This gist discusses the development of steepness node for fishnets.
Steepness is generally defined as the fraction of recruitment from an unfished population (R_0) when the spawning stock biomass is at 20% of its unfished (equilibrium) level (B_0). Given a functional form of the stock-recruit function f, steepness can be tied to the parameters of that function (at least in the case of the Ricker and Beverton-Holt (B-H) S-R functions), such that parameters of f can be expressed in terms of steepness. However, depending on the form of f, the bounds on steepness are different: the Beverton-Holt S-R function admits steepness values in [0.2;1] while the Ricker S-R function admits values between 0.2 and infinity (i.e., recruitment can increase above R_0 at 0.2 B_0).
Given that the range of steepness varies among the B-H and Ricker S-R functions, steepness estimated from one S-R function is not directly applicable to the other, which poses a non-trivial problem for the development of a general purpose prior for steepness. Ideally, the steepness node should have a method for Ricker and B-H steepness. However, empirical approaches to the specification of prior distributions for steepness have generally focused on either one S-R model, with Michielsens & McAllister (2004) being the exception. Myers et al. (1999) first provided estimates of B-H steepness for a large number of species and families based on Myers' S-R database. The estimates were simple transformations of the maximum lifetime reproductive rate, which was estimated from a meta-analysis using the Ricker S-R model across all stocks. Michielsens & McAllister (2004) pointed out that fits of the Ricker and B-H functions lead to different estimates for the slope at the origin and different expectations for steepness. The direct transformation from Ricker model parameters to B-H steepness applied in Myers et al. 1999 therefore gives misleading (i.e., overly conservative) results.
Short of doing a new meta-analysis of steepness from the RAM legacy DB or other data source, Myers et al. could nevertheless be a good starting point for an empirical node; the reported estimates for the maximum lifetime reproductive rate allow to re-calculate the Ricker steepness (instead of the Beverton-Holt steepness reported in their paper). However, given the potentially strong selection bias in the study (for managed, VPA assessed stocks), it seems less than ideal to use species or family level estimates from reported values directly. Rather, these estimates should be combined with life-history to build a predictive model for steepness (as suggested in issue 8) that also accounts for taxonomic patterns.
Theory suggests that recruitment variability and natural mortality M set the lower bounds for steepness (He et al. 2006). However, at low recruitment variability and M, there is little information in these parameters about steepness from these parameters alone. Given an age structured model, Mangel et al. (2010; 2013) showed that steepness depends on growth, fecundity and mortality rates, and Rose et al. 2001 provided evidence that empirical estimates of steepness follow expectations from life-history theory. An empirical node should therefore consider von Bertlanaffy growth and fecundity in addition to recruitment variability and natural mortality. Furthermore, given the complex relationships between life-history parameters and their constraints on steepness, a non-parametric node based on brter or svmer may be the most practical start.
The biggest issue with an empirical node is that the only data that span a range of families and orders remains the somewhat outdated Myers et al 1999. The Myers dataset is the largest, but only gives usable estimates for the Ricker model steepness (i.e., after back-transforming and using equations in Michielsens & McAllister (2004) to calculate Ricker steepness). Shertzer & Conn (2012) obtain priors for demersal fish, but only provide an ad-hoc fit of truncated normal and beta distributions to estimates of steepness from stock assessments. Other datasets that provide more formally derived priors for steepness (e.g., Michielsens & McAllister (2004), Dorn (2002) and Forrest et al. 2010) are most likely too narrow to inform over broad enough a taxonomic range for an empirical node; although they could be included in time as estimates for Atlantic salmon and US/Canada west-coast rockfish, respectively, and/or to give more robust estimates of empirical relationships. However, the fundamental problem of inadequate sources of empirical estimates of steepness for the B-H function steepness means that it will be difficult to construct an empirical node for the B-H.
An alternative to using meta-analysis estimates to inform an empirical node would be to repeat simulations in Mangel et al. 2010 with species for which sufficient data is available. This subset is probably small, and the simulations would require assumptions about larval/early-juvenile survival. This would be a much trickier start but give consistent priors for both Ricker and B-H. A regression against life history parameters from fish-base could then make it possible to get a predicted distribution from the fishnet.
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Dorn, M. W. (2002). Advice on West Coast rockfish harvest rates from Bayesian meta-analysis of stock− recruit relationships. North American Journal of Fisheries Management, 22(1), 280-300.
Forrest, R. E., McAllister, M. K., Dorn, M. W., Martell, S. J., & Stanley, R. D. (2010). Hierarchical Bayesian estimation of recruitment parameters and reference points for Pacific rockfishes (Sebastes spp.) under alternative assumptions about the stock-recruit function. Canadian Journal of Fisheries and Aquatic Sciences, 67(10), 1611-1634.
Mangel, Marc, Jon Brodziak, and Gerard DiNardo. "Reproductive ecology and scientific inference of steepness: a fundamental metric of population dynamics and strategic fisheries management." Fish and Fisheries 11.1 (2010): 89-104.
Mangel, Marc, et al. "A perspective on steepness, reference points, and stock assessment." Canadian Journal of Fisheries and Aquatic Sciences 70.6 (2013): 930-940.
Michielsens, C. G., & McAllister, M. K. (2004). A Bayesian hierarchical analysis of stock recruit data: Quantifying structural and parameter uncertainties. Canadian Journal of Fisheries and Aquatic Sciences, 61(6), 1032-1047.
Myers, R. A., Bowen, K. G., & Barrowman, N. J. (1999). Maximum reproductive rate of fish at low population sizes. Canadian Journal of Fisheries and Aquatic Sciences, 56(12), 2404-2419.
Rose, K. A., Cowan, J. H., Winemiller, K. O., Myers, R. A., & Hilborn, R. (2001). Compensatory density dependence in fish populations: importance, controversy, understanding and prognosis. Fish and Fisheries, 2(4), 293-327.
Shertzer, K. W., & Conn, P. B. (2012). Spawner-recruit relationships of demersal marine fishes: Prior distribution of steepness. Bulletin of Marine Science, 88(1), 39-50.