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\% $ of topological variance, revealing that ecological attractor geometry is fundamentally low-dimensional. Realm membership explains less than $ 0.01\% $ 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 \leq \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.