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Economists have relied on convexity assumptions for a long time but many natural resources that stem from ecosystems have non-convexities in their dynamics. This article illustrates the consequences of relaxing convexity assumptions for management and the role of fast and slow variables. The paper presents a general framework to handle systems with fast and slow variables, and illustrates the method using a model of coral reefs subject to fishing pressure. The insights obtained are used to discuss alternative management strategies.