Cyclicity of slow-fast cycles in a Lotka–Volterra model with weak Allee effect and Holling III function response

We extend previous work on Lotka–Volterra predator–prey models by incorporating a weak Allee effect in the predator population alongside a Holling III functional response (Li et al. in Chaos, Interdiscip. J. Nonlinear Sci. 33(7), 2023). Through a singular perturbation approach, we reduce the model to a slow–fast system and apply geometric singular perturbation theory to characterize its equilibria, bifurcations, and limit cycles. Under the condition $0 < \beta < \frac{1}{27}$ , the critical manifold assumes an S-shaped geometry, giving rise to singular Hopf bifurcations, canard explosions, and relaxation oscillations. Employing slow divergence integrals, we rigorously establish that both “headed” and “headless” canard cycles emerging near the fold points have cyclicity one. Numerical simulations corroborate the analytical findings and illustrate the abrupt transition from a small-amplitude limit cycle to a large-amplitude oscillation via canard explosion. Our results underscore the key role of the predator’s Allee effect in shaping population dynamics and offer novel theoretical insights into ecological oscillations.