Stability and Bifurcation Analysis of a Predator–Prey Model With Generalized Allee Effect on the Prey Population

This research describes a predator–prey system that takes into account the generalized Allee effect, aiming to derive general conclusions applicable to specific Allee effect functions through the use of a generalized function. To make sure the suggested model was accurate from a mathematical perspective, we first investigated the solutions to determine whether they were positive and whether they were bounded. We then assessed the existence of equilibria and the stability behaviors they exhibit and confirmed the emergence of both Hopf and transcritical bifurcations. We discovered that the trivial equilibrium is a saddle or saddle-node in the case of the population subjected to the weak Allee effect and a stable node in the case of the strong Allee effect. Regardless of whether the population is influenced by the weak Allee effect or the strong Allee effect, the system undergoes a transcritical bifurcation, and under certain conditions, a Hopf bifurcation may also occur. In addition, when the sign of the derivative of the Allee effect function with respect to its threshold is positive, the Allee effect contributes to the growth of the predators, and if the sign is negative, the Allee effect leads to a drop in the number of predators at the coexisting equilibrium but have no effect on the equilibrium density of prey.