Analysis and Patterns of Nonlocal Klausmeier Model

TL;DR

This paper extends the Klausmeier vegetation model to include nonlocal diffusion, analyzing its mathematical properties and demonstrating through simulations how nonlocal interactions affect vegetation pattern formation.

Contribution

It introduces a nonlocal extension of the Klausmeier model, establishing well-posedness and exploring the impact of nonlocal diffusion on vegetation patterns.

Findings

Nonlocal diffusion alters vegetation pattern structures.

Nonlocal interactions lead to richer spatial patterns.

Simulation results show significant effects of kernel size.

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.