Predicting the emergence of localised dihedral patterns in models for dryland vegetation

TL;DR

This paper investigates how localized vegetation patterns, such as fairy circles, emerge in dryland models through Turing instabilities, providing a predictive framework and numerical validation across multiple models.

Contribution

It introduces a systematic method to predict localized dihedral patterns in reaction-diffusion models of dryland vegetation, linking key quantities to pattern emergence.

Findings

Localized patterns emerge from Turing instabilities in vegetation models.

Models with similar key quantities exhibit similar pattern types.

Numerical simulations confirm the transient and complex nature of these patterns.

Abstract

Localised patterns are often observed in models for dryland vegetation, both as peaks of vegetation in a desert state and as gaps within a vegetated state, known as `fairy circles'. Recent results from radial spatial dynamics show that approximations of localised patterns with dihedral symmetry emerge from a Turing instability in general reaction--diffusion systems, which we apply to several vegetation models. We present a systematic guide for finding such patterns in a given reaction--diffusion model, during which we obtain four key quantities that allow us to predict the qualitative properties of our solutions with minimal analysis. We consider four well-established vegetation models and compute their key predictive quantities, observing that models which possess similar values exhibit qualitatively similar localised patterns; we then complement our results with numerical simulations of various localised states in each model. Here, localised vegetation patches emerge generically from Turing instabilities and act as transient states between uniform and patterned environments, displaying complex dynamics as they evolve over time.