We employed four standard models of stock-recruitment dynamics (Cushing, Ricker, Beverton-Holt and Shepherd) to quantify the relationship between recruitment and spawner biomass. We also included environmental variables in the models to evaluate whether these variables could improve the model fits (Iles and Beverton 1998) . The models were fitted using linear regression after loge transformation and used all years for which spawner biomass and recruitment data were available (i.e., 1974–1999).
The statistical fits of these models (Table B1) showed that either (a) the spawner biomass terms were insignificant (regardless of whether an environmental term was included); (b) the overall significance of the fitted model was low (P > 0.05); (c) model coefficients could not be defined or (d) the fitted parameters were strongly inter-correlated (i.e., high dependencies) indicating that the data should be described with simpler models. In general these models performed poorly. We instead fitted simple linear regression models of log recruitment vs. spawner biomass and including environmental variables (Table B1). The results are described in the manuscript.
Table B1. Statistical results of fitting spawner biomass–recruitment–environment models to Baltic Sea sprat data with and without environmental input terms.
|
Model name |
Fitted Model |
R2adj.(%) |
Px |
Poverall |
1 SEest |
DW |
|
…
|
Ln R = 16.55 + 0.35 × T |
28 |
0.003 |
0.003 |
0.724 |
2.23 |
|
…
|
Ln R = 18.61 0.0045 × ICE |
24 |
0.0054 |
0.005 |
0.740 |
2.33 |
|
…
|
Ln R = 17.61 + 0.26 × NAOJF |
22 |
0.008 |
0.008 |
0.751 |
2.35 |
|
…
|
Ln R = 17.3 + 7.5x10-7 SSB |
13 |
0.040 |
0.040 |
0.806 |
2.16 |
|
…
|
Ln R = 16.4 + 4.3x10-7 SSB + 0.29 × T |
29 |
0.218; 0.020 |
0.008 |
0.729 |
2.29 |
|
…
|
Ln R = 18.1 + 4.9x10-7 SSB 0.0037 × ICE |
27 |
0.160; 0.027 |
0.011 |
0.738 |
2.41 |
|
…
|
Ln R = 17.2 + 6.2x10-7 SSB + 0.23 × NAOJF |
30 |
0.059; 0.015 |
0.006 |
0.722 |
2.53 |
|
Cushing |
Ln R = 12.6 + 0.39ln SSB |
6 |
0.116 |
0.114 |
0.836 |
2.05 |
| Ln R = 14.7 + 0.15ln SSB + 0.32T |
25 |
0.525; 0.015 |
0.014 |
0.748 |
2.20 |
|
|
Ln R = 16.0 + 0.19ln SSB 0.0040ICE |
22 |
0.425; 0.023 |
0.021 |
0.761 |
2.32 |
|
|
Ln R = 13.6 + 0.30ln SSB + 0.24NAOJF |
23 |
0.118; 0.016 |
0.012 |
0.751 |
2.40 |
|
|
Ricker |
Ln R = 12.6 + ln SSB 0.61 × SSB |
6 |
0.017 |
0.114 |
0.836 |
2.05 |
|
Ln R = 4.3 + ln SSB 9.9x10-7 SSB + 0.27T |
8 |
0.017; 0.051 |
0.156 |
0.830 |
1.90 |
|
|
Ln R = 5.81 + ln SSB 9.3x10-7 SSB 0.0034ICE |
5 |
0.023; 0.072 |
0.207 |
0.840 |
1.98 |
|
|
Ln R = 4.9 + ln SSB 8.1x10-7 SSB + 0.22NAOJF |
10 |
0.025; 0.028 |
0.085 |
0.817 |
2.14 |
|
|
Beverton-Holt |
Ln R = ln(SSB/(b + a × SSB)) |
0 |
- |
- |
||
|
Ln R = ln(SSB/(6.6x10-8 3.4x10-4 × SSB)) + 0.35T |
24 |
0.959; 0.007 |
0.017 |
0.754 |
2.17 |
|
|
Ln R = ln(SSB/(b a × SSB)) + c × ICE |
0 |
- |
- |
|||
|
Ln R = ln(SSB/(b + a × SSB)) + c × NAOJF |
0 |
- |
- |
|||
|
Shepherd |
Ln R = a + lnSSB ln(1 + (SSB/b)c) |
0 |
- |
- |
||
|
Ln R = a + lnSSB ln(1 + (SSB/b)c) + d × T |
0 |
- |
- |
|||
|
Ln R = a + lnSSB ln(1 + (SSB/b)c) + d × ICE |
0 |
- |
- |
Notes: Year classes included in the analyses are those for which both spawner biomass and recruitment data were available (1974–1999). R2adj., P, SEest and DW represent respectively the variation explained by the model (adjusted for the number of degrees of freedom in the model), P = statistical significance of the variables and model, and 1 SEest = standard error of the estimated ln recruitment, and DW = the Durbin-Watson test statistic for autocorrelation. T = May temperature in the Bornholm Basin (see Fig. 1 in article) from the depth layer 45–65 m, ICE = the maximal areal extent of ice coverage in the winter preceding spawning, NAOJF = the North Atlantic Oscillation index averaged for the months of January and February, and SSB = spawning stock biomass.
Literature Cited
Iles, T. C., and R. J. H. Beverton. 1998. Stock, recruitment and moderating processes in flatfish. Journal of Sea Research 39:41–55.