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

• Published in: Nonlinear analysis: real world applications 2008-12, Vol.9 (5), p.2055-2067
• Main Authors: Yafia, Radouane, El Adnani, Fatiha, Alaoui, Hamad Talibi
• Format: Article
• Language: English
• Subjects: Delay differential equations; Hopf bifurcation; Periodic solutions; Predator–prey system; Stability/unstability
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2006.12.017

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.