Abstract Global change may induce changes in savanna and forest distributions, but the dynamics of these changes remain unclear. Classical biome theory suggests that climate is predictive of biome distributions, such that shifts will be continuous and reversible. This view, however, cannot explain a widely observed mismatch between climate and tree cover, which some argue results from fire-vegetation feedbacks maintaining savanna and forest as bistable states, such that, instead, shifts will be discontinuous and irreversible. This bistable model, however, cannot reproduce the spatial aggregation of biomes. Here, we suggest that both models are limited in that they ignore spatial processes, such as dispersal. We examine the contributions of dispersal to determining savanna and forest distributions using a reaction-diffusion model, comparing results qualitatively to empirical savanna and forest distributions in Africa. The diffusion model induces spatially aggregated distributions, separated by a stable savanna-forest boundary. The equilibrium position of that boundary depends not only on precipitation but also on the curvature of precipitation contours with some history dependence (although less than in the bistable model). This model predicts different dynamics in response to global change: the boundary continuously tracks climate, recovering following disturbances, unless remnant biome patches are too small.