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

摘要

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.

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