A general framework for assembly cycles in ecology

Theory predicts that indirect interactions embedded in ecological networks maintain species diversity through cycles leading to oscillatory population dynamics. However, a frame-work that links the structure of species interactions (i.e., the interaction matrix) to the presence, type, and complexity of these cycles remains absent. Here, we first develop an analytical toolbox that combines invasion graphs, a method well-rooted in ecological the-ory, with mathematical approaches that decompose the species interaction matrix into its symmetric and anti-symmetric components. We discover that assembly cycles — sequences of species invasions forming closed loops in the invasion graph — are suppressed under the dominance of symmetric interactions. This ecological scenario corresponds to the condition where intraspecific competition exceeds interspecific competition. Conversely, cycles such as the classic rock-paper-scissors emerge when the anti-symmetric component dominates, which reflects strong competitive asymmetries between species. As these asymmetries increase, we uncover previously unreported cycles involving both sequential and simultaneous multispecies invasions. Then, applying our analytical toolbox to 21 empirically-determined interaction matrices of plant communities, we find that these natural communities exhibit few cycles because of the prevalence of self-limiting effects. Our work provides a tractable platform for investigating when indirect interactions are tightly linked to the emergence of cycles. This platform underscores that a simple ratio, assessing the relative importance between the symmetric and the anti-symmetric components of an interaction matrix determines the limits to the emergence of cycles in nature and the number of species they can maintain.