Fear-driven predator-prey dynamics with prey refuge: analytical framework and physics-informed neural network...

Ecological communities are governed not only by direct consumption but also by indirect behavioral responses triggered by perceived predation risk. Predator-induced fear substantially suppresses prey reproductive output and foraging efficiency even when lethal predation is absent, a mechanism documented across a wide range of taxa including songbirds, ungulates, and marine invertebrates. Motivated by this observation, we formulate a deterministic two-species model that simultaneously incorporates fear-mediated prey growth reduction, partial prey refuge, density-dependent intraspecific regulation, and predator self-interference. The proposed model is distinguished from existing fear-refuge frameworks by jointly embedding four ecological mechanisms within a single functional-response denominator 1 + kv + αu, producing qualitatively novel stability thresholds absent in models incorporating only subsets of these effects. Biological admissibility is rigorously established through positivity and uniform boundedness proofs. The boundedness condition $c\beta \left(1-\delta \right)<2\sqrt{a\eta }$ is derived from first principles by applying Sylvester's criterion to the cross-interaction quadratic form. Three ecologically meaningful equilibria are identified and their local stability is characterized via carefully re-derived Jacobian linearization and the Routh-Hurwitz criterion. Numerical experiments via the fourth-order Runge-Kutta method reveal convergence to a stable coexistence equilibrium across the explored parameter ranges, with the approach transitioning from a stable node to a stable focus as predation intensifies; no sustained oscillations are observed. A physics-informed neural network (PINN) is constructed with four hidden layers of 64 neurons each, tanh activations, Adam followed by L-BFGS training over 10,000 iterations, and 200 collocation points, achieving maximum absolute errors of 7.98 × 10^-3 (prey) and 5.83 × 10^-3 (predator) relative to the RK4 reference. Comparison with a data-driven neural network of identical architecture shows a fivefold accuracy improvement from the physics-informed loss. Numerical evidence for global stability is reported; rigorous Lyapunov-based analysis is identified as future work.