This paper is concerned with a predator–prey model with hyperbolic mortality and prey harvesting. The parameter regions for the stability and instability of the unique positive constant solution of ODE and PDE are derived, respectively. Especially, the global asymptotical stability of positive constant equilibrium of the diffusive model is obtained by iterative technique. The stability and direction of periodic solutions of ODE and PDE are investigated by center manifold theorem and normal form theory, respectively. Numerical simulations are carried out to depict our theoretical analysis.