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

Changrong Zhu; Lei Kong

doi:10.3934/dcdss.2017065

Article Contents

2017, Volume 10, Issue 5: 1187-1206. Doi: 10.3934/dcdss.2017065

This issue Previous Article Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term Next Article

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

Changrong Zhu and
Lei Kong^,

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

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

Received: November 2016

Revised: January 2017

Published: October 2017

Acknowledgement: This work was supported by NSFC grant 11671058.

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Abstract

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.

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Mathematics Subject Classification: Primary: 70K05, 34D20; Secondary: 34C23.

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Figure 1. The number of interior equilibriums of system (4)

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Figure 2. There is no interior equilibrium

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Figure 3. A unique interior equilibrium $E_2$

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Figure 4. A unique interior equilibrium $E_3$

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Figure 5. The bi-stability occurred

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Figure 6. A stable limit cycle

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Figure 7. Two limit cycles

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Figure 8. An unstable limit cycle

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Figure 9. A cusp of codimension 2

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Figure 10. An unstable limit cycle

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Figure 11. An unstable homoclinic loop

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Figure 12. A saddle and a stable focus

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