Paulik revisited: Statistical framework and estimation performance of multistage recruitment functions

Multiple processes act at different stages and different intensities within the timeline between spawning and the age designated as “recruitment”. However, common practice is to model only a single step between spawning stock and recruits. Reasons for this practice include lack of data on the intermediate stages, lack of understanding of the mechanisms and functional form governing intermediate stages, and lack of computational resources to model a multistage process in the appropriate statistical framework. We develop a state-space framework and, using a simulation study, we explore the estimation of multistage stock-recruit functions. We evaluated four different functions (Ricker, Beverton-Holt, Shepherd, and Generalized), and examined the effects on estimation of several factors, including the form of density dependence, the magnitude of measurement error associated with each stage, type of prior on measurement error, and the magnitude of process error between stages. Three-parameter stock-recruit functions (Shepherd, Generalized) correctly identified the form of density dependence in each stage, although the Shepherd model exhibited problems with convergence. Model misspecification resulted in bias, especially in parameters specifying measurement and process error; an informative prior on measurement error improved precision and bias. The Deviance Information Criterion selected the Ricker model too often, even when the true model was Beverton-Holt. Sequential density-dependent stages, even multiple overcompensatory stages, lead to an overall function that appears fairly flat, suggesting that a function capable of producing asymptotic dynamics is a practical default. The common practice of bypassing stages between the first (spawning stock) and the last (recruits) worked reasonably well, except when fitting a Ricker model, most likely because the true function was nearly flat over most of the range of the first stage. An application to data on North Sea herring illustrates that a multi-stage stock-recruit model can generate a stock-recruit function that is intermediate between Ricker and Beverton-Holt models, and that does not match existing three-parameter forms.