A general framework for cycles in ecology

Theory predicts that indirect interactions in ecological networks sustain species diversity through oscillatory dynamics. However, a framework linking interaction structure to these cycles is lacking. We develop an analytical toolbox combining assembly graphs with a mathematical decomposition of interaction matrices into symmetric and anti-symmetric components. We find that assembly cycles--closed loops of species invasions--are suppressed when symmetric interactions dominate, reflecting strong intraspecific competition. Conversely, anti-symmetric dominance, indicating competitive asymmetries, leads to cycles like rock-paper-scissors and novel multispecies invasion patterns. As asymmetries increase, new cycles emerge involving both sequential and simultaneous invasions. Applying this approach to 21 plant community interaction matrices, we find few cycles due to prevalent self-limiting effects. Our work clarifies when indirect interactions drive cycles and introduces a simple ratio assessing symmetric versus anti-symmetric contributions, constraining cycle emergence, and species coexistence in nature. Significant statementUncertainties of the fundamental blocks that maintain biodiversity hinder our understanding of the dynamics we can observe in ecological communities. By applying a well-known mathematical approach to decomposing interactions among species within ecological communities, we discover great potential for observing rock-paper-scissors and more complex cyclic dynamics in nature. However, we predict and confirm analyzing multiple datasets that cyclic dynamics are rare because of two phenomena that pervade ecological communities. These are differences in species performance and the dominance of self-limiting effects. Our framework underscores the importance of studying simple properties of biotic interactions to predict complex ecological dynamics.