Population cycles have been remarkably resistant to explanation, in part because crucial experiments are rarely possible on appropriate spatial and temporal scales. Here we show how new approaches to nonlinear time-series analysis can distinguish between competing hypotheses for population cycles of larch budmoth in the Swiss Alps: delayed effects of budmoth density on food quality, and budmoth-parasitoid interactions. We re- examined data on budmoth density, plant quality, and parasitism rates. Our results suggest that the effect of plant quality on budmoth density is weak. By contrast, a simple model of budmoth-parasitoid interaction accounts for 90% of the variance in budmoth population growth rates. Thus, contrary to previous studies, we find that parasitoid-budmoth interaction