Analysis of a Leslie–Gower Model with Allee Effect on Birth Rate and Saturated Fear Effect

The stability of the ecosystem is essential for the sustainable development of the earth. Studying the population model’s dynamic behavior, including the Allee and anxiety effects, can better represent the ecosystem’s working mechanism, which is crucial for preserving ecological balance. In light of this, the objective of this paper is to build a Leslie–Gower model that incorporates Allee effect on the birth rate and the saturated fear effect on the predator, then analyze its dynamic behavior and the impact of the saturated fear effect on population density. In the process of analysis, the existence and stability of boundary and positive equilibria are established, demonstrating that the origin is an attractor using the blow-up method. By varying the saturated fear effect parameter, the corresponding system will undergo supercritical, sub-critical, and even degenerate Hopf bifurcations. The existence of Bogdanov–Takens bifurcation of codimension-2 (or codimension-3) is demonstrated near the unique positive equilibrium. In light of these bifurcation phenomena, the validity of the theoretical results is confirmed through graphical representations via numerical simulations. The results show that while the fear effect on the predator favorably contributes to the ecological stability, high levels of either the Allee effect or the saturated fear effect pose a hazard to the stability of the ecosystem.