Slow–fast dynamics and diffusion-driven pattern formation in a predator–prey model with Holling type IV response and strong predator Allee effect

This paper investigates a singularly perturbed predator–prey model featuring logistic prey growth, a non-monotone Holling type IV response, and a strong predator Allee effect. Using geometric singular perturbation theory and the Krupa–Szmolyan framework, we derive a slow–fast normal form and explicit asymptotic formulas for singular Hopf and maximal canard curves. These results identify parameter regimes for canard explosions and global bifurcations governing transitions between population extinction and large-amplitude boom-bust cycles. To explore the interface between temporal and spatial dynamics, we extend the model to a reaction–diffusion framework and specifically investigate how timescale separation modulates spatial self-organization. We demonstrate that the timescale-separation parameter ɛ acts as a key determinant in dictating Turing instabilities; it not only shifts the instability thresholds but also significantly influences the morphology selection of spatial patterns, enabling destabilization even in non-standard diffusion regimes. A weakly nonlinear analysis yields coupled Ginzburg–Landau amplitude equations, predicting hexagonal pattern selection and transitions between spot-like and stripe-like morphologies. Our findings reveal that the disparity in timescales is fundamental to the robustness and selection of spatial structures, providing a theoretical explanation for spatial diversity and system resilience in complex ecosystems. Numerical simulations corroborate the analytical predictions, linking canard-mediated critical transitions to timescale-driven pattern formation.