Slow passage through the Busse balloon -- predicting steps on the Eckhaus staircase

TL;DR

This paper analyzes the dynamics of vegetation pattern changes under climate stress using a Ginzburg-Landau model, predicting sudden wavenumber drops and patch annihilations driven by complex resonances.

Contribution

It introduces a novel prediction method for sudden pattern transitions and patch disappearances in vegetation models near the Eckhaus boundary.

Findings

Predicts timing of sudden wavenumber drops

Identifies complex resonances as key mechanisms

Supports results with numerical simulations

Abstract

Motivated by the impact of worsening climate conditions on vegetation patches, we study dynamic instabilities in an idealized Ginzburg-Landau model. Our main results predict time instances of sudden drops in wavenumber and the resulting target states. The changes in wavenumber correspond to the annihilation of individual vegetation patches when resources are scarce and cannot support the original number of patches. Drops happen well after the primary pattern has destabilized at the Eckhaus boundary and key to distinguishing between the disappearance of 1,2, or more patches during the drop are complex spatio-temporal resonances in the linearization at the unstable pattern. We support our results with numerical simulations and expect our results to be conceptually applicable universally near the Eckhaus boundary, in particular in more realistic models.