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Data S1. Binomial test.

Data S2. Derivation of expected recruitment.

Figure S1. α parameter estimates (dots) for a) North Sea cod and b) North Sea herring.

Figure S2. β parameter estimates (dots) for a) North Sea cod and b) North Sea herring. Parameters were estimated using the optim function in R (R Development Core Team, 2007).

Figure S3. η0 parameter estimates (dots) for a) North Sea cod and b) North Sea herring.

Figure S4. Violin plots of η1 parameter estimates (dots) for a) North Sea cod and b) North Sea herring, based on 100 replicates from time series of 43 and 47 data points, respectively.

Table S1. ‘True’ parameter values and parameter estimates for the heteroscedastic stock-recruitment model (equation (1)), for North Sea cod and herring.

Table S2. Parameter estimates for theBeverton–Holt stock-recruitment model (with‘dummy’ variance parameter σ2) forNorth Sea cod and herring.

Table S3. True parameter values and parameter estimates for the Minto stock-recruitment model (equation (1)), for North Sea cod and herring.

Table S4. ‘True’ parameter values and parameter estimates for the Minto stock-recruitment model (equation (1)), for North Sea cod and herring.

Table S5. ‘True’ parameter values and parameter estimates for the Minto stock-recruitment model (equation (1)), for North Sea cod and herring.

Table S6. Maximum sustainable yields for NorthSea cod and North Sea herring, for the Beverton–Holt andheteroscedastic parameter estimates and management models.

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