Allee effects, invasion pinning, and species' borders

All species' ranges are the result of successful past invasions. Thus, mode ls of species' invasions and their failure can provide insight into the for mation of a species' geographic range. Here, we study the properties of inv asion models when a species cannot persist below a critical population dens ity known as an "Allee threshold." In both spatially continuous reaction-di ffusion models and spatially discrete coupled ordinary-differential-equatio n models, the Allee effect can cause an invasion to fail. In patchy landsca pes (with dynamics described by the spatially discrete model), range limits caused by propagation failure (pinning) are stable over a wide range of pa rameters, whereas, in an uninterrupted habitat (with dynamics described by a spatially continuous model), the zero velocity solution is structurally u nstable and thus unlikely to persist in nature. We derive conditions under which invasion waves are pinned in the discrete space model and discuss the ir implications for spatially complex dynamics, including critical phenomen a, 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 pl ausible ecological conditions, species' ranges may be limited by an Allee e ffect. Several example systems appear to fit our general model.