Full length articleExtending integrated stock assessment models to use non-depensatory three-parameter stock-recruitment relationships
Introduction
Many fishery jurisdictions conduct stock assessments in which population dynamics models are fit to monitoring data. These assessments provide various types of information for use in management. For example, they inter alia provide estimates of current spawning stock biomass, unfished spawning stock biomass, current spawning stock biomass relative to reference points, and the inputs for harvest control rules selected to provide scientific recommendations regarding limits on catch and effort. Several jurisdictions rely on estimates of the biomass and fishing mortality corresponding to Maximum Sustainable Yield (BMSY and FMSY respectively). BMSY defines the target biomass (e.g., in the USA; Punt et al., 2006), or forms the basis for calculating the target biomass (e.g., in Australia; Rayns, 2007), while the target (or limit) fishing mortality is often set equal to FMSY or some fraction thereof.
It is well known that the performances of management strategies are not very sensitive to the choice of the proxies for BMSY in harvest control rules (e.g., Clark, 1991, Clark, 2002; Hilborn, 2010, Punt et al., 2014a). Many of the “integrated” methods of stock assessment (Fournier and Archibald, 1982, Maunder, 1998, Maunder and Punt, 2013) are based on population dynamics models include an underlying stock-recruitment relationship (SRR; Dichmont et al., 2016), and can provide estimates of BMSY and FMSY. However, these estimates are rarely used when applying harvest control rules. This can be concerning to stakeholders, especially when the estimates of BMSY and FMSY from the assessment differ substantially from the proxies used to provide management advice. In principle, it is possible to parameterize population dynamics models in terms of the value of BMSY (usually expressed relative to the average unfished B0, i.e., BMSY/B0) and FMSY (expressed relative to natural mortality M) (i.e., by adopting a “leading parameters” approach (Schnute and Kronlund, 1996, Schnute and Richards, 1998, Martell et al., 2008)).
Martell et al. (2008) show how it is possible to parameterize population dynamics models based on FMSY and MSY when the SRR is of the Beverton-Holt or Ricker form. Unfortunately, simply parameterizing conventional stock assessments in terms of FMSY and MSY (see Dichmont et al., 2016 for a review of the packages on which model-based approaches to stock assessment used in the US are based) will not address several other issues related to how productivity is modelled in the population dynamics models on which stock assessments are based.
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Most integrated stock assessments are based on the Beverton-Holt or Ricker form of the SRR (with constant recruitment a special case of a Beverton-Holt SRR).1 Unfortunately for these functional forms, specifying BMSY/B0 is the same as determining the productivity of the population as quantified by FMSY (see Fig. 1 for examples of the relationship between FMSY and BMSY/B0 for the three example species of this paper; Section 2). Thus, the use of 2-parameter SRRs restricts the potential range of values for BMSY/B0.
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Methods of assessment wherein the population size at which MSY is achieved relative to the unfished size is pre-specified are often conducted for cetaceans (e.g., de la Mare, 1989, Punt, 1999), and the results of these assessments often suggest that BMSY/B0 (usually expressed in numbers rather than biomass) is poorly determined when attempts as made to estimate this quantity from monitoring data (e.g., Givens et al., 1995, Punt and Wade, 2012). In contrast, the use of 2-parameter SRRs, as is typically the case for assessments of fish stocks, means that it is inferred that BMSY/B0 is well determined by the data if FMSY is well-determined by the data (or that BMSY/B0 is actually pre-specified if FMSY is pre-specified).
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Many stock assessments, particularly for data-poor scenarios, are now based on methods of stock assessment that quantify uncertainty using Bayesian methods (e.g., Simple Stock Synthesis – SSS: Cope, 2013; Depletion-Based Stock Reduction Analysis – DB-SRA: Dick and MacCall, 2011). Many assessment methods based on age- or size-structured population dynamics models that include integrated SRRs are parameterized in terms of the steepness of the SRR (h: the proportion of unfished recruitment, R0, when spawning stock biomass is reduced to 20% of B0: Francis, 1992) and some measure of unfished biomass. However, the relationship between steepness and BMSY/B0 is non-linear (Fig. 1) so placing (for example) a uniform prior on steepness is not equivalent to the placing such a prior on BMSY/B0. Moreover, the range for BMSY/B0 possible for either of the Beverton-Holt or Ricker SRRs is only a subset of the possible U[0,1] range (Punt et al., 2014a, Mangel et al., 2013; Fig. 1).
The issues outlined above suggest the need for stock assessments to include 3- as well as 2-parameter SRRs so a) there are models with no implicit relationship between BMSY/B0 and FMSY/M, b) it is possible for the population dynamics model to be parameterized in terms of any values of BMSY/B0 and FMSY/M, and c) priors can be placed on these ‘leading parameters’. This paper therefore describes thirteen potential SRRs (two 2-parameter and eleven 3-parameter SRRs), some of which have been proposed in the literature in the past. It then evaluates these relationships in terms of their capacity to achieve a broad range of values for FMSY (expressed in this paper as relative to natural mortality, and thus dimensionless) and BMSY/B0, and whether they exhibit desirable properties of SRRs, such as being defined for the entire range of possible values for B/B0. The paper finally explores the implications of conducting data-poor (catch only) assessments based on the Beverton-Holt SRR and the “best” of the eleven 3-parameter SRRs for three stocks representing contrasting life history types.
Section snippets
Example species
The analyses in this paper are based on information for three example species: aurora rockfish (Sebastes aurora), petrale sole (Eopsetta jordani), and cabezon (Scorpaenichthys marmoratus) off the US west coast. These three species cover the range from relatively short-lived and fast growing (cabezon) to very long-lived and slow growing (aurora rockfish). Table 1 lists the values for the biological parameters (growth, natural mortality and fecundity) for these species. Fishery selectivity is
Candidate stock-recruitment relationships
Table 2 lists the thirteen SRRs considered in this paper. It contains two 2-parameter SRRs (Beverton-Holt and Ricker), along with eleven 3-parameter SRRs. The SRRs are parameterized in terms of the average unfished spawning stock biomass, B0, and the steepness of the SRR. Thus, all of the SRRs in Table 2 satisfy R0 = f(B0) and hR0 = f(0.2B0), where f denotes the stock-recruitment relationship. The SRRs in Table 2 include ten forms that generalize the Beverton-Holt and Ricker SRRs by raising
Computing MSY
In equilibrium, the fully-selected fishing mortality corresponding to MSY is defined by:where , is yield-per-recruit as a function of fully-selected fishing mortality, and R(F) is recruitment when fully-selected fishing mortality2 is F (Sissenwine and Shepherd, 1987). The
Use in stock assessments
The results in the previous section suggest that the Ricker-Power SRR (and to a lesser extent the Pella-Tomlinson SRR) are able to capture a wide range of possible shapes for the yield function. The Pella-Tomlinson SRR has the undesirable feature that recruitment can increase as B → 0 (i.e., when BMSY/B0 is low). It also predicts negative recruitment for some values of B > B0 (Table 2). The Ricker-Power appears to exhibit no undesirable properties for 0 < B ≤ B0, but is undefined for all values
Selection of a SRR
This paper introduced eleven potential 3-parameter SRRs, five of which are newly formulated relationships and not found elsewhere. Six SRRs, including four of the five newly formulated relationships, were excluded because they include depensation over part of the range of biomasses between 0 and B0, and consequently there is not a one–one relationship between spawning biomass-per-recruit and recruitment, which is needed to calculate γ and h given FMSY/M and BMSY/B0. Although this problem could
Acknowledgements
AEP was partially funded by the Joint Institute for the Study of the Atmosphere and Ocean (JISAO) under NOAA Cooperative agreement No. NA15OAR4320063, Contribution No. 2017-097. Malcolm Haddon (CSIRO), Owen Hamel and Lewis Barnett (both NOAA Fisheries), and three anonymous reviewers are thanked for their comments on an earlier version of this paper.
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2020, Fisheries ResearchCitation Excerpt :To illustrate the general applicability of the methods, reference points were calculated for data from the 2019 Northeast Arctic cod assessment (ICES, 2019) using 12 recruitment functions. The recruitment models included two time-series models, four two-parameter models, four three-parameter models, and two spline models; thereby, representing a wide range of model classes (see, e.g., Punt and Cope, 2019 for a list of other models not considered here). Estimates for reference points that do not directly depend on recruitment (i.e., FStatus Quo, F0.1, FMax, and F35%) were practically identical for all 12 models.
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2020, Fisheries ResearchCitation Excerpt :The OM (and the actual assessment for Indian Ocean blue shark) was based on the Beverton–Holt stock-recruitment relationship for which BMSY/K is almost always below 0.5 (Punt et al., 2014). Further work could explore alternative stock-recruitment relationships (e.g., Punt and Cope, 2019) for which the production function is more representative of long-lived species such as sharks for which BMSY/K is believed to be larger than 0.5, and hence for which the prior for Pella–Tomlinson p parameter would need to include values larger than 1. Finally, the purpose of this paper is not to denigrate the use of SPMs, particularly in data-poor situations.