Large amplitude radially symmetric spots and gaps in a dryland ecosystem model

TL;DR

This paper constructs and analyzes large-amplitude radially symmetric vegetation spots and gaps in a dryland ecosystem model, revealing their stability properties and connections to front solutions using advanced mathematical techniques.

Contribution

It introduces a novel construction of large-amplitude spot and gap solutions in a dryland ecosystem model and links their geometry to front solutions, analyzing their stability.

Findings

Large spots are unstable due to sideband instabilities.

Spots are unstable to intermediate wavelength perturbations unless small.

Numerical simulations support the theoretical stability analysis.

Abstract

We construct far-from-onset radially symmetric spot and gap solutions in a two-component dryland ecosystem model of vegetation pattern formation on flat terrain, using spatial dynamics and geometric singular perturbation theory. We draw connections between the geometry of the spot and gap solutions with that of traveling and stationary front solutions in the same model. In particular, we demonstrate the instability of spots of large radius by deriving an asymptotic relationship between a critical eigenvalue associated with the spot and a coefficient which encodes the sideband instability of a nearby stationary front. Furthermore, we demonstrate that spots are unstable to a range of perturbations of intermediate wavelength in the angular direction, provided the spot radius is not too small. Our results are accompanied by numerical simulations and spectral computations.