Steepness is a slippery slope

The Beverton-Holt and Ricker stock-recruit functions were derived in terms of two pre-recruit mortality parameters. Mace, & Doonan, (1988, A generalized bioeconomic simulation model for fish population dynamics) reparameterized the stock-recruit function in terms of steepness, which combines pre-recruit mortality with post-recruit biological parameters defining unfished spawning biomass per recruit. Their parameterization explicitly assumes a stable age distribution at unexploited conditions, but also implicitly assumes that unexploited spawning biomass per recruit is time invariant. Temporal variation could occur in either pre-recruit mortality rates or post-recruit biological parameters, but different dynamics are produced. The former results in variation in both the stock-recruit curve and the population equilibria, whereas variation in the latter only changes the points of equilibria on the curve. Thus, variation in either pre- or post-recruit parameters will result in variation in steepness, maximum sustainable yield and associated management reference points. Empirical measures of components of post-recruit productivity (maturity and mass at age, e.g.) are available for many managed fish stocks and often exhibit temporal variability. Yet use of the steepness parameterization requires an analyst to specify one set of post-recruit biological parameters to define unexploited spawning biomass per recruit for the full time series. This misspecification leads to misperception of biological reference points and has implications for meta-analyses of steepness, and interpretation of the dynamic B0 concept. Returning to the original parameterization allows the isolation of any temporal or inter-population variation in pre- and post-recruit productivity and reduces the potential for mechanistic bias in stock-recruit parameters.

Automated summary

The Beverton-Holt and Ricker stock-recruit functions can be derived based on two pre-recruit mortality parameters, while Mace & Doonan reparameterized it as steepness, combining pre-recruit mortality with post-recruit biological parameters. Changes in pre-recruit mortality rates or post-recruit biological parameters can lead to variations in steepness, maximum sustainable yield, and associated management reference points. It is crucial to accurately specify post-recruit biological parameters to avoid misperception of biological reference points and potential bias in stock-recruit parameters.