Can we predict wavelength changes of banded vegetation patterns?

Banded vegetation patterns are a common feature in drylands. The ability to self-organise into alternating stripes of vegetated and bare soil areas is thought to be a resilience mechanism that prevents catastrophic tipping of dryland plant ecosystems. Several mathematical models exist that describe the dynamics of dryland vegetation bands as periodic travelling waves (PTWs). Models predict that if environmental stress increases, dryland vegetation bands undergo cascades of wavelength changes that progressively increase the characteristic distances between stripes before a transition to desert. It is thus of crucial importance to understand when (i.e., at what parameter values) and how (i.e., to which new wavelength) PTW wavelength changes occur.
In this talk, I show that the traditionally used method of using Busse balloon boundaries to predict parameter values at which wavelength changes occur is often insufficient. Instead, I show that model solutions enter a (potentially long) transient after crossing a stability boundary and present a method to estimate the order of magnitude of the length of this transient. I further review our current knowledge of PTW wavelength selection, a problem that remains unsolved except for special cases, and will present new numerical evidence of selection principles in the context of dryland vegetation patterns.