Summary: The model analyzed in this paper is based on the model set forth by

*M.A. Aziz-Alaoui* and

*M. Daher Okiye* [Appl. Math. Lett. 16, No. 7, 1069–1075 (2003;

Zbl 1063.34044)];

*A.F. Nindjin, M.A. Aziz-Alaoui, M. Cadivel*, Analysis of a a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay, Nonlinear Anal. Real World Appl., in Press.] with time delay, which describes the competition between predator and prey. This model incorporates a modified version of Leslie-Gower functional response as well as that of the Holling-type II. In this paper, we consider the model with one delay and a unique non-trivial equilibrium

${E}^{*}$ and the three others are trivial. Their dynamics are studied in terms of the local stability and of the description of the Hopf bifurcation at

${E}^{*}$ for small and large delays and at the third trivial equilibrium that is proven to exist as the delay (taken as a parameter of bifurcation) crosses some critical values. We illustrate these results by numerical simulations.