A general framework for invasion cycles in ecology.

Theory predicts that indirect interactions in ecological networks sustain species diversity through oscillatory dynamics. However, a framework linking interaction structure to the presence, type, and complexity of these cycles is lacking. Here, we develop an analytical toolbox combining invasion graphs with a mathematical decomposition of interaction matrices into symmetric and antisymmetric components. We find that invasion cycles-closed loops of species invasions-are suppressed when symmetric interactions dominate, reflecting strong self-limitation. Conversely, antisymmetric dominance, indicating competitive asymmetries, leads to the well-known cycles of single-species invasion such as rock-paper-scissors as well as novel multispecies invasion patterns, in which several species simultaneously invade each transition of the cycle. As asymmetries increase, more complex cycles involving both sequential and simultaneous invasions emerge. Yet this potential for cycles is suppressed as variability in intrinsic growth rates increases. Our work clarifies when interactions drive cycles and introduces a simple ratio that assesses symmetric versus antisymmetric contributions in the interaction matrix, constraining cycle emergence and the number of species they can sustain.