Bifurcation analysis of an additional food-provided predator-prey system with anti-predator behavior

This article analyzes the qualitative behavior of a predator-prey system where the predator receives extra food and the prey engages in anti-predator behavior to defend itself against attacks by the predator. The positivity and the boundedness of solutions to the system have been examined. The biologically well-posed equilibrium points of the proposed system are derived, and an analysis of their local stability is conducted. In specific situations, it is observed that the solutions of the proposed system are significantly dependent on the initial values. The emergence of several bifurcations in the system, including the saddle-node, Bogdanov-Takens, and Hopf-Andronov, is also shown. Through numerical simulation, the rise of a homoclinic loop is shown. The analytic results are verified by numerical simulations and phase portrait sketches.