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Zbl 1156.34342
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)
[J] Nonlinear Anal., Real World Appl. 9, No. 5, 2055-2067 (2008). ISSN 1468-1218

Summary: The model analyzed in this paper is based on the model set forth by {\it M.A. Aziz-Alaoui} and {\it M. Daher Okiye} [Appl. Math. Lett. 16, No. 7, 1069--1075 (2003; Zbl 1063.34044)]; {\it 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 2000:
*34K13 Periodic solutions of functional differential equations
92D25 Population dynamics
34K60 Applications of functional-differential equations

Keywords: predator-prey system; delay differential equations; stability/unstability; Hopf bifurcation; periodic solutions

Citations: Zbl 1063.34044

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