Impacts of time-varying productivity on estimated stock-recruitment parameters and biological reference points

11 Models with time-varying parameters are increasingly being considered in the assessment of fish stocks, 12 but their reliability when used to derive biological reference points or benchmarks has not been 13 thoroughly evaluated. Here, we evaluated stock-recruitment models with and without time-varying 14 productivity in a simulation framework for sockeye salmon ( Oncorhynchus nerka ) under different 15 scenarios of productivity and exploitation. Ignoring trends in productivity led to overestimates of 16 productivity and underestimates of capacity when both exploitation rates and productivity declined over 17 time, resulting in an underestimation on average of benchmarks of biological status. Despite being less 18 biased, time-varying models had relatively poor fit based on AICc and BIC model selection criteria. Our 19 simulation results were compared with empirical analyses of 12 Fraser River sockeye salmon stocks in 20 British Columbia, Canada. Although benchmarks were less biased based on time-varying models, 21 underlying true benchmarks based on spawner abundances at maximum sustainable yield, S MSY , trend 22 downwards when productivity declines, which may not be aligned with conservation objectives. We 23 conclude with best practices when adapting biological benchmarks to time-varying productivity. to indicate a persistent shift.

117 We then calculated two biological benchmarks from the estimated stock-recruitment parameters, S MSY 118 and S gen . S MSY is the spawner abundance required to achieve maximum sustainable yield,

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(4) , where W is the Lambert W function (Scheuerell 2016 , where the scalar is set to 0.1 so that the standard deviation in h t ' (0.1 0.06) is approximately • = D r a f t 9 exploitation rates to 5 and 85%, respectively. Following the implementation of this exploitation rate, 160 spawner abundances in the subsequent generation, t+1, were calculated as: .
Model parameterization

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To parameterise the operating model of the simulation framework, we selected productivity ( ) and to compare model fit against observed data without knowledge of true parameters (Peterman and true productivity for both standard and time-varying models. When productivity varies over time, the 206 recent performance metrics may be most useful for informing future application of these models. We 207 therefore evaluated the bias (percent error, PE) in productivity of the last generation, or year (t=60) 208 against the true productivity of that year for both models. 233 Data

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To illustrate the relevance of the simulation results for real-world applications, we applied the two D r a f t 13 (high precision fence counts to lower precision visual surveys).

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When we fit stock-recruitment models to simulated spawner and recruitment data for each Monte Carlo 251 trial, the resulting parameter estimates varied due to the distribution of true underlying data. Figure 1 252 illustrates the impact of data at high spawner abundance on the models' abilities to account for density 253 dependence and produce reliable estimate of S MAX , for two Monte Carlo trials under two different 254 scenarios. Under the first scenario (Fig. 1a), the simulated stock-recruit data contain a limited number of 255 years with high spawner abundance above S MAX , but a relatively high portion of these years resulted in 256 recruitment above R MAX (75%). When fitting a standard Ricker model to these data (Fig 1a, red  D r a f t presented in Figure 1. Under the scenario of declining productivity and constant high exploitation, the 315 overestimation of S MAX (Fig. 1a, Fig. 2c) by the standard model caused both benchmarks to be 316 overestimated in the most recent year relative to the true benchmark (Fig. 4, (Fig. 1b, Fig. 2c) and overestimated productivity on average over the time series (Fig. 1b, Fig. 2b

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Although it is not possible to identify true underlying parameters for Fraser River sockeye stocks, we can 344 compare the relative difference in parameter estimates between standard and time-varying models for 345 the simulation and empirical results to identify potential risks of estimation biases in benchmarks given 346 known biases in the simulation results (Fig. 7). We found that for the 7 stocks where estimated 347 productivity and exploitation rates had declined, the relative difference between parameters of the 348 time-varying versus the standard model for the empirical data were the same as for simulated data 349 when evaluated over the entire time series (Fig. 7) and for the most recent year (Supp. Mat 1, Fig. S5). If 350 the underlying dynamics causing differences in parameter estimates are similar between the simulated 351 and the empirical data, then this convergence highlights possible risk of estimation biases in empirical 352 benchmarks derived from the standard model. In particular, for these stocks, estimates of productivity 353 from the standard model were higher than those for the time-varying model averaged over the time 354 series, and benchmarks were lower (Fig. 7, middle column), possibly due to negative estimation biases 355 as found in the simulation results ( Fig. 6c and d).  Table S1). For stocks where productivity was relatively constant or increasing, the empirical deviations in 359 parameters between model forms also tended to converge with those from the simulation (Fig. 7).

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 Demonstrate that the change in the state of the productivity will persist long enough in 484 comparison to the management plan, so that changing benchmarks will indeed respond to 485 management needs. This step requires an understanding of the mechanism driving those D r a f t 24 harvest control rules) and uncertainty in future trends in productivity (as shown for Pacific of evidence for temporal trends in productivity and/or changes in capacity with a transparent 499 understanding of the impacts of those changes on the ability to achieve management objectives. In the 500 absence of information on plausible drivers, identifying benchmarks or reference points that are robust 501 to plausible changes in productivity is favoured over the application of benchmarks that vary annually.