Bifurcations of Invariant Tori in Predator-Prey Models with Seasonal Prey Harvesting

In this paper we study bifurcations in predator-prey systems with seasonal prey harvesting. First, when the seasonal harvesting reduces to constant yield, it is shown that various kinds of bifurcations, including saddle-node bifurcation, degenerate Hopf bifurcation, and Bogdanov--Takens bifurcation (i.e., cusp bifurcation of codimension 2), occur in the model as parameters vary. The existence of two limit cycles and a homoclinic loop is established. Bifurcation diagrams and phase portraits of the model are also given by numerical simulations, which reveal far richer dynamics compared to the case without harvesting. Second, when harvesting is seasonal (described by a periodic function), 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, including bifurcation diagrams, phase portraits, and attractors of Poincare maps, are carried out to demonstrate the ex...