Paulik revisited: Statistical framework and estimation performance of multistage recruitment functions
Introduction
A large body of research focuses on the survival mechanisms and vital rates of early life-history stages within the concept of recruitment dynamics. Much of this information was simplified into stock-recruit relationships during the 1950s, mainly for stock assessment purposes and recruitment forecasting. Essentially the early life-history dynamics were collapsed into one process between spawning stock size and recruitment. The classical stock and recruitment relationships are generally referred to by author name (e.g., Beverton and Holt, Ricker, Shepherd).
Paulik (1973) recognized that the resultant recruitment was a consequence of processes acting during multiple stages with various levels of density dependence in the early life history. He used a priori knowledge and assumptions about relationships between stanzas (early life-history stages) to model the likely stock to recruitment progression. Here was a recognition that the stock to recruitment process could be complex. However, the model still had single relationships (stationarity) in each stanza that would only produce single relationships for the recruitment. Even with these simplifications, Paulik’s model illustrated how complex recruitment dynamics could arise, such as multiple equilibrium points (locally stable maxima) for a stock to recruitment dynamic. Defining recruitment as a series of nonlinear functions of spawning stock, and allowing for non-stationarity, Solari et al. (1997) also showed how multiple equilibrium points could arise.
Many of Paulik's ideas remained unexplored until Rothschild (1986) brought the concept again to the fore. Rothschild (1986) focused on the diagramatic aspect of Paulik's work as a way of illustrating the early life-history dynamics and provided an explanation of how certain types of stock-recruitment relationships, e.g., Beverton-Holt or Ricker, could arise. For our purpose here, it is necessary to make the distinction between the ‘Paulik diagram’ as presented by Paulik (1973) and Rothschild (1986), and Paulik’s (1973) multistage life-history model. The diagram was designed to illustrate the contribution of early life-history dynamic processes to the functional form of a stock-recruit relationship, while Paulik’s (1973) multistage life-history model was designed to graphically estimate recruitment.
Ulltang (1996) revisited Paulik's ideas and investigated reasons for the lack of biological data being used in stock assessment. The Paulik diagram constructed by Ulltang again used the quadrants introduced by Paulik and Rothschild. However, rather than a series of density-dependent relationships (graphs), he populated the quadrants with words describing the important factors that needed to be considered in each stanza (life-history stage). Within the multistage framework, Brooks and Powers (2007) derived analytical results for how the timing of stage-specific density-independent and density-dependent mortality affects equilibrium statistics (e.g., maximum excess recruitment), and how that can inform management advice. Minto et al. (2008) similarly showed that a multistage model for recruitment can result in heteroscedasticity (where recruitment variance is a nonlinear function of spawning biomass) because process errors prior to density dependence are suppressed when density dependence is strong (e.g., at spawning biomass near equilibrium in a Beverton-Holt model) and not otherwise.
Historically, much of the literature exploring multistage recruitment processes has focused on biological aspects, with far less attention on statistical methodology to estimate the multistage function. Regardless of the number of stages being modelled, it is important to recognize that the observations have error in both the independent variable (the quantity on the x-axis, typically spawner counts or some proxy thereof) and the dependent variable (the quantity on the y-axis, recruitment). For example, if using an egg or larval survey for the independent variable, those annual observations have measurement error associated with them that should not be overlooked. A more egregious oversight could occur if using stock assessment model estimates of spawning biomass and recruitment to fit a stock-recruit function post-hoc. Although common, this practice carries additional concerns: i) both quantities are model output, not observations, and have estimation error associated with them; ii) both quantities are conditional on specifications, or misspecifications, within the model that produced those output; and iii) correlation and autocorrelation exist between/among those quantities as a result of the model framework (Ludwig and Walters, 1981; Walters and Ludwig, 1981; Brooks and Deroba, 2015).
Ludwig and Walters (1981) addressed the statistical bias that results from ignoring measurement error and described an errors in variables (EV) approach to estimating those quantities, provided the ratio between measurement and process error was known. de Valpine and Hilborn (2005) compared the EV approach with state-space models where the error ratio was constrained or unconstrained, and generally concluded that state-space approaches were less biased, had lower root mean square error (RMSE), and more informative confidence regions, although separation of process and measurement error was very difficult and knowing the ratio improved performance. Although de Valpine and Hilborn (2005) tested the Schaeffer production model (Schaeffer, 1954, 1957) rather than a classical stock-recruitment function, we expect their conclusions about difficulty separating process and measurement error hold generally for state-space modeling, particularly when measurement error is much larger than process error (Auger-Méthé et al., 2016). Authors have overcome this difficulty either by using replicated samples for a given state variable or by using available data to estimate the measurement error for data in a state-space model (Knape et al., 2013).
We investigated the Paulik approach to multistage stock-recruitment functions. Our primary goals were to develop methodology that could potentially serve as a general modeling framework, to estimate the timing (or timings) of density dependence, and to identify whether any stock-recruit stages were over-compensatory (producing decreasing recruits with increasing abundance in that stage). Using a state-space framework, we simulate multistage stock-recruitment data with different underlying density-dependent forms. We evaluate 1) the ability to identify the correct stock-recruit functional form (asymptotic or over-compensatory) and whether 3-parameter models reliably distinguish this, 2) parameter estimation (bias, precision, identifiability), 3) performance of full versus reduced models (i.e., estimating all variance parameters or imposing constraints on the number of estimated variance parameters), and 4) estimation performance with and without an informative prior on measurement error. We also provide an example application to North Sea herring, to demonstrate practical considerations and limitations when assembling and fitting data from multiple research surveys that sample different early life stages. Lastly, we compare fits between the multistage framework and the status quo fit (estimating a single transition directly from stage 1 to the last simulated stage), and conclude with a discussion of management implications.
Section snippets
State-space formulation
The traditional stock-recruit functions assume two stages, where the first stage is the total number of eggs (or a proxy of total egg production, such as spawning stock biomass), and the second stage is typically the number of fish at some age (referred to as recruits). In this development, we allow for multiple stages between egg and recruit, and denote Ri as the number of recruits at stage i = 1, 2, …, S. Transitions between each stage are defined by stage-specific stock-recruit functions,
Model convergence
There was strong evidence of convergence for all models except the Shepherd. The Shepherd models converged for Case A (three sequential Ricker SR functions), but for the other cases we observed iterations with poor mixing in the trace plots, non-stationary cumulative quantiles, as well as potential scale reduction exceeding 1.1, particularly for the parameters γi and κi (Fig. S1 in Supplementary material).
Correlations
For all models considered, there was high negative correlation (≤−0.75) between the
Discussion
We developed a state-space modeling framework for estimating multistage recruitment, tested that framework with a simulation study, and demonstrated its application to North Sea herring. Overall, our simulations suggest that estimation of multistage recruitment models is possible. However, we recommend specifying informative priors on measurement error when possible, as they reduced bias in both measurement and process error. In practice, these priors could be informed by the standardization
Acknowledgements
This investigation contributes towards the work plan of the ICES Working Group on Recruitment Forecasting in a variable Environment (WGRFE). We gratefully acknowledge the NOAA Office of Science & Technology for the support that allowed NOAA personnel to travel to WGRFE meetings, via funded proposals through the Stock Assessment Analytical Methods program. The views expressed herein are those of the authors and do not necessarily reflect the views of NOAA or any of its sub-agencies. We
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