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Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response

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Abstract

A discrete predator-prey system with Holling type-IV functional response obtained by the Euler method is first investigated. The conditions of existence for fold bifurcation, flip bifurcation and Hopf bifurcation are derived by using the center manifold theorem and bifurcation theory. Furthermore, we give the condition for the occurrence of codimension-two bifurcation called the Bogdanov-Takens bifurcation for fixed points and present approximate expressions for saddle-node, Hopf and homoclinic bifurcation sets near the Bogdanov-Takens bifurcation point. We also show the existence of degenerated fixed point with codimension three at least. The numerical simulations, including bifurcation diagrams, phase portraits, and computation of maximum Lyapunov exponents, not only show the consistence with the theoretical analysis but also exhibit the rich and complex dynamical behaviors such as the attracting invariant circle, period-doubling bifurcation from period-2,3,4 orbits, interior crisis, intermittency mechanic, and sudden disappearance of chaotic dynamic.

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Authors and Affiliations

  1. Department of Applied Mathematics, College of Science, China Agriculture University; Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China

    Ji-cai Huang

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  1. Ji-cai Huang
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Correspondence to Ji-cai Huang.

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Supported by Chinese Academy Sciences (KZCX2-SW-118) and by the National Natural Science Foundation of China (No.10371037).

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Huang, Jc. Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response. Acta Mathematicae Applicatae Sinica, English Series 21, 157–176 (2005). https://doi.org/10.1007/s10255-005-0227-x

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  • DOI: https://doi.org/10.1007/s10255-005-0227-x

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