All species' ranges are the result of successful past invasions. Thus, models of species' invasions and their failure can provide insight into the formation of a species' geographic range. Here, we study the properties of invasion models when a species cannot persist below a critical population density known as an "Allee threshold." In both spatially continuous reaction-diffusion models and spatially discrete coupled ordinary-differential-equation models, the Allee effect can cause an invasion to fail. In patchy landscapes (with dynamics described by the spatially discrete model), range limits caused by propagation failure (pinning) are stable over a wide range of parameters, whereas, in an uninterrupted habitat (with dynamics described by a spatially continuous model), the zero velocity solution is structurally unstable and thus unlikely to persist in nature. We derive conditions under which invasion waves are pinned in the discrete space model and discuss their implications for spatially complex dynamics, including critical phenomena, in ecological landscapes. Our results suggest caution when interpreting abrupt range limits as stemming either from competition between species or a hard environmental limit that cannot be crossed: under a wide range of plausible ecological conditions, species' ranges may be limited by an Allee effect. Several example systems appear to fit our general model.

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Popular religion in late medieval and early modern Europe posited an invisible world densely populated with both personal spirits and impersonal forces that interacted constantly with the natural and social realms in ambiguous and unpredictable ways. This represented a stark contrast to elite theological discourse, which insisted on dividing the spirit world into a strict moral dichotomy of good and evil spirits, operating within a rigid causal taxonomy of natural, preternatural, and supernatural. During this period, deviations from this elite theological consensus were increasingly labeled “superstitious.” The cognitive science of religion postulates that beliefs about demons and spirits are constrained by the evolved cognitive architecture of our species, and in this article I show that the patterns of religious belief and practice targeted by the critics of superstition are compatible wth key hypotheses in the cognitive science of religion.