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Population cycles have long fascinated ecologists. Even in the most‐studied populations, however, scientists continue to dispute the relative importance of various potential causes of the cycles. Over the past three decades, theoretical ecologists have cataloged a large number of mechanisms that are capable of generating cycles in population models. At the same time, statisticians have developed new techniques both for characterizing time series and for fitting population models to time‐series data. Both disciplines are now sufficiently advanced that great gains in understanding can be made by synthesizing these complementary, and heretofore mostly independent, quantitative approaches. In this paper we demonstrate how to apply this synthesis to the problem of population cycles, using both long‐term population time series and the often‐rich observational and experimental data on the ecology of the species in …