<![CDATA[This study develops a two-dimensional predator-prey model incorporating logistic prey growth, prey group defense, alternative food, and ecological costs to the predator. The novelty of the model lies in the introduction of an alternative food quality parameter, which distinguishes the availability of alternative food from its biological effectiveness in supporting predator growth. The analysis focuses on the existence of equilibrium points, local stability, and forward bifurcation. The analytical results show that the model admits a trivial equilibrium, a predator-free equilibrium, and an interior equilibrium under certain existence conditions. The trivial equilibrium is always unstable, whereas the predator-free equilibrium is locally asymptotically stable when the quality of alternative food is below a critical value and loses its stability when the quality parameter exceeds this threshold. At the critical value, the system undergoes a forward bifurcation characterized by a stability transition from the predator-free equilibrium to a positive interior equilibrium. Numerical simulations confirm the analytical findings and illustrate three main scenarios: predator extinction when the alternative food quality is low, stable coexistence when the alternative food quality is high, and a stability transition through a bifurcation diagram as the alternative food quality parameter crosses its critical value. These results indicate that alternative food quality acts as a threshold parameter determining predator persistence and the transition from predator extinction to stable coexistence.]]>
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