Supplementary Figure 1. Model selection simulation results based on time-varying Graham-Schaefer models with stochastic random walk parameters.

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1 Supplementary Figure 1. Model selection simulation results based on time-varying Graham-Schaefer models with stochastic random walk parameters. Top panels (A, B) represent model selection results for models with time-varying, while bottom panels (C,D) represent models with time-varying. The model specification for is exactly analogous to that given in the Methods of the main text for. Simultaneously time-varying and was not included in the formal estimation due to inherent statistical non-indentifiability. Two model selection criteria were applied: Left panels represent model selection results using the difference in the deviance information criterion 1 ( DIC) between fits of the static vs. time-varying model under a simulation of true underlying change in the parameter. The right panels show the analogous plots using the Watanabe-Akaike Information Criterion 2 ( WAIC). Results represent model selection experiments under an ensemble of simulations (N=1000) where parameters r, K,,,,, and were sampled from uniform densities across biologically reasonable parameter ranges. The proportion of model selection simulations classified correctly as nonstationary within the Bayesian analysis is given in each plot.

2 Supplementary Figure 2. Impact of stochastic carrying capacity on forecasted recovery probabilities. Panels A-B are repeated from Fig.4 of the main text where the carrying capacity is assumed static with posterior probability. The remaining panels investigate the sensitivity of biomass forecasts to varying degrees of stochastic random walk variation in (and therefore also the recovery target ). The degree of variation in is set by its random walk variance where is the mean of.

3 Supplementary Table 1. List of stocks analyzed in the paper. The stock name is given as the stocklong identifier in the RAM Legacy Stock Assessment Database, along with the best mean estimate of stock status (B 0 /B MSY ). # Stock B 0 /B MSY 1 Pacific herring Prince William Sound Pacific herring Sitka Alaska plaice Bering Sea and Aleutian Islands Arrowtooth flounder Bering Sea and Aleutian Islands Arrowtooth flounder Gulf of Alaska Atka mackerel Bering Sea and Aleutian Islands Cabezon Northern California Cabezon Southern California Dover sole Gulf of Alaska Dusky rockfish Gulf of Alaska Flathead sole Bering Sea and Aleutian Islands Flathead sole Gulf of Alaska Greenland turbot Bering Sea and Aleutian Islands Northern rockfish Bering Sea and Aleutian Islands Northern rockfish Gulf of Alaska Northern rock sole Eastern Bering Sea and Aleutian Islands Pacific cod Bering Sea and Aleutian Islands Pacific cod Gulf of Alaska Pacific Ocean perch Eastern Bering Sea and Aleutian Islands Pacific ocean perch Gulf of Alaska Rex sole Gulf of Alaska 2 22 Rougheye rockfish Bering Sea and Aleutian Islands Rougheye rockfish Gulf of Alaska Red king crab Bristol Bay Sablefish Eastern Bering Sea / Aleutian Islands / Gulf of Alaska Snow crab Bering Sea Shortraker rockfish Bering Sea and Aleutian Islands Walleye pollock Aleutian Islands Walleye pollock Eastern Bering Sea Walleye pollock Gulf of Alaska Yellowfin sole Bering Sea and Aleutian Islands Capelin Barents Sea Atlantic cod coastal Norway Atlantic cod Northeast Arctic Greenland halibut Northeast Arctic Golden Redfish Northeast Arctic 0.7

4 37 Haddock Northeast Arctic Pollock Northeast Arctic Northern shrimp Gulf of Maine Antarctic toothfish Ross Sea Southern bluefin tuna Southern Oceans Anchovy kilka Caspian Sea Bight redfish Southeast Australia Deepwater flathead Southeast Australia common gemfish Southeast Australia Jackass morwong Southeast Australia New Zealand ling Eastern half of Southeast Australia New Zealand ling Western half of Southeast Australia Orange roughy Cascade Plateau Orange roughy Southeast Australia Patagonian toothfish Macquarie Island Silverfish Southeast Australia School whiting Southeast Australia Tiger flathead Southeast Australia Blue Warehou Eastern half of Southeast Australia Blue Warehou Western half of Southeast Australia Atlantic cod NAFO 5Zjm Haddock NAFO-5Zejm Haddock NAFO-4X5Y Atlantic cod NAFO 2J3KL inshore Atlantic cod NAFO 3Ps English sole Hecate Strait Pacific herring Central Coast Pacific herring Prince Rupert District Pacific herring Queen Charlotte Islands Pacific herring Straight of Georgia Pacific herring West Coast of Vancouver Island Pacific cod Hecate Strait Pacific cod West Coast of Vancouver Island Rock sole Hecate Strait Pollock NAFO-4X5YZ Atlantic cod NAFO 3Pn4RS Herring NAFO 4R spring spawners Atlantic cod NAFO 4TVn Herring NAFO 4T fall spawners Herring NAFO 4T spring spawners Herring ICES IIIa 0.8

5 78 Herring Northern Irish Sea Herring North Sea Herring ICES VIa Herring ICES VIa-VIIb-VIIc Albacore tuna North Atlantic Bluefin tuna Eastern Atlantic Bigeye tuna Atlantic Skipjack tuna Eastern Atlantic Skipjack tuna Western Atlantic Swordfish Mediterranean Sea Swordfish North Atlantic Swordfish South Atlantic Yellowfin tuna Atlantic Argentine anchoita Northern Argentina Argentine anchoita Southern Argentina Argentine hake Northern Argentina Argentine hake Southern Argentina Patagonian grenadier Southern Argentina Southern blue whiting Southern Argentina Pacific halibut North Pacific Anchovy South Africa Shallow-water cape hake South Africa South African west coast rock lobster South Africa Areas South African west coast rock lobster South Africa Areas South African west coast rock lobster South Africa Areas South African west coast rock lobster South Africa Area South African west coast rock lobster South Africa Area Cape horse mackerel South Africa South coast Deep-water cape hake South Africa Kingklip South Africa Patagonian toothfish South Africa Subantarctic Prince Edward Islands South African abalone South Africa Sardine South Africa Southern spiny lobster South Africa South coast American Plaice NAFO-3LNO American Plaice NAFO-3M Atlantic cod NAFO 3M Atlantic cod NAFO 3NO Greenland halibut NAFO 23KLMNO Redfish species NAFO 3LN Redfish species NAFO 3M 0.55

6 119 Yellowtail Flounder NAFO 3LNO American Plaice NAFO-5YZ Atlantic Halibut NAFO-5YZ Bluefish Atlantic Coast Black sea bass Mid-Atlantic Coast Atlantic cod Georges Bank Atlantic cod Gulf of Maine Haddock NAFO-5Y Haddock Georges Bank Northern shortfin squid Northwestern Atlantic Coast Monkfish Gulf of Maine / Northern Georges Bank Monkfish Southern Georges Bank / Mid-Atlantic Pollock NAFO-5YZ Ocean quahog Atlantic Coast Sea scallop Georges Bank Sea scallop Mid-Atlantic Coast Spiny dogfish Atlantic Coast Summer flounder Mid-Atlantic Coast Silver hake Gulf of Maine / Northern Georges Bank Silver hake Southern Georges Bank / Mid-Atlantic Atlantic surfclam Mid-Atlantic Coast Tilefish Mid-Atlantic Coast White hake Georges Bank / Gulf of Maine Windowpane flounder - Gulf of Maine / Georges Bank Windowpane Southern New England-Mid Atlantic Winter Flounder NAFO-5Z Winter Flounder Southern New England-Mid Atlantic Yellowtail flounder Cape Cod / Gulf of Maine Yellowtail flounder Georges Bank Yellowtail Flounder Southern New England-Mid Atlantic Australian salmon New Zealand Orange roughy New Zealand Mid East Coast Atlantic menhaden Atlantic Arrowtooth flounder Pacific Coast Blackgill rockfish Pacific Coast Black rockfish Northern Pacific Coast Black rockfish Southern Pacific Coast Blue rockfish California Bocaccio Southern Pacific Coast Chilipepper Southern Pacific Coast Cowcod Southern California 0.08

7 160 Canary rockfish Pacific Coast Darkblotched rockfish Pacific Coast English sole Pacific Coast Kelp greenling Oregon Coast Longnose skate Pacific Coast Longspine thornyhead Pacific Coast Pacific hake Pacific Coast Pacific ocean perch Pacific Coast Petrale sole Northern Pacific Coast Petrale sole Southern Pacific Coast Sablefish Pacific Coast Shortspine thornyhead Pacific Coast Widow rockfish Pacific Coast Yelloweye rockfish Pacific Coast Yellowtail rockfish Northern Pacific Coast Capelin Iceland Atlantic cod Faroe Plateau Atlantic cod Iceland Haddock Faroe Plateau Haddock Iceland Pollock Faroe Plateau Black oreo West end of Chatham Rise Smooth oreo Chatham Rise Smooth oreo West end of Chatham Rise Hoki Eastern New Zealand Hoki Western New Zealand New Zealand snapper New Zealand Area Trevally New Zealand Areas TRE Red rock lobster New Zealand area CRA Red rock lobster New Zealand area CRA Red rock lobster New Zealand area CRA Red rock lobster New Zealand area CRA Red rock lobster New Zealand area CRA Red rock lobster New Zealand area CRA Red rock lobster New Zealand area CRA common gemfish New Zealand New Zealand ling New Zealand Areas LIN 3 and New Zealand ling New Zealand Areas LIN 5 and New Zealand ling New Zealand Area LIN 6b New Zealand ling New Zealand Area LIN New Zealand ling New Zealand Area LIN 7WC - WCSI 1.76

8 201 Southern blue whiting Campbell Island Rise Southern hake Chatham Rise Southern hake Sub-Antarctic New Zealand abalone species New Zealand Area PAU 5A New Zealand abalone species New Zealand Area PAU 5B New Zealand abalone species New Zealand Area PAU 5D New Zealand abalone species New Zealand Area PAU American lobster Rhode Island Tautog Rhode Island Winter flounder Rhode Island Bonnethead shark Atlantic Blacktip shark Atlantic Blacktip shark Gulf of Mexico Gag Gulf of Mexico Greater amberjack Gulf of Mexico King mackerel Gulf of Mexico King mackerel Southern Atlantic Coast Gulf menhaden Gulf of Mexico Red grouper Gulf of Mexico Red porgy Southern Atlantic coast Sandbar shark Atlantic Snowy grouper Southern Atlantic coast Spanish mackerel Southern Atlantic Coast Tilefish Southern Atlantic coast Vermilion snapper Southern Atlantic coast Walleye pollock Northern Sea of Okhotsk Albacore tuna South Pacific Ocean Bigeye tuna Western Pacific Ocean Skipjack tuna Central Western Pacific Striped marlin Southwestern Pacific Ocean Yellowfin tuna Central Western Pacific Chilean jack mackerel Chilean EEZ and offshore Dover sole Pacific Coast Gopher rockfish Southern Pacific Coast Pacific sardine Pacific Coast Shortbelly rockfish Pacific Coast Starry flounder Northern Pacific Coast Starry flounder Southern Pacific Coast Tasmanian giant crab Tasmania Walleye pollock Western Bering Sea Atlantic cod Baltic Areas 22 and

9 242 Atlantic cod Baltic Areas Atlantic cod Kattegat Herring ICES Herring ICES Herring ICES Herring Iceland (Summer spawners) Herring ICES common European sole ICES Kattegat and Skagerrak Sprat ICES Baltic Areas Fourspotted megrim ICES VIIIc-IXa Hake Northeast Atlantic North Megrim ICES VIIIc-IXa common European sole Bay of Biscay Mackerel ICES Northeast Atlantic Whiting Northeast Atlantic Atlantic cod Irish Sea Atlantic cod West of Scotland Haddock Irish Sea Haddock West of Scotland European Plaice Irish Sea common European sole Irish Sea Atlantic cod North Sea Haddock ICES IIIa and North Sea Haddock Rockall Bank Norway pout North Sea Pollock ICES IIIa, VI and North Sea Sandeel North Sea common European sole ICES VIId Whiting ICES IIIa, VIId and North Sea Haddock ICES VIIb-k European Plaice ICES VIIf-g European Plaice ICES VIIe common European sole Celtic Sea common European sole Western English Channel Whiting ICES VIIe-k 0.79

10 Supplementary References 1. Gelman, A., Carlin, J. B., Stern, H. S. & Rubin, D. B. Bayesian data analysis. (Chapman & Hall/CRC, 2004). 2. Vehtari, A. & Gelman, A. WAIC and cross-validation in Stan. arxiv 1 15 (2014).