Dynamics on a Leslie–Gower Model with Simplified Holling IV Function Response, Allee Effect and Predator Cannibalism

In this paper, the stability and bifurcation analysis for a Leslie–Gower model with simplified Holling IV functional response, strong Allee effect and predator cannibalism are investigated. First, we examine the existence and stability of possible equilibria, as well as the boundedness of the system. Specifically, the asymptotic dynamics near the origin is performed using a blow-up transformation. Furthermore, various bifurcations are explored, including saddle-node bifurcation, supercritical and subcritical Hopf bifurcations, Bautin bifurcation and Bogdanov–Takens bifurcations of codimensions 2 and 3. The system exhibits diverse dynamical phenomena such as the emergence of a semi-stable limit cycle, the coexistence of a homoclinic loop and a limit cycle and the coexistence of two distinct limit cycles. In particular, with a specific set of parameters, two semi-stable limit cycles appear simultaneously, indicating that the system enters a dual-critical state. Finally, numerical simulations are carried out to validate the theoretical results.