W. D. Kissling, Carsten F. Dormann, Juergen Groeneveld, Thomas Hickler, Ingolf Kühn, Greg J. McInerny, Jose M. Montoya, Christine Römermann, Katja Schiffers, Frank Martin Schurr, Alexander Singer, Jens-Christian Svenning, Niklaus E. Zimmermann, Robert B. O'Hara

• Aim Biotic interactions within guilds or across trophic levels have widely been ignored in species distribution models (SDMs). This synthesis outlines the development of species interaction distribution models (SIDMs), which aim to incorporate multispecies interactions at large spatial extents using interaction matrices. Location Local to global. Methods We review recent approaches for extending classical SDMs to incorporate biotic interactions, and identify some methodological and conceptual limitations. To illustrate possible directions for conceptual advancement we explore three principal ways of modelling multispecies interactions using interaction matrices: simple qualitative linkages between species, quantitative interaction coefficients reflecting interaction strengths, and interactions mediated by interaction currencies. We explain methodological advancements for static interaction data and multispecies time series, and outline methods to reduce complexity when modelling multispecies interactions. Results Classical SDMs ignoreAim Biotic interactions within guilds or across trophic levels have widely been ignored in species distribution models (SDMs). This synthesis outlines the development of species interaction distribution models (SIDMs), which aim to incorporate multispecies interactions at large spatial extents using interaction matrices. Location Local to global. Methods We review recent approaches for extending classical SDMs to incorporate biotic interactions, and identify some methodological and conceptual limitations. To illustrate possible directions for conceptual advancement we explore three principal ways of modelling multispecies interactions using interaction matrices: simple qualitative linkages between species, quantitative interaction coefficients reflecting interaction strengths, and interactions mediated by interaction currencies. We explain methodological advancements for static interaction data and multispecies time series, and outline methods to reduce complexity when modelling multispecies interactions. Results Classical SDMs ignore biotic interactions and recent SDM extensions only include the unidirectional influence of one or a few species. However, novel methods using error matrices in multivariate regression models allow interactions between multiple species to be modelled explicitly with spatial co-occurrence data. If time series are available, multivariate versions of population dynamic models can be applied that account for the effects and relative importance of species interactions and environmental drivers. These methods need to be extended by incorporating the non-stationarity in interaction coefficients across space and time, and are challenged by the limited empirical knowledge on spatio-temporal variation in the existence and strength of species interactions. Model complexity may be reduced by: (1) using prior ecological knowledge to set a subset of interaction coefficients to zero, (2) modelling guilds and functional groups rather than individual species, and (3) modelling interaction currencies and species effect and response traits. Main conclusions There is great potential for developing novel approaches that incorporate multispecies interactions into the projection of species distributions and community structure at large spatial extents. Progress can be made by: (1) developing statistical models with interaction matrices for multispecies co-occurrence datasets across large-scale environmental gradients, (2) testing the potential and limitations of methods for complexity reduction, and (3) sampling and monitoring comprehensive spatio-temporal data on biotic interactions in multispecies communities.…