In this work, a bidimensional continuous-time differential equations system is analyzed which is derived from Leslie type predator-prey schemes by considering a nonmonotonic functional response and Allee effect on population prey. For ecological reason, we describe the bifurcation diagram of limit cycles that appear only at the first quadrant in the system obtained. We also show that under certain conditions over the parameters, the system allows the existence of a stable limit cycle surrounding an unstable limit cycle generated by Hopf bifurcation. Furthermore, we give conditions over the parameters such that the model allows long-term extinction or survival of both populations. (C) 2008 Elsevier Ltd. All rights reserved.