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The coastal dune model describes the temporal evolution of the sand surface elevation h (x, y, t)—defined relative to the MHWL—and the cover fraction ρ_veg (x, y, t) for a single generic grass species. x is the cross-shore distance to the shoreline (x = 0), which separates the foreshore (x < 0) from the backshore (x > 0), and y is the alongshore coordinate. A complete description of the coastal dune model is provided in the Supplementary Methods and includes further details on the calculation of the aeolian transport and vegetation dynamics as well as the initial and boundary conditions used to integrate the model.

Aeolian sand transport.

In the absence of storms, the sand flux q_a (x, y, t) is calculated from the bed wind shear stress, which depends on the surrounding topography and the vegetation cover, and the local sand transport threshold, which we assume is primarily controlled by the sand moisture content. For simplicity in the formulation, sand transport is described by volume, not mass, flux.

Storms.

Storms are defined in the model as HWEs with total water elevation R above the MHWL (R is defined relative to the MHWL). HWEs are considered periodic with period T_HWE and have constant duration. R is randomly distributed following an Erlang distribution ( is mean total water elevation), which has an exponential tail, in agreement with ref. 28, and filters tidal events (that is, there are no events for R 0).

Storm-induced sand transport.

We derive a phenomenological expression to calculate the cross-shore sand flux q_st (x, y, t) for elevations between MHWL and R during HWEs. On the basis of ref. 29 (see ref. 30 for the validation of a similar formulation), we assume the net sand flux q_st over many swash cycles is proportional to the cube of the average speed U of the uprushing wavefront, times the time δt this particular location is submerged: q_st ∝ U³δt/(gT), with gravity g and timescale T. The net sand flux is weighted by a downslope contribution (1 − ∂_xh/tan(α)) that represents the tendency of the surface to reach the equilibrium foreshore slope tan (α). From energy conservation, the kinetic energy of the uprushing wavefront is approximately balanced by its potential energy. In this case and (ref. 29), which leads to

We choose the rescaled duration of HWEs to qualitatively reproduce the main erosional regimes described in ref. 18 (Supplementary Fig. 4). Crucially, the predicted bistable behaviour of barrier islands depends only on the existence of these erosional regimes and not on the details of the sand flux formulation.

Surface dynamics.

At the foreshore (x < 0), we assume an equilibrium defined by a constant slope of angle α. This assumption implies that aeolian erosion is balanced by accretion in the swash zone. As a result, the simulated foreshore acts as a sand reservoir supplying an unlimited amount of sediment to the backshore, effectively feeding dune formation and post-storm recovery. For the shoreline (x = 0), we assume, as a first approximation, that under RSLR it follows Bruun’s rule and migrates landward at a rate proportional to the rate S of RSLR. At the backshore (x > 0), we calculate the change in the sand surface elevation h from mass conservation as

during HWEs, and

otherwise. The last two terms at the right-hand side describe the effects of RSLR and appear because we define the surface elevation relative to the sea level and the shoreline position.

Vegetation dynamics.

As a first approximation, we assume a single generic grass species with a cover fraction ρ_veg that is sensitive to sand erosion and accretion and that can increase up to the maximum cover ρ_veg = 1 during a characteristic time t_veg. We further assume plant growth is also sensitive to frequent saltwater inundation and thus to the proximity of the shoreline and to the elevation above MHWL, such that plants can effectively grow only landward of the vegetation limit L_veg and in places higher than a minimum elevation Z_c. Thus,

where the Heaviside function Θ(s) (1 for s > 0; 0 otherwise) defines the regions where plants can grow.

Parameters.

We investigate model outcomes as function of the parameters characterizing the external forcing and the response of the system (the explored range is in parenthesis): the vertical vegetation limit Z_c (0.02–0.2 m), frequency of HWEs T_HWE (0–20 yr), mean total water elevation (1–2 m), rate S of RSLR (0–0.02 m yr⁻¹) and the imposed onshore wind, characterized by the ratio of the undisturbed shear velocity u_∗0 and the transport threshold u_t (1.5–2.5), and by the fraction r_t of the time the wind is above the transport threshold and sand is available.