered and S. fasciatus as threatened in both management Units (COSEWIC 2010). Despite of the very low spawning stock biomass (SSB), after >25 years of low recruitment, the Fisheries and Oceans Canada (DFO) trawl survey recorded strong cohorts belonging to the 2011, 2012, and 2013 year-classes. Recent DFO surveys also showed that these cohorts are growing in biomass (DFO 2018a). These fish were expected to recruit significantly to the fishery from 2019 to 2020, which could lead to a rapid increase of SSB. We present the results of the MSE for the redfish stocks to assess the trade-off between maximizing the long-term harvest under an inventory management strategy and keeping the risks of growth and recruitment overfishing acceptably low under multiple sources of uncertainty. The MSE needed to consider that the two target species cannot be distinguished on the commercial fishing grounds and have different productivities. Less abundant cohorts, as predicted for S. fasciatus, could be depleted faster and could limit fishing due to conservation objectives. Besides these issues, uncertainties about the population and fishery dynamics also needed to be considered. These are related to past and future trends in recruitment and productivity (e.g., recruitment compensation, natural mortality, and growth), data-related due to species identification and discarding, as well as changes in vulnerability and implementation error (see Selection of uncertainties). 83 Table 4.1. Conservation and fishery objectives and their respective performance metrics (PMs) agreed by the redfish working group. The parenthesis indicates the probability of passing criteria for the PM. Type Objective Performance metric(s) Conservation 1) Increase spawning stock biomass (SSB) for each species above the lower reference point (LRP) and upper stock reference (USR) in 10 years Proportion of simulations where SSB > LRP for each species (>0.95) Proportion of simulations where SSB > USR for each species (>0.95) 2) Once in USR (healthy zone), maintain SSB of each species above LRP (critical zone) and healthy zone after 10 years Proportion of simulations where SSB > LRP for each species (>0.95) Proportion of simulations where SSB > USR for each species (>0.75) 3) Maintain the harvest rate (Ut) of each species below the harvest rate at maximum sustainable yield (Umsy) all 40-years Proportion of years where Ut is below Umsy (>0.5) Fishery 4) Maximize the number of years where the small fish protocol (SFP) is not triggered Average years where fish <22cm represent <15% of catch over: Simulation years 2018-2022 (0.85) all 40-years (0.85) 5) Maximize the number of years where the SFP is not triggered Average years where fish <25cm represent <15% of catch over: Simulation years 2018-2022 (0.85) all 40-years (0.85) 6) Maximize duration of high annual catch Average annual catches over: Simulation years 2028-2037 (10 years) Simulation years 2028-2057 (30 years) Proportion of simulations where CLs â¥ 40kt by 2028-2057 Average number of years where CLs â¥ 40kt between 2028-2057 Proportion of years with 4kt (avg. catch retained in 2015-2017) 7) Maximize catch of large fish (>27cm) Proportion of years where large fish comprise >80% of catch over: Simulation years 2018-2022 (>0.5) all 40-years (>0.75) 8) Maintain fishery stability between years Proportion of years where CLs changes less than 15% between years (>0.75) Average annual variation (AVV) in CL over: Simulation years 2028-2037 (<0.15) Simulation years 2028-2057 (<0.15) 84 4.3 Materials and methods 4.3.1 Framework The first step in the MSE framework was to define the conservation and fishery objectives (Punt et al. 2016). These objectives were established via consultation with the redfish working group (RWG) (i.e., industry stakeholders, DFO fishery managers, non-government organizations, and scientists) (Table 4.1). The agreed-upon objectives had to be translated in terms of quantifiable measures known as performance metrics (PMs) (Table 4.1). PMs enable the quantification of how well management procedures (MPs) meet the different management and fishery objectives. Another crucial aspect of the MSE approach is the identification of key sources of uncertainty (related to the biology of the stock, fishery, data collection, and analyses to determine a catch limit, CL) where the MPs are to be tested for robustness (Punt et al. 2016). The RWG identified five main axes of uncertainty (see âSelection of uncertaintiesâ). 4.3.2 Conservation and fishery objectives and performance metrics For conservation objectives 1 and 2 (Table 4.1), an empirical method was used to establish the reference points (RPs) for each species. In particular, the RPs were based on the average simulated SSB of the operating models (OMs) for the periods when the biomass was high (i.e., 1984-1990 and 1984-1992 for S. mentella and S. fasciatus, respectively). These were considered proxies for the SSBmsy, the SSB at maximum sustainable yield (MSY). The limit reference point (LRP) and upper stock reference (USR) were set at 40% and 80% for the reference periods mentioned above, respectively. The conservation objective 3, which used the harvest rate at MSY (Umsy) in future years, required the re-computation of Umsy for each simulation. This was because the parameters values of the OMs were based on draws from their respective posterior 85 distribution (see Operating model description). Also, when the OMs had parameters that changed in a future year and that determined Umsy (e.g., natural mortality, vulnerability, and growth parameters), the Umsy was recomputed in that year. Many fishery PMs were based on the inputs of industry stakeholders (Table 4.1). Of particular interest was maximizing catches of large fish (e.g., >27 cm), the average annual catch, and avoiding the small fish protocol (SFP) (Objectives 4-7). The SFP is triggered in a sub-area of one of the management Units (i.e., Unit 1 or Unit 2) when fish <22cm represent more than 15% of the catch. This produces a temporary closure of that sub-area. For example, the industry was interested in evaluating the average retained catch (e.g., objective 6a) for the period 2028-2037 (10 years) and 2028-2057 (30 years) and the average number of years where the CLs are â¥40kt in the years 2028-2057 (e.g., objective 6c). Similarly, because the strong cohorts of 2011-2013 had not yet recruited to the fishery, the industry stakeholders were interested in the length-based metrics for the stocks for the next five years (e.g., 2018-2022). For example, objective 4 examined the number of years when the SFP was triggered in the next five years. Passing criteria were suggested for each of the conservation PMs. It was not possible to agree upon and justify a passing criteria for objective 6 only (Maximize duration of high annual catch). As strong cohorts 2011-2013 are expected to fully recruit to the fishery by 2028, most of the fishery PMs were evaluated after this year. 86 Figure 4.3. Example of the harvest control rule (HCR) applied in the management procedures (MPs) for Sebastes mentella (and S. fasciatus). For each species, the raw catch limit (CL) increases or decreases depending on an index (Jt) which measures the level of the biomass of the DFO trawl survey index during the three last years (as a geometric mean) relative to a geometric mean for a reference period (i.e., 1984-2017) (Tables 4.3.1â4.3.2). Here, Jt is called âthe fishery indexâ because it only uses commercially sought after fish sizes for each species. The raw CL decreases linearly as Jt decreases, but the raw CL decreases faster when Jt reaches a threshold due to quadratic term in the HCR (Table 4.3.3). The raw CL is zero at low biomass levels. The HCR was tuned so that the raw CLs were similar to the average historical catch retained (dot) when the stock was at low levels (e.g., 2007-2017). The arrow indicates the average Jt for the last decade. Also, the HCR was tuned to avoid overfishing in periods of high abundance. After calculating the raw CL, a series of rules determine the realized CL for each species (see Fig. 4.4). A full mathematical description of the HCR is in Table 4.3. 87 Figure 4.4. Example for determining the catch limits (CL in kt) for each species. The lines are the raw CL for S. mentella (twodash) and S. fasciatus (long dash), CL caps (dotted), and the realized CL (solid gray) for MP1 under the base case operating model. The raw CL for each species was obtained after applying the HCR (Tables 4.3.1â4.3.3), and then the raw CLs were combined (Table 4.3.4). After that, a CL cap is applied (Table 4.3.5). For MP1, the CL cap has a ramp cap starting in 2020 (at 14.5kt) and ending in 2027 (at 40kt), followed by a maximum cap at the end of the ramp up period (i.e., 40kt). Note that when the combined CL exceeded the cap, the CL corresponds to the cap for that year. Also, note that this MP has a status quo for 2018-2019 (at 3kt). The catch biomass realized for each stock is obtained after catch split implementation (Table 4.3.6) and implementation error (Table 4.3.7). These last two steps are not shown in the plot. 4.3.3 Management procedures The MPs included an empirical (or data-based) HCR for each species to prescribe the CL which follows a âtarget-basedâ approach; i.e., the CL is adjusted up or down each year in relation to the extent to which the recent average of an index of abundance is above or below a target level (Fig. 4.3) (Rademeyer and Butterworth 2011). We used the DFO biomass survey conducted 88 since 1984 in Unit 1 across the northern GSL using a bottom trawl (DFO 2018a), which generates an index for each species. An additional industry biomass survey (performed in Unit 2) was not used in the MP because it represented a much shorter time series (only beginning in 2000). The empirical HCR was based on the MP used for Western component Pollock (Rademeyer and Butterworth 2011). A full mathematical description of the HCR is in Tables 4.2â4.3. Figure 4.5. Flow chart illustrating steps to calculate the realized catch limit (CL) for Sebastes mentella (S.m) and S. fasciatus (S.f.) under management procedure #1 (MP): (i) HCR, (ii) combined CL, (iii) capped CL and population and fishery processes represented in the operating model (OM). Jt is the âfishery indexâ calculated from the DFO trawl survey data (see Fig. 4.3). Note that the OM simulates the catch biomass (CB) removed by species. In the catch split, the CB split by species was drawn randomly from the set of species split ratios observed for 1984-2017 in the Unit 1 bottom trawl survey (DFO has used meristic counts to distinguish between the two species). When simulating implementation error, the catch biomass discarded dead by species in excess of the total CL is taken into account. 89 Table 4.2. Harvest control rule (HCR) parameters for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). Symbol Value Sm; Sf Description axï 4; 4 Tuning parameter for HCR: intercept (kt) bxï 2; 2 Tuning parameter for HCR: slope (kt/year) b No species-specific Slope multiplier cxï 1.2; 1.2 Penalty (kt/year) for HCR 0xJ 1.5; 1.5 Threshold parameter when , ,x t gJ decreases at low biomass levels in the HCR , ,x t gJ Relative fishery index for the species x , , 3x t gI ï½ Historic fishery index for unit 1 for species x; fish > 30 and 29 cm for Sm and Sf, respectively) ,capt pCL No species-specific Catch biomass caps (kt) in time t for the MP=p. ,x tCL Raw catch limit (kt) derived from the HCR for species x ctCL Catch limit (kt) combined for Sm and Sf. *,x tCL Catch limit (kt) after catch split implementation for species x ,ratiox tkillCL Catch limit (kt) after implementation error for species x xï° See Table 4.3.6 Proportion for catch split implementation for species x ratiokillprjD 1.1; 1.1 fish killed:retained ratio for the projections In order to dampen the effects of extreme survey values, the HCR uses the ratio of the geometric mean of the trailing 3 years of the survey biomass index to a reference period (i.e., 1984-2017). The DFO biomass survey index was transformed into a âfishery indexâ by summing only the lengths of the population of most interest to the industry (i.e., â¥30 and 29 cm for S. mentella and S. fasciatus, respectively). Similarly, the simulated âfishery indexâ for the projections also included these lengths and stochastic components (Table 4.5.20). The HCR parameters of the empirical MP (Table 4.2) were tuned using the historical âfishery indexâ so that the prescribed CLs would be similar to the observed retained catch biomass during periods of low abundance but dropping to zero if the âfishery indexâ reached a threshold (Fig. 4.3). 90 Besides, the parameter that controls the rate of harvest (i.e. bxï , Table 4.3.1) in the HCR was also tuned so that the median values generated for the OM would produce no more than ~75% of the Umsy for both species in the 40-year simulation horizon. Table 4.3. Description for the empirical (target-based) harvest control rule (HCR) for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). Description Equation (1) Raw CL , , , 0( )a bx t x x x t g x xCL b J J penï© ï¹ï½ ï ï« ï ï­ ï­ï« ï» ; 3t T gï¾ ï½ (2) Trailing average for fishery index , ,2, , 2017, ,19841exp ln( )31exp ln( )34tx t gtx t g tx t gtIJIï­ï½ï½ï© ï¹ï¢ïª ïºï« ï»ï½ï© ï¹ïª ïºï« ï»ï¥ï¥ (3) Penalty ï¨ ï©, , 02, , 00 x t g xxcx x t g xif J JpenJ J otherwiseï¬ ï¼ï¯ï½ ï­ï ï­ï¯ï® (4) Combined CL 2,1ct x txCL CLï½ï½ï¥ (5) Cap implementation ,, c c capt t t pct capt pCL if CL CLCLCL otherwiseï¬ ï£ï¯ï½ ï­ï¯ï® (6) Catch split implementation ï¨ ï©1,2, 1,~ 0.39,0.441x tx t x tUnifï°ï° ï°ïï ïï½ ï­ ,*,x tct x tCL CL ï°ï½ (7) implementation error , ,*ratio ratiox t x tkill killprjCL CL Dï½ (8) Output control ,, max,min ,ratiox tkillx t kx tCLu UVBï¦ ï¶ï½ ï§ ï·ï§ ï·ï¨ ï¸ (9) Small fish protocol ,,0 x t x tif small fish protocoluu otherwiseï¬ï½ ï­ï® 91 Besides the HCR, the MP included six other components that were applied depending on the MP (see Table 7.4C and Table 7.5C for the complete list of MPs). 1) A ramp cap may be applied, with an initial and maximum CL with starting and ending years (Fig. 4.4). This component generated two kinds of MPs, i.e., âcappedâ and âuncappedâ. Note that the caps were applied to the two species combined because the fishery does not distinguish the between species (Tables 4.3.4â4.3.5). After the cap implementation, the HCR added a âcatch split implementationâ which takes into account the error in species identification in the DFO surveys. The values corresponded to the proportion of S. fasciatus in the catch retained and reported in 2000-2017 (Table 4.3.6). After the âcatch split implementationâ, we applied a multiplier of 1.1 which takes into account âimplementation errorâ due to catch biomass not retained (e.g., lost from nets, spoiled, or caught as unreported bycatch in other fisheries) to be consistent with the recent fishing practices (Table 4.3.7). 2) A status quo period may be applied before the year in which the HCR starts (Fig. 4.4). During the status quo, it was assumed that the catch taken (i.e., not derived from the HCR) is either the average catch retained in 2015-2017 (i.e., 2 838 tons) or some pre-specified catch of interest to managers and industry members. 3) A maximum allowed CL or percentage of change in CL may be applied between years. 4) The SFP (starting in 2018) with CL set to zero may be applied if it is triggered fishery-wide in a given year (Table 4.3.9). 5) A minimum cap may be applied. 6) A multiplier to the parameter that controls the rate of harvest may be applied but only in âuncappedâ MPs (Table 4.3.1). A suit of MPs was proposed and tested in the MSE framework (Tables 7.4Câ7.5C). See Fig. 4.5 for a summary of the steps for calculating the realized CL under MP1. 92 4.3.4 Conditioning the operating model The OM describes the most plausible dynamics of the current stocks and fisheries and is used to project the stocks into the future to assess the performance of the MPs (Rademeyer et al. 2007). We used the available historical data to obtain estimates of the model parameters for the OMs. The historical information included the abundance indices and length compositions from the DFO survey (1984-2017) and the biannual industry survey (2000-2016) carried out in Units 1 and 2, respectively. The DFO and industry surveys generate information (e.g., indices and length compositions) by species based on the number of anal fin soft rays and also genetic analysis (Gascon 2003). The fishery information included the catch biomass retained (1960-2017) and the commercial length compositions from Units 1 and 2 (1984-2017). This information was split by species using the composition estimates from the DFO survey. More information about the survey sampling design and methods for splitting the commercial catches can be found in DFO (2018a). 4.3.5 Operating model description (base case) In order to condition the OMs on data, an age-structured population dynamics model was used. The OMs for the two species included the same sets of equations. Due to the paucity of data, the fisheries in Units 1 and 2 were aggregated into a single fishery in the OM for each species. Similarly, the fishing fleet data in Units 1 and 2 also were combined (e.g., catches and retained length compositions). The survey information was not aggregated. A Bayesian statistical catch-at-length stock reduction analysis (SRA) framework was applied for parameter estimation (McAllister and Ianelli 1997). Parameters were estimated separately for each of the OMs for the two species (see Tables 4.4â4.5; 7.1Câ7.3C, for a full description of the equations and 93 parameters values for the base case). The model was coded in ADMB (Fournier et al. 2012). The parameters estimated included, for example, the long-term average of unfished recruitment, the recruitment compensation ratio, age-1 recruitment deviates from 1952 to 2017, constants of proportionality for the survey biomass, and selectivity parameters (see Table 7.1.1C). A Beverton-Holt stock-recruit model was used to predict the recruitment of age-1 fish (Table 7.1.3C). An informative prior distribution formulated in terms of the Beverton-Holt steepness stock-recruit parameter was applied (Table 7.2.2C; Table 7.3C) (Forrest et al. 2010). We included informative priors for recruitment deviates in years where early reports on redfish have shown evidence of strong cohorts (CAFSAC 1984b; Sévigny et al. 2007) and the historical length composition could not inform the cohort strengths. An empirical-Bayes type meta-analysis of stock-recruit data was carried out to quantify the mean value across Sebastes stocks for large recruitment deviates (DFO 2018b). Informative priors on recruitment deviates were applied for three years (Table 7.2.5C) in the initial abundance at age 3 (Table 7.2.4C) and two early years (i.e., 1956 and 1958) (Table 7.1.2Câ7.1.3C). 94 Table 4.4. Base case operating model (OM1) for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). Note that we omitted subscript x because both species have the same sets of equations. However, we included the subscript x to clarify the differences between the two species for some equations. The parameter with a â^â symbol indicates that the value was estimated in the OM (Table 7.1C), otherwise it was pre-specified. Symbol Value (Sm; Sf) Description Indices x 1; 2 Index for species: 1:S. mentella; 2: S. fasciatus A 1,...,A; 1,...,A Age-year class T 1951,â¦,T; 1951,â¦,T Annual time step L 1,..,L; 1,...,L Length classes (cm) g 1,2,3,4; 1,2,3,4 Index for gear type g: 1: Survey Unit 1 (total biomass or length compositions) 2: Survey Unit 2 (total biomass or length compositions) 3: âfishery indexâ for Unit 1 (see text for details) 4: Index for fishery (combined length compositions retained for Units 1 and 2) p See Tables 7.4Câ7.5C Index for Management Procedures (MPs) Model settings A 57; 57 Plus-group T 2017; 2017 Last year before the management procedure starts (MP)(i.e., 2018) T2 2053; 2053 Year when the MP ends L 57; 57 Number of size classes blk1 1951,â¦,1993;1951,â¦,1993 Vulnerability time-block 1 blk2 1994,â¦, 2017;1994,â¦2017 Vulnerability time-block 2 Parameters Lï¥ 45.82; 41.24 Mean asymptotic length (cm) k 0.096; 0.106 Brody growth parameter (cm/yr) t0 -0.5; -0.5 Theoretical age at length zero (yr) 95 aw 0.00762; 0.00762 Scaling constant for weight-at-length (cm*gr) bw 3.193; 3.193 Allometric factor Ma 0.10; 0.125 Instantaneous natural mortality at age (yr-1) ?Ì?0 0.420; 0.623 Mean unfished recruitment (age-1) x [1e9] ï«Ë 3.082; 6.522 Compensation ratio h 0.435; 0.620 Steepness 3Ë Ë;a tï· ï·ï½ï¢ ï¢ï¢ 1952,..., ;1952,...,T T Fig. 4.10C; Fig 4.11C Estimated deviates for age 3 in the initial abundance and recruitment deviates in year t tï·ï¢ï¢ï¢ 2 2,..., ; ,...,t T T t T Tï¾ ï¾ Simulated recruitment deviates in year t ÏR 1.0; 1.0 Standard deviation of the recruitment variation Ω1 7.889; 7.256 Age-at-50% maturity4 Ω2 1.98; 1.58 Age-at-maturity slope4 501,2Ëga ï½ 2.022, 7.017; 1.676, 5.224 Age-at-50% vulnerability for survey index g 1,2Ë sdga ï½ 0.145, 2.731; 0.201, 0.609 Age-at-vulnerability slope for survey index g 150, 4Ë rett blk ga ï ï½ 9.245; 8.545 Age-at-50% retained for index g and time block 1 1 , 4Ë retsdt blk ga ï ï½ 0.866; 0.471 Age-at-retained slope for index g and time block 1 250, 4Ë rett blk ga ï ï½ 8.184; 8.329 Age-at-50% retained for index g and time block 2 2 , 4Ë retsdt blk ga ï ï½ 0.735; 0.534 Age-at-retained slope for index g and time block 2 1 , 4sdt blk ga ï ï½ 1 , 4retsdt blk ga ï ï½ Age-at-vulnerability slope for index g and time block 1 2 , 4sdt blk ga ï ï½ 2 , 4retsdt blk ga ï ï½ Age-at-vulnerability slope for index g and time block 2 501,2,3lsga ï½ 19.0; 18.0 Age-at-50% vulnerability or retention for the left side of the double logistic vulnerability 96 1,2,3lssdga ï½ 1.0; 1.0 Age-at-vulnerability or retention slope for the left side of the double logistic vulnerability _minvul reta 7; 7 Minimum age-at-retention or vulnerability for projections minmata 5; 4 Minimum maturity-at-age uinit 0; 0 Initial exploitation rate for t=1951 Umax 0.95; 0.95 Maximum exploitation rate 1,2,3Ëgï´ ï½ 0.469, 0.452, 0.407; 0.501, 0.382, 0.519; 0.501, 0.382, 0.516 Standard deviation for survey index g 1,2,3Ëgq ï½ 0.666, 1.745, 0.638; 0.533, 2.495, 0.444; 0.533, 2.495, 0.444 Catchability coefficient for survey index g 1,2,3,4gn ï½ 34, 8, 34, 33; 34, 8, 34, 33 Number of years with index data Derived variables 0ï¦ Unfished equilibrium spawning biomass per recruit la Survivorship-at-age per recruit aL Mean length-at-age (cm) sa Natural survival-at-age (yr) wa Weight-at-age (gr) wl Weight-at-length (gr) ma Fig. 7.1C Proportion mature-at-age SSB0 Unfished spawning biomass (kt) 50, 4t ga ï½ Fig. 7.1C Age-at-50% vulnerability for fishery in year t , 4a gv ï½ Vulnerability-at-age, for index g , , 4a t gv ï½ï¢ Vulnerability-at-age, in year t for the fishery (domed-shaped) , 1,2a gv ï½ï¢ï¢ Vulnerability-at-age for the survey (domed-shaped) ( | )P l axΦ ( , )La af L CV Transition matrix for species x ktC Fig. 4.6; Fig. 4.7 Catch biomass killed in year t (kt) ,reta tf Fraction retained-at-age, a, in year t for the fishery 97 ux,t Fig. 4.6; Fig. 4.7 Harvest rate in year t for species x Nx,l,t Number-at-length, l, in year t for species x SSBt Spawning biomass in year t (kt) ,kx tVB Vulnerable biomass killed in year t (kt) for species x rettVB Vulnerable biomass retained in year t (kt) Observations , , 1,2,3x t gI ï½ Fig. 4.6; Fig. 4.7 Observed survey biomass index g in year t (kt) for species x , , 3x t gI ï½ï¢ Simulated survey fishery index (length -based) in year t (kt) for species x rettC Fig. 4.6; Fig. 4.7 Catch biomass retained in year t for the fishery (kt) ratiokilltD No species-specific (Fig. 7.3C) Catch biomass killed to catch biomass retained ratio in year t offsetta No species-specific (Fig. 7.2C) Offset in year t (ages) _, , 1,2survey obsl t gN ï½ Figs. 7.4C-7.5C; Figs. 7.7-7.8C Observed numbers-at-length for the survey in Units 1 and 2 _, , 4ret obsl t gN ï½ Fig. 7.6C; Fig. 7.9C Observed numbers-at-length (retained) for the fishery in Units 1 and 2 1,2,4gess ï½ 25, 10, 5; 25, 10, 5; Sample size for index g 1,2gCV ï½ 0.25, 0.25; 0.25, 0.25 Coefficient of variation for survey index g LaCV 1 10.12 ,...,0.05 ;0.12 ,...,0.05a A a Aï½ ï½ Coefficient of variation for length-at-age (linear increase) The observed retained catch-at-length data were predicted using a fraction retained at age probability represented by a logistic function (Table 7.1.4C). Discarding data was not available; therefore, the age-at-50% vulnerability for the fishery was calculated by subtracting a time-varying offset parameter (Fig. 7.1C-7.2C), as an input vector to the estimated age-at-50% 98 retained (Tables 7.1.5C-7.1.6C). Two time-blocks were used, i.e., prior to 1994, and then from 1994 onward (Table 7.1.4C) (Fig. 7.1C). In order to calculate the catch biomass killed, the observed catch biomass retained was multiplied by a time-varying input vector representing the ratio of catch biomass killed to catch biomass retained in each historic year (Table 7.1.11C) (Fig. 7.3C). The values were formulated by an interview-based study on historic discarding practices (Duplisea 2018). It was assumed that in 1950, the initial abundance was close to the average unfished equilibrium state (Table 7.1.2C). A linear extrapolation was made to extend the catch time series back to 1951 from the early 1960s (Fig. 4.6 and Fig. 4.7). The likelihood functions applied were conventional ones for statistical catch-at-length stock assessment models (Tables 7.2.6C-7.2.7C) (McAllister and Ianelli 1997). The values that were applied for the coefficient of variation (CV) for survey indices, effective sample sizes for length composition likelihoods, and prior density functions can be found in Table 4.4 and Table 7.3C. In order to characterize parameter uncertainty in the OMs (Punt et al. 2016) in the conditioning and projections, we sampled the parameters from a multivariate normal (MVN) distribution using the posterior mode and posterior variance-covariance matrix obtained from the Hessian matrix. When the OM structure (e.g., a different vulnerability function) or input values (e.g., life history parameters) was modified, a new MVN was obtained for that OM (i.e., the modified model was again fitted to the data). One thousand draws were obtained for the key parameters for the OMs (i.e., indicated by a â^â symbol in Table 4.4). The parameter values drawn were applied in the OMs from 1951 to 2017. If the modelled stock abundance dropped to zero in any year up to 2017, the draw was discarded. 99 Table 4.5. Equations for the base case operating model (OM1) for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). Note that we omitted subscript x because both species have the same sets of equations. However, we included the subscript x to clarify the differences between the two species for some equations. Life history schedules Equations (1) Natural survival-at-age ( )aMas eï­ï½ (2) Survivorship per recruit 111111(1 ) (1 )1 1 1 / [1a initAainitAaAa aAs us uvval l l a Al lï­ï­ï­ï­ï­ï­ï½ï½ ï½ ï­ ï¼ ï¼ï½ ï­ (1 )] A initAus v a Aï¬ï¯ï¯ï­ï¯ï¯ï®ï­ ï­ ï½ (3) Mean length-at-age 0( ( ))[1 ]k a taL L eï­ ï­ï¥ï½ ï­ (4) Proportion mature-at-age 1 2min( ) /0 1 1mataaif a amotherwiseeï­ ï­ï ïï¬ ï£ï¯ï½ ï­ï¯ï«ï® (5) Weight-at-age ww ba aw a Lï½ (6) Vulnerability-at-age 50, ,_min, ,1 2( ) /0 1; 41 , sdt g t gvul reta t ga a aotherwiseif a at blk blk gevï­ ï­ï£ï½ï ï½ï«ï¬ï¯ï­ï¯ï® Stock-recruitment relationship (7) Spawning biomass per recruit 01a Aa a aal m wï¦ï½ï½ï½ï¥ (8) Unfished spawning biomass 0 0 0ËSSB R ï¦ï½ (9) Beverton-Holt stock-recruit parameters 00Ë 11SSBï«ï¡ ï«ï¦ï«ï¢ï½ ï¾ï­ï½ Population dynamics (10) Initial condition 2,1 0Ë 0.5( ),1 0 3 ~N(0,1) 3a Rpr Rpraa aaaN R l aN R l e aï· ï³ ï³ï·ï¢ ï­ï½ ïï¢ï½ ï½ (11) Recruitment deviates see OMs sectionË Ë~ ( , ) tttt t notherwiset Tbootstrap OMï·ï§ï·ï· ï·ï¢ï¢ ï¼ï¬ï½ ï­ï¢ï¢ï¢ï®ï¢ï¢ï¢ ï¢ï¢ (12) Recruitment (t > 1; a = 1) 20.5( )11,11t R RtttSSBN eSSBï§ ï³ ï³ï¡ï¢ï­ï­ï­ï½ï« 100 (13) Abundance dynamics ( 1)t ï¾ 1, 1, 1, 1, 11 1, 11 1, 1, 1 1 (1 )(1 )(1 )a ta t A tA ta a t tA A t tA A t tN a AN NN a As v us v us v uï­ ï­ï­ ï­ï­ï­ ï­ ï­ï­ ï­ ï­ï­ï­ ï¼ ï¼ï½ ï­ï« ï­ ï½ï¬ï¯ï­ï¯ï® (14) Spawning biomass ,1At a t a aaSSB N m wï½ï½ï¥ (15) Vulnerable biomass killed , , ,1 4Akt a t a t g aaVB N v w gï½ï½ ï½ï¥ , , , , , 21 Akt a t a t g a a t gaVB N v w t T v blkï½ï½ ï¾ ïï¥ (16) Vulnerable biomass retained , , , ,1 4Aret rett a t a t g a t aaVB N v f w gï½ï½ ï½ï¥ , , , , , , , 21 ;Aret ret rett a t a t g a t a a t g a taVB N v f w t T v f blkï½ï½ ï¾ ïï¥ (17) Catch biomass killed k kt t tC uVBï½ (18) Catch biomass retained ret rett t tC uVBï½ Fishery index (length-based) (19) Number-at-length ( | ), , , ,1P l ax l t x a t xA LAN Nï´ï´ï½ Φ (20) Fishery index 22Ë Ë0.5, , , , , ,30Ë Ë0.5, , , , , ,29Ë ~ (0,1) 3, 1Ë ~ (0,1) 3, 2xt g gxt g gLxx t g x g x l t x l tl cmLxx t g x g x l t x l tl cmI q N w e N g xI q N w e N g xï¥ ï´ ï´ï¥ ï´ ï´ï¥ï¥ï­ï³ï­ï³ï¢ ï½ ï½ ï½ï¢ ï½ ï½ ï½ï¥ï¥ 4.3.6 Selection of uncertainties The main sources of uncertainty were over how to characterize past and future recruitment and the productivity of S. mentella and S. fasciatus, species composition of historic catches, discarding, changes in vulnerability, and life history parameters (a full list of the OMs is in Table 4.6). The set of OMs was grouped in âcoreâ and âstress-testâ OMs. Here, the core OMs represent the most credible hypotheses for how the fishery and stocks have behaved or will behave and are credible from the standpoint of scientists and stakeholders. The stress-test OMs are plausible alternatives for fishery and stock behaviors but have less scientific credibility than 101 the core models. Thus, in terms of acceptability (e.g., stakeholders) and support (data), the core and stress-test have different weights. Following the common practices in the MSE framework, a base case OM representing the most credible set of specifications for an OM was formulated (Rademeyer et al. 2007). To test the sensitivity of results to alternative inputted values and prior values (e.g., historical strong cohorts), a set of âsensitivity-testâ OMs were also evaluated. We reported AIC values to provide an indication of how well the OM fitted the data (Tables 7.6C-7.7C); the AIC however was not (and should not) used to rank the plausibility of OMs. AIC measures only statistical performance; in contrast, working group knowledge of the fishery was taken to be more appropriate for judging plausibility (Guthery et al. 2005). 102 Table 4.6. Core, stress, and sensitivity test operating models (OMs) formulated for the redfish management strategy evaluation (MSE). Full description of the base case operating model (OM1) is given in Tables 7.1-7.3C. OM Type Description Concerns addressed 1 Core (base case) Logistic fishery selectivity, in two-blocks (1951-1993 and 1994-2017). Future recruitment as seen in the past, simulated with a non-parametric bootstrap using the recruitment estimates from 1970-2016. HCR uses a future fish killed:retained ratio ( )ratiokillprjD equal to 1.1 The base case was developed based on the current state of knowledge about redfish stocks in Units 1+2 6 Core Like the base case but no strong cohorts for the 20 years of projections It may be possible that after the strong cohorts of 2011-2013 there will be no more strong cohorts for the next 20 years. As a result, the expected projected biomass would be lower than expected 8 Core Like the base case but the M is age-specific following a Lorenzen function (M is higher for smaller fish) for historic and future projections The strong cohorts 2011-2013 may introduce density-dependent effects due to cannibalism and juvenile natural mortality 9 Core Like the base case but future recruitment is simulated with a parametric bootstrap, with variance and autocorrelation parameters generated from the recruitment estimates from 1970-2016 The future recruitment is unpredictable so the recruitment dynamics may not follow the observed recruitment in the past 10-11 Core Like the base case but the historic catch biomass retained and species composition of landings is biased towards either S. mentella (OM10) or S. fasciatus (OM11) The catch split is based on DFO survey carried out in Unit 1. This assumes that the species composition in Unit 1 is similar to Unit 2 2 Stress Like the base case but the historic and future fishery and survey vulnerability are dome-shaped The vulnerability may not be asymptotic due to temporal gear/vessel changes or spatial distribution of fish/fleet 3 Stress Like the base case but future natural mortality (M) is doubled for the first 20 years of projections The strong cohorts of 2011-2013 may introduce density dependent effects due to cannibalism and juvenile natural mortality. As a result, the expected projected biomass would be lower than expected 4 Stress Like the base case but for the first 20 years of projections, the growth rate is reduced by reducing the asymptotic length (Lâ) to 2/3 as large as in the base case, while fixing the other von Bertalanffy growth parameters The strong cohorts of 2011-2013 may introduce density-dependent effects on individual growth rates via, for example, competition and food limitation. As a result, the expected projected biomass would be lower than expected 5 Stress Like the base case but no strong cohorts for the 40 years of projections It may be possible that after the strong cohorts of 2011-2013 there will be no more strong cohorts for the next 40 years. As a result, the expected projected biomass would be lower than expected 103 14 Stress Like the base case but the historic fish killed:retained ratio ( ratiokilltD ) is either increased or decreased by 50% No quantitative information was available to estimate discarding rates. The catch biomass retained would be more or less to the reported landings 15-16 Stress Like the base case but historic and future M is lower by a factor of 0.75 (OM15) or higher by a factor of 1.25 (OM16) M was input parameter in the OMs. This productivity parameter has a large impact on management strategy performance. It has been recommended to consider this uncertainty in any MSE (Punt et al., 2014) 17-18 Stress Like the base case but the historic and future steepness (h) is lower by a factor of 0.75 (OM18) or higher by a factor of 1.25 (OM17) respect to the base case h was input parameter in the OMs. This productivity parameter has a large impact on management strategy performance. It has been recommended to consider this uncertainty in any MSE (Punt et al., 2014) 22 Stress Like the base case but the fishery vulnerability for 2017-2021 reverts back to the estimated for the first time-block (1951-1993) It is possible that the estimated fishery vulnerability for the first time-block is a better presentation for the period when the strong cohorts 2011-2013 are still not recruited 23 Stress Like the base case but the ratiokillprjD was increased from 1.1 to 2 from 2018-2020 The historic high discarding in the late 1980s would occur again in 2018-2020 due to the strong cohorts 2011-2013 but the discarding will return to 1.1 for 2021-2057 24 Stress Like the base case but the ratiokillprjD was increased from 1.1 to 2 from 2018-2020 and the future M is doubled for the first 20 years of projections (OM3) The historic high discarding in the late 1980s would occur again in 2018-2020 due to the strong cohorts of 2011-2013 but the discarding will return to 1.1 for 2021-2057. Moreover, the strong cohorts of 2011-2013 may introduce density dependent effects due to cannibalism and juvenile natural mortality for the first 20 years of projections 13 Sensitivity Like the base case but it uses an alternative input offset vector (offsetta ) to calculate the historic age-at-50% vulnerability for fishery. The offset was increase/decrease 0.5 respect to the base case. No information was available to estimate the vulnerability-at-age. It was estimated indirectly from age-at-50% retained minus a time-varying offset. The offset take into consideration changes in fishing vulnerability due to discarding 19- 20 Sensitivity Like the base case but it assumes a lower (OM19) and higher (OM20) prior mean for the strong historic cohorts respect to the base case No data were available to estimate earlier strong cohorts. These historic strong cohorts were included as informative priors. Priors have an impact in model estimates 104 4.3.7 Present and future recruitment It was important to obtain credible historic recruitment estimates for the redfish stocks and to develop a credible model to generate future recruitment. Future recruitment deviate sequences were simulated as they were observed in the past (Table 4.6 OM1). This was done using a nonparametric bootstrap which sampled from the set of historical recruitment deviate estimates (DFO 2018b). However, fishery stakeholders were interested in having MPs tested against a variety of scenarios for future recruitment patterns (Table 4.6 OM5-6). We also developed a conditional parametric bootstrap protocol (Table 4.6 OM9). In the case of long-lived species such as redfish, incomplete time-series of fishery-independent/dependent data limit the estimation of past cohort strengths (in terms of magnitude and frequency). Past strong redfish cohorts have been identified based on literature (e.g., 1952, 1980; CAFSAC 1984b). Unfortunately, earlier length compositions do not exist to estimate these cohort strengths. We examined the effect of including these early strong cohorts as informative priors as was described in the âOperating model descriptionâ section. We also examined their impact using alternative (higher and lower) prior means for a strong recruitment deviate for these strong cohorts (Table 4.6 OM19-20). 4.3.8 Species identification Stock productivity determines the recovery of depleted stocks but also the magnitude of sustainable catches (Hilborn and Walters 1992). In the case of S. mentella and S. fasciatus, these species are difficult to tell apart and are not distinguished in commercial fishery data (DFO 2018a). This situation has important implications for the reconstruction of historical abundance trends, estimates of cohort strengths, and stock productivity. We addressed this issue applying 105 different catch split hypotheses to the data (i.e., retained catch biomass and length compositions; Table 4.6 OM10-11). The length compositions (e.g., from the DFO bottom trawl surveys) contain juveniles of S. fasciatus that do not contribute to the adult population to the Units 1 or 2 but instead to the Grand Banks stock (3LNO) (Valentin et al. 2015). As a result, the bottom trawl survey length compositions have produced spurious estimates of abundance and harvest rates in the model fitting (e.g., Duplisea et al. 2016). We applied a simple method to remove these cohorts from the bottom trawl survey length compositions (see Fig. 7.4C). 4.3.9 Discarding The credibility of the estimates of cohort strength and stock productivity are also compromised due to past (in the 1980s and 90s) and present practices of discarding small fish (Duplisea 2018). As a result, the commercial length compositions are quite uninformative in terms of cohort strengths, vulnerability, and fishing mortality. Several OMs were developed to address this issue. For example, to explore the hypothesis that discarding was higher or lower compared with the base case, we applied two stress-test OMs, one with a 50% increase and another with a 50% decrease for the multiplier to calculate the catch killed biomass. (Table 4.6 OM14). Following suggestions from industry participants, we examined, using two other OMs, the hypotheses that high discarding rates may occur for 2018-2020 because the fishery cannot avoid the small fish from the 2011-2013 cohorts. We increased the âimplementation errorâ from 1.1 to 2 in the HCR but we reverted it back to 1.1 from 2021-2057 (Table 4.3.7). In two stress test OMs, we applied this scenario under the base case (Table 4.6 OM1) and under high levels of natural mortality (Table 4.6 OM24). 106 4.3.10 Change in vulnerability The commercial trawl gear has changed considerably over the past several decades (DFO 2018a). We used two time-blocks to describe the changes in vulnerability. This hypothesis was represented in the OM1 (base case). In a stress test OM, we examined a further hypothesis where the future fishery vulnerability for 2017-2021 reverts back to the fishery vulnerability estimated for the first time-block (Table 4.6 OM22). This represents the hypothesis that fishery vulnerability changes when strong cohorts are not completely recruited (i.e., small fish are discarded) in contrast the vulnerability patterns that favor the capture of larger fish in periods of low adult abundance and low recruitment. Changes in fishing gear/vessels can further affect the vulnerability at age, which can be asymptotic or dome-shaped depending on whether older fish are more or less vulnerable to the fishing gear (Table 4.6 OM2). Stress test OM2 had dome-shaped vulnerability for both the fishery and the bottom trawl survey. Changes in discarding practices also affect fishery vulnerability. In two other stress test OMs, we used different input values for the offset vector to calculate the age-at-50% vulnerability (Table 4.6 OM13). 4.3.11 Life history parameters Stakeholders were concerned about the potential density-dependent effect on individual growth and natural mortality rates (M) from the high predicted abundance for the 2011-2013 cohorts. Under this hypothesis, the strong cohorts 2011-2013 would generate lower biomass than anticipated due to low growth, cannibalism, lack of food, and intra- or inter-specific competition. In one stress test OM, the future low growth rates (e.g., for the next 20 years) were implemented by reducing the asymptotic length (Lâ) to 2/3 of the base case value (Table 4.4) while retaining 107 the other von Bertalanffy growth parameters (Table 4.6 OM4). For M, we examined several hypotheses as follows: in a core OM, the value for M is higher for smaller fish (Table 4.6 OM8); in some additional stress test OMs, M increases during periods with strong cohorts (e.g., for the next 20 years; Table 4.6 OM3), both historic and future M are lower or higher than the values for M specified under the base case (Table 4.6 OM15-16), and M increases for the next 20 years and there are high discarding rates for 2018-2020 (Table 4.6 OM24). Finally, in some other stress test OMs, we examined alternative values for the recruitment compensation parameter with respect to the base case (Table 4.6 OM17-18). When the parameter values were modified (e.g., increasing or decreasing M), the SRA model was re-run to condition the OM. 4.4 Results 4.4.1 Operating model fit and diagnostics For both species, the base case OMs showed high harvest rates in the 1990s, as high as about 45-50%. After 1993, the harvest rates decreased gradually to be lower than 5% in the most recent years (Fig. 4.6 and Fig. 4.7). SSB showed a marked decline to the mid-1970s, an increase, because of the biomass generated from the strong cohorts in 1980, and a marked decline caused by the high harvest rates in the 1990s, likely targeting the 1980 cohorts. SSB remained low for both species until the last few years when the SSB started to increase because of maturing fish from 2011-2013 cohorts. Survey indices for both species followed the estimated SSB pattern. Survey indices showed an increase prior to that in the SSB because the surveys included juvenile fish (particularly for DFO survey in Unit 1), which have increased in abundance in the surveyed area (DFO 2018a). Recruitment estimates showed spasmodic dynamics; a few strong large 108 positive recruitment deviates were interrupted by many years of relatively small positive and negative recruitment deviates. The strong cohorts from 2011-2013 for S. mentella were the largest ever estimated for the redfish stocks. Also, the strong cohorts for both species occurred at the same time (Figs. 4.6-4.7; 7.10C-7.11C). For both species, the survey length compositions fitted as well or better than in previous statistical catch-length models (e.g., Duplisea 2018). The OMs predicted very closely the length compositions when they belonged to large cohorts (i.e., the strong cohorts 2011-2013) and were able to predict the strong 1980 cohorts when they started to appear in the survey length compositions in 1984 (Figs. 7.4C-7.7C). The OMs were not able to predict the retained length compositions for S. mentella very well, but did so for S. fasciatus (Figs. 7.6C-7.9C). For both species, there were no concerning residual patterns for fits to the survey indices for either Units 1 or 2 (Figs. 7.10C-7.11C). A retrospective analysis showed no evidence of retrospective patterns for the time series of SSB, recruitment, and harvest rates (Fig. 7.12C). 4.4.2 Operating model outputs Estimates of parameters of the 15 OMs are listed in Table 7.6C and Table 7.7C for S. mentella and S. fasciatus. Umsy estimates for S. mentella were lower than those for S. fasciatus while the unfished spawning biomass was higher for S. mentella. The spawning biomass depletion in 2017 (D2017) was higher for S. fasciatus than S. mentella. For the base case, Umsy, MSY, and D2017 were 0.04, 25kt, 0.27 for S. mentella and 0.09, 34kt, and 0.15 for S. fasciatus. 109 Figure 4.6. Plots (from top to bottom) for the base case operating model (OM1) for S. mentella of the 1) catch biomass killed and retained, 2) harvest rates, 3) observed (dots) and predicted fits (lines) for the Units 1 and 2 survey indices, and 4) age 1 recruitment (long dash) and spawning stock biomass (SSB)(solid line). The shaded bands are 95% confidence intervals. 110 Figure 4.7. Plots (from top to bottom) for the base case operating model (OM1) for S. fasciatus of the 1) catch biomass killed and retained, 2) harvest rates, 3) observed (dots) and predicted fits (lines) for the Units 1 and 2 survey indices, and 4) age 1 recruitment (long dash) and spawning stock biomass (SSB)(solid line). The shaded bands are 95% confidence intervals. 111 4.4.3 Management procedure performance 4.4.3.1 Conservation objectives (1-3) For both species, all 22 MPs passed conservation objectives under the base case OM1 (Table 4.1). When catch splits were biased towards S. mentella (OM10), the uncapped MPs 20, 25, 26, 27, and 45 failed to meet conservation objective 3 (e.g., maintain harvest rates below Umsy for both species) for S. fasciatus. Also, we found that when the discarding rates were higher than those in the base case (OM14), two capped MPs (18 and 22) and the same uncapped MPs mentioned for OM10 failed to meet conservation objective 3 (Fig. 7.13C). 4.4.3.2 Fishery objectives for avoiding SFP (4-5) In the short term (five years), all the MPs failed to meet the fishery objectives of avoiding the 22 (i.e., SFP) and 25 cm fish (i.e., objective 4 and 5), under all of the OMs. The starting year of the HCR therefore had no impact on avoiding the SFP. Considering the 40 years of projections and the core OMs, only OM11 where the catch split was biased to S. fasciatus avoided the SFP for 22 cm fish. However, for OM1 the capped MPs 1, 4, 14, 24, 34, and 40 passed the PM criteria while for OM11 all the MPs passed (Fig. 7.14C). All the MPs that passed the different PM criteria under all OMs had caps between 14, 5, and 40kt (Table 7.4C). Under the stress-test OMs, most of the uncapped MPs did not pass the SFP except under the OMs that assumed no strong cohorts for the next 40 years (OM5), had lower discarding rates (OM14), and lower M (OM15). However, some capped MPs (e.g., 1, 4, 14, 24, 34, and 40) passed the SFP under different levels of recruitment compensation (OM17-18) and higher discarding rates for 2018-2020 (OM23) (Fig. 7.14C). 112 4.4.3.3 Fishery objectives for maximizing catch biomass and fish size (6-7) Uncapped MPs had better performance in terms of retained catch biomass than capped MPs, but uncapped MPs showed more variability in annual catches (Fig. 4.8). Uncapped MPs showed lower variability for the first ten years after that the 2011-2013 cohorts were assumed entirely recruited (i.e., for the simulation years 10-20) compared with 10-40 years of simulations. During the simulation years 10-20, the retained catch biomass was ~75kt for uncapped MPs. For capped MPs, the retained catch biomass increased with the MP cap (Table 7.4C). All the capped MPs with 40kt caps reached the cap independently of the OMs except in OMs that assumed double M or age-specific M (OM3 and 8), dome-shaped vulnerability (OM2), and low growth (OM4) (Fig. 4.8 and Fig. 7.15C). The highest retained catch biomass with capped MPs was under MP-16 with a cap of 80kt with ~60-65kt (depending on the core OM). The uncapped MPs with HCR-generated catch limits reduced by a factor 80% (i.e., MP12, 29, 43, and 44) produced slightly inferior catch biomass than those that did not have reduced catch limits. The OM with recruitment simulated with parametric bootstrapping (OM9) had a higher retained catch biomass in the 10-40 year horizon than in the 10-20 year horizon, especially with uncapped MPs. OMs with lower M and higher recruitment compensation had a larger effect on the performance of uncapped MPs, generating ~85-90kt in the 10-20 year horizon. Note that this PM was not affected by the starting cap because of the horizon used to calculate the retained catch biomass. Uncapped MPs with the SFP (such that CL = 0 if SFP is violated) and a maximum increase/decrease in catch limit of 5kt (i.e., MP18 and 22) had the highest proportions of simulations (i.e., >0.9) with CLs â¥40kt across all of the core OMs (except OM8) and most of the stress OMs (Fig. 7.16C). Similarly, for the core OMs, uncapped MPs with maximum increase/decrease of 15% but without the SFP (i.e., MP14 and 34) also scored high (i.e., >0.8) 113 except for the OM with no strong cohorts for the next 40 years (OM5). MP14 and 34 also scored high under the OM15 and 18 (Fig. 7.16C). Figure 4.8. Average catch retained (in kt) for the 10-20 and 10-40 year horizon for the 21 management procedures (MPs) under the core operating models. Dots show medians and bars shown the 90% confidence intervals. The x-axis indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR. 114 MP18 and 22, on average, are expected to produce 30 years with CLs â¥40kt for most of the OMs (except for OMs 2, 3, 4, and 8). All the capped MPs under the OM9 showed â¥28 years with CLs â¥40kt (Fig. 4.9). Similarly, many MPs under OM15 and 18 also scored high (e.g., â¥26 years) (Fig. 4.17C). There was a trade-off between the number of years with CLs â¥40kt and the average catch retained; uncapped MPs had high catches but with fewer years with CLs >40kt, especially when 100% of the slope for the HCR was applied (e.g., MP45). MP18 and 22 maximized the number of years with CLs â¥40kt and the average catch under most of OMs. All the MPs scored high under the OM9 (Figs. 7.18C-7.19C). All MPs performed well (e.g., â¥0.9) for the proportion of years with CL â¥4kt (i.e., landings in 2017) except for MP18 and 22 under OM2, 3, 4, 8, and 24. All of the MPs failed to pass the criterion in which the proportion of years where large fish (27cm) comprise >80% of catch in the 5 and 40-year horizons. Some uncapped MPs had â¥0.5 of the years with large fish under OM11 and 15. 4.4.3.4 Fishery objectives for stable catches (8) All the MPs passed the PM criteria of changes in CL between years less than 15%, but uncapped MPs scored lower than capped MPs (Fig. 7.20C). Similarly, all MPs passed criteria (â¤15%) for the average annual variation between years (AAV), but uncapped MPs scored lower than capped MPs in the 10-40 and 10-20 year horizons. 115 Figure 4.9. Average number of years with catch limits (CL) â¥40kt in the 2028-2057 year horizon for the 21 management procedures (MPs) under the core operating models. Dots show medians and bars shown the 90% confidence intervals. The x-axis indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR. 116 4.5 Discussion In this paper we examined key challenges to managing fisheries for stocks that show spasmodic recruitment. This recruitment pattern is found in Sebastes species and many other fishes such as hakes, sardines, capelins, mackerels, and anchovies (de Moor et al. 2011; Needle 2016; DFO 2018c). Strongly spasmodic recruitment can make fishery management particularly challenging. Under spasmodic recruitment and particularly when recruitment shows only a weak dependence on the parental stock, the classical assumptions of steady-state models with conventional patterns in recruitment variability (e.g., where the largest cohorts are no more than a few times the long-term average and the HCRs considered are often directly based on MSY-based reference points) cannot be applied here. In fact, there is growing evidence that this situation is quite common. For example, when examining patterns of variability in fish stocks, Spencer and Collie (1997) found that only three of the 30 stocks conformed to steady-state behaviour. Recent studies have shown that many fish stocks around the world demonstrate non-steady state behaviours, e.g., non-stationary stock-recruitment functions (Dorner et al. 2008; de Moor et al. 2011; Vert-pre et al. 2013; Minto et al. 2014; Britten et al. 2016). The two redfish stocks examined in this paper are extreme examples of spasmodic recruiters; the MSE carried out on them provides information on how to sustainably manage fisheries for such stocks. We note characteristics of models and management strategies that are most promising for developing sustainable management of these stocks: 1. Management harvest control rules should be triggered only by biomass increase for sizes of fully mature individuals larger than just post-recruit sizes. 117 2. A simple empirical stock size indicator (e.g., bottom trawl survey biomass) has advantages both for modeling and for communication with stakeholders as the main indicator of stock status. 3. Capped harvest control rules provide stability in catch and can extend the number of years with large harvests following spasmodic recruitment events. 4. A range of recruitment modeling methods including parametric and non-parametric resampling ensures robustness of the management strategy to the main perceived uncertainty. 4.5.1 Harvest control rules responsive to the biomass of large-sized fish We identified several desirable properties of an empirical approach to managing SRS such as redfish: First, a âtarget-basedâ empirical approach was computationally and conceptually more tractable than a model-based approach (i.e., simulation of a stock assessment in the closed-loop). OMs fitted to data already captured the strong 2011-2013 cohorts, which were projected and propagated with uncertainty in the OMs. Thus, the problem was reduced to finding MPs that best extended the harvest benefits of these strong cohorts (e.g., number of years with large catches and larger fish landed) while avoiding high discard rates. Second, the parameters of an empirical MP do not depend on MSY reference points. The HCR parameters were tuned in the base case OM using Umsy estimates to avoid high harvest rates for the less abundant species (i.e., S. fasciatus). Third, a significant fraction of the catches will be small fish when strong cohorts are recruiting to the fishery. Applying an empirical MP based on a total biomass index (e.g., the Unit 1 DFO bottom trawl survey) is problematic because it will be highly âreactiveâ to the juvenile fish biomass. However, this survey index was modified to a âfishery indexâ by including only 118 larger fish (i.e., >30 and >29 cm for S. mentella and S. fasciatus, respectively) which dampened the variability in CLs by reducing the effect of the strong cohorts. This âfishery indexâ also has other benefits. It only âreactsâ to changes in the surveyed biomass of large fish. More importantly, it allows harvest of the strong cohorts a few years later when they have a larger commercial size, and reduces the risk of growth overfishing. Basing the âfishery indexâ on large-sized fish of both species, avoided high harvest rates on the stock component with weaker recruitment. In earlier simulations, when the total biomass index was based on smaller fish, a strong cohort led to high CLs and high harvest rates on the weaker stock. 4.5.2 Use an empirical index to formulate harvest control rules For short-lived species that show a clear boom and bust recruitment dynamics (e.g., sardines and anchovies) empirical MPs have been applied (e.g., de Moor et al., 2011; de Moor and Butterworth 2016). Model-based MPs also have been applied with a rectilinear or empirical HCR (e.g., Roel and De Oliveira 2007; Hicks et al., 2016; Needle 2016). Some species of gadoids, for example, as Pacific hake and haddock, appear to have low average recruitment with occasional large year-classes (e.g., Needle 2016). However, it is not clear whether a rectilinear HCR will outperform an empirical MP when the fishery targets strong cohorts; e.g., it should be sufficiently reactive when the biomass increases rapidly and should reduce fishing mortality when the strong cohorts are exhausted. In the case of sardine and anchovy, de Moor et al., (2011) and de Moor and Butterworth (2016) showed that including a âtwo-tierâ system with an empirical MP allowed large catch limits during âboomsâ in abundance and decreased catch limits once the biomass started to decline, but still produced reasonable stability for the industry and low risk of depletion. 119 In the case of spasmodically recruiting species caught together and that cannot be told apart on the fishing grounds, the control parameters of the âtarget-basedâ MP had an important role in determining the CLs and avoiding overfishing. S. fasciatus has high recruitment compensation but the strong 2011-2013 cohorts were weaker for this species. Therefore, the risk of rapidly exhausting these cohorts is higher for S. fasciatus as preliminary simulations predicted. Tuning the control parameters so that the harvest rates were on average lower than the estimated Umsy levels avoided the risk of overfishing for the less abundant species (e.g., S. fasciatus). The tuned parameters also prevented the SSB from falling below the LRP at the end of the 40-yr projection. 4.5.3 Capped harvest control rules The ramp cap period in the MPs was a critical aspect identified to regulate CLs following large recruitment events because strong cohorts can generate a rapid increase in biomass mainly from immature fish. Note that most of the MPs evaluated in the literature impose a single cap (e.g., Hicks et al., 2016). A set of capped and uncapped MPs with a suite of rules and constraints (e.g., changes in rate of harvest in the HCR, maximum change in CL between years) were developed to extend the duration of high projected biomass and reduce the discarding of small fish (e.g., SFP, starting years for the MP, status quo, minimum caps). The suite of MPs exposed the trade-off of âinventory planningâ. The aim was to maximize the number of years with "good catches" instead of maximizing total catches as is traditionally done in MSE. The notion of âgood catchesâ (e.g., 40kt) was an industry objective that was rooted in its processing capacity and the market for redfish fillets. We showed that depending on the MP, stable catches of about 20-60kt could be caught in the next two decades while still avoiding overfishing. Uncapped MPs 120 produced the largest catches, but capped MPs produced the longest periods of large catches (e.g., >40kt) and more stable catches than uncapped MPs. Capped MPs lowered the risk of growth overfishing but the use of a large fish âfishery indexâ and tuning the HCR parameters also helped. Stakeholders mentioned that the current market for redfish is limited (i.e., mostly sold for bait). The suite of capped MPs limited the future catches taken so that the industry could build the fishing processing capacity required and develop better access to markets for higher-valued redfish products. HCRs within MPs commonly evaluated for the steady-state stocks are not suitable for SRS. It is interesting to note that if a threshold-type HCR had been applied (e.g., the 40-10 control rule used by the Pacific Fishery Management Council), the Unit 2 fishery would have been closed also for more than 25 years (the Unit 1 fishery was closed in 1995, but a quota of 2kt was subsequently allowed to maintain an index fishery). This hypothetical situation would have had caused even greater economic hardship. Maintaining low catches under low stock sizes indicates that harvest rates fixed at low levels may be suitable for SRS under periods of low abundance. Recent literature has shown that threshold-type HCR produces unnecessary hardship to the industry in mixed-stock fisheries and when stock-recruit parameters are nonstationary. In this situation, a better alternative is fixed harvest rates rules that allow fishing even when the stock is low (e.g., Hawkshaw and Walters 2015; Walters et al. 2019). Similarly, fixed escapement HCRs may not be suitable because they might not allow for maximizing the number of years with large catches (instead it will maximize total catch, Hilborn and Walters 1992) and it may produce undesirably high harvest rates. For example, before the current management Units were established in the mid-1990s, the harvest rates were over 10%, peaking around 50% in 1993. Probably, the early scientific advice given did not consider the 121 spasmodic dynamics of the redfish stocks or was based on the rules of thumb, âtake what you can when [abundance is] highâ and âfish [for] something else in bad periodsâ, that were suggested for these kinds of stocks at the time (Caddy and Gulland 1983). Either way, as a result, the CLs applied to the last strong cohorts in the 1980s exhausted these cohorts quickly. Typical MPs used for steady-state stocks seem too ârigidâ to accommodate the spasmodic stock dynamics. For example, the reference harvest rate (or Ftarget) in the fixed harvest rate rules (or in threshold-type rules) have little meaning when the stock changes rapidly from very low to very high abundance owing to a single very abundant year class. Moreover, as these HCRs depend on estimates of stock size (e.g., a model-based approach), the prescribed CLs would be too risky from a conservation perspective, simply because the biomass estimates may be biased and imprecise (Hilborn and Walters 1992). A âtarget-basedâ HCR, when carefully tuned, is both more flexible and safer. For example, this HCR still prescribes a CL when the stock is at low stock sizes with catches similar to those when the stock was low, thus, accommodating the possibility that a long run of weak year-classes could occur in the future for these stocks. However, the HCR can also protect the stocks when the âfishery indexâ drops below a threshold (de Moor et al. 2011; de Moor and Butterworth 2016). When the stock is in an abundant period, the capped MPs will extend the years with large catches. Our MPs followed the empirical approach of the âboomâ and âbustâ dynamics of de Moor et al. (2011) and de Moor and Butterworth (2016) but with a more extreme âboomsâ and more extended âbustâ periods than is typical. Roel and De Oliveira (2007) also used two-slope strategies (i.e., an empirical HCR) that included a minimum CL to ensure that fishery closures are minimized. 122 4.5.4 Test harvest strategies against a variety of recruitment assumptions For SRS, the projections of the strong cohorts require a careful study of the historical recruitment pattern and reliable estimates of the cohort strengths. Long periods of weak recruitment have been documented for S. mentella in other areas (Templeman 1976; Sévigny et al. 2007; Planque et al. 2012) and in others Sebastes (e.g., Wetzel et al. 2017). According to our OMs, the estimated strong cohorts for 2011-2013 are the largest observed (for both species) in GSL-LCH. Preliminary projections showed that including more than four strong cohorts (here defined as recruitment deviates greater than 2.5 in logarithm space) in the projection horizon produced unrealistically high (combined) biomasses. We also found that the typical approach of generating future recruitment from a stock-recruitment relationship (i.e., expected recruitment plus error) or even selected ranges of years of historical recruitments (e.g., Punt and Methot 2005) produced future stock sizes which were not at all representative of what was seen previously in these stocks. We found that redfish stocks showed five key features in the recruitment dynamics. First, the strong recruitments events for S. mentella and S. fasciatus were highly correlated: as one species had a strong recruitment, so did the other (Fig. 7.10C and Fig. 7.11C). Second, cohort strength was consistently higher for S. mentella (e.g., the magnitude of the recruitment deviate was higher). The last strong cohort in 1980 also contained both species, but S. mentella dominated according to genetics and clustering analysis (Valentin et al. 2015). Third, we found that strong cohorts were produced in a pulse of no fewer than eight years apart following the previous strong cohort (same as Planque et al. 2012). Fourth, the large recruitment deviates occurred within no more than three consecutive years (or two according to Templeman 1976) but only two large deviates could occur within such sequences. Finally, for the period where cohort 123 strengths could be estimated (i.e., 44 years), there were no more than four strong cohorts (Fig. 7.10C and Fig. 7.11C). As such, we developed both parametric and non-parametric bootstrap approaches to mimicking these recruitment dynamics. Both methods had a similar performance in term of conservation and fishery objectives, though the non-parametric approach was preferred by the RWG; however, it has proven difficult to replicate redfish recruitment dynamics using a parametric approach. The parametric method seems to be too optimistic regarding the proportion of simulations and numbers of years with CLs >40kt across OMs and MPs. The non-parametric approach, although perhaps more realistic, is more complex to implement and is limited by the time series length of observations which cover less than about two generation times for the stock. It may, however, better replicate the particular recruitment dynamics of these stocks. We used a Beverton-Holt stock-recruitment function to simulate future recruitment. It can be argued that a Ricker stock-recruitment function may be a better option, as it takes into account density-dependent effects at large stock sizes (e.g., Roel and De Olivera 2007). Also, instead of sampling recruitment deviates, a better option could be sampling directly from the historical recruitment (e.g., Punt and Methot 2005) to avoid using a stock-recruitment function. For example, the strong redfish cohorts can be produced at very low levels of SSB and after a long period of weak recruitment events. We used a stock-recruitment function mainly because DFO required MSY reference points to avoid overfishing. However, both concerns were addressed in OM5 and OM6 where no strong cohorts were simulated for the 40-years and the first 20 years of projections, respectively. Therefore, these OMs simulated 1) a strong density-dependent effect due to a large SSB (as in a Ricker function) and 2) no strong recruitments despite including a stock-recruitment function. Our MPs still were robust to these two OMs (both passed 124 conservation objectives). Even if the non-parametric approach was not able to fully mimic the redfish dynamics (e.g., OM1), the better performing MPs still avoided overfishing in this OM. Other studies with SRS have also simulated the recruitment consistent with historical observations. For example, Needle (2016) simulated one large recruitment in a random year within each 10-year simulation period, where the large year-classes were separated by at least two years. Low recruitments were simulated by a log-normal distribution about the geometric mean of low historical observations. de Moor et al., (2016) used a hockey-stick stock-recruitment model with autocorrelation in recruitment deviates. For periods of low recruitment, Roel and De Olivera (2007) used a Ricker model with autocorrelation and a cap so that recruitment does not exceed the maximum estimated in the stock assessment. A large cohort was generated about once in every 20 years as long as the SSB had as minimum historical threshold. 4.5.5 Other considerations 4.5.5.1 Data quantity and quality (including survey and catch data e.g., discarding) For the redfish fishery and SRS in general, an essential aspect was the availability of fishery-independent abundance surveys to detect and monitor strong recruitment events. OMs fitted to only commercial length compositions are uninformative for estimating future strong year classes. It can thus be argued that effective management of SRS can be achieved more easily in data-rich situations as it is the case for redfish stocks. In a data-limited situation, of course, potential strong cohorts could have been simulated as a possible OM, but without the knowledge of when the cohort(s) was produced, the magnitude, and whether both species produced strong cohorts at the same time. This approach could generate less credibility and added uncertainty in the MSE process because the redfish species also have different productivity. 125 Careful analysis of survey samples was essential to eliminate the Grand Banks cohorts. S. fasciatus has generated strong year classes that disappear from research surveys after a few years from Units 1 and 2 (Valentin et al. 2015). For this study, genetic analyses were fundamental to discriminate the Grand Banks cohorts from those that will contribute to the redfish fishery. For example, when strong cohorts are detected in fish <22cm in future surveys, genetic analyses can help determine their potential contribution to the fishery. Also, it is important to monitor strong cohorts to define PMs. DFO trawl surveys have tracked these strong cohorts since 2011-2013, which are expected to materialize in the fishery by 2028. As a result, stakeholders were interested in PMs after this period when they start to contribute to the catch biomass. Moreover, there were no strong cohorts before 2011-2013. Thus, it was inappropriate to evaluate PMs (e.g., average catch biomass) before 2028. In times of very strong recruitment, high percentages of small fish in the catch and high discarding rates are possible before the fish grow into marketable sizes; catches with large percentages of small fish have recently been observed on the Unit 1 and 2 fishing grounds (DFO 2018a). For the redfish stocks, there were no data available to estimate the vulnerability at age and discard rates in the OMs. This likely resulted in the poor fit to the retained compositions, especially for S. mentella. Uncertainty in the vulnerability at age and discarding were addressed in the OMs (e.g., OM1, 13, 14, 22, 23, and 24). The results showed that in the long term the vulnerability at age and discarding had a low impact on the fishery and conservation objectives compared with changes in productivity parameters (e.g., individual growth rates, recruitment compensation, and M). For example, a change in vulnerability at age towards younger fish (i.e., OM22) for the less abundant species, S. fasciatus, still passed the conservation criteria. However, all of the OMs showed that the SFP would be triggered unavoidably for the next two years 126 independently of the MPs. This finding agreed with the Unit 1 DFO bottom trawl survey where the biomass of S. mentella and S. fasciatus juveniles was 60 and ten times higher, respectively, than their mean biomass for 1995-2015. The number of times that the SFP will be triggered will depend significantly on growth rates of the strong 2011-2013 cohorts. Current individual growth estimates have shown that the strong 2011-2013 cohorts are growing as expected (DFO 2018a). Projections showed that under the OM4 (i.e., low individual growth) the catch biomass retained would be significantly reduced. We believe that this is unlikely. Under this hypothesis, the last strong 1980 cohorts would not have produced the observed catch biomass retained if these cohorts had suffered a strong density-dependent effect. A similar situation can be inferred by doubling M as the fishery representatives had suggested. Monitoring the strong 2011-2013 cohorts will provide more information about density-dependent effect in SRS. 4.5.5.2 Stakeholder participation and buy-in This work showed the importance of the MSE process in helping to refocus efforts from selecting the âbest stock assessment modelâ for redfish stocks (e.g., McAllister and Duplisea 2011; Duplisea et al. 2016; Duplisea 2018) to selecting MPs that can meet objectives and are robust to uncertainty (Rademeyer et al. 2007; Punt et al. 2016). This situation was facilitated because of the prospective strong 2011-2013 cohorts, which are expected to be fully recruited to the fishery by 2028 (DFO 2018a). This situation âforcedâ the stakeholders to focus on both generating management objectives for this fishery and acceptable trade-offs. After a series of meetings and workshops (in total nine), the stakeholders, at first skeptical, got to recognize the value of participating fully in the MSE process. As the process progressed, the RWG received 127 more requests from the industry to assess new OMs and candidate MPs (e.g., we ended up with more than 17 OMs and 22 MPs). We believe that the MSE was facilitated by the dynamic, open, and participatory process of the MSE. The entire process of conditioning the OMs, MP and PM development, projections, and peer-review took one and a half years, with results in-progress presented at every meeting. As the models were being developed it was appropriate to communicate to stakeholders that the results as presented were provisional and would be updated with further analysis before upcoming meetings. Therefore, we recommend that to engage participation, analysts be prepared to present MSE results in-progress throughout the MSE process, rather than waiting until the MSE models are fully developed and results are finalized. We opted for an empirical HCR instead of a model-based approach to prescribing TACs. Empirical MPs have the benefit that they are more transparent and easier to explain to the industry (Geromont and Butterworth 2015b) compared to model-based MPs. For example, over the meetings, an industry member noticed that the initial HCRs could not meet their objectives when the stocks were at low abundance. The control parameters to the HCRs were revised and modified to address the concerns. This situation would have been less easily resolved under a model-based approach. 4.5.6 Conclusion: recommendations for managing spasmodically recruiting stocks Only rather general and imprecise recommendations have been provided for managing stocks with spasmodic recruitment (SRS)(e.g., Caddy and Gulland 1983), despite being not uncommon. We thus offer based on our recent experience with Atlantic redfish, the following new recommendations. 128 1) Fishery-independent survey data are needed to estimate trends in abundance and cohort strength and model future recruitment. For example, well-designed fishery-independent trawl surveys that cover the known spatial extent of the stock(s) can enable reliable detection of strong recruitment events and monitoring of changes in abundance. The trawl survey should provide age or length composition samples of pre-recruits to give advance warning of strong recruitment events. It is less critical to attempt to gather age or length composition data from the fishery, since possible increases in discarding rates when stronger cohorts are recruiting to the fishery compromise the usefulness of these data. 2) Devoting resources and time to obtain credible records of harvested catch biomass (i.e., retained + discarded) is essential for stocks with spasmodic recruitment; adjusting catch biomass records for past discarding practices (e.g., Duplisea 2018) will reduce bias in reconstructions of stock biomass and productivity. 3) For SRS of longer-lived species such as gadoids and Sebastes, it may be advantageous to adopt an inventory management harvest strategy, which takes a longer-term view of the fishery when strong cohorts are predicted to recruit to the fishery. However, management procedures should also be designed to allow for continued low harvests during extended periods of low abundance. For example, in the Unit 1 and 2 Atlantic redfish fishery, the largest cohorts for both stocks originated from the lowest estimated abundances in the sixty year history of the fishery and while non-negligible harvests were allowed to be taken by industry. 4) Management procedures for long-lived species such as redfish could include the following features: (i) An empirical HCR based on fishery-independent abundance indices based on desirable fish sizes. This allows harvest of the strong cohorts a few years later and reduces the risk of growth overfishing (i.e., new recruits). (ii) Ramp caps in catch limits when abundance is 129 building and a maximum cap when harvestable biomass has peaked. Also ramp-ups should be compatible with feasible expansions of industrial capacity and foreseeable marketing opportunities for the expanded harvests. 5) Bayesian approaches are suitable for fitting operating models for SRS because they provide formalized methods to include results from meta-analyses of key parameters from other similar stocks (e.g., Forrest et al. 2010) and anecdotal information about, e.g., trawl survey catchability (DFO 2018b), historic discarding practices from fisher interviews (Duplisea 2018) and strong cohorts from early fishery management briefs. 6) It is critical to formulate operating models of multi-stock complexes targeted by a single fishery which simultaneously simulate the capture of the main fish stocks of interest to the fishery. This can help the analysis become more credible to industry and managers and allow the evaluation of trade-offs between the conservation of weaker stocks and fishery yields, especially following spasmodic recruitment events for one or more stocks. 7) Where scientific and managerial capacity are available and harvest potential is significant, we would recommend that fishery managers consider adopting MSE as a collaborative approach. Close collaboration between scientists, fishery managers, and industry stakeholders is essential for developing management procedures that enable well-controlled responses of industrial capacity to large systematic changes in harvestable biomass. In addition, this helps industry to establish and adapt effectively to better understanding of stock dynamics. For example, awareness of âboom-and-bustâ stock dynamics can help industry to develop long-term plans to avoid overcapacity during periods of low productivity and achieve long-term profitable levels of capacity especially after large cohorts are detected. 130 5 Chapter 5: Conclusion 5.1 Conclusion This dissertation has three studies that aim to advance understanding on methods for statistically assessing time-varying recruit parameters in fish populations and harvest control rules (HCRs) that could cope with the sometimes large associated changes in fish production. I examined the performance of current state-space methods to track changes in fish productivity and carrying capacity. State-space methods such as the Kalman filter (Kalman 1960; Durbin and Koopman 2001) have been used in global meta-analysis of marine stocks to assess changes in carrying capacity (Britten et al. 2016). In Chapter 2, I evaluated the ability of the Kalman filter to detect non-stationarity in stock-recruit parameters from VPA and SCA outputs using simulated data. No studies have addressed this issue before. The selection of the best stock assessment method to test stock-recruit parameters is a matter of current discussion. Dickey-Collas et al. (2015) suggested that statistical stock assessment models (i.e., SCA) with a stock-recruitment relationship are more appropriate if the final goal is to estimate MSY reference points, perform projections, and closed-loop simulations. Maunder and Thorson (2019) stated that modelling the stock-recruitment relationship outside the population dynamics (e.g., VPA) ignores precision and autocorrelation in the recruitment deviates, and ignores error in the spawning stock biomass. Furthermore, it does not allow the stock-recruitment relationship to inform the population dynamics and does not allow for propagating uncertainty into the management outcomes and model parameters (Maunder and Punt 2013; Dickey-Collas et al. 2015). Dickey-Collas et al. (2015) indicated that if the final goal is to perform fish productivity analyses, the best approach is to avoid parametric and distributional assumptions in time-series of stock-recruits. 131 The results in Chapter 2 showed that simple stock assessment methods like VPA could generate model outputs similar in quality to those from SCA. Thus, the stated disadvantages of formulating a stock-recruitment relationship external to a stock assessment model (e.g., Maunder and Punt 2013; Maunder and Thorson 2019) could not be verified in my analyses. Besides, for some scenarios, the VPA outperformed the SCA in tracking regime shifts. VPA was often better than SCA at propagating uncertainty and autocorrelated recruitment deviates (e.g., regime shifts) in the evaluation of management outcomes. Methods to evaluate time-varying stock-recruit parameters have not been widely used. For example, in the world conference on stock assessment methods (Deroba et al. 2015), this problem was almost not considered. This problem has also captured relatively little attention in past and recent stock assessments. This dissertation supports the notion of focusing on time-varying stock-recruit parameters, as they determine fish productivity and sustainable harvest rates. In Chapter 2, I showed that the VPA could be more practical and reliable for obtaining the stock-recruit data, especially if these data exhibit non-stationarity in stock-recruit parameters. Chapter 2 showed the importance of performing a simulation-evaluation to assess the direction of the bias and precision when the time-varying stock-recruit parameters are estimated using state-space methods. The simulation-evaluation also showed that non-stationarity in stock-recruit parameters is overall challenging to estimate. Given evidence that changes in fish productivity and carrying capacity are frequent (Wada and Jacobson 1998; Dorner et al. 2008; Barange et al. 2009b; Cahuin et al. 2009; de Moor et al. 2011; Peterman and Dorner 2012; Vert-pre et al. 2013; Minto et al. 2014; Britten et al. 2016; Malick and Cox 2016; de Moor and Butterworth 2016), 132 this dissertation suggests that the types of analyses in Chapter 2 should be performed more often in stock assessments. State-space models, based on the Kalman filter, can detect the overall trend in the stock-recruit parameters, but the estimability of fish productivity and carrying capacity (α and β stock-recruit parameters, respectively) depended on the historical removal pattern. I found that across stocks (jack mackerel, anchovy, and hake) and catch histories (informative and uninformative), under a regime shift in carrying capacity, the Kalman filter correctly tracked the stock-recruit parameters. Consequently, the time-series of MSY reference points were correctly identified. However, when the catch history was informative, the Kalman filter generated a positive bias for the productivity trend for the most recent regime shifts. Unfortunately, the simulation-evaluations suggested that time-varying fish productivity was more difficult to track. In this scenario, the Kalman filter performance depended on the stock and catch history. For hake, for example, the Kalman filter correctly identified both stock-recruit parameters when the regime shift was in productivity and when both stock-recruit parameters were non-stationary and negatively correlated. The Kalman filter correctly identified the regime shifts under both catch history patterns. The Kalman filter performance was different for jack mackerel and anchovy and depended on the catch history. Under a productivity regime shift, the Kalman filter generated a positive bias for productivity and carrying capacity for the most recent period. Consequently, the time-series of Umsy showed positive bias for jack mackerel and anchovy. The time-series of Bmsy had a lower bias. Surprisingly, the bias for the most recent regime shifts was less severe when the stock-recruitment data came from the VPA. This situation occurred across stocks. The 133 informative catch history reduced the bias for productivity, but not for the carrying capacity trend. In the scenario when both stock-recruit parameters were non-stationary and negatively correlated, the carrying capacity trend was more challenging to track than the productivity across stocks. In this situation, the carrying capacity showed a positive bias for the most recent period. Overall, estimations for jack mackerel performed the worst in tracking productivity changes followed by anchovy when the regime shift was in productivity. The estimability of the stock-recruit parameters has implications for fishery management. For example, if stock productivity changes over time, harvest rates need to be updated accordingly with periods of low or high productivity (Walters and Martell 2004). While Peterman et al. (2000) strongly recommended to scientists to use the Kalman filter methods to track time-varying productivity in salmon stocks, this dissertation found that this method should be used with caution. Kalman filter methods (and state-space models in general) can fail even with the use of relatively simple models (Auger-Méthé et al. 2016). For example, the Kalman filter performance depended on the time-series pattern in historical catch removals, life history type of the harvested population, regime shift type, and stock assessment method. Chapter 2 stated that this method should be used under intensive simulation-evaluation if Kalman filter-based results are to be used for management advice. Simulation analysis could help to further identify stock assessment configurations and management options robust to non-stationarity in the stock-recruitment relationship. It is concerning to notice that despite the extensive evidence of non-stationarity in stock-recruitment relationships, only a few studies have examined the robustness of HCRs to changes 134 in fish productivity and carrying capacity (e.g., Walters and Parma 1996; Hawkshaw and Walters 2015). Thus, there are few guidelines to approach changes in fish productivity. Closed-loop simulations or management strategy evaluations (MSEs) (Aâmar et al. 2009a, 2009b; Ianelli et al. 2011; Szuwalski and Punt 2013a, 2013b) have used methods such as âDynamic B0â (MacCall et al. 1985), âsliding windowsâ (e.g., Aâmar et al. 2009b), and the âSTARSâ approach (e.g., Szuwalski and Punt 2013b) to test HCRs under changes in productivity. These methods have shown little or no improvement (in some instances deterioration) in achieving management goals under changes in fish productivity. For example, Szuwalski and Punt (2013b) concluded that the âSTARSâ approach only performed well if the population dynamics was truly regime-like, and the shifts went from high to low productivity. Similarly, Berger (2019) found little differences between the âDynamic B0â and the static B0 for 18 West Coast US groundfish stocks in triggering appropriate management actions. In Chapter 3, in contrast to previous studies, the HCRs were examined through stochastic optimization in policy space (SOPS) (Moxnes 2003, 2005; Walters and Martell 2004). This method is appealing because it shares features with stochastic dynamic programming (SDP), and it allows incorporation of closed-loop simulation (or MSE) components as well. Findings were obtained about optimal HCRs under regime shifts. First, the analysis showed the importance of including both observation and process errors in the evaluation of optimal HCRs (e.g., HCRs that maximize a utility function). Most of the SDP models only incorporate process error except for those in Memarzadeh and Boettiger (2019). HCR performance was sensitive to different levels of observation error (and process errors). Second, the optimal HCR parameters depended on how the population dynamics were modelled, the utility function, and stock productivity (e.g., the intrinsic population growth, r). A comparison of the three surplus production models (SPMs) 135 showed that some model configurations (e.g., continuous vs. discrete) were more robust to observation error. For example, highly productive stocks (e.g., anchovy) showed less robustness in the optimal HCR parameters to observation error than stocks with lower productivity. I showed that the traditional SPM (i.e., the discrete model with production before harvest) did not perform well in closed-loop simulation for stocks with high productivity. For these stocks, a better alternative was to use a continuous SPM. The fixed escapement rule (FER) was maximized under the risk-neutral utility function. The FER produced the best performance in terms of average biomass yield across stocks, SPMs, and biomass estimates, but at the cost of frequent closures. The optimal FER could be used as a benchmark for average biomass yield. The risk-averse utility function and MSY HCRs produced less or similar average biomass yield compared with the optimal FER. This situation also was true across stocks and regime shifts. If managers want to avoid frequent closures, the risk-averse utility function could be an option. For example, for the base case scenario (high observation and process error) in anchovy, the continuous SPM under the risk-averse utility function had the best average biomass yield under the carrying capacity and productivity regime shifts. The resulting average biomass yield was not far from the optimal FER. For the autoregressive regime shift, the best performance was under the discrete SPM with harvest before production (dHPM). For stocks with lower productivity, e.g., hake and redfish, both the risk-neutral and risk-averse utility functions had an average biomass yield that was similar when the regime shift was in carrying capacity and productivity. For the risk-averse utility function and the autoregressive regime shift, the performance depended on the SPM. For hake, the average biomass yield was higher under the dHPM while for redfish this occurred under the continuous SPM. The risk-136 averse utility function may or may not produce a fixed harvest rate (FHR) strategy. When the stocks had low productivity, the FHR was supported more often. I compared optimal HCRs versus MSY HCRs under regime shifts and the continuous SPM. For the base case and across stocks, the time-invariant and time-varying optimal HCRs (both utility functions) had an average biomass yield similar to that for the MSY HCRs under the carrying capacity and productivity regime shifts. For anchovy, the time-varying Umsy HCR had the best performance under the productivity regime shift, but its performance did not differ much from that under the time-varying optimal HCRs (under both utility functions). For the autoregressive regime shift, across stocks, the time-invariant risk-neutral HCR and the time-varying MSY escapement HCR had the highest average biomass yield. In a recent review of theory and practice in recruitment modelling and robust HCRs (Maunder and Thorson 2019) the authors stated that âconstant harvest rate strategies may not be preferable under regime shiftsâ. As I showed in Chapter 3, the time-invariant and time-varying risk-averse HCRs performed well and their performance did not differ much from escapement HCRs. It was important to define the regime shift type. If only the carrying capacity changes, a fixed optimal harvest rate or a MSY harvest rate (e.g., Umsy) could perform well if the population growth parameter can be roughly determined (Walters and Parma 1996). Maunder and Thorson (2019) stated that âconstant escapement strategies are optimal when recruitment is stationary and uncorrelatedâ. First, a constant (or fixed) escapement strategy is optimal when a risk-neutral utility function is maximized (Reed 1979). I found this rule still can be optimal under regime shifts, including an autoregressive process. Second, I demonstrated that the FER with a risk-averse utility function can be supported under several conditions and regime shift types. For 137 example, a FER could be supported under a risk-averse utility function when the observation error was low. This situation occurred across regime shifts for productive stocks. The results from Chapter 3 showed that to establish optimal harvest strategies it was necessary to define how the fishery harvest occurs (e.g., discrete or continuous fishery), when the stock biomass was measured (i.e., after of before growth production), and the level of observation error and stock productivity. When the observation error was high and the stock had low productivity, I did not find much benefit in using complex biomass estimates. For the base case scenario, under a risk-neutral utility function, and a dHPM, I found that more complex biomass estimates increased the minimum based stock for anchovy. In this situation, the fishery closures occurred at larger stock sizes (i.e., under high process and observation errors). The SOPS showed that optimal HCRs that make direct use of trawl survey indices (e.g., data-based HCRs, in MSE terminology, Rademeyer et al. 2007) had performance in average yield similar to that for model-based HCRs (e.g., HCRs that depend on a statistical method to derive biomass estimates). This finding has implications for MSE configurations; e.g., it is not necessary to simulate complex stock assessments (as many MSEs currently do) as part of the management procedure (MP). By simulating model-based approaches, model assumptions and configurations may incorrectly interpret productivity changes as complex non-linear model parameter combinations. Instead, data-based HCRs could track more directly changes in biomass and productivity. Contrary to what has been reported about HCRs (e.g., Deroba and Bence 2008), the [optimal] fixed escapement HCR performed the best in terms of average yield under different regime shifts, but at the cost of high variability in catches. âA prioriâ or âad hocâ HCRs could generate results contrasting these ones (see Deroba and Bence 2008) because the HCRs are not 138 optimized to an objective function. As demonstrated in Chapter 3, HCR performance depended on model configurations and other sources of uncertainty. Even when an âa prioriâ HCR is similar across studies (e.g., a threshold HCR), the performance will be conditioned on the selection of the HCR parameters. Finally, Chapter 3 addressed some as yet unresolved issues with the identification of HCRs robust to regime shifts with the use of optimal control theory, i.e., optimal HCRs vs âa prioriâ HCRs. For example, Deroba and Bence (2008) mentioned âpolicy parameters unadjusted or adjusted for uncertaintyâ in their study. However, most of their literature review dealt with âa prioriâ HCRs. HCR parameters need to be optimized to deal with uncertainty appropriately. Furthermore, Deroba and Bence (2008) classified the studies according to âobjective functionsâ. Again, just a few studies in their literature review had an objective function. This dissertation suggests that the HCRs should be categorized according to whether they maximize an objective function (e.g., SDP, POMP, and SOPS) or do not (e.g., âa prioriâ HCR). Chapter 3 intended to clarify these issues. In Chapter 4, data-based HCRs were examined in an extreme situation of non-stationarity in stock-recruit parameters, using Atlantic redfish stocks as a case study. In these redfish stocks, the recruitment strength most often does not depend on the spawning biomass amount. Long-term trawl survey abundance indices showed that strong recruitment events can occur even at very low levels of spawning biomass (DFO 2018). In this situation, HCRs that depend on MSY reference points cannot be used. The recruitment in these species has a spasmodic pattern characterized by a very long series of low recruitments (e.g., >20 years) followed by a few strong recruitment events (Valentin et al. 2015). These few strong recruitment events have supported the redfish fishery for many years because redfish are long-lived (>40 years) (Valentin et al. 139 2015). Using an MSE approach, the results showed that the data-based HCR performed well in terms of conservation and fishery objectives under the âbustâ and âboomâ dynamics of the redfish stocks. Chapter 4 suggested that data-based HCRs could be applied to similar stocks and those with less extreme recruitment dynamics. This Chapter showed the usefulness of fishery independent surveys for tracking biomass changes. Chapter 4 complemented the analysis in Chapter 2. I showed that HCRs that directly used trawl survey indices performed well over a large number of different operating models, and were found to be highly robust to many different forms of uncertainty and that due to this it was not necessary to evaluate model-based HCRs. This dissertation demonstrated some highly significant benefits from the direct use in HCRs of results from research surveys that track changes in fish productivity over the more common addition of model complexity to model-based HCRs that attempt to simulate future stock assessments. It is likely that the poor performance of the HCRs examined under regime shifts (e.g., Aâmar et al. 2009a, 2009b; Szuwalski and Punt 2013a, 2013b) was due to the use of sophisticated model-based approaches. Regime shifts can be overlooked when complex model-based HCRs are used to track changes in productivity. Instead, a simpler approach, for example, that uses results from a well-designed survey could be more effective in responding to changes in stock biomass (and productivity). 5.2 Future research In Chapter 2, the Kalman filter estimated the variance components for productivity, carrying capacity, and for the observation equation. Estimating variances in state-space models is difficult to accomplish. Future analyses could fix some variance parameters in the Kalman filter to 140 evaluate if the estimation performance can be improved under the regime shifts examined. For example, for a productivity regime shift, the variance for the observation equation can be fixed to some true value, while allowing the variance for productivity to be estimated. In Chapter 2, the simulation model had a time-invariant selectivity when in fact it is common for selectivity-at-age to vary considerably over short time sequences (Bence et al. 1993; Butterworth et al. 2014; Ichinokawa et al. 2014; Maunder et al. 2014). Future analyses could test the effects of including time-varying selectivity on non-stationarity in stock-recruit parameter estimates. More research is needed to evaluate the robustness of HCRs to changes in productivity and carrying capacity. Recent studies show relatively little progress in this area. In Chapter 3, I used a simple surplus production model to test the robustness of HCRs. Chapter 3 could be extended to models with multiple dynamic states such as age-structured models. Furthermore, research could be extended to apply simpler population models that included a stock-recruitment relationship (e.g., delay-difference models). These models could explore the robustness of HCRs to other processes such as recruitment overfishing, growth overfishing, time-varying selectivity, and cohort strength that cannot be incorporated in surplus production models. In Chapter 3, the analyses used simulated data, but the analyses can use real data as well (Mohn 1999; Brooks and Legault 2015). This study found, like other studies, that as observation error increases the performance in terms of average yield decreases. Instead, a recent study using partially observed Markov decision process (POMDP) achieved a higher economic yield when the observation error increased (Memarzadeh and Boettiger 2019). Future simulations should investigate these discrepancies. Chapter 4 examined the application of data-based HCRs to spasmodic stocks. However, it could be interesting to evaluate similar data-based HCR under the regime shift scenarios 141 performed in Chapter 3. 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Investigación del status y posibilidades de explotación biológicamente sustentables anchoveta centro-sur, año 2014. Instituto de Fomento Pesquero (IFOP). 153 7 Appendices 7.1 Appendix A Chapter 2: Tables and Figures Table 7.1A. Equilibrium functions to calculate the time-varying MSY reference points. Description Equation (1) Estimated parameters Ë ( )tref MSYUï ï½ (2) Selectivity-at-age ?Ì?ð = {ð£ð(ÎÌð ðð)?Ì ?ð(ÎÌð£ðð) (3) Survivorship-at-age a ðð¥ð ={ 1â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð = 1ðð¥ðâ1ð ðâ1(1 â ?Ì?ðâ1ð¢ð)â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡1 < ð < ð´ðð¥ðâ1ð ðâ1(1 â ?Ì?ðâ1ð¢ð)1 â (1 â ?Ì?ðð¢ð)ð ðâ¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡â¡ð = ð´ (4) Equilibrium eggs per-recruit 1a Ae a a aalx m wï¦ï½ï½ï½ï¥ (5) Equilibrium recruitment {?Ì?ð¡ , ?Ì?ð¡} = {ÎÌð¾ð¹ âµ (ÎÌð ðð)ÎÌð¾ð¹ âµ (ÎÌð£ðð) ð ð¡ð = (?Ì?ð¡ + ln(ðð))/?Ì?ð¡ðð (6) Equilibrium yield per-recruit ðð¦ = ð¢ð âðð¥ðð=ð´ð=1?Ì?ðð¤ð (7) Equilibrium yield ( )e e yt e tY u R ï¦ï½ (8) Maximum Equilibrium yield 0 1( ) max ( )teet MSY t euMSY U Y uï¼ ï¼ï½ (9) Equilibrium spawning stock biomass at MSYU te MSYteMSY t eu UB R ï¦ï½ï½ 154 Table 7.2A. Median relative error (MRE) and median absolute relative error (MARE). ð ð¸ð¡ð,ð = (ð¸ð¡ð,ð â ðð¡ð,ððð¡ð,ð)100 ð´ð ð¸ð¡ð,ð = |ð¸ð¡ð,ð â ðð¡ð,ððð¡ð,ð| 100 ð ð¸ð¡ð,ð is the relative error for metric ð during time ð¡ for the trial ð, ðð¡ð,ð is the true value from the OM, and ð¸ð¡ð,ð is the estimated quantity ð during time ð for the trial ð. The calculations were similar for ð´ð ð¸ð¡ð,ð. The median of 100 trials was calculated over the years and then over simulations (MRE and MARE, respectively). 155 Table 7.3A. Median relative error (MRE) and median absolute relative error (MARE) for recruitment, spawning biomass, stock-recruit parameters, and MSY management reference points for jack mackerel, calculated from VPA and SCA model outputs under five regime shifts scenarios and two Kalman filter estimators (KFÏ0,0 and KFÏÎ±Î²). The results are for the uninformative catch history. ?Ì?ð ðºðºð©?Ì? ð¶?Ì? ð·?Ì? ð´ðºðð ð©ð´ðºðð ð¼ð´ðºðð Filters/Scenarios VPA SCA VPA SCA VPA SCA VPA SCA VPA SCA VPA SCA VPA SCA MRE (KFÏ0,0) Stationary-SRR -0.03 -0.7 1.12 0.64 -3.82 -6.46 -1.92 -3.09 -1.49 -3.4 0 0.71 -4.88 -4.76 Productivity -0.12 0.81 1.05 0.7 63.63 62.09 50.91 59.52 18.26 19.41 -19.19 -16.45 31.81 38.19 Density-dependence -0.29 1.01 1.15 1.13 -1.41 3.69 -1.06 0.91 -2.22 3.29 0.26 0.70 -2.02 3.10 Non-stationary-SRRâ 0.02 1.41 0.93 1.72 15.84 37 9.74 17.28 -3.96 8.07 -2.25 0.99 11.81 32.98 Non-stationary-SRR+ -0.19 -0.75 0.95 0.82 63.64 53.45 66.25 60.96 -5.65 -6.1 -25.57 -23.23 39.86 35.31 MRE (KFÏ0,0) Stationary-SRR - - - - -1.75 -6.4 -1.39 -2.84 -1.42 -3.29 1.21 0.71 -5.12 -4.76 Productivity - - - - 26.26 51.77 32.7 56.74 -4.31 6.84 -2.7 -12.24 10.09 24.73 Density-dependence - - - - -9.20 0.22 -6.24 -0.05 -1.32 5.18 3.99 2.80 -6.79 -0.83 Non-stationary-SRRâ - - - - 17.35 36.26 13.51 14.85 1.10 10.22 -3.11 -1.32 6.12 24.45 Non-stationary-SRR+ - - - - 25.69 9.63 35.9 28.68 -3.82 -1.84 -25.11 -5.85 37.18 16.71 MARE (KFÏÎ±Î²) Stationary-SRR 7.35 5.6 3.21 1.34 17.58 11.08 11.02 6.47 5.18 3.79 5.98 3.52 12.14 7.86 Productivity 7.37 5.53 3.3 1.57 64.02 62.09 50.91 59.52 21.28 20.33 21.79 18.48 36.25 38.38 Carrying-capacity 7.48 5.79 3.22 1.54 22.15 9.76 17.86 11.87 15.03 10.35 15.55 10.78 16.19 7.02 Non-stationary-SRRâ 7.46 6.34 3.3 1.93 32.21 39.03 21.74 21.17 22.95 24.52 12.92 13.13 22.72 33.34 Non-stationary-SRR+ 7.26 5.71 3.19 1.51 63.64 53.45 68.27 60.96 17.33 16.97 26.59 23.56 40.70 35.31 MARE (KFÏÎ±Î²) Stationary-SRR - - - - 20.36 11.44 13.39 6.85 4.28 3.78 5.96 3.68 10.95 8.45 Productivity - - - - 56.08 67.72 43.88 62.03 19.42 18.87 20.45 24.74 35.41 41.08 Density-dependence - - - - 23.69 17.38 18.38 15.35 16.71 12.98 15.12 11.87 19.05 10.83 Non-stationary-SRRâ - - - - 30.48 37.43 20.44 18.61 19.43 24.01 12.87 13.56 19.75 26.13 Non-stationary-SRR+ - - - - 54.85 38.73 74.48 55.33 18.85 19.06 26.21 25.08 37.18 26.81 156 Table 7.4A. Median relative error (MRE) and median absolute relative error (MARE) for recruitment, spawning biomass, stock-recruit parameters, and MSY management reference points for anchovy, calculated from VPA and SCA model outputs under five regime shifts scenarios and two Kalman filter estimators (KFÏ0,0 and KFÏÎ±Î²). The results are for the uninformative catch history. ?Ì?ð ðºðºð©?Ì? ð¶?Ì? ð·?Ì? ð´ðºðð ð©ð´ðºðð ð¼ð´ðºðð Filters/Scenarios VPA SCA VPA SCA VPA SCA VPA SCA VPA SCA VPA SCA VPA SCA MRE (KFÏ0,0) Stationary-SRR -0.36 0.38 0.66 0.04 -5.73 -6.30 -3.66 -3.84 -2.14 -2.03 0.47 1.29 -4.11 -3.38 Productivity -0.49 1.21 0.38 -0.41 30.11 77.78 38.48 52.34 2.68 43.85 -8.56 -6.35 23.30 53.21 Density-dependence -0.52 0.13 0.48 -0.63 -3.53 -0.50 -3.16 -0.41 -0.95 -1.14 1.07 0.01 -2.42 -0.81 non-stationary-SRRâ -0.34 7.96 0.31 2.44 -0.24 5.59 2.00 -11.93 -5.66 1.46 -0.20 11.14 -0.09 -5.39 non-stationary-SRR+ -0.48 1.78 0.37 0.8 31.46 53.61 76.52 60.38 -5.86 3.84 -14.56 -12.64 22.20 35.48 MRE (KFÏ0,0) Stationary-SRR - - - - -3.79 -2.02 -1.06 -0.22 -1.91 -1.46 0.15 -0.84 -2.82 -0.81 Productivity - - - - 54.46 75.02 44.98 57.90 11.48 42.25 -7.93 -6.83 32.67 49.54 Density-dependence - - - - -5.63 -3.65 -7.69 -7.37 0.82 2.44 3.15 2.54 -4.91 -3.78 Non-stationary-SRRâ - - - - -0.46 12.32 3.77 -3.06 -1.64 8.72 0.52 6.73 -2.59 -0.52 Non-stationary-SRR+ - - - - 12.52 43.14 31.03 48.20 -4.78 2.99 -11.97 -9.14 14.20 25.21 MARE (KFÏÎ±Î²) Stationary-SRR 8.52 7.14 10.19 4.29 12.38 9.88 11.37 8.24 5.50 3.61 5.73 4.18 7.73 5.23 Productivity 8.82 8.01 9.75 4.62 40.67 77.78 38.48 52.34 28.85 43.85 14.85 13.52 28.52 53.21 Density-dependence 8.91 7.35 9.74 4.67 15.70 12.48 18.12 14.26 11.71 10.11 14.18 12.16 7.41 6.84 Non-stationary-SRRâ 8.87 12.02 9.63 7.62 24.39 32.42 31.15 30.55 26.17 33.12 20.36 23.47 18.01 19.10 Non-stationary-SRR+ 8.51 7.56 9.97 4.46 33.14 53.61 76.52 60.38 21.71 20.75 19.75 18.18 28.74 35.48 MARE (KFÏÎ±Î²) Stationary-SRR - - - - 16.67 12.84 13.40 11.23 7.17 4.20 7.27 5.65 9.74 7.17 Productivity - - - - 59.09 75.02 45.04 57.90 34.68 42.25 16.26 14.61 37.74 49.54 Density-dependence - - - - 19.86 18.28 21.51 17.81 13.92 12.58 15.97 14.60 9.82 8.94 Non-stationary-SRRâ - - - - 19.82 27.10 22.77 24.47 23.97 25.75 18.19 20.71 14.93 16.33 Non-stationary-SRR+ - - - - 31.82 43.14 50.67 48.98 23.10 22.69 20.59 17.66 22.13 26.37 157 Table 7.5A. Median relative error (MRE) and median absolute relative error (MARE) for recruitment, spawning biomass, stock-recruit parameters, and MSY management reference points for hake, calculated from VPA and SCA model outputs under five regime shifts scenarios and two Kalman filter estimators (KFÏ0,0 and KFÏÎ±Î²). The results are for the uninformative catch history. ?Ì?ð ðºðºð©?Ì? ð¶?Ì? ð·?Ì? ð´ðºðð ð©ð´ðºðð ð¼ð´ðºðð Filters/Scenarios VPA SCA VPA SCA VPA SCA VPA SCA VPA SCA VPA SCA VPA SCA MRE (KFÏ0,0) Stationary-SRR -0.41 -0.1 0.64 0.97 -4.17 -5.74 -0.56 -2.87 -1.58 -3.49 -0.51 2.24 -5.2 -4.3 Productivity -0.24 1.43 0.59 1.36 19.32 22.01 24.96 15.81 7.73 9.85 -11.57 -6.21 10.35 10.17 Density-dependence -0.38 1.14 0.91 1.41 1.14 4.76 1.15 0.90 -0.98 2.81 -0.75 0.04 -1.92 2.71 Non-stationary-SRRâ -0.53 1.58 0.53 1.83 13.03 29.93 23.56 33.84 -12.37 3.64 -5.68 -9.92 10.82 19.88 Non-stationary-SRR+ -0.34 0.5 0.9 1.87 47.91 92.40 88.42 96.15 -5.06 -5.29 -33.16 -34.25 34.45 60.81 MRE (KFÏ0,0) Stationary-SRR - - - - -3.14 -5.29 -0.42 -2.36 -1.78 -3.43 0.48 1.72 -4.98 -4.19 Productivity - - - - 5.20 -1.67 -0.17 -6.66 -6.83 -4.03 3.46 5.31 -7.35 -3.92 Density-dependence - - - - -0.30 0.33 0.52 -1.91 -2.43 4.10 0.45 0.78 -2.49 -0.57 Non-stationary-SRRâ - - - - 18.84 38.82 18.38 21.96 5.26 24.75 -7.88 -5.62 9.32 22.29 Non-stationary-SRR+ - - - - 64.17 91.28 85.15 99.00 -8.18 -1.69 -31.53 -35.03 42.35 57.34 MARE (KFÏÎ±Î²) Stationary-SRR 7.41 5.54 5.02 3.2 11.91 7.78 8.51 5.68 5.30 4.38 6.56 4.50 9.05 5.32 Productivity 7.63 5.95 4.78 3.14 31.67 26.50 34.58 22.95 21.28 18.75 18.61 13.89 19.20 17.80 Density-dependence 7.56 5.78 4.74 3.03 14.68 11.40 15.93 13.78 13.94 10.90 14.96 12.59 11.09 7.24 Non-stationary-SRRâ 7.6 6.05 5.06 3.13 35.11 40.41 32.21 34.06 30.80 39.03 19.41 17.64 25.78 26.08 Non-stationary-SRR+ 7.72 5.74 5.06 3.29 49.63 92.40 88.42 96.15 22.83 21.85 33.63 34.25 35.58 60.81 MARE (KFÏÎ±Î²) Stationary-SRR - - - - 12.09 7.81 9.31 6.02 5.33 4.03 6.37 4.56 8.60 5.09 Productivity - - - - 41.89 29.48 31.20 26.78 23.10 20.43 17.10 16.72 26.18 22.40 Density-dependence - - - - 18.08 15.98 16.77 14.50 14.29 14.03 15.75 12.32 12.90 9.50 Non-stationary-SRRâ - - - - 29.72 49.91 26.38 27.03 25.82 35.38 17.09 16.56 21.30 26.01 Non-stationary-SRR+ - - - - 64.17 91.28 90.30 99.00 23.68 22.50 32.81 36.57 42.99 57.34 158 Figure 7.1A. Outputs from the SCA and VPA models for jack mackerel. Panel rows show recruitment, spawning biomass, and exploitation rates. Black lines indicate true trends. The median and 95% confident intervals for the estimates are the thick line and shaded regions, respectively. The scenario corresponds to non-informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 159 Figure 7.2A. Outputs from the SCA and VPA models for anchovy. Panel rows show recruitment, spawning biomass, and exploitation rates. Black lines indicate true trends. The median and 95% confident intervals for the estimates are the thick line and shaded regions, respectively. The scenario corresponds to non-informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 160 Figure 7.3A. Outputs from the SCA and VPA models for jack mackerel. Panel rows show recruitment, spawning biomass, and exploitation rates. Black lines indicate true trends. The median and 95% confident intervals for the estimates are the thick line and shaded regions, respectively. The scenario corresponds to an informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 161 Figure 7.4A. Outputs from the SCA and VPA models for anchovy. Panel rows show recruitment, spawning biomass, and exploitation rates. Black lines indicate true trends. The median and 95% confident intervals for the estimates are the thick line and shaded regions, respectively. The scenario corresponds to an informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 162 Figure 7.5A. Outputs from the SCA and VPA models for hake. Panel rows show recruitment, spawning biomass, and exploitation rates. Black lines indicate true trends. The median and 95% confident intervals for the estimates are the thick line and shaded regions, respectively. The scenario corresponds to an informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 163 Figure 7.6A. Stock-recruit parameter estimates from the Kalman filter obtained from the SCA and VPA models for jack mackerel; subscripts indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The median and 95% confident intervals are the thick line and shaded regions, respectively. Black lines indicate true trends. The scenario corresponds to an informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 164 Figure 7.7A. Stock-recruit parameter estimates from the Kalman filter obtained from the SCA and VPA models for anchovy; subscripts indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The median and 95% confident intervals are the thick line and shaded regions, respectively. Black lines indicate true trends. The scenario corresponds to an informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 165 Figure 7.8A. Stock-recruit parameter estimates from the Kalman filter obtained from the SCA and VPA models for hake; subscripts indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The median and 95% confident intervals are the thick line and shaded regions, respectively. Black lines indicate true trends. The scenario corresponds to an informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 166 Figure 7.9A. MSY reference points calculated from outputs from the Kalman filter, SCA, and VPA models for jack mackerel. Gray and blue-red lines indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The medians and 95% confident intervals are thick line and shaded regions, respectively. Black lines indicate true trends. The scenario corresponds to an informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 167 Figure 7.10A. MSY reference points calculated from outputs from the Kalman filter, SCA, and VPA models for anchovy. Gray and blue-red lines indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The medians and 95% confident intervals are thick line and shaded regions, respectively. Black lines indicate true trends. The scenario corresponds to an informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 168 Figure 7.11A. MSY reference points calculated from outputs from the Kalman filter, SCA, and VPA models for hake. Gray and blue-red lines indicate whether the filter estimated (ÏaÎ²) or not (Ï0,0) the correlation parameter, respectively. The medians and 95% confident intervals are thick line and shaded regions, respectively. Black lines indicate true trends. The scenario corresponds to an informative catch history. Columns show the regime shift (RS) scenarios (see Fig. 2.1 for a description). 169 7.2 Appendix B Chapter 3: Tables and Figures Table 7.1B. Kalman filter equations for the assessment model using a surplus production model. Description Equation (1) Estimated parameters ï¨ ï©2 2Ë Ë Ë Ë Ë, , ,R IK r ï³ ï³ï ï½ (2) Initial states and variance ( |0)( |0)0.011ttxPï½ï½ (3) Prediction phase for state variable ( | 1) 1 1 1 1 1 1( | 1) 1 1 1 1 1( ) ( )(1 ( ) / ) 1 3.2.2(1 ( ) / ) 1 3.2.3t t t t t t t tt t t t t t tx B Q r B Q B Q K t ifx B rB B Q K Q t ifï­ ï­ ï­ ï­ ï­ ï­ ï­ï­ ï­ ï­ ï­ ï­ ï­ï½ ï­ ï« ï­ ï­ ï­ ï½ï½ ï« ï­ ï­ ï­ ï½ (4) Prediction phase for variance 2 2( | 1) ( | 1) ( | )(1 2 / ) 1t t t t t t RP r x K P tï³ï­ ï­ï½ ï« ï­ ï« ï½ (5) Updated phase for the state variable (posterior biomass estimate) ï¨ ï©( | 1)( | ) ( | 1) ( | 1)2( | 1)t tt t t t I t t t I tt t IPx x q B x q BP ï³ï­ï­ ï­ï­ï¢ ï¢ï¢ï½ ï« ï­ ï½ï« (6) Updated phase for variance ( | 1)( | ) ( | 1) 2( | 1)1t tt t t tt t IPP PP ï³ï­ï­ï­ï¦ ï¶ï½ ï­ï§ ï·ï§ ï·ï«ï¨ ï¸ (7) innovation (error) ( | 1)t I t t te q B x ï­ï¢ï½ ï­ (8) Kalman gain ( | 1)2( | 1)t ttt t IPKP ï³ï­ï­ï½ï« (9) 2( | 1)kft t t IF P ï³ï­ï½ ï« (10) log likelihood 210.5 0.5 lnT Tfktkft tenll FFï½ï½ ï­ ï­ï¥ ï¥ 170 Table 7.2B. Kalman filter equations for the assessment model using a local model, Description Equation (1) Estimated parameters ï¨ ï©2 2Ë Ë Ë Ë, ,R IB ï³ ï³ï ï½ (2) Initial states and variance ( |0) ( | )2 2 2 2( |0)1 0.011 0.41t I t tt I RIx q BP qqï³ï½ ï½ ïï½ ï½ï½ (3) Prediction phase for state variable ( | 1) 1 1t t I tx q B tï­ ï­ï½ ï½ (4) Prediction phase for variance 2( | 1) 1 1t t t RP P tï³ï­ ï­ï½ ï« ï½ (5) Updated phase for the state variable (posterior biomass estimate) ï¨ ï©( | 1)( | ) ( | 1) ( | 1)2( | 1)t tt t t t I t t t I tt t IPx x q B x q BP ï³ï­ï­ ï­ï­ï¢ ï¢ï¢ï¢ï½ ï« ï­ ï½ï« (6) Updated phase for variance ( | 1)( | ) ( | 1) 2( | 1)1t tt t t tt t IPP PP ï³ï­ï­ï­ï¦ ï¶ï½ ï­ï§ ï·ï§ ï·ï«ï¨ ï¸ (7) innovation (error) ( | 1)t I t t te q B x ï­ï¢ï½ ï­ (8) Kalman gain ( | 1)2( | 1)t ttt t IPKP ï³ï­ï­ï½ï« (9) 2( | 1)kft t t IF P ï³ï­ï½ ï« (10) log likelihood 210.5 0.5 lnT Tfktkft tenll FFï½ï½ ï­ ï­ï¥ ï¥ 171 Figure 7.1B. Average closures for anchovy for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. The red line represents an arbitrary minimum threshold. 172 Figure 7.2B. Average overfished status for anchovy for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. Here, values below 0.8 (Bt/Bmsy) indicate overfished (bottom red line). 173 Figure 7.3B. Average overfishing status for anchovy for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. Here, values over 1.2 (Ut/Umsy) indicate overfishing (upper red line). 174 Figure 7.4B. Average variation in catch yield (standard deviation) for anchovy for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. The red line represents an arbitrary minimum threshold. 175 Figure 7.5B. Optimal harvest parameters, and B* (minimum based-stock, black) and Q* (slope, red), under a risk-neutral and risk-averse utility functions for hake. The operating models (OMs) included different regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. 176 Figure 7.6B. Performance in average yield for hake from optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. . 177 Figure 7.7B. Average closures for hake for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. The red line represents an arbitrary minimum threshold. 178 Figure 7.8B. Average overfished status for hake for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. Here, values below 0.8 (Bt/Bmsy) indicate overfished (bottom red line). 179 Figure 7.9B. Average overfishing status for hake for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. Here, values over 1.2 (Ut/Umsy) indicate overfishing (upper red line). 180 Figure 7.10B. Average variation in catch yield (standard deviation) for hake for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. The red line represents an arbitrary minimum threshold. . 181 Figure 7.11B. Optimal harvest parameters, and B* (minimum based-stock, black) and Q* (slope, red), under a risk-neutral and risk-averse utility functions for redfish. The operating models (OMs) included different regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. 182 Figure 7.12B. Performance in average yield for redfish from optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. 183 Figure 7.13B. Average closures for redfish for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. The red line represents an arbitrary minimum threshold. 184 Figure 7.14B. Average overfished status for redfish for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. Here, values below 0.8 (Bt/Bmsy) indicate overfished (bottom red line). 185 Figure 7.15B. Average overfishing status for redfish for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. Here, values over 1.2 (Ut/Umsy) indicate overfishing (upper red line). 186 Figure 7.16B. Average variation in catch yield (standard deviation) for redfish for optimal harvest strategies (risk-neutral and risk-averse utility functions) from different operating models (OMs) including regime shifts, surplus production models (dHP, dPH, and cPH models), and biomass estimates (DBM, EFK, and LM). The numbers in OMs indicate process error, autocorrelation in process error, and observation error, respectively. The red line represents an arbitrary minimum threshold. 187 Figure 7.17B. Performance in average closures for optimal harvest strategies (risk-neutral and risk-averse utility functions) and MSY harvest control rules for three stocks under different regime shifts, biomass estimates (DBM and LM) for the continuous surplus production model (cPHM). The red line represents an arbitrary minimum threshold. 188 Figure 7.18B. Average overfished status for optimal harvest strategies (risk-neutral and risk-averse utility functions) and MSY harvest control rules for three stocks under different regime shifts, biomass estimates (DBM and LM) for the continuous surplus production model (cPHM). Here, values below 0.8 (Bt/Bmsy) indicate overfished (bottom red line). 189 Figure 7.19B. Average overfishing status for optimal harvest strategies (risk-neutral and risk-averse utility functions) and MSY harvest control rules for three stocks under different regime shifts, biomass estimates (DBM and LM) for the continuous surplus production model (cPHM). Here, values over 1.2 (Ft/Fmsy) indicate overfishing (upper red line). 190 Figure 7.20B. Average variation in catch yield (standard deviation) for optimal harvest strategies (risk-neutral and risk-averse utility functions) and MSY harvest control rules for three stocks under different regime shifts, biomass estimates (DBM and LM) for the continuous surplus production model (cPHM). The red line represents an arbitrary minimum threshold. 191 7.3 Appendix C Chapter 4: Tables and Figures Table 7.1C. Description for the Bayesian statistical catch-at-length stock reduction analysis (SRA) model (OM1) for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). Note that we omitted subscript x because both species have the same sets of equations. Description Equation (1) Parameters estimated 1 1 2 22017 500 3 1952 1,2 1,250 504, , ,{ } ,{ , } ,{ , }Ë{ , , , }ret ret ret rett sda t t g g g g g gsd sdblk blk blk blk gR q a aa a a aï« ï· ï· ï´ï½ï½ ï½ ï½ ï½ï½ï¢ ï¢ï¢ï¦ ï¶ï§ ï·ï ï½ï§ ï·ï¨ ï¸ Population dynamics (2) Initial abundance 20.5( )0,0 3 3Rpr Rpraaa taR l e aNR l aï³ ï³ï· ï­ï¢ïï½ïï¬ï¯ï­ï¯ï® (3) Recruitment 22111,110.5( ) 0.5( ) 1951, 19531 1t Rpr Rprt R RtttttSSBe ifSSBNSSBe otherwiseSSBï· ï³ ï³ï· ï³ ï³ï¡ï¢ï¡ï¢ï­ï­ï­ï­ï¢ï¢ ï­ï¢ï¢ ï­ï«ï½ï«ï¬ï¯ï¯ï­ï¯ï¯ï® (4) Fraction retained for the fishery (g = 4) 501, 1,502 2 ,( ) /( ) /1 2,0 5; 71111 sdblk g blk gsdblk blk gret retret reta aa areta tif t blk a t blk aeef otherwiseï­ ï­ ïï­ ï­ ïï ï£ ï ï£ï½ï«ï« ï¬ï¯ï¯ï¯ï­ï¯ï¯ï¯ï® (5) Vulnerability for the fishery (g = 4) 50, 150,50, retretoffsett g toffsett g tt ga a t blkaa aïï½ï­ï­2 t blkïï¬ï¯ï­ï¯ï®, 1,, retretsdt gsdt gsdt ga t blkaaïï½2 t blkïï¬ï¯ï­ï¯ï® (6) Logistic vulnerability for fishery (g =4) 50, ,1 2, ,( ) /0 5; 71 4 1sdt g t ga t ga a aif t blk a t blk avgeï­ ï­ï ï£ ï ï£ï¬ï¯ï½ ï­ï½ï¯ï«ï® ( 1951; 0)initt uï½ ï½192 (7) Vulnerability for the survey (total biomass) 50( ) /,0 11 1sdg ga a aa gavotherwiseeï­ ï­ï½ï½ï«ï¬ï¯ï­ï¯ï® (8) Option for double logistic fishery (retained) 50 50, ,, ,( ) / ( ) /, ,, ,, ,41 11 1 1 max ( )sd sdls lst g t g g ga t ga a a a a aa t ga t ga a t ggve evvvï­ ï­ ï­ ï­ï½ï¦ ï¶ï¢ ï½ ï­ï§ ï·ï§ ï·ï« ï«ï¨ ï¸ï¢ï¢ ï½ï¢ (9) Option for double logistic vulnerability for survey 50 50,( ) / ( ) / ),,,1,21 11 1 1+ max ( )sd sdls lsg g g ga ga a a a a aa ga ga a ggve evvvï­ ï­ ï­ ï­ï½ï¦ ï¶ï¢ï¢ ï½ ï­ï§ ï·ï§ ï·ï« ï¨ ï¸ï¢ï¢ï¢ï¢ ï½ï¢ï¢ (10) Harvest rates lim limmaxlog (1 log ) /k kt t tu U C VBï½ ï« ï­ (11) Catch killed ratiokillk rett t tC C Dï½ (12) Vulnerable biomass killed , , , ,14 Akt g a t a t g aagVB N v wï½ï½ï½ï¥ (13) Predicted vulnerable biomass for the survey , , ,1 1, 2 At g a t a g aagVB N v wï½ï½ï½ï¥ (14) Predicted vulnerable biomass for the fishery index , ,10 3 At g a t aagVB N wï½ï½ï½ï¥ (15) Catch-at-age (killed) for the fishery , , , , , 4ka t g a t a t g tC N v u gï½ ï½ (16) Predicted vulnerable numbers-at-length for the survey , , , , 1, 2P l al t g a t a gN N v gï½ ï½( | )Φ (17) Predicted proportions-at-length for the survey , ,, ,, ,5 1,2l t gl t g Ll t gl cmNpNgï¾ï½ ï½ï¥ (18) Observed proportions-at-length for the survey _, ,, ,_, ,5 1,2survey obsl t gl t g Lsurvey obsl t gl cmNoNgï³ï½ ï½ï¥ (19) Predicted numbers-at-length (retained) for the fishery , , , , 4ret k ret P l al t g a t g a,tN C f gï½ ï½( | )Φ 193 (20) Predicted proportions-at-length (retained) for the fishery , ,, ,, ,18 4retl t gretl t g Lretl t glNpNgï³ï½ ï½ï¥ (21) Observed proportions-at-length (retained) for the fishery _, ,, ,_, ,18 4ret obsl t gretl t g Lret obsl t glNoNgï³ï½ ï½ï¥ Table 7.2C. Residuals and negative log-likelihood function (NLL) for fitting the Bayesian statistical catch-at-length stock reduction analysis (SRA) model for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). Note that we omitted subscript x because both species have the same sets of equations. Description Equation (1) Conditional likelihood estimates , , ,,ln2 2,1,2,3ln( ) ln( )ln 1/1( 2)gt g t g t gg g t gt gqgg t gt gggI VBq nq enï¨ï¨ï´ ï¨ïïï½ï½ ï­ï½ï½ï½ï­ï¥ï¥ Priors (2) 224( )2( )hhhh hPï«ï«ï³ï½ï«ï­ï½ (3) 22( ) 1, 22( )g gqg qgq qP gï³ï­ï½ ï½ï¥ (4) 232( )2( )aRprPï·ï· ï·ï³ï½ï¢ï¢ ï¢ï­ï½ (5) 21951 1951221953 19532221952,1953:2017ln( ) 2( )ln( ) 2( )ln( ) 2( )RprRprtt RprPï·ï· ï·ï³ï· ï·ï³ï· ï·ï³ï¢ï¢ïï½ï¬ï¢ï¢ ï¢ï¢ï­ï¯ï¯ï¯ï¢ï¢ ï¢ï¢ï­ï¯ï­ï¯ï¯ ï¢ï¢ ï¢ï¢ï­ï¯ï¯ï®ï¥ 194 (6) Likelihood components 2,21,2, , , ,1,2, , , ,4( ) 2( )( ln )( ln ) t gg t gg l t g l t gg t lret retg l t g l t gg t lLCVL ess o pL ess o pï¨ï¨ïï¢ïï¢ï¢ïï½ï½ ï­ï½ ï­ï¥ï¥ï¥ï¥ï¥ï¥ï¥ï¥ (7) NLL ( ) h qnll L L L P P P Pï¨ ï· ï·ï¢ ï¢ï¢ ï¢ ï¢ï¢ï ï½ ï« ï« ï« ï« ï« ï« Table 7.3C. Priors used for the Bayesian statistical catch-at-length stock reduction analysis (SRA) model for redfish Sebastes mentella (Sm) and S. fasciatus (Sf). The priors for the historical recruitment deviates that could not be estimated in the SRA derived from an empirical meta-analysis of stock-recruit data across Sebastes stocks for large recruitment deviates (DFO 2018b). Symbol Value (Sm;Sf) Description ï·ï¢ï¢ 2.1; 2.1 Mean prior for recruitment deviates ï·ï¢ 2.1; 2.1 Mean prior for deviates in initial abundance Rprï³ 1.0; 1.0 Standard deviation of the prior for recruitment deviates and initial abundance 1,2gq ï½ 0.2; 0.2 Mean prior for catchability for survey index g 1,2gqï³ï½ 1.0; 1.0 Standard deviation of the prior for catchability for survey index g h 0.67; 0.67 Mean prior for steepness (Forest et al. 2010) hï³ 0.17; 0.17 Standard deviation of the prior for steepness (Forest et al. 2010) 195 Table 7.4C. Capped management procedures (MPs) tested in the management strategy evaluation (MSE). MPs marked with an asterisk (*) were chosen to be forwarded for peer review in spring 2018. The MPs have a starting and ending year associated with minimum and maximum caps in kt. MPs may or not have additional rules, e.g., status quo, small fish protocol (SFP), and catch limits (CL) constraints. See text for details. The status quo assumes approx. 3kt per year. The p indicates the original number assigned to the MPs. p Cap start Cap end Min. cap Max. cap Status quo/ SFP/CL constraints Notes on HCR 1* 2020 2027 14.5 kt 40 kt yes/no/no 4 2018 2025 14.5 kt 40 kt no/no/no 6 2018 2029 17 kt 60 kt no/no/no 8 2020 2047 14.5 kt 120 kt yes/no/no 14* 2020 2027 14.5 kt 40 kt yes/no/yes annual change of 15% (after 2028) 16 2018 2035 14.5 kt 80 kt no/no/no 17 2020 2037 14.5 kt 80 kt yes/no/no 18 2018 2029 17 kt 60 kt no/yes/yes maximum increase or decrease in CL of 5kt. 19 2018 2020 20 kt 60 kt no/no/no 22 2020 2031 17 kt 60 kt yes/yes/yes maximum increase or decrease in CL of 5kt. 24 2022 2029 14.5 kt 40 kt yes/no/no 34 2022 2029 14.5 kt 40 kt yes/no/yes annual change of 15% (after 2028) 40 2020 2027 14.5 kt 40 kt yes/no/no Minimum cap of 2.5 kt. Table 7.5C. Uncapped management procedures (MPs) tested in the management strategy evaluation (MSE). MPs marked with an asterisk (*) were chosen to be forwarded for peer review in spring 2018. The HCR start indicates the year when the multiplier for the slope of the HCR was applied. Before that year, the MP assumes status quo/pre-specified catches in kt. The status quo assumes approx. 3kt per year. The p indicates the original number assigned to the MPs. p HCR Start Status quo/ pre-specified CL Multipler slope Notes on HCR 12 2020 yes/no 80% 20 2020 yes/no 100% 25 2018 no/yes 100% Set CLs at 5, 5, 10, and 10 kt for 2018-2021 26 2022 yes/no 100% 27 2022 no/yes 100% Set CLs at 10.5 kt for 2018-2021(i.e., current unit 1+2 CL). 29 2022 yes/no 80% 43* 2018 no/yes 80% Set CLs at 7.5, 10, 15, and 20 kt for 2018-2021 44* 2018 no/yes 80% Set CLs at 5 kt for 2018-2019 45* 2018 no/yes 100% Set CLs at 5 kt for 2018-2021 196 Table 7.6C. Parameter estimates for the Core and Stress-test operating models (OMs) fitted to S. mentella. Estimates for models 3, 4, 5, and 9 are identical to OM1 because the modifications were in the projected population dynamics. Unfished spawning biomass (SSB0) in kt, h (Steepness), maximum sustainable yield (MSY), harvest rate at MSY (Umsy), spawning biomass depletion in 2017 (D2017), number of estimated parameters in the OM (#p), negative log-likelihood (NLL), and Akaike information criterion (AIC). A full description for the OMs is given in Table 7.1C. OM Type Description SSB0 h MSY Umsy D2017 #p NLL AIC 1 Core Base case 1509 0.44 25 0.041 0.265 82 3621.8 7407.7 8 Core M is age-specific 1509 0.34 20 0.031 0.307 82 3681.0 7526.0 10 Core More S. mentella in catch split 1714 0.45 29 0.040 0.272 82 3629.0 7422.1 11 Core Less S. mentella in catch split 1287 0.43 21 0.041 0.260 82 3622.3 7408.6 2 Stress Domed fishery selectivity 1488 0.46 29 0.081 0.158 82 3525.3 7214.6 13 Stress Lower offset vector 1500 0.43 25 0.042 0.265 82 3619.7 7403.5 13 Stress Higher offset vector 1519 0.44 25 0.041 0.266 82 3623.7 7411.3 14 Stress Lower catch killed: retained ratio 910 0.45 16 0.043 0.312 82 3619.8 7403.7 14 Stress Higher catch killed: retained ratio 2110 0.43 35 0.041 0.247 82 3623.7 7411.4 15 Stress Low M 1478 0.57 26 0.048 0.244 82 3618.7 7401.5 16 Stress High M 1612 0.33 20 0.030 0.280 82 3628.5 7420.9 17 Stress High steepness 1374 0.54 29 0.058 0.296 82 3622.1 7408.4 18 Stress Low steepness 1623 0.33 18 0.025 0.223 82 3621.9 7407.7 19 Stress Low prior strong cohorts 1582 0.43 26 0.041 0.253 82 3621.8 7407.7 20 Stress High prior strong cohorts 1412 0.44 24 0.042 0.284 82 3621.8 7407.7 197 Table 7.7C. Parameter estimates for the Core and Stress-test operating models (OMs) fitted to S. fasciatus. Estimates for models 3, 4, 5, and 9 are identical to OM1 because the modifications were in the projected population dynamics. Unfished spawning biomass (SSB0) in kt, h (Steepness), maximum sustainable yield (MSY), harvest rate at MSY (Umsy), spawning biomass depletion in 2017 (D2017), number of estimated parameters in the OM (#p), negative log-likelihood (NLL), and Akaike information criterion (AIC). A full description for the OMs is given in Table 7.1C. OM Type Description SSB0 h MSY Umsy D2017 #p NLL AIC 1 Core Base Case 1116 0.62 34 0.094 0.153 82 3483.7 7131.3 8 Core M is age-specific 1140 0.47 30 0.079 0.143 82 3464.8 7093.5 10 Core More S. mentella in catch split 958 0.62 29 0.094 0.150 82 3488.2 7140.4 11 Core Less S. mentella in catch split 1380 0.60 40 0.091 0.152 82 3495.4 7254.9 2 Stress Domed fishery selectivity 1488 0.46 29 0.081 0.158 82 3525.3 7214.6 13 Stress Lower offset 1107 0.62 34 0.096 0.155 82 3484.2 7132.3 13 Stress Higher offset 1120 0.62 33 0.093 0.152 82 3483.4 7130.8 14 Stress Lower catch killed: retained ratio 701 0.63 22 0.096 0.176 82 3481.0 7126.1 14 Stress Higher catch killed: retained ratio 1533 0.62 46 0.093 0.144 82 3486.1 7136.1 15 Stress Low M 1183 0.73 32 0.090 0.140 82 3492.4 7148.8 16 Stress High M 1078 0.53 34 0.096 0.116 82 3478.4 7120.7 17 Stress High steepness 807 0.78 29 0.129 0.218 82 3483.8 7131.5 18 Stress Low steepness 1323 0.47 30 0.062 0.126 82 3483.6 7131.2 19 Stress Low prior mean for strong cohorts 1192 0.61 36 0.092 0.144 82 3483.7 7131.5 20 Stress High prior mean for strong cohorts 1009 0.63 31 0.097 0.169 82 3483.5 7131.1 198 Figure 7.1C. Vulnerability at age for the fishery and maturity at age calculated for Sebastes mentella (upper panel) and S. fasciatus (lower panel) from the base case operating model. The two time-blocks for the fishery were calculated from the estimated fraction retained at age and an offset vector (see Fig. 7.2C; Table 7.1.4C-7.1.5C). Note that a minimum age of vulnerability and maturity were applied to the proportion at age (see Table 4.4). These values were based on literature (e.g., maturity) and empirical observations from the retained catch at length data. 199 Figure 7.2C. Offset input vector (red line) used to calculate the vulnerability at age (green line) based on the estimated fraction retained at age (blue line) for S. mentella (upper panel) and S. fasciatus (lower panel) from the base case operating model (Tables 7.1.4C-7.1.5C). Two time-blocks were applied, the first up to 1993 and the 2nd following 1993. Note that both species had the same offset vector. 200 Figure 7.3C. Input vector for the catch biomass killed to catch biomass retained ratio for S. mentella and S. fasciatus from the base case operating model (see Table 7.1.11C). The values were formulated by an interview-based study on historic discarding practices (see Duplisea 2016 for details). 201 Figure 7.4C. Observed catch at length (bars) for S. mentella from the DFO survey in Unit 1 and fit (red line) based on the base case operating model (OM1). S. fasciatus cohorts belonging to the Grand Banks (Valentin et al. 2015) were removed from the composition data. The method employed was relatively simple. Essentially the abundance in length classes between 7 and 17 cm (inclusive) were set equal to the mean abundance in those classes for years adjacent to the arrival and departure of the cohorts. For example, the 2003 year class is present from about 2005-2008 and between lengths of about 7 and 17 cm. Therefore, the abundance for 7 cm individuals in all years from 2005 to 2008 was set equal to the mean abundance of 7 cm individuals in 2004 and 2009. 202 Figure 7.5C. Observed catch at length (bars) for S. mentella from the Groundfish Enterprise Allocation Council (GEAC) survey in Unit 2 and fit (red line) based on the base case operating model (OM1). The survey was designed for the DFO. 203 Figure 7.6C. Observed catch at length retained (bars) for S. mentella from in Unit 1+2 and fit (red line) based on the base case operating model (OM1). Length compositions for 2017 were not included in the model because records were incomplete at the time of this study. 204 Figure 7.7C. Observed catch at length (bars) for S. fasciatus from DFO survey in Unit 1 and fit (red line) based on the base case operating model (OM1). S. fasciatus cohorts belonging to the Grand Banks (Valentin et al. 2015) were removed from the composition data (see Fig. 7.4C for a description). 205 Figure 7.8C. Observed catch at length (bars) for S. fasciatus from the GEAC survey in Unit 2 and fit (red line) based on the base case operating model (OM1). The survey was designed for DFO. 206 Figure 7.9C. Observed catch at length retained (bars) for S. fasciatus from in Unit 1+2 and fit (red line) based on the base case operating model (OM1). Length compositions for 2017 were not included in the model because records were incomplete at the time of this study. 207 Figure 7.10C. Recruitment deviates (upper panel) and standardized residuals for fit of the survey biomass indices in Unit 1 (blue dot) and 2 (red dot) for S. mentella (lower panel) based on the base case operating model (OM1). Note strong cohorts were defined as recruitment deviates greater than 2.5 in logarithm space. 208 Figure 7.11C. Recruitment deviates (upper panel) and standardized residuals for fit of the survey biomass indices in Unit 1 (blue dot) and 2 (red dot) for S. fasciatus (lower panel) based on the base case operating model (OM1). Note strong cohorts were defined as recruitment deviates greater than 2.5 in logarithm space. 209 Figure 7.12C. Retrospective analysis for the time series of spawning stock biomass, recruitments, and harvest rates for S. mentella (upper panel) and S. fasciatus (lower panel), based on the base case operating model (OM1). 210 211 Figure 7.13C. Average proportion of the simulations where the ratio of the harvest rate (Ut) to the harvest rate at MSY (Umsy) is less than 1 for S. mentella and S. fasciatus for the 40 years of projections and the 21 management procedures (MPs) under the core (upper panel) and stress operating models (lower panel)(see Table 4.6). Dots show medians. The label on the x-axis indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR (see Tables 7.4C-7.5C). The horizontal line indicates the pass criteria. 212 213 Figure 7.14C. Average number of years where fish < 22cm represent <15% of the catch over the 40 years of projections for the 21 management procedures (MPs) under the core (upper panel) and stress operating models (lower panel)(see Table 4.6). Dots show medians and bars shown the 90% intervals. The label on the x-axis indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR (see Tables 7.4C-7.5C). The horizontal line indicates the pass criteria 214 Figure 7.15C. Average catch retained (in kt) for the 10-20 and 10-40 year horizon for the 21 management procedures (MPs) under the stress operating models (see Table 4.6). Dots show medians and bars shown the 90% intervals. The label on the x-axis indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR (see Tables 7.4C-7.5C). 215 216 Figure 7.16C. Average proportion of the simulations where the catch limit (CL) exceeded 40kt for the 2028-2057 year horizon for the 21 management procedures (MPs) under the core (upper panel) and stress operating models (lower panel) (see Table 4.6). Dots show medians. The label on the x-axis indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR (see Tables 7.4C-7.5C). 217 Figure 7.17C. Average number of years where the catch limit (CL) exceeded 40kt for the 2028-2057 year horizon for the 21 management procedures (MPs) under stress operating models (see Table 4.6). Dots show medians and bars shown the 90% intervals. The label on the x-axis indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR (see Tables 7.4C-7.5C). 218 Figure 7.18C. Trade off plot between the average catch retained (in kt) for the 2028-2057 year horizon where the catch limit (CL) exceeded 40kt for the 2028-2057 for the 21 management procedures (MPs) under the core (see Table 4.6). Dots show medians. The label on panels indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR (see Tables 7.4C-7.5C). 219 Figure 7.19C. Trade off plot between the average catch retained (in kt) for the 2028-2057 year horizon where the catch limit (CL) exceeded 40kt for the 2028-2057 for the 21 management procedures (MPs) under the stress test operating models (see Table 4.6). Dots show medians. The label on panels indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR (see Tables 7.4C-7.5C). 220 221 Figure 7.20C. Average proportion of years where the catch limit (CL) change less than 15% between years for the 40 year of projection for the 21 management procedures (MPs) under core (upper panel) and stress operating models (see Table 4.6). Dots show medians and bars shown the 90% intervals. The label on the panels indicates: the assigned number for the MP, capped (C) or uncapped MPs, and the starting year for the HCR (see Tables 7.4C-7.5C
Thesis/Dissertation
eng
Oceans and Fisheries
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Fisheries harvest management procedures under time-varying productivity
Text