Cyclicity of the Limit Periodic Sets for a Singularly Perturbed Leslie–Gower Predator–Prey Model with Prey Harvesting

• Published in: Journal of dynamics and differential equations Vol. 36; no. 2; pp. 1721 - 1758
• Main Authors: Yao, Jinhui, Huzak, Renato
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2024
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; Ordinary Differential Equations; Partial Differential Equations; Slow–fast normal form; Entry–exit function; Predator–prey model; Slow divergence integral; Slow–fast system; Blow-up
• ISSN: 1040-7294; 1572-9222
• DOI: 10.1007/s10884-022-10242-2

In this paper, we study the Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting. Our main focus is on the cyclicity of diverse limit periodic sets, including a generic contact point, canard slow–fast cycles, transitory canards, slow–fast cycles with two canard mechanisms, singular slow–fast cycle, etc. We develop new techniques for finding the maximum number of limit cycles produced by slow–fast cycles containing both the generic and degenerate contact point away from the origin (such slow–fast cycles are the transitory canards and cycles with two canard mechanisms). It can be applied not only to the Leslie–Gower predator–prey model, but more general systems as well. The main tool is geometric singular perturbation theory including cylindrical blow-up and the notion of slow divergence integral. We also study dynamics near the origin using non-standard techniques (constructing generalized normal sectors). The uniqueness and stability of a relaxation oscillation is shown using the notion of entry–exit function. Some interesting dynamical phenomena, such as relaxation oscillation and canard explosion, are simulated to illustrate the theoretical results.