A diffusive plant-herbivore system with Neumann boundary conditions and Dirichlet boundary conditions is investigated. Local and global stability of spatially homogeneous steady states are established. We derive the conditions for the occurrence of Hopf bifurcation and steady state bifurcation and provide geometrical methods to locate the bifurcation values. A large variety of different types of short-term behavior, including oscillations both in space and in time, or oscillations with different amplitudes, are discussed.