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Complex Dynamics in Predator-prey Models with NonmonotonicFunctional Response and Harvesting

Published online by Cambridge University Press:  17 September 2013

J. Huang ,
J. Chen ,
Y. Gong  and
W. Zhang
J. Huang*
Affiliation:
School of Mathematics and Statistics, Central China Normal University Wuhan, Hubei 430079, P. R. China
J. Chen
Affiliation:
Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA
Y. Gong
Affiliation:
School of Mathematics and Statistics, Central China Normal University Wuhan, Hubei 430079, P. R. China
W. Zhang
Affiliation:
School of Mathematics and Statistics, Northeast Normal University Changchun, Jilin 130024, P. R. China
*
⋆ Corresponding author. E-mail: hjc@mail.ccnu.edu.cn
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Abstract

In this paper we study the complex dynamics of predator-prey systems with nonmonotonicfunctional response and harvesting. When the harvesting is constant-yield for prey, it isshown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopfbifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. Theexistence of two limit cycles and a homoclinic loop is established by numericalsimulations. When the harvesting is seasonal for both species, sufficient conditions forthe existence of an asymptotically stable periodic solution and bifurcation of a stableperiodic orbit into a stable invariant torus of the model are given. Numerical simulationsare carried out to demonstrate the existence of bifurcation of a stable periodic orbitinto an invariant torus and transition from invariant tori to periodic solutions,respectively, as the amplitude of seasonal harvesting increases.

Keywords

predator-prey modelconstant-yield harvestingseasonal harvestingBogdanov-Takens bifurcationdegenerate Hopf bifurcationperiodic solutioninvariant torus
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 95 - 118
Copyright
© EDP Sciences, 2013

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