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Nonlinear Dynamics

, Volume 73, Issue 1–2, pp 815–826 | Cite as

Periodic solutions and homoclinic bifurcation of a predator–prey system with two types of harvesting

  • Mingzhan Huang
  • Shouzong Liu
  • Xinyu Song
  • Lansun Chen
Original Paper

Abstract

In this paper, a predator–prey model with both constant rate harvesting and state dependent impulsive harvesting is analyzed. By using differential equation geometry theory and the method of successor functions, the existence, uniqueness and stability of the order one periodic solution have been studied. Sufficient conditions which guarantee the nonexistence of order k (k≥2) periodic solution are given. We also present that the system exhibits the phenomenon of homoclinic bifurcation under some parametric conditions. Finally, some numerical simulations and biological explanations are given.

Keywords

Predator–prey system Order k periodic solution Successor function Orbitally asymptotically stable Homoclinic bifurcation 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (11171284), the Innovation Scientists and Technicians Troop Construction Projects of Henan Province (104200510011), program for Innovative Research Team (in Science and Technology) in University of Henan Province (2010IRTSTHN006) and Sci-tech program project of Henan Province (122300410034).

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Mingzhan Huang
    • 1
  • Shouzong Liu
    • 1
  • Xinyu Song
    • 1
  • Lansun Chen
    • 2
  1. 1.College of Mathematics and Information ScienceXinyang Normal UniversityXinyangP.R. China
  2. 2.Institute of Mathematics, Academy of Mathematics and System SciencesAcademia SinicaBeijingP.R. China

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