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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 to functional-differential equations |

92D25 | Population dynamics (general) |

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

### Keywords:

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

Zbl 1063.34044
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\textit{R. Yafia} et al., Nonlinear Anal., Real World Appl. 9, No. 5, 2055--2067 (2008; Zbl 1156.34342)

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### References:

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