Vigilance and Defensibility in Discrete Predator-Prey Dynamics: Bifurcations, Strong Resonances, and Empirical Validation with Moose-Wolf Data

Behavioral defenses can shape predator-prey dynamics as strongly as direct consumption. A two-dimensional discrete-time model is developed that separates adaptive vigilance by prey (reducing effective intrinsic growth) from baseline defensibility against predation (attenuating interaction strength). The positivity of solutions is established and constructed a forward-invariant region, then a complete local analysis of equilibria is presented. On the prey axis, the restriction reduces to a logistic map, yielding a flip (period-doubling) bifurcation of the boundary equilibrium; at the predator-invasion threshold the boundary and coexistence states exchange stability via a transcritical bifurcation. For the interior equilibrium, the closed-form stability thresholds are derived and proved the occurrence of a Neimark--Sacker bifurcation; the first Lyapunov coefficient is computed explicitly and is negative, implying a supercritical transition to small-amplitude invariant circles. Furthermore, it is located and unfolded codimension-two strong resonances (1:2, 1:3, 1:4), providing normal-form coefficients and transversal intersection conditions. Parameter-plane computations and time series corroborate the theory, revealing regimes of cycles, quasiperiodicity, and chaos (as indicated by positive maximal Lyapunov exponents). In addition, the system is calibrated to a well-studied, long-term moose-wolf interaction, discussing parameter identifiability and recovering characteristic oscillations. Collectively, these findings present an empirical and unified theoretical framework for how defensibility and vigilance regulate persistence, stability, and the occurrence of chaotic dynamics in ecological models. Numerical simulation is presented to illustrate the theoretical discussion.