Topological characterization of ecological dynamics via persistent homology

Conventional linear methods provide valuable insights into ecological population dynamics but may not fully capture their underlying geometric complexity. We present a large-scale topological characterization of ecological population dynamics using persistent homology applied to 500 time series from the BioTIME database, spanning marine, terrestrial, and freshwater ecosystems. Time-delay embedding and Vietoris–Rips filtration yield two classes of topological invariants: Betti numbers ${\beta }_k$, which count persistent topological features, and persistence entropy $H_k$, which quantifies their distributional complexity. These invariants quantify multiscale cyclic organization in a manner that complements spectral and autoregressive approaches. Three principal components capture $92.6\mathrm{\%}$ of topological variance, revealing that ecological attractor geometry is fundamentally low-dimensional. Realm membership explains less than $0.01\mathrm{\%}$ of this variance, demonstrating that habitat type imposes negligible constraints on dynamical complexity relative to within-realm heterogeneity, a finding that challenges the widely assumed structuring role of environmental context. An exceptionally strong coupling ( $\rho =0.989$) between ${\beta }_k$ and $H_k$ reflects an information-theoretic bound $H_k\le {\log }_2({\beta }_k)$. These results support shared dynamical mechanisms, including density dependence, predator-prey interactions, and life history trade-offs, as primary determinants of attractor topology, and they establish persistent homology as a noise-robust complement to conventional methods for comparative ecological analysis.