In this paper, we consider a predator-prey model with Cosner type
functional response and combined harvesting. First, we explore the
existence and stability of the equilibria. Then using the center
manifold theorem and normal form theory, we investigate codimension one
and codimension two bifurcations of the model. The analysis shows that
the system has a variety of bifurcation phenomena including
transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation,
Bogdanov-Takens bifurcation and homoclinic bifurcation. Our findings
indicate that the dynamics with harvesting are significantly richer than
the system without harvesting. Finally, numerical simulations are
provided to support the analytical results.