Large Amplitude Radially Symmetric Spots and Gaps in a Dryland Ecosystem Model

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.

Automated summary

In this study, far-from-onset radially symmetric spot and gap solutions were constructed in a two-component dryland ecosystem model using spatial dynamics and geometric singular perturbation theory. The geometry of these solutions was compared with that of traveling and stationary front solutions in the same model. The instability of spots with large radius was demonstrated, and it was shown that spots are unstable to a range of perturbations of intermediate wavelength in the angular direction.