Prager: A nonequilibrium surplus-production model 



381 



between matching the observed effort (Fig. 2B) and 

 matching the index (Fig. 2C). The tuned analysis 

 gave lower estimates of MSY, f MSY , and a less opti- 

 mistic impression of the current level of the stock, as 

 represented by the ratio B 1992 /B MSY (Table 2). It also 

 estimated that the recent fishing mortality rate, as 

 represented by the ratio F W91 /F MSY , was somewhat 

 higher. 



Estimated median biases from each analysis were 

 small. In the base analysis, no management bench- 

 mark was estimated to have a bias exceeding 1.5% 



Table 1 



Data used in two production model analyses loosely 

 based on swordfish, Xiphias gladius, in the North 

 Atlantic Ocean. Yield and standardized fishing-ef- 

 fort-rate data are from Hoey et al. (1993) with mi- 

 nor revisions. Hypothetical abundance index data 

 are the mean of ages 3 through 5 + in numbers from 

 Scott et al. ( 1992). The index was constructed solely 

 for illustrative purposes, and is designated "hypo- 

 thetical" because it probably is not a good indicator 

 of total-stock biomass. 



Year 



Yield (t) 



Fishing 



effort rate 



(10 6 hooks/yr) 



Hypothetical 



abundance 



index 



1962 

 1963 

 1964 

 1965 

 1966 

 1967 

 1968 

 1969 

 1970 

 1971 

 1972 

 1973 

 1974 

 1975 

 1076 

 1977 

 1978 

 1979 

 1980 

 1981 

 1982 

 1983 

 1984 

 1985 

 1986 

 1987 

 1988 

 1989 

 1990 

 1991 



5,342 



10,189 



11,258 



8,652 



9,338 



9,084 



9,137 



9,138 



9,425 



5,198 



4,727 



6,001 



6,301 



8,776 



6,587 



6,352 



11,797 



11,859 



13,527 



11,126 



12,832 



14,423 



12,516 



14,255 



18,278 



19,959 



19,137 



17,008 



15,594 



13,212 



6.45 

 8.54 

 24.45 

 25.30 

 31.39 

 28.90 

 40.11 

 43.23 

 38 47 



19.22 

 22.97 

 21.17 

 18.14 

 20.40 

 40.13 

 35.44 

 34.85 

 40.73 

 55.10 

 49.44 

 59.55 

 80.75 

 98.91 

 97.08 

 90.46 

 85.86 

 69.86 



1.000 

 0.816 



DISS 



0.483 

 0.526 

 0.411 

 0.377 

 0.368 

 0.359 

 0.352 



of the corresponding uncorrected estimate (Table 2). 

 Estimated median biases for the tuned analysis were 

 only slightly higher; with only the estimated bias in 

 /" MSY slightly exceeding 2%. Estimates of median bias 

 in individual model parameters (such as r and K) 

 were slightly higher yet, but only for B 1 was bias 

 estimated as higher than about 2.5%. 



Approximate 80% nonparametric confidence inter- 

 vals computed by Equations 14 and 15 were derived 

 from the bootstrap. These were computed for the in- 

 dividual model parameters, management bench- 

 marks, indicators of stock position, and for each 

 year's relative stock size estimate (Table 2; Fig. 3). A 

 unitless nonparametric measure of the precision of 

 estimates was constructed by dividing the bias-cor- 

 rected 50% confidence interval (interquartile range; 

 not shown here) by the corresponding median bias- 

 corrected estimate. The resulting statistic, the rela- 

 tive interquartile range (RIR) is a nonparametric 

 analog of the coefficient of variation. The RIR was of 

 similar magnitude for both models, and was small- 

 est in MSY and f MSY , the benchmarks that do not 

 depend on q. Estimates of the quantities that depend 

 on q, and that thus involve absolute scaling, exhib- 

 ited relative IQ ranges of about 50% (Table 2). 



Estimates of relative biomass (B z scaled to B MSY ) 

 and fishing mortality rate (F z scaled to F MSY ) were 

 also similar from the two models (Fig. 3). They show 

 a declining biomass through 1991, with an increase 

 expected thereafter (at the projected harvest rate of 

 12,000 t/yr, which is less than the MSY estimates). 

 As expected, the precision of estimates during the 

 projection period was less than during the period for 

 which data were available. 



In summary, this example demonstrates that much 

 more than MSY can be estimated from a production 

 model. Biomass trajectories can be computed easily, 

 as can estimated confidence intervals derived 

 through the bootstrap. If an independent index of 

 abundance is available, the model can be tuned to 

 that index. Another useful feature is that projections 

 can be used to estimate the probable effects of quo- 

 tas or other management measures. 



Discussion 



The modeling framework described here is based on 

 the logistic population model. The history of this 

 model was summarized by Kingsland (1982), who 

 pointed out that the model originated in the work of 

 Verhulst (1845) and Robertson (1923), was popular- 

 ized by Pearl and Reed (1920), and was also studied 

 by Lotka (1924). The model was introduced to fish- 

 ery science by Graham (1935) and Schaefer (1954, 



