Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-24T00:04:44.383Z Has data issue: false hasContentIssue false
  • English
  • Français

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: [email protected]
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrews, J. F.. A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng., 10 (1968), 707-723. CrossRefGoogle Scholar
Bogdanov, R.. Bifurcations of a limit cycle for a family of vector fields on the plane. Selecta Math. Soviet. 1 (1981), 373-388. Google Scholar
Bogdanov, R.. Versal deformations of a singular point on the plane in the case of zero eigen-values. Selecta Math. Soviet. 1 (1981), 389-421. Google Scholar
Brauer, F.. Periodic solutions of some ecological models. J. Theor. Biol. 69 (1977), 143-152. CrossRefGoogle ScholarPubMed
Brauer, F., A Sánchez, D.. Periodic environments and periodic harvesting. Natural Resource Modeling. 16(3) (2003), 233-244. CrossRefGoogle Scholar
Collings, J. B.. The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model. J. Math. Biol., 36 (1997), 149-168. CrossRefGoogle Scholar
S.-N. Chow, J. K. Hale. Methods of Bifurcation Theory. Springer-Verlag, Berlin-Heidelberg-New York, 1982.
Etoua, R. M., Rousseau, C.. Bifurcation analysis of a Generalissed Gause model with prey harvesting and a generalized Holling response function of type III. J. Differential Equations, 249 (2010), 2316-2356. CrossRefGoogle Scholar
M. W. Hirsch, S. Smale, R. L. Devaney. Differential Equations, Dynamical Systems and An Introduction to Chaos. Elsevier, California, 2004.
Hammill, E., Petchey, O. L., Anholt, B. R.. Predator functional response changed by induced defenses in prey. The American Naturalist, 176(6) (2010), 723-731. CrossRefGoogle ScholarPubMed
Lamontagne, Y., Coutu, C., Rousseau, C.. Bifurcation analysis of a predator-prey system with generalized Holling type III functional response. J. Dynam. Differential Equations. 20 (2008), 535-571. CrossRefGoogle Scholar
May, R., Beddington, J. R., Clark, C. W., Holt, S. J., Laws, R. M.. Management of multispecies fisheries. Science, 205 (1979), 267-277. CrossRefGoogle ScholarPubMed
L. Perko. Differential Equations and Dynamical Systems. Springer, New York, 1996.
Ruan, S., Xiao, D.. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math., 61(4) (2001), 1445-1472. Google Scholar
Takens, F.. Forced oscillations and bifurcation, in “Applications of Global Analysis I”. Comm. Math. Inst. Rijksuniversitat Utrecht. 3 (1974), 1-59. Google Scholar
R. J. Taylor. Predation. Chapman and Hall, New York, 1984.
Wolkowicz, G. S. K.. Bifurcation analysis of a predator-prey system involving group defence. SIAM J. Appl. Math. 48 (1988), 592-606. CrossRefGoogle Scholar
Xiao, D., Zhu, H.. Multiple focus and hopf bifurcations in a predaotr-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 66 (2006), 802-819. CrossRefGoogle Scholar
Zhu, H., Campbell, S. A., Wolkowicz, G. S. K.. Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math. 63 (2002), 636-682. CrossRefGoogle Scholar
Z. Zhang, T. Ding, W. Huang, Z. Dong. Qualitative Theory of Differential Equation. Transl. Math. Monogr. Vol. 101, Amer. Math. Soc. Providence, RI, 1992.