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

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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.

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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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Authors and Affiliations

  1. College of Mathematics and Information Science, Shaanxi Normal University, Xi’an, 710062, P.R. China

    Guangyao Tang & Sanyi Tang

  2. School of Science, Hubei University for Nationalities Enshi, Hubei, 445000, P.R. China

    Guangyao Tang

  3. Natural Resources Institute, University of Greenwich at Medway, Central Avenue, Chatham Maritime, Chatham, Kent, ME4 4TB, UK

    Robert A. Cheke

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  1. Guangyao Tang
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  3. Robert A. Cheke
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Correspondence to Sanyi Tang.

Appendix

Appendix

Latin Hypercube Sampling/Partial Rank Correlation Coefficient (LHS/PRCC) sensitivity analysis is an efficient tool often employed in uncertainty analysis to explore the entire parameter space of a model with a minimum number of computer simulations. It involves the combination of two statistical techniques, Latin Hypercube Sampling (LHS) and Partial Rank Correlation Coefficient (PRCC) analysis. The goal of LHS/PRCC sensitivity analysis is to identify key parameters whose uncertainties contribute to prediction imprecision and to rank these parameters by their importance in contributing to this imprecision.

The LHS procedure is implemented by dividing the range of values for a given parameter into equally probable intervals, and the sampling is performed independently for each parameter. LHS can result in an unbiased estimate of the average model output, with the advantage that it requires fewer samples than simple random sampling to achieve the same accuracy. PRCC is a robust sensitivity measure for nonlinear but monotonic relationships between inputs and output, which is considered to be more powerful at determining the sensitivity of a parameter.

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Tang, G., Tang, S. & Cheke, R.A. Global analysis of a Holling type II predator–prey model with a constant prey refuge. Nonlinear Dyn 76, 635–647 (2014). https://doi.org/10.1007/s11071-013-1157-4

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