Methods for calculating coexistence mechanisms: Beyond scaling factors

How do species coexist? A framework known as modern coexistence theory can 'measure coexistence' by partitioning invasion growth rates into coexistence mechanisms, terms which correspond to classes of explanations for coexistence. There are several reasonable ways to define coexistence mechanisms, each depending on exactly how a species perturbed to low density (the invader) is compared to other species that remain at their typical densities (the residents). Using conceptual arguments and two case studies, we compare five methods for calculating coexistence mechanisms: 1) scaling factors, the traditional approach which attempts to eliminate the linear effects of regulating factors; 2) the simple comparison, which gives equal weight to all resident species; 3) generation time scaling, a novel method which corrects for intrinsic differences in population-dynamical speed; 4) beta scaling, where resident growth rates are scaled by a measure of relative sensitivity to competition; and 5) the invader-invader comparison, a previously obscure method in which a focal species is compared to itself at high versus low density. We find that the conventional scaling factors can lead to nonsensical results when species have strong and asymmetric niche differences; though scaling factors can be useful in certain theoretical studies, they are not recommended for explaining coexistence in real communities. Invader-invader comparisons are also problematic, as they do not effectively measure specialization or niche differentiation. The universally-applicable simple comparison often works well, but can give counterintuitive results when species have disparate generation times. The beta scaling method often works well in simple models, but faces implementation problems in complex models. We tentatively recommend generation time scaling as the all-purpose method for calculating coexistence mechanisms.