Mathematical Modelling of Natural Phenomena
Predator-prey model, Constant-yield harvesting, Seasonal harvesting, Bogdanov-Takens bifurcation, Degenerate Hopf bifurcation, Periodic solution, Invariant torus
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.
Huang, Jicai; Chen, Jing; Gong, Yijun; and Zhang, Weipeng, "Complex Dynamics in Predator-prey Models with Nonmonotonic Functional Response and Seasonal Harvesting" (2013). Mathematics Faculty Articles. 245.