In many existing predator-prey or plant-herbivore models, the numerical response is assumed to be proportional to the functional response. In this paper, without such an assumption, we consider a diffusive plant-herbivore system with Neumann boundary conditions. Besides stability of spatially homogeneous steady states, we also derive conditions for the occurrence of Hopf bifurcation and steady-state bifurcation and provide geometrical methods to locate the bifurcation values. We numerically explore the complex transient spatio-temporal behaviours induced by these bifurcations. A large variety of different types of transient behaviours including oscillations in one or both of space and time are observed.
