A general framework for cycles in ecology

Understanding the mechanisms that promote cyclic dynamics among species is critical for predicting biodiversity patterns and community assembly. Here, we present a general framework for studying cycles in ecological systems. Using invasion graphs derived from the Lotka-Volterra equations, we discovered that cycles are primarily driven by the antisymmetric component of the interaction matrix, which captures directional imbalances in interaction strength and promotes non-hierarchical dynamics. We also generalized the concept of cycles to include multi-species equilibria and transitions between states with varying species compositions. Additionally, we found that intrinsic growth rates modulate the likelihood of cycles: homogeneous growth rates promote cycles, while greater variability, particularly with negative growth rates, inhibits them. Our results suggest that cycles are constrained to ecological systems with pronounced interaction asymmetries and relatively uniform growth rates, providing a theoretical basis for their absence in natural communities.