Analysis and Patterns of Nonlocal Klausmeier Model

Abstract: This work studies a nonlocal extension of the Klausmeier vegetation model in $\mathbb{R}^N$ $(N \ge 1)$ that incorporates both local and nonlocal diffusion. The biomass dynamics are driven by a nonlocal convolution operator, representing anomalous and faster dispersal than the standard Laplacian acting on the water component. Using semigroup theory combined with a duality argument, we establish global well-posedness and uniform boundedness of classical solutions. Numerical simulations based on the Finite Difference Method with Forward Euler integration illustrate the qualitative effects of nonlocal diffusion and kernel size on vegetation patterns. The results demonstrate that nonlocal interactions significantly influence the spatial organization of vegetation, producing richer and more coherent structures than those arising in the classical local model.