Work

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

Year: 2025

Type: article

Abstract: 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.

Source: Advances in Continuous and Discrete Models

DOI: 10.1186/s13662-025-04039-y

Authors: Kaiqi Tang, Jun Li, Kexin Chen, Mingju Ma

Institutions: Xidian University, Xi'an Polytechnic University

Language: English

Topic: Mathematical and Theoretical Epidemiology and Ecology Models

Subfield: Public Health, Environmental and Occupational Health

Field: Medicine

Domain: Health Sciences

Open Access status: hybrid

Estimated APC paid: $1,890

Funder: Xi'an Polytechnic University