Predicting changes in species' distributions is a crucial problem in ecology, with leading methods relying on information about species' putative climatic requirements. Empirical support for this approach relies on our ability to use observations of a species' distribution in one region to predict its range in other regions (model transferability). On the basis of this observation, ecologists have hypothesized that climate is the strongest determinant of species' distributions at large spatial scales. However, it is difficult to reconcile this claim with the pervasive effects of biotic interactions. Here, we resolve this apparent paradox by demonstrating how biotic interactions can affect species' range margins yet still be compatible with model transferability. We also identify situations where small changes in species' interactions dramatically shift range margins.

One of the key problems in ecology is our need to anticipate the set of locations in which a species will be found (hereafter species' distributions). A major source of uncertainty in these models is the role of interactions among species (hereafter biotic interactions). Unfortunately, it is difficult to directly study this problem at large spatial scales and we lack a clear understanding of when biotic interactions shape species' distributions. We show a simple, direct link between the ease of species' coexistence and the importance of competition for shaping species' distributions. We show that increasing the ease of species' coexistence reduces the influence of biotic interactions. Changing the spatial scale of the analysis can reduce the influence of species interactions, but only when it promotes regional coexistence. Using these ideas, we analyze the conditions under which biotic interactions alter species' distributions in a Lotka–Volterra model of competition along an environmental gradient and argue that coexistence theory, rather than scale alone, provides a guide to the influence of species interactions. Our results provide a guide to the facets of biotic interactions that are necessary to anticipate their effects on species distributions. As such, we expect our work will help the development of more realistic distribution models.

This paper investigates the exact rate of convergence in Ulam's method: a well-known discretization scheme for approximating the invariant density of an absolutely continuous invariant probability measure for piecewise expanding interval maps. It is shown by example that the rate is no better than O( log n n ), where n is the number of cells in the discretization. The result is in agreement with upper estimates previously established in a number of general settings, and shows that the conjectured rate of O( 1n ) cannot be obtained, even for extremely regular maps.

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