1. Perturbation analyses of population projection matrices predict the response of a population's growth rate to changes in lifestage-specific vital rates. Such predictions have been widely used in population management but their reliability remains hotly debated. 2. We grew replicate populations of the water flea Daphnia magna in controlled, density-independent conditions and subjected treatment populations to harvesting of the largest lifestage. We predicted the growth rate of treatment populations using sensitivity analysis (a linear approximation), and transfer function analysis (TFA; which captures nonlinear responses) applied to projection matrix models parameterized from the control populations. 3. When perturbation analyses considered only the direct effect of harvesting on adult survival, the growth rate of harvested populations (averaging 0.051) was significantly overestimated (average of 0.112) by TFA and non-significantly underestimated (average of 0.012) by sensitivity. 4. When the indirect effects of harvesting on other vital rates were accounted for in a structured perturbation, TFA gave accurate predictions (average growth rate of 0.068), while sensitivity gave significant underestimates (average of -0.043). 5. Our results demonstrate two crucial sources of error that may influence predictions of the impacts of demographic perturbations on population dynamics. First, impacts of stage-specific harvesting are inherently nonlinear, hence predictions based on sensitivity must be treated with caution. Second, stage-specific perturbations can change non-target demographic rates, even in the absence of adaptation. 6. Population managers should consider both nonlinear and indirect effects of perturbations when designing management interventions. We encourage the development of methods to assess the robustness of predictions to unforeseen perturbation structures and indirect harvesting impacts.