Increasing temporal variance leads to stable species range limits.

What prevents populations of a species from adapting to the novel environments outside the species' geographic distribution? Previous models highlighted how gene flow across spatial environmental gradients determines species expansion versus extinction and the location of species range limits. However, space is only one of two axes of environmental variation-environments also vary in time, and we know temporal environmental variation has important consequences for population demography and evolution. We used analytical and individual-based evolutionary models to explore how temporal variation in environmental conditions influences the spread of populations across a spatial environmental gradient. We find that temporal variation greatly alters our predictions for range dynamics compared to temporally static environments. When temporal variance is equal across the landscape, the fate of species (expansion versus extinction) is determined by the interaction between the degree of temporal autocorrelation in environmental fluctuations and the steepness of the spatial environmental gradient. When the magnitude of temporal variance changes across the landscape, stable range limits form where this variance increases maladaptation sufficiently to prevent local persistence. These results illustrate the pivotal influence of temporal variation on the likelihood of populations colonizing novel habitats and the location of species range limits.