Chaotic dynamics and its linear feedback control in a predator–prey interaction model with predation fear and wind flow effect using Physics Informed Neural Networks

In this study, we develop and analyze a tri-trophic predator prey model that incorporates predator-induced fear in the prey population and wind-driven modulation of predation efficiency. The model represents interactions among insects (prey), birds (predators), and feral cats (top predators), where fear induces non-consumptive effects that suppress prey growth, while wind acts as an environmental disturbance influencing intermediate hunting efficiency. We establish the positivity, boundedness, and existence of biologically feasible equilibria and derive sufficient conditions for both local and global stability using Jacobian analysis and Lyapunov functions. Conditions for saddle–node, transcritical, and Hopf bifurcations are obtained, revealing critical thresholds that govern stability loss and the emergence of oscillatory dynamics. Comprehensive numerical investigations including bifurcation diagrams and iso-spike patterns reveal a wide range of dynamical behaviors, from steady states and periodic oscillations to chaos. The presence of chaos is confirmed through the computation of maximum Lyapunov exponents. To suppress chaotic oscillations, a linear feedback control strategy is designed around the interior equilibrium, and its effectiveness is verified analytically via eigenvalue analysis and numerically through time-series simulations and control-parameter bifurcation diagrams. Global sensitivity analysis based on Partial Rank Correlation Coefficients (PRCC) identifies the most influential parameters governing long-term system dynamics. In addition, a Physics-Informed Neural Network (PINN) framework is employed for robust parameter estimation, representing a novel application of PINNs to chaotic ecological systems. The results indicate that increased predation fear has a stabilizing effect by suppressing chaos, whereas strong wind flow tends to destabilize the system by promoting oscillatory and chaotic dynamics. PINNs method found four unknown parameters with only 0.51% error on average. The neural network predictions matched the true population dynamics with R2=0.99, meaning it explained 99% accuracy. This worked 100 to 500 times faster than traditional optimization methods. Overall, the study highlights the combined roles of behavioral responses, environmental disturbances, control strategies, and machine learning tools in shaping complex ecological dynamics.