A major debate in ecology concerns the relative importance of intrinsic factors and extrinsic environmental variations in determining population size fluctuations^1-6. Spatial correlation of fluctuations in different populations caused by synchronous environmental shocks^2,7,8 is a powerful tool for quantifying the impact of environmental variations on population dynamics^8,9. However, interpretation of synchrony is often complicated by migration between populations^8,10. Here we address this issue by using time series from sheep populations on two island in the St Kilda archipelago^11-13. Fluctuations in the sizes of the two populations are remarkably synchronized over a 40-year period. A nonlinear time-series model shows that a high and frequent degree of environmental correlation is required to achieve this level of synchrony. The model indicates that if there were less environmental correlation, population dynamics would be much less synchronous than is observed. This is because of a threshold effect that is dependent on population size; the threshold magnifies random differences between populations. A refined model shows that part of the required environmental synchronicity can be accounted for by large-scale weather variations. These results underline the importance of understanding the interaction between intrinsic and extrinsic influences on population dynamics¹⁴.