In this paper, we develop a diffusive plant invasion model with delay under the homogeneous Neumann boundary condition. First, the existence and uniqueness of a non-negative solution, persistence property, and local asymptotic stability of the constant steady states are established. Then by analyzing the associated characteristic equation, the stability of steady states and the existence of Hopf bifurcation are demonstrated. Under special circumstance, the discontinuous Hopf bifurcation is also investigated. Furthermore, the existence and non-existence of nonconstant positive steady states of this model are studied through considering the effect of large diffusivity. Finally, in order to verify our theoretical results, some numerical simulations are also included. It shows that the numerically observed behaviors are in good agreement with the theoretically proposed results. On the basis of numerical simulations, we provide some useful comparisons for the readers between our results and the existing ones, and discuss the effects of the delay and the diffusion term on dynamic behaviors. Numerical results indicate that diffusion can make the system unstable and increasing delay may cause the plant to go extinct.
