Bifurcation and control for a discrete-time prey-predator model with Holling-IV functional response

Qiaoling Chen; Zhidong Teng; Zengyun Hu

International Journal of Applied Mathematics and Computer Science (2013)

  • Volume: 23, Issue: 2, page 247-261
  • ISSN: 1641-876X

Abstract

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The dynamics of a discrete-time predator-prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.

How to cite

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Qiaoling Chen, Zhidong Teng, and Zengyun Hu. "Bifurcation and control for a discrete-time prey-predator model with Holling-IV functional response." International Journal of Applied Mathematics and Computer Science 23.2 (2013): 247-261. <http://eudml.org/doc/257117>.

@article{QiaolingChen2013,
abstract = {The dynamics of a discrete-time predator-prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.},
author = {Qiaoling Chen, Zhidong Teng, Zengyun Hu},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {discrete prey-predator model; flip bifurcation; Hopf bifurcation; saddle-node bifurcation; OGY chaotic control},
language = {eng},
number = {2},
pages = {247-261},
title = {Bifurcation and control for a discrete-time prey-predator model with Holling-IV functional response},
url = {http://eudml.org/doc/257117},
volume = {23},
year = {2013},
}

TY - JOUR
AU - Qiaoling Chen
AU - Zhidong Teng
AU - Zengyun Hu
TI - Bifurcation and control for a discrete-time prey-predator model with Holling-IV functional response
JO - International Journal of Applied Mathematics and Computer Science
PY - 2013
VL - 23
IS - 2
SP - 247
EP - 261
AB - The dynamics of a discrete-time predator-prey model with Holling-IV functional response are investigated. It is shown that the model undergoes a flip bifurcation, a Hopf bifurcation and a saddle-node bifurcation by using the center manifold theorem and bifurcation theory. Numerical simulations not only exhibit our results with the theoretical analysis, but also show the complex dynamical behaviors, such as the period-3, 6, 9, 12, 20, 63, 70, 112 orbits, a cascade of period-doubling bifurcations in period-2, 4, 8, 16, quasi-periodic orbits, an attracting invariant circle, an inverse period-doubling bifurcation from the period-32 orbit leading to chaos and a boundary crisis, a sudden onset of chaos and a sudden disappearance of the chaotic dynamics, attracting chaotic sets and non-attracting sets. We also observe that when the prey is in chaotic dynamics the predator can tend to extinction or to a stable equilibrium. Specifically, we stabilize the chaotic orbits at an unstable fixed point by using OGY chaotic control.
LA - eng
KW - discrete prey-predator model; flip bifurcation; Hopf bifurcation; saddle-node bifurcation; OGY chaotic control
UR - http://eudml.org/doc/257117
ER -

References

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