Geodesic
Complex Dynamics in Predator-prey Models with Nonmonotonic Functional Response and Harvesting
J. Huang1 ; J. Chen2 ; Y. Gong1 ; W. Zhang3
1 School of Mathematics and Statistics, Central China Normal University Wuhan, Hubei 430079, P. R. China
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
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 95-118.

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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.
DOI : 10.1051/mmnp/20138507

J. Huang 1 ; J. Chen 2 ; Y. Gong 1 ; W. Zhang 3

1 School of Mathematics and Statistics, Central China Normal University Wuhan, Hubei 430079, P. R. China
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 
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J. Huang; J. Chen; Y. Gong; W. Zhang. Complex Dynamics in Predator-prey Models with Nonmonotonic Functional Response and Harvesting. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 95-118. doi : 10.1051/mmnp/20138507. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138507/

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