Analysis of a model for banded vegetation patterns in semi-arid environments with nonlocal dispersal

TL;DR

This paper investigates how replacing local diffusion with nonlocal seed dispersal in a vegetation pattern model affects the formation of stripe patterns in semi-arid environments, revealing that wider dispersal inhibits pattern formation.

Contribution

It introduces a nonlocal dispersal term into the Klausmeier model and analyzes its impact on vegetation pattern formation, providing new insights into seed dispersal effects.

Findings

Wider seed dispersal inhibits pattern formation.

Nonlocal model predicts larger parameter region for patterns.

Pattern formation depends on dispersal kernel decay.

Abstract

Vegetation patterns are a characteristic feature of semi-arid regions. On hillsides these patterns occur as stripes running parallel to the contours. The Klausmeier model, a coupled reaction-advection-diffusion system, is a deliberately simple model describing the phenomenon. In this paper, we replace the diffusion term describing plant dispersal by a more realistic nonlocal convolution integral to account for the possibility of long-range dispersal of seeds. Our analysis focuses on the rainfall level at which there is a transition between uniform vegetation and pattern formation. We obtain results, valid to leading order in the large parameter comparing the rate of water flow downhill to the rate of plant dispersal, for a negative exponential dispersal kernel. Our results indicate that both a wider dispersal of seeds and an increase in dispersal rate inhibit the formation of patterns. Assuming an evolutionary trade-off between these two quantities, mathematically motivated by the limiting behaviour of the convolution term, allows us to make comparisons to existing results for the original reaction-advection-diffusion system. These comparisons show that the nonlocal model always predicts a larger parameter region supporting pattern formation. We then numerically extend the results to other dispersal kernels, showing that the tendency to form patterns depends on the type of decay of the kernel.