Two types of predator–prey models with harvesting: Non-smooth and non-continuous

This article investigates continuous and impulsive threshold harvesting strategies on the predator which needs to be applied only when the predator population is above or reaches the harvesting threshold. For the continuous threshold model, the system is nonsmooth and has complex dynamics with multiple internal equilibria, limit cycle, homoclinic orbit, saddle–node bifurcation, transcritical bifurcation, subcritical and supercritical Hopf bifurcation, Bogdanov–Takens bifurcation and discontinuous Hopf bifurcation. In order to prevent the predator population being above the threshold, we further extend our model with impulsive threshold harvesting strategies. The model is non-continuous and the existence and stability of positive order-1 and order-2 periodic solutions were obtained by using the Poincaré map. It is seen that the impulsive threshold harvesting strategies are more effective than the continuous. Furthermore, some numerical simulations are given to illustrate our results.