Supplementary material from "Water-limited vegetated ecosystems driven by stochastic rainfall: feedbacks and bimodality"

In arid or semi-arid ecosystems, water availability is one of the primary controls on vegetation growth. When subsurface water resources are unavailable, the vegetation growth is dictated by the rainfall, and the random nature of the rainfall arrivals and quantities induces a probability distribution of soil moisture and vegetation biomass via the coupled dynamic equations of biomass balance and water balance. We have previously obtained an exact solution for these distributions under certain conditions, and shown that the mapping of rainfall variability to observed biomass variability can be successfully applied to a field site. Here, we expand upon our earlier theoretical work to show how the dynamics can give rise to more complicated, bimodal (and multimodal) structures in the biomass distribution when positive feedbacks between growth and water availability are included. We also derive some new analytical results for the crossing properties of this system, which enable us to determine on what timescale the effects of these feedbacks will be felt, and relatedly, how long will the system take to cross between different modes.