Introduction
Society is challenged to steward the exploitation of marine resources effectively for the health of the planet. Forecasting recruitment is a commonly recognized, but elusive, goal for the sustainable stewardship and rational management of exploited fish populations. Reliable estimation of recruitment is critically important to assessment of the exploitable segment of those populations (Needle 2001). Furthermore, industry constantly must prepare for changes in abundance, age structure, and spatial distribution of commercial species to optimize economic return and maintain a sustainable resource. Managers must make meaningful and effective decisions based on assumptions of the future state of fishery resources. Although the need for accurate recruitment forecasts is great, there are few studies that have focused on this specific problem. An Aquatic Science and Fisheries Abstracts literature search returned only 22 peer-reviewed papers with titles containing the words "recruitment" and either "prediction" or "forecast" (date of search: 4 April 2007). This is a surprisingly small number of papers given the fact that the estimation of recruitment has been a persistent preoccupation of fisheries scientists for more than a century (Gushing 1988; Kendall and Duker 1998).
One source of frustration in predicting recruitment is the seemingly unexplainable complex relationship between fish and their environment (Bailey et al. 2005; Ciannelli et al. 2007). All recruitment models assume that recruitment inevitably will be reduced if there is insufficient spawning biomass (Beverton and Holt 1957; Ricker 1975; Schnute 1985). Often, however, spawner-recruit models fail to adequately reveal processes and mechanisms affecting recruitment and, hence, are consequently unable to predict future recruitment with satisfactory precision. The environmental factors that influence recruitment in a complex, nonlinear, dynamical system often obscure the relationship between spawners and recruits to such a degree that any patterns or relationships, even if present, are not easily identified (Bailey et al. 2005). Furthermore, imprecise measurements of recruitment, parental stock, and environmental factors add additional uncertainty (Walters and Ludwig 1981). We still have only a limited understanding of the recruitment process despite our best efforts to understand recruitment dynamics because of the complex, nonlinear interactions within and between physical and biological variables.
Another obstacle to forecasting recruitment is the lack of established protocols for selecting appropriate statistical approaches. Traditional linear modeling approaches such as multiple linear regressions (MLRs) often have been used to relate external variables to recruitment variability. With the development of powerful personal computers, newer modeling techniques such as artificial neural networks (ANNs) and generalized additive models (GAMs) also have been tried. These new modeling techniques use nonparametric approaches that do not require inflexible assumptions, as opposed to MLRs that require restrictive assumptions such as functional linear relationships between the variables (Chen and Ware 1999; Huse and Ottersen 2003). Megrey et al. (2005) recently explored the performance of MLRs, GAMs, and ANNs to forecast recruitment from simulated data with known properties, but their conclusions were based on one random realization of simulated data.
The goal of this study is to build on the work of Megrey et al. (2005). We intend to advance that exploration by using the same three forecasting techniques and then applying a robust statistical methodology to evaluate forecasting accuracy. The objectives of this paper are (i) to build recruitment forecasting models using three different statistical methods, two different response variables, and two model constraints, (u) to fit the models to an in-sample data set of environmental covariates thought to influence recruitment variability of Gulf of Alaska (GOA) walleye pollock (Theragra chalcogramma, hereafter referred to as pollock), (iii) to use the best-fit models to forecast recruitment by applying them to portions of the time series reserved according to a Monte Carlo resampling strategy, (iv) to compare the forecast performance of the models, and (v) to recommend forecast modeling techniques based on those comparisons.
Materials and methods
Data sets
We modeled age-2 GOA pollock abundance as a function of spawning biomass and a suite of environmental covariates thought to be influential to recruitment success. At the time of this analysis, estimates of annual recruitment (REC) and spawning stock biomass (SSB) from the annual age-structured stock assessment model (Dorn et al. 2003) were available for the period 1961-2003. We assumed that the recruitment series is not serially correlated or that it is weakly correlated at an ignorable level. This assumption was required because none of the statistical methods used in this study is designed to deal with serial correlations. Also, tests using Monte Carlo resampling are not plausible if there is strong serial correlation. Although no serial correlation was assumed, the degree of serial correlation was examined by employing the sample autocorrelation function on the model residuals at various time lags (1 to 10). The significance of the autocorrelation was assessed with the approximate 95% confidence interval (Brockwell and Davis 2003).
A suite of environmental data was also available for the same period. The recruitment data were lagged 2 years to coincide with SSB and environmental covariates for the birth year, resulting in time series of 41 annual data points spanning the year classes 1961-2001.
Environmental covariates were selected based on a conceptual model of GOA pollock recruitment (Megrey et al. 1996) and the results of an exploratory analysis of the relationship between GOA recruitment success and the physical environment (Megrey et al. 1995). That analysis showed that age-2 recruitment abundance is closely related to precipitation, atmospheric sea-level pressure gradient, and local wind mixing. Guided by this analysis, we chose a subset of these variables as covariates, including local physical parameters, climate-scale indices, and SSB (the only biological variable; variables considered for this study are given in Table 1).
Environmental covariates considered include sea surface temperature (SST), wind mixing energy (WMX), freshwater runoff index (FRN), Northeast Pacific pressure index (NEP), Pacific Decadal Oscillation index (PDO), and Southern Oscillation index (SOI). The environmental data series were obtained as monthly averages. SST and WMX are estimated values, centered on the exit of Shelikof Strait (56[degrees]N, 156[degrees]W) and derived from the National Center for Environmental Prediction (NCEP) data reanalysis. FRN is an index for integrated GOA coastal freshwater discharge anomaly (Royer 1982). NEP is the sea-level pressure difference between points over the north-central Pacific and near Reno, Nevada (Emery and Hamilton 1985). PDO is the first principal component of the North Pacific monthly SST variability, poleward of 20[degrees]N; it describes the decadal variability in cool and warm phases of Pacific environmental regimes (Mantua and Hare 2002). SOI, the anomaly in the sea-level pressure difference between Tahiti (18[degrees]S, 150[degrees]W) and Darwin (10[degrees]S, 130[degrees]E), is a good indicator of tropical variations related to El Nino events (Trenberth 1984). Although biological variables such as predation and prey availability are known to affect recruitment success, the only biological variable considered for the forecasting models in this study was SSB. Other available biological time series were incomplete, short in length, and did not overlap the recruitment time series enough to make them useful.
Environmental covariates considered in this study have different temporal and spatial scales of influence. Some are regional and others are basin scale in their spatial scope. Similarly, in the temporal domain, some factors are important in establishing optimum conditions prior to spawning, some during spawning and larval life stages, and some during early juvenile stages. In building recruitment forecast models, we averaged the monthly environmental covariate data over 3-month periods considered to be important to pollock recruitment. These correspond to pollock prespawning (January-March), spawning (April-June), and early juvenile (July-September) life history periods. We elected to ignore the period from October through December, by which time young-of-the-year pollock are independent free-swimmers and, assuming they have reached critical size to survive the upcoming winter, are much less susceptible to the environmental covariates we chose (Bailey 1989). We treated the average of each 3-month period for each environmental covariate as a separate explanatory variable for the model. Thus, there were 19 covariates, including SSB, available for analysis.
For the purpose of identification and to clarify the presentation, we added to the end of each environmental covariate's acronym a number (1, 2, or 3) that describes the temporal influence of the environmental covariate on the life history period (prespawning, spawning, or early juvenile, respectively). All environmental time series covariates were normalized ((x - [[mu].sub.x])/[[sigma].sub.x]) prior to analysis (Fig. 1).
The recruitment forecast of GOA pollock is currently made at five categorically ordered levels of recruitment strengths (weak to strong) based on a weighted scoring of the assemblage of different biological and physical information in the region (Dorn et al. 2003). The recruitment forecast is used to project the future stock status, consequently to recommend the fishing quota to the fisheries managers.
This is the reason that recruitment forecasts remain a vital information component required by resource management decision-makers dealing with exploited marine ecosystems. In this study, the recruitment forecast is modeled and tested using an abundance scale, rather than subjective ordinal scale, thus providing more information to fisheries managers.
Environment-dependent spawner--recruit model
We adopted the generalized Ricker (1975) spawner--recruit model to specify recruitment as a function of spawning biomass and other environmental covariates as generalized by Hilborn and Walters (1992) to include environmental covariates.
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