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Fear-driven predator–prey dynamics with prey refuge: analytical framework and physics-informed neural network approach
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DOI：10.3389/frai.2026.1868693.png)
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Abstract
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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 cβ(1-δ)<2aη 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.
Keywords:
stability analysis
physics-informed neural networks
bifurcation
fear effect
predator–prey model
prey refuge
Routh–Hurwitz criterion
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