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

Abstract: 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 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 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. [Copyright &y& Elsevier]