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 $O(\frac{\log n}{n})$, 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 $O(\frac{1}{n})$ cannot be obtained, even for extremely regular maps.

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.