Turing patterns in a networked vegetation model

A vegetation model composed of water and plants was proposed by introducing a weighted graph Laplacian operator into the reaction-diffusion dynamics. We showed the global existence and uniqueness of the solution via monotone iterative sequence. The parameter space of Turing patterns for plant behavior is obtained based on the analysis of the eigenvalues of the Laplacian of weighted graph, while the amplitude equation determining the stability of Turing patterns is obtained by weakly nonlinear analysis. We also show that the optimal rainfall is only determined by the density of the water. By some numerical simulations, we examine the individual effect of diffusion term on the formation of regular Turing patterns. We show that the large diffusion induces stable Turing patterns.