Multiple limit cycles in a piecewise-smooth predator-prey model with additive Allee effect

This paper focuses on the existence of limit cycles in a piecewise-smooth predator-prey model with additive Allee effect. We establish two mechanisms for generating limit cycles through bifurcation analysis and geometric singular perturbation theory. Firstly, when the positive equilibrium is located near the discontinuous boundary, the system is composed of a regular subsystem and a singularly perturbed subsystem. The study reveals that there are exactly two nested crossing limit cycles surrounding a boundary focus. Secondly, when the positive equilibrium is away from the discontinuous boundary, the system is a nonlinear piecewise-smooth continuous system. The study reveals that the additive Allee effect can trigger the Hopf bifurcation. In addition, combining with numerical simulations, we further confirm that there can be three or more nested limit cycles. These findings not only reveal multiple stability but also offer a theoretical foundation for the complex dynamics of ecosystems.