A machine learning framework for stage-structured predator-prey systems with fear effect: Sparse Identification, Gaussian Process Regression, and bifurcation analysis

This paper presents a comprehensive machine learning framework for the analysis of a stage-structured predator-prey system incorporating the fear effect, intra-specific competition among juvenile predators, and maturation delay. The three-species model, comprising prey (N), juvenile predators (P ₁), and mature predators (P ₂), is governed by a non-linear ordinary differential equation system in which the prey growth rate is modulated by a fear factor 1/(1+kP ₂). We first establish the existence and local stability of the coexistence equilibrium E ^* = (16.02, 16.09, 32.19) and rigorously identify a supercritical Hopf bifurcation at a critical fear intensity k _crit ≈ 0.067, beyond which the equilibrium becomes unstable and a stable limit cycle emerges. Five machine learning methods are deployed and benchmarked: (i) Sparse Identification of Non-linear Dynamics (SINDy) with a generic polynomial library, which recovers the predator equations exactly (R ² = 1.0) and approximates the prey equation (R ² = 0.9997); (ii) Dynamic Mode Decomposition (DMD), whose linear assumption yields poor reconstruction (R ² = -2.885 for P ₂), demonstrating the inadequacy of linear methods for this non-linear system; (iii) Gaussian Process Regression (GPR) with a periodic-RBF composite kernel, achieving RMSE values of 1.92, 1.03, and 0.84 from sparse noisy observations; (iv) a physics-constrained function approximation for recovering population trajectories from incomplete data; and (v) a polynomial surrogate ODE model for derivative field approximation. Additionally, the effect of maturation delay τ on system stability is examined through delay differential equation analysis with a fourth-order Runge-Kutta integrator, revealing delay-induced destabilization and transitions to limit cycle dynamics at τ ≥ 10. All numerical results are independently verified and all performance metrics are computed from actual analyses. The framework provides a rigorous, reproducible methodology for applying modern data-driven methods to stage-structured ecological models.