Economic impacts of invasive species worldwide are substantial. Management strategies have been incorporated in population models to assess the effectiveness of management for reducing density, with the implicit assumption that economic impact of the invasive species will also decline. The optimal management effort, however, is that which minimizes the sum of both the management and impact costs. The relationship between population density and economic impact (what we call the "density-impact curve") is rarely examined in a management context and could take several nonlinear forms. Here we determine the effects of population dynamics and density-impact curves of different shapes on optimal management effort and discover cases where management is either highly effective or a waste of resources. When an inaccurate density-impact curve is used, the increase in total costs due to over- or underinvestment in management can be large. We calculate the increase in total costs incurred if the density-impact curve is incorrect and find that the greater the maximum impact caused by an invasive species, the more important it is not only to reduce its density, but also to know the shape of the density-impact relationship accurately. Lack of information regarding the relationship between density and economic impact causes the most acute problems for invaders that cause high impact at low density, where management typically will be too little, too late. For species that are only problematic at high density, ignorance of the density-impact curve can lead to overinvestment in management with little reduction in impact.
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