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Computing the heteroclinic bifurcation curves in predator–prey systems with ratio-dependent functional response

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Abstract

Predator–prey models with Michaelis–Menten–Holling type ratio- dependent functional response exhibit very rich and complex dynamical behavior, such as the existence of degenerate equilibria, appearance of limit cycles and heteroclinic loops, and the coexistence of two attractive equilibria. In this paper, we study heteroclinic bifurcations of such a predator–prey model. We first calculate the higher order Melnikov functions by transforming the model into a Hamiltonian system and then provide an algorithm for computing higher order approximations of the heteroclinic bifurcation curves.

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Authors and Affiliations

  1. Department of Mathematics, University of Miami, Coral Gables, FL, 33124-4250, USA

    Shigui Ruan

  2. Department of Mathematics, Shanghai Jiao Tong University, Minhang, Shanghai, 200240, China

    Yilei Tang

  3. Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan, 610064, China

    Yilei Tang & Weinian Zhang

Authors
  1. Shigui Ruan
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  2. Yilei Tang
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  3. Weinian Zhang
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Corresponding author

Correspondence to Shigui Ruan.

Additional information

S. Ruan research was supported by NSF grant DMS-0715772.

W. Zhang research was supported by NSFC(China), TRAPOYT, CPSF and China MOE Research Grants.

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Ruan, S., Tang, Y. & Zhang, W. Computing the heteroclinic bifurcation curves in predator–prey systems with ratio-dependent functional response. J. Math. Biol. 57, 223–241 (2008). https://doi.org/10.1007/s00285-007-0153-z

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  • DOI: https://doi.org/10.1007/s00285-007-0153-z

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