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

Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

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

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

Acta Mathematicae Applicatae Sinica, English Series Vol. 21, No. 1 (2005) 157–176 Bifurcations and Chaos in a Discrete Predator-prey System with Holling Type-IV Functional Response Ji-cai Huang Department of Applied Mathematics, College of Science, China Agriculture University; Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China (E-mail: hjc@amss.ac.cn) 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. Keywords predator-prey system; bifurcations; Bogdanov-Takens bifurcation; chaos 2000 MR Subject Classification 92B05; 34D05 1 Introduction An extensive system of the familiar Lotka-Volterra system can be modeled by x ˙ = xρ(x) − yp(x), y˙ = y γp(x) − ψ(y) , (A) where x and y represent the prey density and predator density, respectively. The specific growth rate, ρ(x)= 1 − k x − k x , governs the growth of the prey in the absence of predators, the 1 2 function...
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Publisher
Springer-Verlag
Copyright
Copyright © 2005 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
D.O.I.
10.1007/s10255-005-0227-x
Publisher site
See Article on Publisher Site

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.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Jan 1, 2005

References

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