Stability and Hopf bifurcation analysis of a predator-prey system with multiple delays and generalized Allee effect

In this paper, we develop a predator-prey system with a parameterized generalized Allee effect function and multiple discrete delays. One delay accounts for the negative feedback in the prey, while the other represents the gestation period in the predator population. First, we demonstrate the positivity and boundedness of solutions for the non-delayed system and establish conditions for the existence and stability of equilibria. For the delayed model, we assess the impact of varying delays on the stability of equilibria, discovering that the system exhibits Hopf bifurcations for both delays. Additionally, we determine the crossing curves to explore the stability transitions of equilibria within the delay parameter space. By computing the normal form, we determine the direction, stability, and period of bifurcating periodic solutions. Finally, numerical simulations are conducted to validate the theoretical findings. These simulations reveal that for the Allee effect function considered in this paper, the stability of the system remains unaffected when the delay is comparatively minor. Nonetheless, as the delay grows, the system shifts from a state of stability to one of instability, which even leads to chaotic dynamics. Additionally, the combination of the two delays makes the oscillation frequency of the original chaos higher.