This paper develops a method to infer changing ecological rules from long-term moose and wolf counts on Isle Royale National Park. The authors use 61 years of population data (1959–2019) and a physics-aware deep learning tool to recover both constants and year-by-year changes in birth, death, and predation rates. Their approach also successfully reproduces a large observed drop in moose numbers that occurred after the training period.

The researchers model the system as a non-autonomous prey–predator system. “Non-autonomous” means some model parameters can change with time to represent shifting environments. They use a θ-logistic rule for moose growth, which is a flexible form of density-dependent growth that is thought to suit large mammals better than simple logistic growth. For predation they compare two common choices. Holling type‑II assumes the kill rate depends mainly on prey density. Ratio‑dependent response assumes the kill rate depends on the ratio of prey to predators, which captures strong competition or interference among wolves. Past studies of Isle Royale have favored ratio dependence, and the authors test both forms here.

To estimate time-varying and constant parameters from noisy, irregular time series, the team uses Physics-Informed Neural Networks (PINNs). A PINN is a neural network trained not only to fit data but also to satisfy the underlying differential equations by using automatic differentiation. The authors use a backward‑compatible PINN (bc‑PINN), add a self‑adaptive weighting scheme to balance different training objectives, and apply transfer learning to improve stability and speed. Before fitting, they run a structural identifiability analysis to check that the available data can in principle determine the model parameters.

Their framework reconstructs the historical population trajectories well for both functional responses, and it estimates both time-dependent and constant parameters. When asked to predict beyond the fitted period, the ratio‑dependent model showed a better trend. The authors report that their method was able to predict the sudden decline in moose observed in 2020, suggesting the time-varying parameter approach can capture important shifts in the system.