Dynamics of a Predator–Prey System in the Chihuahuan Desert: A Discrete-Time Model with Empirical Validation

Understanding predator–prey dynamics in arid ecosystems is essential for anticipating how habitat transformation (e.g. shrub encroachment) reshapes species interactions. We formulate a discrete-time, four-species model for jackrabbits (Lepus californicus), cottontails (Sylvilagus audubonii), coyotes (Canis latrans), and kit foxes (Vulpes macrotis) that couples logistic prey growth, predation, and intraguild suppression between predators. We derive biologically feasible equilibria and study local dynamics and bifurcations. We introduce and analyze a four-dimensional (4D) predator–prey map with explicit intraguild suppression, and show that the Jacobian at the interior equilibrium factorizes into linear and cubic parts, enabling stability analysis via cubic Jury/Schur inequalities. Building on this structure, we provide closed-form, eigenvalue-free criteria specialized to our 4D map for detecting Neimark–Sacker (NS) and period-doubling (PD) bifurcations, and locate flip–Neimark–Sacker (FNS) codimension-two points by solving intersecting algebraic conditions. We chart two-parameter dynamics in the [Formula: see text]-plane, combining periodicity maps with full Lyapunov-spectrum classification (fixed point, limit cycle, 2-torus, chaos, hyperchaos) and the Kaplan–Yorke fractal dimension, thereby quantifying the geometry and intensity of complex dynamics. We carry out an empirical calibration using 2014–2020 camera-trap data from the Jornada Basin LTER site, estimate parameters by nonlinear least squares, and assess fit with RMSE, [Formula: see text], Durbin–Watson, and Shapiro–Wilk diagnostics. The model reproduces observed multi-species trajectories with high accuracy, supporting the ecological realism of the proposed interactions. Collectively, these advances deliver an analytically transparent and data-informed framework for diagnosing thresholds (NS, PD, FNS) that govern transitions from stable coexistence to multi-frequency oscillations and chaos/hyperchaos. The results clarify how modest shifts in growth/predation and intraguild effects can reorganize community dynamics under ongoing dryland habitat change.