Stochastic dynamics of barrier island elevation.

Barrier islands are ubiquitous coastal features that create low-energy environments where salt marshes, oyster reefs, and mangroves can develop and survive external stresses. Barrier systems also protect interior coastal communities from storm surges and wave-driven erosion. These functions depend on the existence of a slowly migrating, vertically stable barrier, a condition tied to the frequency of storm-driven overwashes and thus barrier elevation during the storm impact. The balance between erosional and accretional processes behind barrier dynamics is stochastic in nature and cannot be properly understood with traditional continuous models. Here we develop a master equation describing the stochastic dynamics of the probability density function (PDF) of barrier elevation at a point. The dynamics are controlled by two dimensionless numbers relating the average intensity and frequency of high-water events (HWEs) to the maximum dune height and dune formation time, which are in turn a function of the rate of sea level rise, sand availability, and stress of the plant ecosystem anchoring dune formation. Depending on the control parameters, the transient solution converges toward a high-elevation barrier, a low-elevation barrier, or a mixed, bimodal, state. We find the average after-storm recovery time-a relaxation time characterizing barrier's resiliency to storm impacts-changes rapidly with the control parameters, suggesting a tipping point in barrier response to external drivers. We finally derive explicit expressions for the overwash probability and average overwash frequency and transport rate characterizing the landward migration of barriers.

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 time scale the effects of these feedbacks will be felt, and, relatedly, how long the system will take to cross between different modes.