Species persistence can be threatened by substantial temporal variation in food resources over time. On the other hand, spatial heterogeneity in resources at the landscape scale might allow mobile consumers to compensate for temporal variability in resource availability at the local scale. We evaluated this hypothesis, using an extensive data set on foraging, grass growth, and movement by Thomson's gazelles living on the Serengeti Plains. Here we show that modelled populations of Thomson's gazelles can only persist under Serengeti conditions in the face of observed levels of rainfall stochasticity by making adaptive movements to take advantage of ephemeral spatial distributions of food resources. More importantly, our models suggest that Thomson's gazelles in Serengeti require unrestricted access to relatively large areas of grassland (> 1600 km2) to guarantee long-term persistence, particularly when there is positive spatial autocorrelation in resource abundance, as is the case in Serengeti. If this proves to be true for other species and/or other systems, then understanding of complex behavioural responses to spatially and temporally heterogeneous food supplies may be essential to successful conservation of grazing herbivores.