We define diversity measures that take account of the varying similarities between species, and show how they can be used. We state an unexpected theorem on maximizing diversity: there is a single abundance distribution that maximizes diversity from all viewpoints simultaneously. There follows a broad-brush survey of magnitude, which is closely related to maximum diversity and is defined in the very wide generality of enriched categories. In the case of metric spaces, magnitude encodes fundamental geometric invariants of size (such as volume, surface area and dimension) and is related to the concept of capacity in potential theory.

Chapter 6 analyses the ecological mechanisms, and implications, of intraspecific trait variability (ITV) and some key approaches to take ITV properly into account in modern trait-based analyses. The different sources of ITV, genetic variation, epigenetic effects and phenotypic plasticity, are discussed and put in the context of species evolution, adaptation to environmental conditions, species distribution potential (including invasive species) and the effects of species on multiple ecosystem properties and trophic interactions. Different tools are provided to quantify how strong ITV affect ecological patterns. A comparison of within- vs between-species trait variability in a community is discussed. Tools showing how strong the effect of changes in species composition (turnover) compared to ITV along environmental gradients are provided. Finally, methods considering ITV to quantify trait differences between species, via trait overlap in trait probability distributions, are discussed in the light of modern tools measuring functional diversity across different scales

Chapter 3 provides an overview of the concepts and approaches needed to assess ecological and phenotypic differentiation between organisms. First, an historical perspective on earlier systems ‘classifying’ species in terms of their traits into different ‘types’ is provided. Second, other schemes such as the r/K continuum, the C-S-R scheme and the leaf economic spectrum are introduced. These approaches, aimed at defining different ‘types’ of organisms, are discussed in terms of their importance for interpreting ecological patterns and for communication with non-experts. A further distinction between response and effect functional groups is provided, with a guideline on how to define these groups with ‘a priori’ ecological hypotheses or ‘a posteriori’ data-driven approaches. The Gower distance is introduced as a useful way to characterize the differences between organisms in terms of multiple types of traits. At the same time, a number of often overlooked problems with this distance metric are discussed. The R material for this chapter illustrates these issues with practical examples.