Local-manifold-distance-based regression: an estimation method for quantifying dynamic biological interactions with empirical time series.

Quantifying species interactions based on empirical observations is crucial for ecological studies. Advancements in nonlinear time-series analyses, particularly S-maps, are promising for high-dimensional and non-equilibrium ecosystems. S-maps sequentially perform a local linear model fitting to the time evolution of neighbouring points on the reconstructed attractor manifold, and the coefficients can approximate the Jacobian elements corresponding to interaction effects. However, despite that the advantages in nonlinear forecasting with noise-contaminated data, these methodologies have a limitation in the Jacobian estimation accuracy owing to non-equidistantly stretched local manifolds in the state space. Herein, we therefore introduced a local manifold distance (LMD) concept, a non-equidistant measure based on the multi-faceted state dependency. By integrating LMD with advanced computation techniques, we presented a robust and efficient analytical method, LMD-based regression (LMDr). To validate its advantages in prediction and Jacobian estimation, we analysed synthetic time series of model ecosystems with different noise levels and applied it to an experimental protozoan predator-prey system with established biological information. The robustness to noise was the highest for LMDr, which also showed a better correspondence to expected predator-prey interactions in the protozoan system. Thus, LMDr can be applied to study complex ecological networks under dynamic conditions.