Estimating "Steepness" of the Stock-Recruitment Relationship, a Key Measure of Fish Stock Productivity

The productivity and resilience of a fish stock is commonly measured by a parameter of the stock-recruitment function, the relationship between the stock's adult biomass (spawning stock) and corresponding production of young fish (recruitment). The crucial parameter is the "steepness" parameter, h, widely defined as the ratio of 2 recruitment levels: the recruitment obtained when the spawning stock is at 20% of its virgin level, and the recruitment at the virgin population level (i.e., the population in the absence of fishing). The higher h is, the more resilient the population is, the more robust the stock is to harvesting, and the sooner the stock is likely to rebuild after fishing pressure is relaxed.

The steepness parameter has proved to be one of the most difficult assessment model parameters to estimate. But Hui-Hua Lee, a researcher with the University of Hawaii Joint Institute for Marine and Atmospheric Research and a member of the PIFSC Fisheries Research and Monitoring Division joined with PIFSC colleagues to meet the challenge in a recent article: "Can steepness of the stock-recruitment relationship be estimated in fishery stock assessment models?" This is the title of a manuscript the group submitted to the ICES Journal of Marine Science. In most stock assessments, steepness is not estimated from reliable data; rather, for purposes of the assessment a value is specified based on judgment of the modeler, or assigned a range of possible values. The PIFSC group's paper resulted from recent interest in trying to estimate steepness inside fishery stock assessment models.

Using simulation methods, the authors evaluated the ability to estimate steepness of the Beverton-Holt stock-recruitment relationship for 12 Pacific Coast groundfish species. These species had been assessed using the fully integrated stock assessment model Stock Synthesis II (SS2) and the results were rigorously reviewed by independent experts. The PIFSC group conducted simulation analyses to evaluate the ability to estimate h within these assessment models. The major advantage of using the assessment model as both simulator and estimator was to control any misspecification in the model. The underlying assumption is that these assessment models reflect characteristics of the "true" population.

The simulation results indicated that, in most cases, the steepness parameter h was estimated with moderate to low precision and moderate to high bias. A high proportion of steepness estimates from the simulated data and the original data occurred at the upper or lower bounds for steepness (as shown in the accompanying figure).

Distribution of steepness estimates from the simulation data (based on the maximum gradient less than 
                 one and unbounded estimates). Vertical dashed lines represent true/assumed values of steepness from 
                 original stock assessments. Dots represent estimated values of steepness from original data sets and 
                 horizontal bold dashed lines represent confidence intervals around the estimates.  Many of the estimates 
                 were near the bounds for <em>h</em>.
Distribution of steepness estimates from the simulation data (based on the maximum gradient less than one and unbounded estimates). Vertical dashed lines represent true/assumed values of steepness from original stock assessments. Dots represent estimated values of steepness from original data sets and horizontal bold dashed lines represent confidence intervals around the estimates. Many of the estimates were near the bounds for h.

The poor estimates of steepness indicated that often there is little information in the data about this parameter. The analysis suggested that steepness should not be estimated inside the stock assessment model unless there is good contrast in spawning stock biomass, that is, data were available at both high levels of stock biomass and at low levels. For stocks depleted to low spawning biomass levels, rebuilding from those levels should decrease the bias in estimated h. And for stocks without good contrast, specified fixed values or informative Bayesian priors on h may still be required. The paper concludes that with the appropriate data collected at the right population sizes (levels of spawning biomass), h may be estimable. But given our typical inability to estimate h reliably, management strategies robust to uncertainty in h should be adopted.