The geometry of the distance-decay of similarity in ecological communities

Understanding beta-diversity has strong implications for evaluating the extent of biodiversity and formulating effective conservation policy. Here, we show that the distance-decay relationship, an important measure of beta-diversity, follows a universal form which we call the piecewise quadratic model. To derive the piecewise quadratic model, we develop a new conceptual framework which is based on geometric probability and several key insights about the roles of study design (e.g., plot dimensions and spatial distributions). We fit the piecewise quadratic model to six empirical distance-decay relationships, spanning a range of taxa and spatial scales, including surveys of tropical vegetation, mammals, and amphibians. We find that the model predicts the functional form of the relationships extremely well, with coefficients of determination in excess of 0.95. Moreover, the model predicts a phase transition at distance scales where sample plots are overlapping, which we confirm empirically. Our framework and model provide a fundamental, quantitative link between distance-decay relationships and the shapes of ranges of taxa.