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Fishery managers need to establish harvesting strategies that balance economic benefits against ecological objectives, including avoiding unintended catastrophic consequences. Our results show that classical assumptions for fisheries management can yield severe instabilities in our quantified views of socioecological tradeoffs, making their ability to inform stakeholder preferences questionable. The complex ecological interactions implied by different parameterizations of such systems yield highly complex and nonlinear dynamic properties with multiple distinct basins of attraction. We show that small changes in our deeply uncertain representations of predator-prey systems can fundamentally shift their dynamics and the validity of candidate management strategies for harvest. 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