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Complex dynamics in predator-prey models with nonmonotonic functional response and harvesting. (English) Zbl 1307.34068
The authors study the complex dynamics of predator-prey systems with nonmonotonic functional response and harvesting. When the harvesting is constant, 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 the bifurcation of a stable periodic orbit into a stable invariant torus 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.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C45 Invariant manifolds for ordinary differential equations
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