© 2018 Elsevier Ltd We study the consequences of an Allee effect acting on the prey species in a Leslie–Gower predator–prey model. For this goal we make extensive use of analytical tools from dynamical systems theory complemented with a numerical bifurcation analysis. By studying the dynamics at infinity under a suitable compactification we prove that the model is well-posed in the sense that all the solutions are bounded. We provide a thorough analysis of the number and stability of equilibrium points. In particular, the origin is a non-hyperbolic equilibrium and presents different regimes of local (un)stability depending on certain conditions on the model parameters. In addition, we find curves of homoclinic, Hopf, and saddle–node bifurcations around a Bogdanov–Takens point. In this process, our findings indicate that the survival threshold for both populations in the two-dimensional phase space can be either a limit cycle, a homoclinic orbit, or the stable manifold of a saddle-equilibrium.