1. Population cycles in voles are often thought to be generated by one-year delayed density-dependence on the annual population growth rate. In common voles, however, it has been suggested by Turchin (2003) that some populations exhibit first-order cycles, resulting from strong overcompensation (i.e. carrying capacity overshoots in peak years, with only an effect of the current year abundance on annual growth rates). 2. We focus on a common vole (Microtus arvalis) population from western France, that exhibits 3-year cycles. Several overcompensating nonlinear models for populations dynamics are fitted to the data, notably those of Hassell, and Maynard-Smith and Slatkin. 3. Overcompensating direct density-dependence (DD) provides a satisfactory description of winter crashes, and one-year delayed density-dependence is not responsible for the crashes, thus these are not classical second-order cycles. A phase-driven modulation of direct density-dependence maintains a low-phase, explaining why the cycles last three years instead of two. Our analyses suggest that some of this phase-dependence can be expressed as one-year delayed DD, but phase-dependence provides a better description. Hence modelling suggests that cycles in this population are first-order cycles with a low phase after peaks, rather than fully second-order cycles. 4. However, based on the popular log-linear second-order autoregressive model, we would conclude only that negative delayed density-dependence exists. The additive structure of this model cannot show when delayed DD occurs (here, during lows rather than peaks). Our analyses thus call into question the automated use of second-order log-linear models, and suggests that more attention should be given to non-(log)linear models when studying cyclic populations. 5. From a biological viewpoint, the fast crashes through overcompensation that we found suggest they might be caused by parasites or food rather than predators, though predators might have a role in maintaining the low phase and spatial synchrony.