Dynamical Analysis of a Discrete Amensalism System with Beddington–DeAngelis Functional Response and Double Allee Effects

In this paper, a discrete amensalism system with Beddington–DeAngelis functional response and double Allee effects is proposed, and the dynamical behaviors of the system in terms of both strong and weak Allee effects are considered. First, the existence and stability of equilibrium points in the system are discussed. It is shown that under different parameter conditions, the strong Allee effect leads to three locally asymptotically stable equilibrium points, while the weak Allee effect results in two globally asymptotically stable equilibrium points. This finding suggests that populations are more likely to achieve long-term stability and development when influenced by the weak Allee effect rather than the strong Allee effect. Second, the central manifold theorem and bifurcation theory reveal that systems with either strong or weak Allee effect can undergo transcritical bifurcation, fold bifurcation, and flip bifurcation. In view of the adverse effects of chaos on population dynamics, a hybrid control strategy is employed to effectively manage the chaos generated by flip bifurcation. Finally, several numerical simulations are conducted to explore the influence of the Allee effect on the system. Our research indicates that the double Allee effects contribute to system stability to a certain extent. Furthermore, the greater the intensity of the Allee effect, the longer the time required for the system to reach a stable steady-state solution. Notably, the strong Allee effect makes the system take more time to attain the steady-state solution compared to the weak Allee effect.