Blurring the Busse balloon: Patterns in a stochastic Klausmeier model

Abstract

We investigate (in)stabilities of periodic patterns under stochastic forcing in reaction-diffusion equations exhibiting a so-called Busse balloon. Specifically, we used a one-dimensional Klausmeier model for dryland vegetation patterns. Using numerical methods, we can accurately describe the transient dynamics of the stochastic solutions and compare several notions of stability. In particular, we show that stochastic stability heavily depends on the model parameters, the intensity of the noise and the location of the wavenumber of the periodic pattern within the deterministic Busse balloon. Furthermore, the boundary of the Busse balloon becomes blurred under the stochastic perturbations.