Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes

The model analyzed in this paper is based on the model set forth by [M.A. Aziz-Alaoui, M. Daher Okiye, Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes, Appl. Math. Lett. 16 (2003) 1069-1075, 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. (c) 2007 Elsevier Ltd. All rights reserved.