Global analysis of a Holling type II predator–prey model with a constant prey refuge
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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.
KeywordsGlobal analysis Refuge Limit cycle Uniqueness Nonsmooth predator–prey model
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).
- 36.Zhang, Z.F., Ding, T.R., Huang, W.Z., Dong, Z.X.: Qualitative Theory of Differential Equations. American Mathematical Society, Providence (2006) Google Scholar