The stability of population oscillations in ecological systems is considered. Experiments suggest that in many cases the single patch dynamics of predator-prey or host-parasite systems is extinction prone, and stability is achieved only when the spatial structure of the population is expressed via desynchronization between patches. A few mechanisms have been suggested so far to explain the inability of dispersal to synchronize the system. Here we compare a recently discovered mechanism, based on the dependence of the angular velocity on the oscillation amplitude, with other, already known conditions for desynchronization. Using a toy model composed of diffusively coupled oscillators we suggest a classification scheme for stability mechanisms, a scheme that allows for either a priori (based on the system parameters) or a posteriori (based on local measurements) identification of the dominant process that yields desynchronization.