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Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model

Journal of Dynamics and Differential Equations volume 30, pages 447–472 (2018)Cite this article

Abstract

Loladze et al. (Bull Math Biol 62:1137–1162, 2000) proposed a highly cited stoichiometric predator–prey system, which is nonsmooth, and thus it is extremely difficult to analyze its global dynamics. The main challenge comes from the phase plane fragmentation and parameter space partitioning in order to perform a detailed and complete global stability and bifurcation analysis. Li et al. (J Math Biol 63:901–932, 2011) firstly discussed its global dynamical behavior with Holling type I functional response and found that the system has no limit cycles, and the internal equilibrium is globally asymptotically stable if it exists. Secondly, for the system with Holling type II functional response, Li et al. (2011) fixed all parameters (with realistic values) except K to perform the bifurcation analysis and obtained some interesting phenomena, for instance, the appearance of bistability and many bifurcation types. The aim of this paper is to provide a complete global analysis for the system with Holling type II functional response without fixing any parameter. Our analysis shows that the model has far richer dynamics than those found in the previous paper (Li et al. 2011), for example, four types of bistability appear: besides the bistability between an internal equilibrium and a limit cycle as shown in Li et al. (2011), the other three bistabilities occur between an internal equilibrium and a boundary equilibrium, between two internal equilibria, or between a boundary equilibrium and a limit cycle. In addition, this paper rigorously provides all possible bifurcation passways of this stoichiometric model with Holling type II functional response.

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Acknowledgments

The third author is partially supported by the NSFC (11571041) and the Fundamental Research Funds for the Central Universities. The fourth author is partially supported by the NSERC.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Tian Xie & Xiong Li

  2. School of Information and Mathematics, Yangtze University, Jingzhou, Hubei, 434023, People’s Republic of China

    Xianshan Yang

  3. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1, Canada

    Hao Wang

Authors
  1. Tian Xie
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  2. Xianshan Yang
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  3. Xiong Li
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  4. Hao Wang
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Correspondence to Hao Wang.

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Xie, T., Yang, X., Li, X. et al. Complete Global and Bifurcation Analysis of a Stoichiometric Predator–Prey Model. J Dyn Diff Equat 30, 447–472 (2018). https://doi.org/10.1007/s10884-016-9551-5

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  • DOI: https://doi.org/10.1007/s10884-016-9551-5

Keywords

  • Stoichiometric predator–prey system
  • Holling type II functional response
  • Limit cycle
  • Bistability
  • Bifurcation