Corresponding author: G. Mariotti, Department of Earth and Environment, Boston University, Boston, MA 02215, USA. (email@example.com)
 We propose a simple model for growth of benthic biofilm subject to variable hydrodynamic disturbances and with a biofilm-dependent erodibility (biostabilization). Model results show that, for disturbances with equal intensity, the biofilm is eroded or not depending on its current biomass, which is a function of the past evolution trajectory. Because of the finite time needed for a biofilm to develop, both the intensity and frequency of periodical disturbances, such as tidal currents, determine whether the biofilm can approach its equilibrium biomass. Spring-neap tidal modulation favors biofilm development, since the reduction of the current shear stress associated with neap tides allows biofilm growth, thus increasing biostabilization and the biofilm's likelihood to withstand the subsequent energetic spring tides. On the other hand, diurnal tidal modulations are negative for biofilm development, because the diel biofilm growth is almost negligible. Under stochastic disturbances associated with wind waves, there are two most-likely states for the biofilm biomass: either close to zero or close to the equilibrium value, depending on wave intensity. If biostabilization is reduced or eliminated, the probability of intermediate values for biofilm biomass becomes also significant. The role of biostabilization is hence to exacerbate the probability of the end-member states. Finally, because of the nonmonotonic relationship between water depth and wave induced bed stresses, only extremely shallow and deep areas favor biofilm persistence. If light attenuation with depth is considered, deep water becomes unsuitable for biofilm growth when water turbidity is high.
 In addition, hydrodynamic disturbances such as currents and waves play a cardinal role, eroding the biofilm and eventually detaching it from the sediment surface [Bryers, 2000; Tolhurst et al., 2006, 2009]. Haynes et al.  showed that, when sediments are covered by biofilm, “the entrainment process does not take place in a manner akin to that of a single rolling particle. Instead, entrainment occurs via biomat failure and carpet-like erosion of the biofilm-sediment composite.” In other words, surface biofilm removal occurs in an “all-or-nothing” fashion. This phenomenon stems from the fact that the thickness of the biofilm (μm to mm [MacIntyre et al., 1996; Decho, 2000; de Brouwer and Stal, 2001]) is much smaller than the thickness of the sediment active layer that is mobilized during recursive erosive events, such as tidal currents and wind waves in intertidal muddy environments (mm to cm [Christie et al., 1999; Andersen et al., 2007]).
 Growth of benthic biofilm (e.g., river periphyton) subject to hydrodynamic disturbances has been previously modeled by Uehlinger et al. , Labiod et al. , and Boulêtreau et al. [2006, 2008, 2010]. However, these studies never took into account biofilm-mediated stabilization. Other studies have parameterized the increase in sediment stability, but they have not coupled it with biofilm development [Le Hir et al., 2007]. Van de Koppel et al.  first combined biofilm growth with biostabilization, introducing a feedback mediated by silt accumulation. Similarly to these authors, we will study biofilm growth limited by hydrodynamic disturbances, the erosion capability of which is controlled by biofilm-enhanced sediment stabilization. However, contrary to van de Koppel et al. , we neglect the coupling with sediment composition, and we focus on the complexity arising from realistic time-dependent hydrodynamic disturbances.
 Because of the increase in sediment stability triggered by biofilms [Le Hir et al., 2007], a hysteresis is predicted, i.e., the state of the system depends not only on its current but also on its past environment. Given the same hydrodynamic disturbances, two possible scenarios occur: 1) bare sediments are constantly eroded and biofilm development is thwarted, or 2) sediments with biofilm-enhanced stability prevent the erosive processes and allow the persistence of a steady biofilm. Hence, the presence of biofilm depends on the previous history and, in particular, on the competition between biofilm growth and sediment stabilization on one side, and the intensity and frequency of the hydrodynamic disturbances on the other.
 In this paper we will explore these complex dynamics using a simple dynamical model. We thus determine what set of hydrodynamic conditions allows biofilm growth and persistence. In particular, we explore two types of disturbances: tidal currents, modeled as a multifrequency deterministic process, and wind waves, modeled as a stochastic process. In addition, for the case of wind waves, we explicitly consider the effect of light attenuation with depth. The model shows that a combination of biological and hydrodynamic disturbances determines the fate of biofilms in shallow coastal areas.
2. Biofilm and Hydrodynamic Modeling
 Here we apply a model that was initially proposed for riverine peryphyton [Uehlinger et al., 1996], and recently modified and tested in the field and in laboratory experiments, using different types of sediment substrate [Labiod et al., 2007; Boulêtreau et al., 2006, 2008, 2010]. The model describes the evolution of the biofilm biomass X, measured as mg Chl-a/m2:
Term 1a is the biofilm growth according to a logistic law, where μmax is the maximum growth rate, and Ks is the half-saturation constant. Term 1b is the light limitation (assuming a photosynthetic biofilm, e.g., diatom dominated), where I is the daily averaged light intensity and KI is the half-saturation constant for the light limitation, while Term 1c is the temperature control, modeled with the Arrhenius law, where σ is the coefficient of temperature dependence. Here we assume that the temperature Te is equal to the reference value Teo, and therefore we set this term equal to unity in the model. Term 2 is the chronic biofilm detachment caused by moderate hydrodynamic disturbances, Term 3 is the self-generated detachment, not associated to hydrodynamics [Boulêtreau et al., 2006], and Term 4 is the catastrophic biofilm removal by extreme hydrodynamic disturbances. The value Xb is the background biofilm biomass, always present even when the biofilm is completely removed, which allows for biofilm recolonization.
 The coefficient for the chronic flow detachment was originally set by Uehlinger et al.  proportional to the flow discharge Q, i.e., εhydro_chronic = CdetQ.Labiod et al.  pointed out that the shear velocity, u*, is a better predictor, thus proposing εhydro_chronic = C′detu*. After performing several experiments with constant discharge and variable u* (induced by the biofilm dependent bed roughness), Labiod et al.  calibrated the value for the coefficient C′det, considering only the terms 1–2 in equation (1). According to their formulation, the biofilm detachment is zero in absence of flow, and hence the biofilm grows indefinitely driven by term 1 in equation (1). This unphysical situation results from neglecting the biofilm decay processes which occur independently of hydrodynamics: senescence, heterotrophic processes [Boulêtreau et al., 2010], and benthic macrofauna grazing [Montserrat et al., 2008] that we include in term 3 in equation (1). Boulêtreau et al.  modeled the coefficient εauto as a function of the active bacterial density Bd, i.e., εauto = CautoBd. Given the uncertainty related to the measurement of chronic hydrodynamic detachment and autogenic decay, and the overlapping of the two processes [Boulêtreau et al., 2006], we assume that the coefficients εhydro_cronic and εauto are constant. This simplification implies either that u* and Bd are constant in time, or that u* is variable in time, but Bd is constant and the coefficient εauto is much greater than εhydro_chronic.
 In addition to chronic flow detachment, extremely high flows are able to mobilize the bed particles and completely remove the biofilm. This process has been modeled analogously to the chronic flow detachment, but with a very high detachment coefficient εhydro_catacstrophic (e.g., 100 days−1 [Uehlinger et al., 1996]). Unfortunately, the quantification of εhydro_catacstrophic is extremely difficult, since it requires the measurement of the loss in biofilm mass in a very short time, e.g., hours or minutes, during extreme events.
 An alternative approach is to assume that the catastrophic erosion events are instantaneous. This simplification is motivated by the observation that the duration of extreme erosive events (∼ hours) are normally much shorter than the time scale of biofilm growth (∼ weeks). The catastrophic erosion (term 4) is therefore modeled as
where δ is the Dirac function, Eo is the intensity of the extreme event, and ti is the time of occurrence. In addition, we assume that the extreme events act on the biofilm as a threshold, “all-or-nothing” process:
where τ is the shear stress associated to the extreme event. If τ is lower than or equal to the critical value for erosion τcr, no effects on the biofilm result. If τ is greater than the critical value, the substrate is eroded and the biofilm is completely detached and it is reduced to the background value Xb. The assumption is supported by the fact that the erosion occurring during extreme hydrodynamic events is on the order of mm-cm [Christie et al., 1999], which is much greater than the biofilm thickness, i.e., μm-mm [MacIntyre et al., 1996]. The model is therefore reduced to
where ε is a global decay parameter, and μe is the effective maximum growth, equal to the product between μmax and the light attenuation term (1b).
 Finally, the presence of biofilm enhances the sediment critical shear stress [Le Hir et al., 2007]. Many relationships between the critical shear stress for erosion and EPS have been proposed, but they are only valid in specific geographic environments [Yallop et al., 1994; Riethmüller et al., 2000; Defew et al., 2002; Friend et al., 2003]. The most common proxies for EPS are Chlorophyll a (Chl-a) which is easy to measure [Underwood and Smith, 1998], or colloidal carbohydrates, which are proportional to the amount of EPS but are more difficult to determine. We assume that the increase in critical shear stress is proportional to biofilm biomass (measured as Chl-a concentration), as suggested by a number of observations [Le Hir et al., 2007]:
where τcr,o is the critical shear stress of bare sediments not colonized by biofilms.
2.2. Model for Light Attenuation
 We assume an exponential decay of the available light with water depth:
where kd is an attenuation factor. In addition to the direct effect of seawater, four water column components are commonly assumed to attenuate light: particulate inorganic matter, particulate organic matter, phytoplankton, and gilvin [Lawson et al., 2007]. For simplicity we assume that these factors are independent of biofilm dynamics and are included in the attenuation constant kd in equation (6).
 We then consider the effect of water level modulation by tide, and we compute the time-averaged light intensity reduction, r [0, 1]:
as a function of tidal range and bed elevation below Mean Sea Level (MSL), where Tω is the tidal period. We find that bed elevation is the dominant parameter determining light intensity reduction (Figure 1). For each fixed bed elevation, an increase in tidal range from 1 to 3 m result in an attenuation of light intensity less than 10%. Therefore we assume that the light intensity attenuation is a function of bed elevation only and independent of tidal range. We hence neglect the light penetration variability induced by changes in water turbidity. The consequences of this assumption are presented in the Discussion section.
2.3. Tidal Currents
 Astronomical tides have a periodical nature: the tide can be reconstructed as the linear superimposition of sinusoidal signals with different amplitude, celerity and phase [Boon, 2004]. The main constituents of the tidal signal are generally the main lunar semidiurnal tide, M2 (TM2 = 12.42 h), and the main solar semidiurnal tide, S2 (TS2 = 12 h). The combination of these two constituents controls the spring-neap modulation, whose period is 14.76 days. The higher the ratio between S2 and M2, here defined as FSN, the higher is the spring neap modulation. In addition, the components K1 and O1 determine the diurnal modulation of the tide (T1 = 24 h). The form number, F, defined as the ratio between the sum of the main diurnal component (A1 = K1 + O1) and the sum of the main semi-diurnal components (A2 = M2 + S2), is often used to quantify the diurnal modulation of the tide [Boon, 2004].
 The tidal constituents clearly define the periodicity of the disturbances (diurnal, semidiurnal, and spring-neap modulation). In order to quantify the hydrodynamic disturbance (i.e., bed shear stress) associated to the various tides, we introduce a simplified model. Assuming a quasi-static tidal propagation, the tidal velocity can be estimated as [Boon, 1975; Fagherazzi et al., 2008]
where L is the drainage length, d is the water depth, and a is tidal amplitude. Assuming a reference drainage length and depth, and a quadric law for the current induced bed shear stress, it results that
After fixing the bed shear stress related to the main tidal constituent M2, i.e., fixing the mean diurnal range, we estimate the variation induced by the higher-harmonics using the scaling of equation (9). We thus compute the spring-neap shear stress modulation as:
Alternatively, fixing the semidiurnal amplitude, we compute the diurnal modulation as:
2.4. Wind Waves
 Locally generated wind waves triggered by storms are able to produce elevated bed shear stresses. Wave conditions, such as significant wave height Hs and peak period Tp, can be estimated in intertidal environments using the semi-empirical relationships of Young and Verhagen  [see Fagherazzi and Wiberg, 2009]:
with A1 = 0.493(gd/[Uwind]2)0.75, B1 = 3.13 10−3(gd/[Uwind]2)0.57, A2 = 0.331 (g χ /[Uwind]2)1.01, B2 = 5.215 10−4(g χ /[Uwind]2)0.73, where g is the gravitational acceleration, where χ is the fetch and Uwind is the reference wind speed. The wave-induced shear stress τwave is then calculated as
where fw is a friction factor, and k is the wave number, computed with the dispersion relationship.
 In order to realistically characterize wind variability, four significant shallow tidal basins in the U.S.A. are considered: Plum Island Sound, Massachusetts (PIE), Virginia Coast Reserve (VCR), Willapa Bay, in Washington State (WB), and Apalachicola Bay in Florida (AB). For all cases, we consider the hourly wind speed measured at NOAA-NDBC buoy stations, located approximately 10–30 km offshore of the tidal basins. The corresponding time series are consistent (i.e., measured in similar conditions), and extensive (∼10 years). To account for the wind speed attenuation from offshore to onshore, the measured wind speed is reduced by 30%. The stations considered are: IOSN3-Isle of Shoals (2001–2009) for PIE; CHLV2- Chesapeake Light (2001–2009) for VCR; 46029-Col River Bar (2001–2009) for WB; SGOF1-Tyndall Tower (2003–2011) for AB.
 We assume a Weibull distribution for the hourly wind speed:
where β is the shape parameter and Wo is the scale parameter of the distribution. The distribution parameters (Wo and β) are: 6.1 m/s and 2.0 for PIE, 7.2 m/s and 1.9 for VCR, 5.6 m/s and 1.9 for WB, and 6.2 m/s and 1.9 for AB. Given the small variability in the shape parameter, we assume that it is constant and equal to 2.0, and we use the wind speed scale as control parameter to describe waves variability.
 The wave processes are simulated in the model using a Monte Carlo approach: equation (4) is computed for a time interval (1 h), at the end of which τcr is estimated from the current value of X (equation (5)). A random value is extracted from the wind probability distribution (equation (14)), and the bed shear stress is computed from equations (12) and (13). If the bed shear stress is greater than the critical value, the catastrophic erosion occurs and X is set equal to Xb. The computation is performed for 50 equivalent-years, corresponding to ∼4 × 105 steps. The frequency distribution of each realization is used as an estimate for the biofilm time-independent probability distribution function p(X). In addition, for the particular case with α = 0, the probability distribution is computed using the steady state master equation (see Appendices A and B).
3.1. Biofilm Dynamics Without Disturbances
 The model in equation (4) can be solved analytically if all parameters are constant in time and E is equal to zero, giving a solution in the following form:
which includes hundreds of polynomial term and it is not reported for brevity (see Appendix B). The steady state solution of the system reads
The model has 6 parameters: μe, Ks, ε, Xb, τcr,o, α. The range of values found in literature and the reference values selected here are reported in Table 1. With the reference values, the steady state equilibrium is equal to about 200 mg Chl-a/m2, which is a commonly found in intertidal environments [Le Hir et al., 2007]. The time needed to reach X = 0.9Xeq is about 21 days, similar to the time needed to reach the biofilm peak in laboratory experiments [Labiod et al., 2007; Haynes et al., 2011]. After fixing the reference values, we investigate the effect of different hydrodynamic conditions on biofilm dynamics.
Table 1. Parameters' Range Found in Literature and Parameters' Value Used as Reference in the Model
3.2. Biofilm Dynamics With Ideal Hydrodynamic Disturbances: Deterministic and Single-Frequency Results
 We first consider the simple case in which the hydrodynamic disturbances are constant in time (i.e., the Dirac function in equation (2) is replaced with a unitary constant). Starting from a condition with no biofilm (X = Xb), the biofilm grows only if τ < τcr,o, reaching Xeq asymptotically. If τ ≥ τcr,o, sediments are continuously eroded, preventing biofilm establishment, i.e., X = Xb.
 If the disturbance intensity varies in time, a hysteresis is introduced. A temporary reduction of the disturbances allows the biofilm to grow and increase the critical shear stress. Once strong disturbances are reintroduced, the biofilm is able to withstand a higher shear stress. Indeed biostabilization extends the range of disturbance intensities for which two stable states are possible: for the same disturbance intensity the biofilm biomass can be either equal to the background or to the equilibrium value.
 It is clear that the biofilm dynamics is controlled by the temporal modulation of the disturbance. We consider a disturbance with period T (i.e., ti = iT, i = 1, 2, 3,…) and intensity τo. Depending on the values of T and τo, different qualitative dynamics are predicted (Figure 2). With high-intensity and rare events (case 1), the biofilm grows during the time span between two consecutive events, but it is destroyed every time a hydrodynamic event occurs. The increased sediment resistance is not enough to prevent the erosion caused by the high-intensity events. With weak and rare events (case 2), the biofilm increases the sediment resistance and no erosion occurs during the hydrodynamic events. As a result, the biofilm asymptotically approaches the equilibrium value. Finally, because of the finite time needed to grow and increase its resistance, a new settled biofilm is periodically detached by small but frequent events (case 3).
 Case 2 and 3 represents a hysteresis. A disturbance with the same intensity is eroding (case 3) or not eroding the biofilm (case 2) depending on biofilm history. This example demonstrates that the presence of biofilm depends not only on the intensity, but also on the frequency of the disturbing events.
 A sensitivity analysis reveals that the parameters μe, Ks, ε, and α are the most important for the determination of the equilibrium configuration (only the variation with μe shown in Figure 3). The first three parameters influence biofilm growth, i.e., are biological parameters, while the last describes the effect of biofilm on sediment strength. The high sensitivity with respect to the biological parameters suggests that variations in temperature, Te, nutrients, μmax, and light availability, I, can catastrophically switch the biofilm state.
3.3. Model With Tidal Disturbances: Deterministic and Multifrequency
 So far the biofilm was subject to single-frequency, deterministic disturbances. Now we study the model results under the effect of tides and waves. The former will be modeled as a multiple-frequency deterministic process, while the latter will be modeled as a stochastic process.
 We approximate a tide with several harmonics and hence multiple time scales with only two disturbances: one occurring during neap and one during spring tide, separated by a half spring-neap cycle (7.4 days) (Figure 4). The biofilm biomass is set equal to Xb in correspondence to a neap tide.
 In order to reach steady state, the biofilm has to withstand both neap tide (smaller disturbance, occurring at the beginning of the biofilm development), and spring tide (higher disturbance, but occurring at a late stage of biofilm development). We define a neap-controlled case when the biofilm is destroyed by the neap tide, and a spring-controlled case when the biofilm is destroyed by the spring tide.
 A non-trivial behavior emerges when both the intensity of the shear stress associated to the M2 tide (τM2), and the ratio FSN are changed in the model (equation (10) and Figure 5). For small values of FSN the biofilm is neap-controlled: the shear stress reduction associated with the neap is not enough to prevent the tide from destroying the newly settled biofilm. For high values of FSN the biofilm is spring-controlled: the weak neap tide allows the biofilm to initially grow, but the strong spring tide destroys the ∼7.4 days old biofilm. There is an intermediate value of FSN, about 0.35 using the reference parameters (Table 1), that identifies the switch between neap and spring controlled regimes. This corresponds to an optimal configuration, at which the biofilm is capable of surviving the highest mean shear stress intensity (i.e., the τM2 shear stress).
 The diurnal modulation is modeled similarly to the spring-neap modulation, but with a shorter period (Figure 4). Differently from the spring-neap, the diurnal modulation has a negative effect on biofilm. Since biofilm growth in a day is minimal, the critical shear stresses during the maximum and minimum diurnal tide are approximately the same. Therefore, an increase in the modulation (i.e., increasing the maximum diurnal tide) triggers biofilm destruction (Figure 6).
3.4. Biofilm Growth and Wind Waves: A Stochastic Approach
 We now explore the stochastic disturbance associated with wind waves. We implicitly assume that wind waves are the dominant process, and hence we restrict the analysis to environments with a small tidal range. We first fix the water depth and we explore the effect of different biostabilization conditions. We then investigate how the water depth influences the model's outcome.
 We begin by exploring the case in which the biostabilization is not present (α = 0). Three wind scenarios are investigated: intense, intermediate, and weak (Figure 7). With intense wind conditions (Wo = 5 m/s) the biofilm is periodically destroyed. The peak of p(X) is found for X ∼ Xb (no biofilm). With weak wind conditions (Wo = 3 m/s), the biofilm is destroyed infrequently. The peak of p(X) is found at X ∼ Xeq. In both cases the probability that the biofilm is at an intermediate state between Xb and Xeq is negligible. With moderate wind conditions (Wo = 4 m/s), the system spends a considerable amount of time close to Xeq, but it is also periodically destroyed. Even though two peaks are present in the probability distribution p(X) for X ∼ Xb and X ∼ Xeq, the probability of an intermediate state is considerable.
 We then introduce biostabilization, and we explore the phase space with axis Wo and α (Figure 8). Three regions are identified: one in which the biofilm expected value is close to zero, one in which the expected value is ∼Xeq, and one in which the expected value is intermediate between the two end-members (Figure 8a). The wind speed scale Wo determining the position of the intermediate region increases with increasing value of α.
 The nature of the intermediate region is explored clustering the biofilm in three states:
where R is the biofilm range, equal to Xeq − Xb. The state which has the highest probability is considered dominant. For low biostabilization potential (α), only one among Sb, Seq, and Sint, is the dominant state (Figure 8b). In particular, the intermediate state is dominant for moderate values of Wo, (as seen in Figure 7b). For higher stabilization potential, the intermediate state becomes less likely and eventually disappears (Figure 8b). Under these conditions, even when the expected value is between Xb and Xeq, the dominant state is either Sb or Seq.
3.5. The Role of Water Depth: Competition Between Wave Stress and Light Attenuation
 Finally, we explore the effect of different water depth, after fixing the biostabilization potential (α = 0.01 Pa/(mg Chl-a/m2)) and the wind speed scale (Wo = 5 m/s).
 For a small attenuation factor (kd ≪ 1 m−1), the biofilm expected value is ∼Xeq for both very shallow (d ∼0.1 m) and deep water (d > ∼0.6 m), while for intermediate water depths (d ∼0.1–0.6 m) the expected value is about zero (Figure 9c). This situation is solely controlled by the wave stress modulation with water depth.
 For higher values of kd (∼1 m−1), the expected biomass value for deep water (2–3 m) decreases and rapidly approaches zero. Even though the bed shear stress remains unvaried, light attenuation decreases the biofilm growth rate μe, and hence reduces the biofilm equilibrium value and the maximum critical shear stress. If we increase the attenuation coefficient above 3 m−1, the biofilm completely disappears in deep water, remaining only in extreme shallow waters.
 Here we have proposed a simple model for the development of benthic biofilms in shallow coastal areas, assuming constant parameters for growth and autogenic decay. The assumption of constant parameters neglects some variability, such those associated to the dynamics of macrofauna grazing and light availability. On the other hand, our assumptions allow to explore time-dependent biofilm dynamics that have been overlooked in the past. This model should therefore be considered as an exploratory tool, paving the ground for more realistic representations of biofilm dynamics.
 The model does not account for biofilm patchiness, which is a common feature [Guarini et al., 1998; Weerman et al., 2010]. Instead, the model undertakes a spatially implicit approach, similar to the model of van de Koppel et al. . Further investigations are necessary to assess the role of patchiness. The simplicity of the proposed model makes it suitable for an implementation in a spatially distributed model.
 The model shows that biostabilization, combined with time-depended disturbances, induces bistability, without the necessity of coupling with sediment composition [van de Koppel et al., 2001]. The system bistability, with the biofilm mass switching between two stable states as a function of biofilm growth and catastrophic hydrodynamic erosion, has some similarities with seagrass dominated environments, often found in lakes and intertidal mudflats [Scheffer et al., 2001; Carr et al., 2010]. The fully developed biofilm with high sediment resistance is equivalent to a tidal flat vegetated by seagrasses. In this case the vegetation maintains low water turbidity, allowing for light penetration and a positive feedback with vegetation growth. On the contrary, in the present model the persistence of biofilm is attributed to its biostabilization effect, and it is not mediated by light penetration. We recognize that because of the biofilm light-dependence (equation (1)) a feedback between biofilm erodibility and turbidity could potentially develop (i.e., affecting the coefficient kd in equation (6)). However, because of the elevated biofilm patchiness [Guarini et al., 1998; Weerman et al., 2010], it is not possible to simply relate water turbidity to the spatially variable biofilm erodibility. This feedback should be investigated considering both biofilm spatial distribution and sediment advection from external sources. It is likely that these feedbacks will increase the resilience of the two states, bare sediment and fully developed biofilm.
 The model considers two distinct environments: one dominated by tidal currents and one dominated by wind waves. The former is representative of areas with small fetch, such as tidal creeks, while the latter is representative of extensive tidal flats with limited tidal range.
 For fixed characteristics of the disturbances, small variations of the biological parameters that control the biofilm growth (i.e., μe) are able to switch the system from periodically destroyed to fully developed biofilm and vice versa. These catastrophic variations could be induced for example by altering nutrients, temperature, or water turbidity, effects that are all included in the parameter μe.
 Our results predict that, for the same mean tidal range, environments with moderate spring-neap modulation are more prone to support a biofilm population, while environments with a diurnal modulation are less prone to support biofilms.
 When wind waves are considered, the effect of biostabilization is to exacerbate the likelihood of finding the biofilm in one of the two end-member states: negligible biofilms or fully developed biofilm mat. This result might explain why intertidal environments exhibit extensive areas of fully developed biofilms, extensive areas of bare sediments, and only little areas of intermediate biofilm condition [Guarini et al., 1998; Orvain et al., 2012; Ubertini et al., 2012], and why the areas of fully developed biofilm persist in time [Guarini et al., 1998; van der Wal et al., 2010].
 Our results can be affected by temporal correlations of wind events. Storms clustered in a narrow time window followed by a long fair weather period favor biofilm recovery, promoting the persistence of the fully developed biofilm mat. Single storms separated by very regular periods of fair weather are likely to create dynamics similar to those associated to deterministic disturbances, such as the tidal currents. However, autocorrelation in wind events would not change the qualitative result that biostabilization promotes two end-member biofilm states.
 The model shows that for a given a set of environmental conditions, such as wave exposure, nutrient availability and grazing pressure, two biofilm states are possible. As a result, careful considerations should be taken when biofilm distribution is correlated with physical variables, such as bed elevation and mud content. In particular, the hydrodynamic disturbances occurring before biofilm sampling should be systematically taken into account.
 A novel model of biofilm growth in shallow coastal areas shows that biostabilization strongly affects biofilm dynamics. This process forces the system to cluster into two end-member cases: a biofilm mass close to zero and a fully developed biofilm.
 The model indicates that not only the intensity of the hydrodynamic disturbances (waves and tides) determines biofilm dynamics, but also the time scale of biofilm growth and the frequency of the disturbances. The model predicts that the spring-neap modulation is favorable to biofilm growth, since the neap period allows the biofilm to develop, increasing the stability of the sediment substrate. Furthermore, we find that the diurnal modulation is always negative for biofilms, because the daily growth is negligible and does not promote biostabilization.
 Our model further predicts that with a stochastic forcing the probability distribution of biofilm concentration is clustered between two end-members: negligible biofilm and biofilm fully developed. A catastrophic switch between the two states can occur depending on external drivers. Biostabilization also decreases the likelihood of states with intermediate biofilm biomass.
 Finally, because of the nonmonotonic relationship between water depth and wave induced bed stresses, only extremely shallow and deep areas are suitable for biofilm persistence. If light attenuation with depth is considered, deep water becomes unsuitable for biofilm growth when the water turbidity is high.
 If biostabilization is neglected (i.e., α = 0 and hence τcr = τcr,o), then the stochastic process can be analyzed in a simplified way. The catastrophic erosion becomes a Poisson process with constant parameter λ, which expresses the probability that an event with τ > τcr,o occurs in a given time interval (1 h).
 In this simplified scenario, the time-independent probability distribution function of biofilm is computed solving the steady state master equation [see Rodriguez-Iturbe et al., 1999]:
where g(X) is the right hand side of equation (4) minus the term E. The term on the right hand side of equation (A1) is a consequence of the independence of occurrence of the erosive events, i.e., the Poisson process. The integration of equation (A1) reads:
where C is an integration constant. The analytical expression of p(X) contains hundreds of terms and it is not reported for brevity (see Appendix B). The master equation constitutes an alternative tool with respect to the Monte Carlo approach, and hence it is used to validate the latter (see Figure 7).
 Here we report a MATLABR script to solve equations (15) and (A2) analytically, using the symbolic toolbox. The parameters used are Ks, me, e, Xb, Wo, beta, tcro, fetch, do corresponding respectively to Ks, μe, ε, Xb, Wo,β, τcr,o, χ and d.