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Nonlinear Dynamics

, Volume 76, Issue 1, pp 635–647 | Cite as

Global analysis of a Holling type II predator–prey model with a constant prey refuge

  • Guangyao Tang
  • Sanyi TangEmail author
  • Robert A. Cheke
Original Paper

Abstract

A global analysis of a Holling type II predator–prey model with a constant prey refuge is presented. Although this model has been much studied, the threshold condition for the global stability of the unique interior equilibrium and the uniqueness of its limit cycle have not been obtained to date, so far as we are aware. Here we provide a global qualitative analysis to determine the global dynamics of the model. In particular, a combination of the Bendixson–Dulac theorem and the Lyapunov function method was employed to judge the global stability of the equilibrium. The uniqueness theorem of a limit cycle for the Lineard system was used to show the existence and uniqueness of the limit cycle of the model. Further, the effects of prey refuges and parameter space on the threshold condition are discussed in the light of sensitivity analyses. Additional interesting topics based on the discontinuous (or Filippov) Gause predator–prey model are addressed in the discussion.

Keywords

Global analysis Refuge Limit cycle Uniqueness Nonsmooth predator–prey model 

Notes

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSFC, 11171199) and by the Fundamental Research Funds for the Central Universities (GK201305010).

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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.College of Mathematics and Information ScienceShaanxi Normal UniversityXi’anP.R. China
  2. 2.School of ScienceHubei University for Nationalities EnshiHubeiP.R. China
  3. 3.Natural Resources InstituteUniversity of Greenwich at MedwayKentUK

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