Interrelationships between two species are a basic theme in Population Dynamics, particularly the interaction between predators and their prey. This importance is due to the fact that it allows a deeper understanding of the behavior of complex food webs. In this paper we extend the analysis of a modified Leslie-Gower predator-prey model by assuming that the functional response is sigmoid or Holling type III and the predator have an alternative food. We show that the system representing the model has up to three positive equilibrium points; we establish conditions to determine the nature of each equilibrium point. In addition, we show the existence of different types of bifurcations, including those of Hopf and the homoclinic. The analytical results are discussed from an ecological perspective.