We introduce a novel framework for conceptualising, quantifying and unifying discordant patterns of species richness along geographical gradients. While not itself explicitly mechanistic, this approach offers a path towards understanding mechanisms. In this study, we focused on the diverse patterns of species richness on mountainsides. We conjectured that elevational range midpoints of species may be drawn towards a single midpoint attractor - a unimodal gradient of environmental favourability. The midpoint attractor interacts with geometric constraints imposed by sea level and the mountaintop to produce taxon-specific patterns of species richness. We developed a Bayesian simulation model to estimate the location and strength of the midpoint attractor from species occurrence data sampled along mountainsides. We also constructed midpoint predictor models to test whether environmental variables could directly account for the observed patterns of species range midpoints. We challenged these models with 16 elevational data sets, comprising 4500 species of insects, vertebrates and plants. The midpoint predictor models generally failed to predict the pattern of species midpoints. In contrast, the midpoint attractor model closely reproduced empirical spatial patterns of species richness and range midpoints. Gradients of environmental favourability, subject to geometric constraints, may parsimoniously account for elevational and other patterns of species richness.

Center for Population Biology University of California Davis CA 95616 USA

College of Forestry Beijing Forestry University Beijing 100083 China

Department of Biology University of Utah Salt Lake City UT 84112 USA

Department of Biology University of Vermont Burlington VT 05405 USA

Department of Botany and Biodiversity Research University of Vienna Rennweg 14 1030 Vienna Austria

Department of Ecology and Evolutionary Biology University of Colorado Boulder CO 80309 USA

Department of Ecology and Evolutionary Biology University of Connecticut Storrs CT 06269 USA

Department of Environmental Science University of Basel Basel Switzerland

Department of Geography University of Marburg 35032 Marburg Germany

Departmento de Ecologia Universidade Federal de Goiás CP 131 Goiânia GO 74 001 970 Brasil

Environmental Futures Research Institute Griffith University Nathan Qld 4111 Australia

Forest Ecology and Conservation Group Imperial College London Silwood Park Campus Buckhurst Road Ascot Berkshire SL5 7PY UK

Institute for Tropical Biology and Conservation Universiti Malaysia Sabah 88400 Kota Kinabalu Sabah Malaysia

Institute of Entomology Biology Centre of the Czech Academy of Sciences and Faculty of Science University of South Bohemia Branišovská 31 370 05 České Budějovice Czech Republic

Institute of Systematic Botany University of Zurich 8008 Zurich Switzerland

Life Sciences Department Natural History Museum South Kensington London SW7 5BD UK

New Guinea Binatang Research Center P O Box 604 Madang Papua New Guinea

Phyletisches Museum Friedrich Schiller Universität Jena 07743 Germany

Program in Ecology Evolution and Conservation Biology Department of Biology University of Nevada Reno NV 89557 USA

School of Natural and Physical Sciences University of Papua New Guinea P O Box 320 National Capital District Papua New Guinea

University of Colorado Museum of Natural History Boulder CO 80309 USA

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