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Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting. (English) Zbl 1259.34035

The paper treats a predator-prey model representing a modified version of Leslie-Gower with Holling-type II functional response, which has already been analyzed previously by several researchers. In particular, the authors also consider prey harvesting of Michaelis-Menten type. The resulting planar vector field in dimensionless form depends on six positive parameters and has up to two equilibria in the open positive quadrant as well as on the positive axes outside the origin. The authors present a detailed analysis of the behavior of this model and thoroughly investigate the possible bifurcations, i.e., saddle-node, transcritical, Hopf, and Bogdanov-Takens bifurcation, the latter one being realized by a series of analytic transformations and making also use of the Malgrange Preparation theorem. Exhaustive numerical simulations suggest some further phenomena such as homoclinic loops and separatrices.

MSC:

34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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