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A fundamental question in ecology is how are species added to the phenotypic or functional trait space when one moves from a species poor to species rich assemblage. Specifically, the functional volume of an assemblage can expand and/or be packed more tightly to accommodate an increasing number of species. Quantifying the packing and filling of trait space therefore provides critical information regarding the generation of species richness gradients, but it is also informative for our understanding of community assembly as the functional volume is expected to be constrained by abiotic factors while the degree of the packing of the volume should be limited by biotic interactions. In this investigation we quantify the packing and filling of trait volumes in tree assemblages in eastern North America. We demonstrate that both the functional volume and the degree of packing both increase with increasing species richness. However, the increase in the functional volume is less than expected suggesting that the overall functional space is constrained. We then show that null models that incorporate a functional volume constraint uncover a greater degree of separation in trait space than expected suggesting the importance of biotic interactions within an abiotically defined functional volume.