Gradual changes in exploitation, nutrient loading, etc produce shifts between alternative stable states (ASS) in ecosystems which, quite often, are not smooth but abrupt or catastrophic. Early warnings of such catastrophic regime shifts are fundamental for designing management protocols for ecosystems. Here we study the spatial version of a popular ecological model, involving a logistically growing single species subject to exploitation, which is known to exhibit ASS. Spatial heterogeneity is introduced by a carrying capacity parameter varying from cell to cell in a regular lattice. Transport of biomass among cells is included in the form of diffusion. We investigate whether different quantities from statistical mechanics—like the variance, the two-point correlation function and the patchiness—may serve as early warnings of catastrophic phase transitions between the ASS. In particular, we find that the patch-size distribution follows a power law when the system is close to the catastrophic transition. We also provide links between spatial and temporal indicators and analyse how the interplay between diffusion and spatial heterogeneity may affect the earliness of each of the observables. We find that possible remedial procedures, which can be followed after these early signals, become more effective as the diffusion becomes lower. Finally, we comment on similarities of and differences between these catastrophic shifts and paradigmatic thermodynamic phase transitions like the liquid–vapour change of state for a fluid like water.