Math. Model. Nat. Phenom.
Volume 8, Number 5, 2013Bifurcations
|Page(s)||95 - 118|
|Published online||17 September 2013|
Complex Dynamics in Predator-prey Models with Nonmonotonic Functional Response and Harvesting
School of Mathematics and Statistics, Central China Normal
University Wuhan, Hubei
430079, P. R.
2 Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250, USA
3 School of Mathematics and Statistics, Northeast Normal University Changchun, Jilin 130024, P. R. China
⋆ Corresponding author. E-mail: email@example.com
In this paper we study the complex dynamics of predator-prey systems with nonmonotonic functional response and harvesting. When the harvesting is constant-yield for prey, it is shown that various kinds of bifurcations, such as saddle-node bifurcation, degenerate Hopf bifurcation, and Bogdanov-Takens bifurcation, occur in the model as parameters vary. The existence of two limit cycles and a homoclinic loop is established by numerical simulations. When the harvesting is seasonal for both species, sufficient conditions for the existence of an asymptotically stable periodic solution and bifurcation of a stable periodic orbit into a stable invariant torus of the model are given. Numerical simulations are carried out to demonstrate the existence of bifurcation of a stable periodic orbit into an invariant torus and transition from invariant tori to periodic solutions, respectively, as the amplitude of seasonal harvesting increases.
Mathematics Subject Classification: 34C23 / 34C25 / 34A26
Key words: predator-prey model / constant-yield harvesting / seasonal harvesting / Bogdanov-Takens bifurcation / degenerate Hopf bifurcation / periodic solution / invariant torus
© EDP Sciences, 2013
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