A machine learning framework for stage-structured predator–prey systems with fear effect: Sparse Identification, Gaussian Process Regression, and bifurcation analysis
- 1Department of Mathematics, Vellore Institute of Technology , , ,
Abstract
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 (P1), and mature predators (P2), is governed by a non-linear ordinary differential equation system in which the prey growth rate is modulated by a fear factor 1/(1+kP2). 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 kcrit ≈ 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 (R2 = 1.0) and approximates the prey equation (R2 = 0.9997); (ii) Dynamic Mode Decomposition (DMD), whose linear assumption yields poor reconstruction (R2 = −2.885 for P2), 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.
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BibTeX
@article{ramraj2026machine,
title = {{A machine learning framework for stage-structured predator–prey systems with fear effect: Sparse Identification, Gaussian Process Regression, and bifurcation analysis}},
author = {G. Ramraj and T. Poornima},
journal = {Frontiers in artificial intelligence},
year = {2026},
month = oct,
doi = {10.3389/frai.2026.1887024},
url = {https://europepmc.org/article/PMC/PMC13719457},
note = {CC BY},
}
RIS
TY - JOUR
TI - A machine learning framework for stage-structured predator–prey systems with fear effect: Sparse Identification, Gaussian Process Regression, and bifurcation analysis
AU - G. Ramraj
AU - T. Poornima
PY - 2026
DA - 2026/10/16
JO - Frontiers in artificial intelligence
DO - 10.3389/frai.2026.1887024
UR - https://europepmc.org/article/PMC/PMC13719457
AB - 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 (P1), and mature predators (P2), is governed by a non-linear ordinary differential equation system in which the prey growth rate is modulated by a fear factor 1/(1+kP2). 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 kcrit ≈ 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 (R2 = 1.0) and approximates the prey equation (R2 = 0.9997); (ii) Dynamic Mode Decomposition (DMD), whose linear assumption yields poor reconstruction (R2 = −2.885 for P2), 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.
SN - 2624-8212
ER -