Abstract: We considered the dynamical properties of a class of plant-herbivore diffusion system. Firstly, we proved the global existence, dissipation and persistence of the solutions of system by using the method of upper and lower solutions and the comparison theorem of parabolic equations. Secondly, based on the principle eigenvalue theory of elliptic operator and Lyapunov function method, we gave the existence, the local and global asymptotic stability of constant steady state solutions, and established the criterion of the Turing instability of the system. Finally, the effectiveness of results was verified by some numerical simulations.

Key words: plant-herbivore diffusion system, constant steady state solution, Lyapunov function, Turing instability, numerical simulation