# American Institute of Mathematical Sciences

October  2017, 10(5): 1187-1206. doi: 10.3934/dcdss.2017065

## Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting

 College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

* Corresponding author: klcqu20132016@163.com (L. Kong)

Received  November 2016 Revised  January 2017 Published  June 2017

Fund Project: Acknowledgement: This work was supported by NSFC grant 11671058.

In the present paper the dynamics of a Leslie-Gower predator-prey model with Michaelis-Menten type predator harvesting is studied. We give out all the possible ranges of parameters for which the model has up to five equilibria. We prove that these equilibria can be topological saddles, nodes, foci, centers, saddle-nodes, cusps of codimension 2 or 3. Numerous kinds of bifurcations also occur, such as the transcritical bifurcation, pitchfork bifurcation, Bogdanov-Takens bifurcation and homoclinic bifurcation. Several numerical simulations are carried out to illustrate the validity of our results.

Citation: Changrong Zhu, Lei Kong. Bifurcations analysis of Leslie-Gower predator-prey models with nonlinear predator-harvesting. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1187-1206. doi: 10.3934/dcdss.2017065
##### References:

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##### References:
The number of interior equilibriums of system (4)
There is no interior equilibrium
A unique interior equilibrium $E_2$
A unique interior equilibrium $E_3$
The bi-stability occurred
A stable limit cycle
Two limit cycles
An unstable limit cycle
A cusp of codimension 2
An unstable limit cycle
An unstable homoclinic loop
A saddle and a stable focus
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