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Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes. (English) Zbl 1156.34342
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.
##### MSC:
 34K13 Periodic solutions of functional differential equations 92D25 Population dynamics (general) 34K60 Qualitative investigation and simulation of models