Bogdanov-Takens bifurcation of codimension 3 of a predator-prey model with constant-yield predator releasing rate

The bifurcation analysis of a predator-prey system with Holling-II functional response function and constant-yield predator releasing rate is carried out in this paper, which is a fundamental model for studying pulse control strategies or switching strategies based on the threshold level of the pest population. We show that the system can have at most two positive equilibria, and can undergo a sequence of bifurcations, including transcritical bifurcation, pitchfork bifurcation, saddle-node bifurcation, Hopf bifurcation, degenerate Hopf bifurcation of codimension 2, and Bogdanov-Takens bifurcation of codimensions 2 and 3. Complex dynamics, such as multi-steady states, the existence of a semi-stable limit cycle, a homoclinic loop, and multiple coexistent periodic orbits, are revealed. These results show that the constant-yield predator releasing rate can be used to organize the complex dynamics and bifurcations of the model. The proposed model with constant-yield predator releasing rate is not only necessary for characterizing many practical problems including integrated pest management, but also can generate more rich bifurcation phenomena and dynamic behaviors.