Fig. 2

Latent gradient regression algorithm enables parameterization of generalized Lotka–Volterra (gLV) network model. a A flowchart showing how linear regression (LR; shaded in light yellow) is expanded to include gradients (\( g \)) as latent parameters in our latent gradient regression (LGR; shaded in light blue) algorithm. \( X\left( t \right) \): observed time series; \( \widehat{X}\left( {\text{t}} \right) \): simulated time series; \( \alpha ,\beta \): gLV model coefficients; \( g \): gradients (i.e., time-derivatives of \( \ln \left( {X\left( t \right)} \right) \); \( J\left( {\alpha ,\beta } \right) \): penalty function; \( \left\| \cdot \right\|_{F} \): Frobenius norm; LM: Levenburg–Marquardt. b, c Benchmark of the LGR algorithm using synthetic data in the absence (b) and presence (c) of noise. The synthetic data was generated by a 3-species gLV network model (b), where solid arrows represent positive (point end)/negative (blunt end) interactions and dashed arrows represent intrinsic population growth (incoming)/decline (outgoing) in the absence of other species (the same as in d,e). The best-fit model predictions (lines) are contrasted with the synthetic data (filled circles) in the lower part of b. MSE: mean squared error. d, e Performance of the LGR algorithm in inferring real ecosystems. d The protozoan predator (Didinium nasutum)-prey (Paramecium aurelia) ecosystem. Unit of abundance in y axis: individuals/mL. e The ecosystem of a rotifer predator (Brachionus calyciflorus) and two algae prey (Chlorella vulgaris). Unit of abundance in y axis: 10 individual females/mL for the rotifer and 10⁶ cells/mL for the algae. In both d and e, the inferred gLV models are shown in the upper part and their predictions (lines), together with the observed data (empty circles), are shown in the lower part. To eliminate the initial transient period, the first 13 and 4 data points of population dynamics in d and e were removed respectively