RESUMEN
Since certain prey hide from predators to protect themselves within their habitats, predators are forced to change their diet due to a lack of prey for consumption, or on the contrary, subsist only with alternative food provided by the environment. Therefore, in this paper, we propose and mathematically contrast a predator-prey, where alternative food for predators is either considered or not when the prey population size is above the refuge threshold size. Since the model with no alternative food for predators has a Hopf bifurcation and a transcritical bifurcation, in addition to a stable limit cycle surrounding the unique interior equilibrium, such bifurcation cases are transferred to the model when considering alternative food for predators when the prey size is above the refuge. However, such a model has two saddle-node bifurcations and a homoclinic bifurcation, characterized by a homoclinic curve surrounding one of the three interior equilibrium points of the model.