The spatial scaling of stability is key to understanding ecological sustainability across scales and the sensitivity of ecosystems to habitat destruction. Here we propose the invariability–area relationship (IAR) as a novel approach to investigate the spatial scaling of stability. The shape and slope of IAR are largely determined by patterns of spatial synchrony across scales. When synchrony decays exponentially with distance, IARs exhibit three phases, characterized by steeper increases in invariability at both small and large scales. Such triphasic IARs are observed for primary productivity from plot to continental scales. When synchrony decays as a power law with distance, IARs are quasilinear on a log–log scale. Such quasilinear IARs are observed for North American bird biomass at both species and community levels. The IAR provides a quantitative tool to predict the effects of habitat loss on population and ecosystem stability and to detect regime shifts in spatial ecological systems, which are goals of relevance to conservation and policy.

We consider a simple model of an open partially expanding map. Its trapped set K in phase space is a fractal set. We first show that there is a well defined discrete spectrum of Ruelle resonances which describes the asymptotisc of correlation functions for large time and which is parametrized by the Fourier component ν on the neutral direction of the dynamics. We introduce a specific hypothesis on the dynamics that we call "minimal captivity". This hypothesis is stable under perturbations and means that the dynamics is univalued on a neighborhood of K. Under this hypothesis we show the existence of an asymptotic spectral gap and a fractal Weyl law for the upper bound of density of Ruelle resonances in the semiclassical limit ν → ∞. Some numerical computations with the truncated Gauss map illustrate these results.