<![CDATA[This paper introduces a new predator-prey model with supplementary food sorces for the predator and the Allee effect on prey. Analytical proof of the existence, uniqueness, non-negativity, and boundedness of the solutions validates the model. Three equilibrium points are found: a trivial point, whose local stability depends on the Allee effect’s strength; a semi-trivial point, and an interior point whose local stability depends on certain conditions. The Lyapunov function and La Salle invariance principle show that each equilibrium point is globally asymptotically stable. The system displays intricate dynamical behaviors, including forward and Hopf bifurcations, along with bistability, which is regulated by critical parameters such as the predation conversion rate, environmental protection rate, and supplementary food sources. Under a weak Allee effect, increasing the predation conversion rate or supplementary food can shift the system from predator extinction to oscillatory coexistence. In contrast, under a strong Allee effect, these increases may instead drive both species to extinction if thresholds are exceeded. Moreover, variations in environmental protection rate yield contrasting outcomes: while a higher rate under weak Allee conditions may stabilize prey and eliminate predators, under strong Allee conditions, it may lead to total extinction. These findings illustrate the intricate relationship between biological and environmental components, highlighting the necessity for preventive measures and supplementary resources in population outcomes. The findings indicate that environmental protection and supplementary food sources influence species persistence and extinction in predator-prey dynamics, which is crucial for ecological conservation.]]>
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