Using data-limited approaches to assess data-rich Indian Ocean bigeye tuna: Data quantity evaluation and critical information for management implications

Using data-limited approaches to assess data-rich Indian Ocean bigeye tuna: Data quantity evaluation and critical information for management implications

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Ya’nan Li¹^,², Jiangfeng Zhu¹^,³^,^*, Xiaojie Dai¹^,³, Dan Fu⁴, Yong Chen²

Acta Oceanologica Sinica | 2022, 41(3) : 11 - 23

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Acta Oceanologica Sinica | 2022, 41(3): 11-23

• Marine Biology •

Using data-limited approaches to assess data-rich Indian Ocean bigeye tuna: Data quantity evaluation and critical information for management implications

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Ya’nan Li¹^,², Jiangfeng Zhu¹^,³^,^*, Xiaojie Dai¹^,³, Dan Fu⁴, Yong Chen²

Affiliations

¹ College of Marine Sciences, Shanghai Ocean University, Shanghai 201306, China

² School of Marine Sciences, University of Maine, Orono, ME 04469, USA

³ Key Laboratory of Oceanic Fisheries Exploration, Ministry of Agriculture and Rural Affairs, Shanghai 201306, China

⁴ Indian Ocean Tuna Commission, Victoria 361, Seychelles

Published: 2022-03-25 doi: 10.1007/s13131-021-1933-9

Outline

Abstract

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The majority of fishery stocks in the world are data limited, which limits formal stock assessments. Identifying the impacts of input data on stock assessment is critical for improving stock assessment and developing precautionary management strategies. We compare catch advice obtained from applications of various data-limited methods (DLMs) with forecasted catch advice from existing data-rich stock assessment models for the Indian Ocean bigeye tuna (Thunnus obesus). Our goal was to evaluate the consistency of catch advice derived from data-rich methods and data-limited approaches when only a subset of data is available. The Stock Synthesis (SS) results were treated as benchmarks for comparison because they reflect the most comprehensive and best possible scientific information of the stock. This study indicated that although the DLMs examined appeared robust for the Indian Ocean bigeye tuna, the implied catch advice differed between data-limited approaches and the current assessment, due to different data inputs and model assumptions. Most DLMs tended to provide more optimistic catch advice compared with the SS, which was mostly influenced by historical catches, current abundance and depletion estimates, and natural mortality, but was less sensitive to life-history parameters (particularly those related to growth). This study highlights the utility of DLMs and their implications on catch advice for the management of tuna stocks.

Key words

stock assessment / bigeye tuna / data-limited / fisheries management / Indian Ocean

Cite this Article

Ya’nan Li, Jiangfeng Zhu, Xiaojie Dai, Dan Fu, Yong Chen. Using data-limited approaches to assess data-rich Indian Ocean bigeye tuna: Data quantity evaluation and critical information for management implications[J]. Acta Oceanologica Sinica, 2022 , 41 (3) : 11 -23 . DOI: 10.1007/s13131-021-1933-9

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1 Introduction

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Fisheries stock assessment provides critical information necessary for the conservation and management of fish stocks. Stock assessment models estimate fish stock parameters, determine stock status, and provide management advice on optimal fishing levels (Hilborn and Walters, 1992). The evolvement of stock assessment methods and the advancement of computing power enabled sophisticated stock assessment models to be built that make use of multiple datasets to inform a wide range of population and fishing processes. Both the richness of data and the complexity of assessment models have increased overtime (Maunder and Punt, 2013). Assessment models range from very simple models that utilize only a single data source (e.g., catch-only method) to highly integrated analysis, which is capable of simultaneously analyzing a large number of data inputs including environmental and ecosystem drivers.

Integrated analysis methods have become the preferred approach for conducting stock assessments since the publication of a seminal paper by Fournier and Archibald in 1982 (Fournier and Archibald, 1982; Fournier et al., 1998; Bull et al., 2012; Methot and Wetzel, 2013; Doonan et al., 2016; Punt et al., 2020). Integrated analyses, such as Stock Synthesis (SS) (Methot and Wetzel, 2013), are commonly employed because they are able to integrate multiple data sources, simultaneously model various processes, and are flexible in terms of model configuration (Cope, 2013; Methot and Wetzel, 2013). Integrated models are based on a coherent mathematical and statistical framework, which governs the population and fishing processes, and links the system dynamics to observational data (Maunder and Piner, 2017). Integrated analysis typically requires more data in order to support the modelling of population dynamics at a finer scale. However, for many stocks, data collected from different sources may have conflicting signals, often due to inadequate sampling processes, resulting in poor model fits. In some instances, conflicts among data sets can be caused by model misspecification as some population processes are not well understood (Maunder et al., 2017; Sagarese et al., 2019). Data conflict can introduce significant bias and uncertainty to the estimates of essential parameters and derived quantities which are difficult to quantify, and potentially result in inadequate management recommendations (Maunder et al., 2017; Griffiths and Fay, 2015; Van Beveren et al., 2017; Zhu and Kitakado, 2019).

Comparing different modelling approaches helps us better understand population dynamics, allows us to evaluate the influence of crucial data inputs that on the assessment, and to identify appropriate data-limited approaches for coping with data limitations (Arnold and Heppell, 2015; Sagarese et al., 2019; Zhu and Kitakado, 2019). As data-limited methods (DLMs) was often used as interim solutions to allow time for data collection (e.g., Berkson and Thorson, 2015; Newman et al., 2015), understanding the impact of data quantity on stock assessment is important for improving stock assessment and developing precautionary management strategies (Cummings et al., 2014; Sagarese et al., 2019).

Tuna are among the world’s most commercially valuable species, and are exploited by fleets from more than 70 countries. The most important species for commercial and recreational tuna fisheries are yellowfin (Thunnus albacares), bigeye (Thunnus obesus), bluefin (Thunnus thynnus), albacore (Thunnus alalunga), and skipjack (Katsuwonus pelamis) (ISSF, 2018). Assessing and managing these highly migratory species has been the focus of regional tuna fisheries management organizations (tRFMO). Integrated models such as MULTIFAN-CL and SS are commonly used by most tRFMOs to assess tuna stock status and provide management advice (ISSF, 2018). These integrated models are known to be able to capture well a range of uncertainty, including input uncertainty (uncertainty about input data or the quality of the information), statistical uncertainty (parameter estimation), and structural uncertainty (uncertainty associated model configurations or assumptions) (ISSF, 2018). One, or a combination of these uncertainties, is usually considered when determining stock status for providing management advice.

The Indian Ocean bigeye tuna (BET) is a large epi- and mesopelagic species distributed in the tropical and sub-tropical waters of the Indian Ocean. BET is a high-value species caught in large volumes by industrial fleets, subject to intense data collection. Thus, there is relatively more information collected on this species that allows the undertaking of fully quantitative stock assessments. Indian Ocean BET has been subject to stock assessment using SS3 (Fu, 2019), on the weight-of-evidence available in 2019, the BET stock is determined to be not overfished but subject to overfishing (IOTC Secretariat, 2020). The assessment has particularly highlighted the input uncertainty with respect to data quality and quantity (Fu, 2019). The research effort to evaluate and reduce the input uncertainty for improving management advice has been recommended by the IOTC Scientific Committee (ISSF, 2018).

In this study, we applied the DLMs to the Indian Ocean BET stock, and quantitatively compare the input data sets to identify their impacts on the stock assessment and the formulation of management strategies. Incorporating multiple sources of input uncertainty in a stock assessment can better account for the risks associated with proposed management options and promote decisions that are more robust to such uncertainties. The results are also relevant to many other commercial target and bycatch species under the IOTC mandate (e.g., neritic tuna, billfish, and shark), with most of these species lacking sufficient biological or exploitation information to produce a defensible quantitative stock assessment, as their data collection and reporting mechanisms are limited to the artisanal and semi-industrial fleets. Thus, another objective of this study is to evaluate if it is possible to use DLMs to provide fisheries management advice for data-limited stocks.

2 Materials and methods

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2.1 Data-rich model: SS

SS (version v.3.30.15; Methot et al., 2020) is an age- and size-structured assessment model in the class of models termed integrated analysis models. The SS model has a population sub-model that simulates a stock’s growth, maturity, fecundity, recruitment, movement, and mortality processes, an observation sub-model estimates expected values for various observed data, a statistical sub-model characterizes the data’s goodness of fit and obtains best-fitting parameters with associated variance, and forecast sub-model projects need management quantities (Methot, 2009; Cope, 2013; Methot et al., 2020). The SS model outputs the quantities with confidence intervals required to implement risk-averse fishery control rules. SS has been applied in a wide variety of fishery assessments globally (Methot et al., 2020). The latest stock assessment for Indian Ocean BET was conducted using SS3 in 2019 (Fu, 2019). The SS3 assessment implements an age- and spatially structured model that reflected the population and fishery dynamics of the species. The assessment model covers the period 1975−2018 with the inclusion of composite longline CPUE indices, length compositions, and tag release/recovery data. To date model development has focused on accounting for the differences in regional exploitation patterns, resolving data conflicts, and exploring seasonal movement patterns.

2.2 Data-limited methods

The Data-Limited Methods Toolkit (DLMtool, version 5.4.5; Carruthers and Hordyk, 2018, 2020) is a software library for evaluating the performance of data-limited MPs. The DLMtool R package offers a robust, transparent approach for comparing, selecting, and applying various data-limited management methods. DLMtool utilizes parallel computing to make powerful diagnostics accessible (Punt et al., 2016; Carruthers and Hordyk, 2020). The DLM tool has two distinct components, a management strategy evaluation (MSE) simulation module and an application module which estimates the target catch using available data input. We used the application portion of DLMtool (and not the MSE), which has a wide range of built-in methods of varying complexity, but also allows users to specify their own options or to modify the existing methods. In this study, various DLMs were applied in setting target catches to the Indian Ocean BET stock, and these results were compared with those obtained with the SS3 assessment model. The data inputs for the SS3 assessment were extracted from IOTC Working Party on Tropical Tuna Meeting Website (https://iotc.org/WPTT/21/Data/14-SA-BET). Specific details on the data sources required for the DLMs are provided in Table 1.

We categorized the DLMs into five categories: catch-based methods, abundance-based methods, index-based methods, length-based methods, and age-based methods. A summary of these methods was highlighted and is presented in Table 2. Catch-based methods have generally been employed where insufficient data exist for determining an overfishing limit (OFL) using more sophisticated methods (Carruthers et al., 2014). Several catch-based methods have been adopted for the neritic and tuna assessments in the past several years and were deemed the best choice for the available data in the IOTC (Zhou et al., 2019). As an alternative to DLMs that rely solely or primarily on catch data and/or depletion estimates, there are also abundance-based and index-based methods. We tested a class of methods relying on estimates of current abundance and the fishing mortality rate at maximum sustainable yield (FMSY). We also explored length-based methods and age-based methods, as length and age composition data are the second-most abundant information held by the IOTC Secretariat, which potentially provides information on fishery status we note that we focus only the DLMs that can be applied to the bigeye tuna fishery and not all the DLMs in the toolkit were tested.

2.3 Species information and data

The data used in the bigeye tuna assessment consist of catch and length composition data, longline CPUE indices, and tag release-recapture data. Figure 1 shows the stock trajectory from stock assessments of bigeye tuna in the Indian Ocean (1975−2020). The age-frequency and length-frequency distributions for bigeye tuna are shown in Figs 2 and 3 (every five years). The Dep, Mort, BMSY_B0, FMSY_M, vbK, vbLinf distributions from stock assessments of bigeye tuna in the Indian Ocean are shown in Fig. 4.

2.4 Sensitivity to data inputs

A sensitivity analysis was conducted for all DLMs to explore which data inputs most affect catch advice (Carruthers and Hordyk, 2020). Sensitivity analysis is a method in DLMtool that determines the inputs for a given DLM of class output and then analyse the sensitivity of catch advice estimates to marginal differences in each input (Carruthers and Hordyk, 2020). The variation assigned to each input determines the range of values over which the sensitivity is evaluated. In this way, the sensitivity test is standardized to be commensurate with the uncertainty ascribed to each parameter. The input data explored included bigeye tuna life history data, fishery data, abundance data, composition data and reference data depending on the different DLMs.

2.5 Comparison of catch advice from data-rich and data-limited assessments

Data-limited catch advice was produced for the years following the terminal year of data and was held constant between assessments. The SS catch advice equivalent to the overfishing limit was determined by the prescribed optimal target reference point and current stock status and was extracted from SS projections (via the forecast submodel) three years after the terminal assessment year (Table 1). To enable comparisons of catch advice from DLMs with SS-derived catch advice, 34 DLMs were used to produce catch advice.

For each data-limited approach, a probability density function of catch advice was derived using 10 000 random draws from parameter distributions defined by the input mean and CV (Table 1). The median of the probability density function was used for the purpose of comparison (Carruthers et al., 2016). The distribution of the catch recommendation from SS was assumed to be normal and was obtained using a maximum likelihood approach. Because catch advice was set for a number of years in advance (to account for time lags caused by data collation and assessment implementation), assumed catches are fixed at predetermined quota levels for the first two years of the projection in SS. Thus, the third year of the projection represents the first year of catch advice; therefore, the forecasted catch (extracted as a point estimate with standard deviation) was used for comparison. Although the years being compared are not identical (e.g., terminal year of data, 2018; data-limited catch advice, 2019; SS catch advice, 2021), the approach to developing catch advice is similar (i.e., produce catch advice for the next possible year).

To quantitatively compare catch advice from each DLMs to the data-rich SS model (i.e., data-rich projection from current stock assessment model; OFL assessment), we calculated the relative absolute error (RAE) for the OFL (Dick and MacCall, 2011) with the following equation:

(1)

$ RAE=\frac{|DLM-{OFL}_{\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}|}{{OFL}_{\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}}} , $

where OFL _assessment was extracted from projections using the base SS assessment model as discussed above. The median of the probability density function was used for the purpose of comparison (Carruthers et al., 2016). Data-limited catch advice was produced for the year following the terminal year of data and was held constant between assessments. Larger RAE values indicate higher data-limited catch advice compared with SS catch advice, whereas smaller RAE values suggest similar catch advice between methods (close to zero). Inherently we assume that derived products and parameters from SS reflect the “known truth” for the purpose of addressing whether simpler models can produce similar results, an assumption that may not be accurate.

3 Results

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3.1 Sensitivity of performance to inputs and value of information

Sensitivity analyses revealed that input data tended to affect the catch advice for all methods included in this study, and detailed information on the input data is provided in Table 1. For almost all DLMs catch advice was sensitive to catch data (Cat, AvC) (Table 3). The majority of DLMs requiring an estimate of Mort in the input, catch advice was particularly sensitive. Other inputs such as depletion estimates, abundance, BMSY_B0 and FMSY_M were also influential in deriving catch advice (Table 3). In instances in which catch data was required as data inputs, these recommendations were seldom sensitive to life-history parameters related to growth such as wla and wlb, and the theoretical age at length zero (vbt0).

For bigeye tuna life-history data (Table 1), except for Mort, the historical life history data of bigeye tuna was not sensitive to catch advice using catch-based methods (Table 3). This result indicated that for most catch-based methods, the bigeye tuna fishery life-history data had little influence on catch advice. However, for Delay-Difference Stock Assessment (DD and DD4010), the sensitivity analysis results showed that the catch advice was more sensitive to the current level of Dep and the shortest length at full selection (LFS), where catch advice was positively correlated to Dep and negatively correlated to LFS (Table 3). For the abundance-based method, the catch advice of Beddington and Kirkwood life history method (BK) was sensitive to life-history parameters vbLinf and vbK, and we also found that catch advice from BK was fairly linearly related to the level of vbLinf and vbK over which sensitivity was tested. Lratio_BHI, Lratio_BHI2, Lratio_BHI3 and age-based (BK_CC) methods have similar results (Table 3).

However, for composition data (Table 1), the sensitivity analysis results of age-based methods showed that age composition data were not sensitive to catch advice (Table 3). For BMSY_B0 and FMSY_M, all methods that require these two parts of the data were sensitive, especially for catch-based and abundance-based methods. The catch advice was positively correlated to FMSY_M, and negatively correlated with BMSY_B0.

3.2 Comparison of catch advice between DLMs and SS

Catch advice derived from data-limited approaches for bigeye tuna in the Indian Ocean was shown in Table 4. We found that catch advice for data-limited approaches was highly variable and uncertain, with standard deviations greatest for BK_CC (244 970 t) and smallest for DynF (60 t). Among the five categories of DLMs, the age-based method has a larger standard deviations than the other four categories, and the corresponding CV was also the largest (Table 4). Among the 34 DLMs, the catch advice varies so much between the different methods, BK (227 622 t) has the highest catch advice and Itarget1 (54 681 t) has the lowest catch advice. The index-based and length-based methods had lower catch advice than the other three categories of methods (Table 4).

We also compared the distributions of relative absolute errors between SS and DLMs for Indian Ocean BET (Fig. 5). The majority of catch-based, index-based, and length-based tested (excluding DBSRA, SPMSY) resulted in RAEs less than 1 (Fig. 5). Most abundance-based and age-based methods resulted in RAEs greater than 1 (Fig. 5). Only Itarget1, DCAC4010, Islope1 and IT5 produced an RAE below 0.1, and relatively similar OFL distributions compared to SS (Figs 5 and 6). The median catch advice of these four DLMs was within 10% of the OFL of SS, and OFL distribution peaked near the OFL distribution of SS (Fig. 6).

The comparison of the OFL estimated by the SS model and DLMs for Indian Ocean BET was showed in Fig. 6. Most methods resulted in wider OFL distributions (median range: 54 681 (Itarget1)–227 622 t (BK)) compared to SS (61 931 t) (Table 4). This indicated a substantial amount of uncertainty when compared to the OFL distribution produced by the data-rich SS model. For catch-based methods, except DBSRA4010, DCAC4010, CC1, SPMSY, the OFL distributions of the other seven methods were relatively narrow (Fig. 6a). For index-based methods, only Itarget1 OFL distribution was relatively close, and the catch advice was smaller than SS (Fig. 6b, Table 4). The OFL distribution of IT5 was similar to SS; OFL distribution peaked near the OFL distribution of SS (Fig. 6b). For the abundance-based methods, Fratio, DepF and Fratio4010 result in high and relatively wide OFL distributions (Fig. 6c). This showed that these three methods have higher uncertainty. The OFL distribution of the Length-based method was more uniform and narrower than the other four types of DLMs methods (Fig. 6d). The OFL distribution of the three age-based methods was wider, and the catch advice was much higher than the catch advice based on SS (Fig. 6e, Table 4).

4 Discussion

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Identifying the impacts of input data quality and quantity was critical for improving stock assessment and developing precautionary management strategies. This analysis aimed to investigate whether similar assessment results could be achieved with DLMs as opposed to more complex conventional stock assessment methods for Indian Ocean BET. We applied a DLM sensitivity analysis to explore which input data most affect catch advice. Catch-based, index-based, and length-based DLMs tended to produce similar results across life-history stages, other methods included abundance-based and age-based DLMs also produced viable results for Indian Ocean BET. This analysis focused on the range of DLMs commonly applied to date. While most methods examined in the study were feasible for bigeye tuna based on available data inputs, the resulting OFL distributions were not necessarily accurate or robust to uncertainty. Many DLMs produced relatively wide OFL distributions, suggesting a substantial amount of uncertainty. For almost all applicable DLMs, catch advice was particularly sensitive to Cat, Mort, Abun, Dep, and FMSY_M with higher data inputs corresponding to higher quotas (positive correlation). In some instances, catch advice was also sensitive to life-history parameters relating to growth (vbLinf, vbK) and BMSY_B0.

In recent years, the IOTC explored various DLMs for some small tuna species, including application of catch-based methods and length-based methods (Dick and McCall, 2011; Martell and Froese, 2013; Cope, 2013; Hordyk et al., 2015; Hordyk, 2019; Froese et al., 2017, 2018; Rudd, 2018; Rudd and Thorson, 2018). There was generally substantial uncertainty in the estimation of stock status, and the results were susceptible to input parameters. The examination of data-rich assessment management frameworks using DLMs has revealed common patterns and highlighted potential challenges in developing catch advice for data-poor stocks. The catch-based, index-based, or length-based methods showed considerable promise. Index-based methods and length-based methods in particular often outperformed other DLMs in reproducing the OFL that is consistent with the SS model. Yet, additional testing using a management strategy evaluation framework is required to adequately evaluate the performance of both methods based on representative stock life histories and fleet characteristics. The closed-loop simulation studies such as MSE should be considered most appropriate to determine the most feasible management strategy. Data-limited applications can provide much-needed insight into stock dynamics within data-poor stocks (such as small tuna or like-species tuna) until data collection improves, time series of abundance lengthen, and/or analytical resources expand.

In this study, the output from SS was taken as the “truth” or more realistic reflection of “true” fisheries dynamics, an approach which sought to determine whether simple models could obtain similar results to a more complex model. Neither the aforementioned assumption nor the statistical procedures necessarily imply that any of the models were correct. In the practice of setting harvest recommendations, complex models were often regarded as more reputable sources. However, for data-poor stocks, complex models may also be biased due to violation of assumptions (e.g., constant fishing efficiency) or model misspecification, and some key parameters (e.g., steepness, natural mortality, etc.) are often inestimable (Carruthers et al., 2014). Therefore, we recommend that more DLMs be explored for data-poor stocks using data-limited assessment methods and MSE.

For data-poor species, the lack of consistent and long-term fishery-independent surveys exacerbates uncertainty in assessing stock dynamics (Cummings et al., 2014). Simple management procedures based on an index of abundance and length have gained momentum in recent years (Geromont and Butterworth, 2015a, b). They thus warrant additional efforts to quantify the relative abundance. For length-based methods, mean length information was relatively easy to obtain even in data-poor fisheries. Closed-loop simulation studies such as management strategy evaluation should be considered to determine the most feasible management. DLMs to bigeye tuna can serve as a learning experience for managing data-limited stocks in the Indian Ocean. With their sensitivity to data inputs in the analyses of results, DLMs can provide much-needed insight into the stock dynamics of data-poor stocks (such as small tuna or like-species tuna) until data collection, time series of abundance length and/or analytical resources expand.

Acknowledgements

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The senior author’s work was conducted at Yong Chen’s lab in the School of Marine Sciences, University of Maine. We thank the developers of DLMtool for their splendid work and technical support, and the data supported by the IOTC secretary. Jessica Chen helped edit the paper, which was supported by the Fisheries Learning Network.

Funding

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The National Natural Science Foundation of China under contract No. 41676120.

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doi: 10.1007/s13131-021-1933-9

Receive Date：2021-01-16
Online Date：2025-11-21
Published：2022-03-25

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Received：2021-01-16
Accepted：2021-03-04

Funding

The National Natural Science Foundation of China under contract No. 41676120.

Affiliations

¹ College of Marine Sciences, Shanghai Ocean University, Shanghai 201306, China

² School of Marine Sciences, University of Maine, Orono, ME 04469, USA

³ Key Laboratory of Oceanic Fisheries Exploration, Ministry of Agriculture and Rural Affairs, Shanghai 201306, China

⁴ Indian Ocean Tuna Commission, Victoria 361, Seychelles

Corresponding:

* E-mail: jfzhu@shou.edu.cn

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表12种不同金属材料的力学参数

科 Family	属数 Number of genus	种数 Number of species	占总种数比例 Percentage of total species (%)	属 Genus	种数 Number of species	占总种数比例 Percentage of total species (%)
鹅膏菌科Amanitaceae	2	11	5.26	鹅膏菌属 Amanita	10	4.78
小菇科 Mycenaceae	2	12	5.74	丝盖伞属 Inocybe	5	2.39
多孔菌科 Polyporaceae	8	14	6.70	蜡蘑属 Laccaria	5	2.39
红菇科 Russulaceae	3	23	11.00	小皮伞属 Marasmius	6	2.87
				小菇属 Mycena	11	5.26
				光柄菇属 Pluteus	5	2.39
				红菇属 Russula	17	8.13
				栓菌属 Trametes	5	2.39

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Table 1. Data extracted from the 2019 Indian Ocean BET SS assessment model file for DLMs

Input	Description	Data
Input	Description	Value	Coefficient of variation (CV)
Year	years corresponding to data	1975−2018	−
t/a	number of years	44	−
Units	metric tonnes	−	−
Life history
MaxAge/a	maximum age	11	−
Mort/a⁻¹	natural mortality rate	0.29	0.20
steep	steepness of the Beverton Holt stock-recruitment relationship	0.8	0.20
vbLinf/cm	Von Bertalanffy Linf parameter	150.91	0.10
vbK	Von Bertalanffy K parameter	0.11	0.10
vbt0	Von Bertalanffy t ₀ parameter	−1.16	0.10
wla	weight-length parameter alpha	2.22×10⁻⁰⁵	0.10
wlb	weight-length parameter beta	3.01	0.10
L50/cm	length at 50 percent maturity	44.25	0.10
L95/cm	length increment from 50 percent to 95 percent maturity	52.64	0.10
Fishery
Cat	annual sum of total catch (1975−2018)	40 020−93 515	0.10
AvC	average catch (Cat) over period with depletion estimates (1975−2018)	960 008.67	0.20
LFC	length at first capture	13.35	0.20
LFS	shortest length fully vulnerable to fishing	30.94	0.20
Cref	reference or target catch set to MSY	86 235.60	0.20
Bref	reference or target biomass set to spawning biomass at MSY	555 249	0.20
Abundance
Ind	relative total abundance index (1975−2018)	Longline CPUE indices: 1.50−0.51	0.20
Dt	depletion over time t SSB _now /SSB _now-t+1	0.34	0.25
Dep	stock depletion SSB _current /SSB _unfished	0.31	0.25
Abun	current abundance (2018) (spawning biomass)	688 529	0.25
Composition
CAA	catch-at-age data (1975−2018)	44 a×11 ages	−
CAL	catch-at-length data (1975−2018)	44 a× 95 length bins	−
CAL_bins	the values delimiting the length bins for the catch-at-length data	10−200 cm, 2 cm bins	−
ML	mean length time series (1975−2018)	121.46−71.90 cm	−
Reference (2019 SS assessment)
BMSY_B0	the most productive stock size relative to unfished	0.25	0.045
FMSY_M	an assumed ratio of FMSY to M	0.90	0.25
Ref	reference OFL(a reference quota level)	61 931.40	−

Note: − represents no data.

Table 2. Description of DLMs applied and model inputs

Type	Method abbreviation	Description	Input	Reference
Catch-based	AvC	average catch over entire time series	Cat	Newman et al. (2014); Carruthers and Hordyk (2020)
	CC1	recent mean catch (last 5 a) Constant catch linked to average catches (TAC = C _average).	Cat	Geromont and Butterworth (2015b); Carruthers et al. (2016); Carruthers and Hordyk (2020)
	DCAC	depletion-corrected average catch Depletion is estimated each management interval and used to update the catch limit recommendation based on the historical catch.	AvC, BMSY_B0, Dt, FMSY_M, Mort	MacCall (2009); Harford and Carruthers (2017); Carruthers and Hordyk (2020)
	DCAC_40	DCAC is assuming current stock biomass to be exactly at 40 percent of unfished levels.	AvC, BMSY_B0, FMSY_M, Mort	MacCall (2009); Harford and Carruthers (2017); Carruthers and Hordyk (2020)
	DCAC4010	The dynamic DCAC is paired with the 40-10 rule that throttles back the OFL to zero at 10 percent of unfished stock size.	AvC, BMSY_B0, Dt, FMSY_M, Mort	MacCall (2009); Harford and Carruthers (2017); Carruthers and Hordyk (2020)
	DBSRA	depletion-based stock reduction analysis	BMSY_B0, Cat, Dep, FMSY_M, L50, vbK, vbLinf, vbt0	Dick and MacCall (2010, 2011); Carruthers and Hordyk (2020)
	DBSRA_40	DBSRA assuming stock depletion is 40% of unfished levels (B _current /B ₀= 0.4).	BMSY_B0, Cat, FMSY_M, L50, vbK, vbLinf, vbt0	Dick and MacCall (2010, 2011); Carruthers and Hordyk (2020)
	DBSRA4010	DBSRA with a 40-10 harvest control rule	BMSY_B0, Cat, Dep, FMSY_M, L50, vbK, vbLinf, vbt0	Dick and MacCall (2010, 2011); Carruthers and Hordyk (2020)
	DD	Delay-Difference Stock Assessment	Cat, Ind, L50, MaxAge, Mort, vbK, vbLinf, vbt0, wla, wlb	Hilborn and Walters (1992); Carruthers et al. (2012); Carruthers and Hordyk (2020)
	DD4010	Delay-Difference Stock Assessment with a 40-10 harvest control rule	Cat, Ind, L50, MaxAge, Mort, vbK, vbLinf, vbt0, wla, wlb	Hilborn and Walters (1992); Carruthers et al. (2012); Carruthers and Hordyk (2020)
	SPMSY	catch trend surplus production MSY method	Cat, L50, MaxAge, vbK, vbLinf, vbt0	Martell and Froese (2013); Carruthers and Hordyk (2020)
Index-based	Islope1	Index Slope Tracking method CPUE slope (Adjust catch advice based on slope in CPUE for last 5 a or 10 a.)	Cat, Ind	Geromont and Butterworth (2015a); Carruthers et al. (2016); Carruthers and Hordyk (2020)
	Itarget1	CPUE target (Adjust catch advice to achieve a target CPUE, where target=1.5×mean CPUE during reference period.)	Cat, Ind	Geromont and Butterworth (2015a); Carruthers et al. (2016); Carruthers and Hordyk (2020)
	IT5	Iterative Index Target method. Maximum annual changes in TAC are 5 percent.	Ind, Iref	Carruthers and Hordyk (2020)
	Iratio	Mean Index Ratio	Cat, Ind	Jardim et al. (2015); ICES (2012)
	SBT1	Make incremental adjustments to TAC recommendations based on index levels relative to target levels (B _MSY /B ₀) and catch levels relative to target levels (MSY).	Cat, Ind	Li (2011); Carruthers and Hordyk (2020)
	SPmod	surplus production based catch-limit modifier	Cat, Ind	Carruthers et al. (2016); Carruthers and Hordyk (2020)
	GB_slope	Geromont and Butterworth index slope Harvest Control Rule	Cat, Ind	Carruthers and Hordyk (2020); Geromont and Butterworth (2015b)
Abundance-based	SPslope	catch trend surplus production MSY	Abun, Cat, Ind	Carruthers et al. (2016); Carruthers and Hordyk (2020)
	Fratio	FMSY /M ratio method Requires an estimate of current abundance.	Abun, FMSY_M, Mort	Gulland (1971); Walters and Martell (2002); Martell and Froese (2013); Carruthers and Hordyk (2020)
	DepF	Depletion Corrected Fratio	Abun, Dep, FMSY_M, Mort	Gulland (1971); Walters and Martell (2002); Martell and Froese (2013); Carruthers and Hordyk (2020)
	DynF	Dynamic Fratio MP	Abun, Cat, FMSY_M, Ind, Mort	Carruthers and Hordyk (2020)
	Fadapt	Adaptive Fratio	Abun, Cat, FMSY_M, Ind, Mort	Carruthers et al. (2016); Maunder (2014)
	Fratio4010	Paired with the 40-10 rule that throttles back the OFL to zero at 10 percent of unfished biomass.	Abun, Dep, FMSY_M, Mort	Gulland (1971); Walters and Martell (2002); Martell and Froese (2013); Carruthers and Hordyk (2020)
	BK	Beddington and Kirkwood life history method	Abun, LFC, vbK, vbLinf	Beddington and Kirkwood (2005); Carruthers and Hordyk (2020)
Length-based	LstepCC1	step-wise constant catch using mean length (Catch adjusted based on ratio of recent to reference mean length.)	Cat, ML	Geromont and Butterworth (2015a); Carruthers et al. (2016); Carruthers and Hordyk (2020)
	Ltarget1	length target (Adjust catch advice to achieve a target mean length, where target=1.05×mean length during reference period.)	Cat, ML	Geromont and Butterworth (2015a); Carruthers et al. (2016); Carruthers and Hordyk (2020)
	Lratio_BHI	mean length-based indicator MP Assumes M/K=1.5 and FMSY/M=1.	CAL, CAL_bins, Cat, LFS, vbLinf	Jardim et al. (2015)
	Lratio_BHI2	More general version that calculates the reference mean length as a function of M, K, and presumed FMSY/M.	CAL, CAL_bins, Cat, FMSY_M, LFS, Mort, vbK, vbLinf	Jardim et al.(2015)
	Lratio_BHI3	A modified version of Lratio_BHI2 where mean length is calculated for lengths>modal length (Lc).	CAL, CAL_bins, Cat, FMSY_M, LFS, Mort, vbK, vbLinf	Jardim et al.(2015)
	DCAC_ML	Depletion-Corrected Average Catch that uses a mean length estimator for current depletion.	AvC, CAL, Cat, Lbar, Lc, Mort, vbK, vbLinf	Gedamke and Hoenig (2006); MacCall (2009); Carruthers and Hordyk (2020)
Age-based	Fratio_CC	Current abundance is estimated using average catch and estimate of fishing mortality from an age-based catch curve.	CAA, Cat, FMSY_M, Mort	Gulland (1971); Walters and Martell (2002); Martell and Froese (2013); Carruthers and Hordyk (2020)
	BK_CC	Beddington and Kirkwood life history method that uses Catch Curve to estimate current abundance based on catches and recent fishing mortality.	CAA, Cat, LFC, vbK, vbLinf	Beddington and Kirkwood (2005); Carruthers and Hordyk (2020)
	YPR_CC	Yield per recruit analysis that uses a Catch Curve to estimate recent abundance.	CAA, Cat, LFS, MaxAge, vbK, vbLinf, vbt0	Beverton and Holt (1993)
Integrated analysis	SS	Stock Synthesis statistical age-structured population model	−	Methot (2009); Methot and Wetzel (2013); Methot et al. (2020); Fu (2019)

Note: − represents no data.

Table 3. Sensitivity analysis (SA) of input data needed for DLMs

Method	Input data
	Life history									Fishery						Abundance				Com^*		Ref^*
	MaxAge	Mort	vbLinf	vbK	vbt0	wla	wlb	L50	L95	Cat	AvC	LFC	LFS	Cref	Bref	Ind	Dt	Dep	Abun	CAA	CAL	BMSY_B0	FMSY_M
Catch-based
AvC	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	−	−	−	−	−	−	−	−
CC1	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	−	−	−	−	−	−	−	−
DCAC	−	SA	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	−	−	SA	SA
DCAC_40	−	SA	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	−	−	−	−	−	SA	SA
DCAC4010	−	SA	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	−	−	SA	SA
DBSRA	−	−	SA	SA	SA	−	−	SA	−	SA	−	−	−	−	−	−	−	SA	−	−	−	SA	SA
DBSRA_40	−	−	SA	SA	SA	−	−	SA	−	SA	−	−	−	−	−	−	−	−	−	−	−	SA	SA
DBSRA4010	−	−	SA	SA	SA	−	−	SA	−	SA	−	−	−	−	−	−	−	SA	−	−	−	SA	SA
DD	SA	SA	SA	SA	SA	SA	SA	SA	−	SA	−	−	−	−	−	SA	−	−	−	−	−	−	−
DD4010	SA	SA	SA	SA	SA	SA	SA	SA	−	SA	−	−	−	−	−	SA	−	−	−	−	−	−	−
SPMSY	SA	−	SA	SA	AS	−	−	SA	−	SA	−	−	−	−	−	−	−	−	−	−	−	−	−
Index-based	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−
Islope1	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	−	−	−	−	−
Itarget1	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	−	−	−	−	−
IT5	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	−	−
Iratio	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	−	−	−	−	−
SBT1	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	−	−	−	−	−
SPmod	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	−	−	−	−	−
GB_slope	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	−	−	−	−	−
Abundance-based
SPslope	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	SA	−	−	−	−
Fratio	−	SA	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	SA	−	−	−	SA
DepF	−	SA	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	SA	SA	−	−	−	SA
DynF	−	SA	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	SA	−	−	−	SA
Fadapt	−	SA	−	−	−	−	−	−	−	SA	−	−	−	−	−	SA	−	−	SA	−	−	−	SA
Fratio4010	−	SA	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	SA	SA	−	−	−	SA
BK	−	−	SA	SA	−	−	−	−	−	−	−	SA	−	−	−	−	−	−	SA	−	−	−	−
Length-based
LstepCC1	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	−	−	−	−	−	−	−	−
Ltarget1	−	−	−	−	−	−	−	−	−	SA	−	−	−	−	−	−	−	−	−	−	−	−	−
Lratio_BHI	−	−	SA	−	−	−	−	−	−	SA	−	−	SA	−	−	−	−	−	−	−	SA	−	−
Lratio_BHI2	−	SA	SA	SA	−	−	−	−	−	SA	−	−	SA	−	−	−	−	−	−	−	SA	−	SA
Lratio_BHI3	−	SA	SA	SA	−	−	−	−	−	SA	−	−	SA	−	−	−	−	−	−	−	SA	−	SA
DCAC_ML	−	SA	SA	SA	−	−	−	−	−	SA	SA	−	−	−	−	−	−	−	−	−	SA	−	−
Age-based	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−	−
Fratio_CC	−	SA	−	−	−	−	−	−	−	SA	−	−	−	−	−	−	−	−	−	SA	−	−	SA
BK_CC	−	−	SA	SA	−	−	−	−	−	SA	−	SA	−	−	−	−	−	−	−	SA	−	−	−
YPR_CC	SA	−	SA	SA	SA	−	−	−	−	SA	−	−	SA	−	−	−	−	−	−	SA	−	−	−

Note: The criteria for sensitive and not sensitive, the range of values over which the sensitivity is evaluated in DLMs; deep color, sensitive; light color, not sensitive; Com^*, Composition; Ref^*, Reference (2019 SS assessment); −, no data.

Table 4. Catch advice (unit, t) derived from data-limited approaches for BET in the Indian Ocean

Method	Percentile					Mean value	SD	CV
Method	0%	25%	50%	75%	95%	Mean value	SD	CV
AvC	42982	83732	95695	109422	132435	97557	19542	0.20
CC1	77643	89424	92103	94884	99042	92174	4089	0.04
DCAC	51772	74312	77754	80632	84277	77326	4647	0.06
DCAC_40	58180	76301	79037	81491	84667	78720	3955	0.05
DCAC4010	7347	44053	57887	73636	83281	57759	17522	0.30
DBSRA	24674	100709	125733	156165	212521	131782	43585	0.33
DBSRA_40	142250	172261	178896	185519	195422	179044	9894	0.06
DBSRA4010	0	52125	86420	135382	211378	98773	60128	0.61
DD	77715	100266	106888	113756	124094	107241	9832	0.09
DD4010	72611	100597	106856	113722	124024	107350	9732	0.09
SPMSY	58118	109687	137981	157536	169613	132010	29725	0.23
Islope1	59962	60259	60321	60384	60473	60322	92	0.00
Itarget1	45508	53019	54681	56334	58776	54709	2454	0.04
IT5	28308	55590	63426	72829	88320	64857	13033	0.20
Iratio	34344	68736	79084	91321	110840	80894	16897	0.21
SBT1	60541	84898	90803	97226	107028	91316	9129	0.10
SPmod	1778	73427	84251	93839	105964	83035	15509	0.19
GB_slope	74811	81136	86904	93106	103026	87610	8453	0.10
SPslope	76927	88790	91864	94984	99445	91918	4544	0.05
Fratio	41906	132359	169883	219604	315231	182374	69922	0.38
DepF	28688	107361	140791	183081	269037	151165	61636	0.41
DynF	92711	92711	92711	92711	92711	92712	60	0.00
Fadapt	109321	115927	117695	119498	122239	117776	2656	0.02
Fratio4010	0	78253	114160	159439	245602	125180	65050	0.52
BK	78369	189727	227622	272895	356358	236085	64401	0.27
LstepCC1	77680	89370	92078	95021	99270	92207	4176	0.05
Ltarget1	71799	82385	84864	87460	91442	84983	3812	0.04
Lratio_BHI	57360	96251	106083	117023	134707	107186	15659	0.15
Lratio_BHI2	46075	75779	84593	94851	111461	85977	14375	0.17
Lratio_BHI3	47747	76026	84869	94684	111086	85959	14020	0.16
DCAC_ML	80079	88373	89603	90663	91889	89410	1751	0.02
Fratio_CC	25896	88877	127001	199889	532951	191280	216492	1.13
BK_CC	49377	119849	162504	251020	644250	238947	244970	1.03
YPR_CC	36783	92241	126980	196721	537264	191021	212107	1.11

Note: SD, standard deviation; CV, coefficient of variation.

Fig. 1. The stock trajectory from the stock assessments of BET in the Indian Ocean (1975−2018).

Fig. 2. Age-frequency distributions form stock assessments of BET in the Indian Ocean (1975−2018).

Fig. 3. Length–frequency distributions form stock assessments of BET in the Indian Ocean (1975−2018).

Fig. 4. Parameter distributions form stock assessments of BET in the Indian Ocean (1975−2018).

Fig. 5. Comparison of relative absolute error of DLMs for Indian Ocean BET.

Fig. 6. Comparison of the overfishing limits (OFL) estimated by the data-rich SS.

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