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The secret gardener:
vegetation and the emergence
of biogeomorphic patterns in
tidal environments
rsta.royalsocietypublishing.org
Cristina Da Lio1,2 , Andrea D’Alpaos3
and Marco Marani1,4
Research
Cite this article: Da Lio C, D’Alpaos A,
Marani M. 2013 The secret gardener:
vegetation and the emergence of
biogeomorphic patterns in tidal environments.
Phil Trans R Soc A 371: 20120367.
http://dx.doi.org/10.1098/rsta.2012.0367
One contribution of 13 to a Theme Issue
‘Pattern formation in the geosciences’.
Subject Areas:
hydrology
Keywords:
tidal environments, marsh, vegetation,
ecosystem engineering
Author for correspondence:
Marco Marani
e-mail: [email protected]
1 Department of ICEA, University of Padova, Via Loredan 20,
Padova, 35131, Italy
2 Institute of Marine Sciences, National Research Council, Arsenale –
Tesa 104, Castello 2737/F, 30122, Venezia, Italy
3 Department of Geosciences, University of Padova, Via Gradenigo 6,
35131, Padova, Italy
4 Nicholas School of the Environment and Pratt School of
Engineering, Duke University, Durham, NC 27708, USA
The presence and continued existence of tidal
morphologies, and in particular of salt marshes,
is intimately connected with biological activity,
especially with the presence of halophytic vegetation.
Here, we review recent contributions to tidal
biogeomorphology and identify the presence of
multiple competing stable states arising from a
two-way feedback between biomass productivity
and topographic elevation. Hence, through the
analysis of previous and new results on spatially
extended biogeomorphological systems, we show that
multiple stable states constitute a unifying framework
explaining emerging patterns in tidal environments
from the local to the system scale. Furthermore, in
contrast with traditional views we propose that biota
in tidal environments is not just passively adapting
to morphological features prescribed by sediment
transport, but rather it is ‘The Secret Gardener’,
fundamentally constructing the tidal landscape. The
proposed framework allows to identify the observable
signature of the biogeomorphic feedbacks underlying
tidal landscapes and to explore the response and
resilience of tidal biogeomorphic patterns to variations
in the forcings, such as the rate of relative sealevel rise.
2013 The Author(s) Published by the Royal Society. All rights reserved.
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1. Introduction
The time evolution of topographic elevation z(x, t) (computed with respect to mean sea level,
hereinafter MSL) at position x (in a planar cartesian reference frame) is jointly determined by the
rate (e.g. in (mm yr−1 )) of inorganic sediment deposition, Qs (B, x, t) (the sum of sediment settling
owing to gravity and trapping by standing biomass), the rate of organic soil production Qo (B, x, t)
(associated chiefly with below-ground biomass production), the erosion rate E(B, MPB, x, t)
(owing to wind waves and, possibly, to tidal flows) and the rate of relative sea-level rise, R
(herafter, RSLR, the algebraic sum of sea-level rise and local subsidence, possibly a function of
time and space):
∂z
(2.1)
= Qs (B, x, t) + Qo (B, x, t) − E(B, MPB, x, t) − R(x, t).
∂t
MPB, accounts for the stabilizing effect of MicroPhytoBenthos, the benthic microalgae whose
polymeric excretion forms a biofilm with high mechanical strength [3,23]. B(x, t) is the biomass
produced by halophytic vegetation, i.e. vegetation adapted to living in the hypersaline and
hypoxic marsh soils [10,24], which affects inorganic deposition (by increasing flow energy
dissipation, reducing the turbulent kinetic energy and by direct trapping, e.g. [25–27]), produces
organic soil (e.g. [4,28,29]) and reduces erosion (by damping wind waves and stabilizing the
sediment, e.g. [30–32]). In turn, vegetation development is strongly linked to marsh elevation
(summarizing the effects of sediment salinity and aeration, the stress factors that are more
directly acting on marsh vegetation), such that a strong feedback exists between geomorphic
characteristics and biomass production. Equation (2.1) inherently refers to a ‘vertical’ accounting
of deposition and erosion contributions, and hence does not include edge erosion processes.
We note, however, that neglecting such processes does not affect the validity of the results
presented here, as ‘vertical’ dynamics and horizontal erosion of tidal landforms are largely
independent [33,34].
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2. Depositional tidal biogeomorphology
rsta.royalsocietypublishing.org Phil Trans R Soc A 371: 20120367
Tidal landforms in lagoons and estuaries exist in a constant pursuit of sea-level rise, offset by an
interplay of inorganic and organic sediment deposition (e.g. [1–4]). An outcome of such pursuit
is the generation of some striking biological and morphological patterns at different scales. At the
large scale, marshes, tidal flats and subtidal areas form the ‘universal’ texture of the tidal zone
[3–7]. At a smaller scale, zonation patterns, a mosaic of sharply bounded and nearly homogeneous
vegetation patches, are widespread in marshes worldwide [8–10]. Indeed, many aspects of tidal
biogeomorphodynamics and of the associated patterns have been seen to be tightly linked to
interactions between biotic and abiotic processes [11–18].
We point out here that intertidal vegetation and microalgae act as ecosystem engineers [19]
by exerting a control on their substrate elevation, by stabilizing the sediment against erosion
and by directly producing organic sediment. We also suggest that ecosystem engineering plays a
fundamental role in determining tidal biogeomorphic patterns, in contrast with their traditional
interpretation as the result of a passive adaptation of biota to controlling physical processes and to
the associated edaphic conditions [1]. In particular, we show that depositional intertidal patterns
emerge from, and are the signature of, two-way feedbacks between accretion rates, inorganic
sediment transport and biological processes controlling organic sediment productivity. To this
end, we first briefly review contributions on biogeomorphic equilibria in a conceptual lumped
setting [20,21]. Subsequently, building on new observational and modelling results [22], we
introduce original analyses on the spatially extended equilibrium configuration of salt marshes,
showing how multiple competing stable states, largely regulated by biota, provide a unifying
framework to explain emerging patterns in tidal environments. Through this framework, we
finally explore how tidal biogeomorphic patterns change with vegetation characteristics and
competition dynamics, as well as their responses and resilience to variations in the forcings, such
as rates of relative sea-level rise.
2
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(a) Modelling framework
Marani et al. [3,20] and D’Alpaos et al. [21] have examined the numerical solution of equation (2.1)
when this is integrated in space to describe the time evolution of the mean elevation of a tidal
platform (scales indicatively larger than 1 km). Hydrodynamic flows and sediment transport in
this lumped approach are parametrized to describe the exchange of water and sediment between
the channel (where the tidal wave propagates and suspended sediments are transported) and the
tidal platform. This is done according to the following conservation equation [3,35]:
dh
d
(DC) = −ws C − αBβ C + C̃ ,
dt
dt
(3.1)
where h(t) = H · sin(2π t/T) is the instantaneous tidal elevation with respect to the local MSL (H
being the tidal amplitude and T = 12 h the main tidal period). D(t) = h(t) − z is the instantaneous
water depth. C is the instantaneous suspended sediment concentration (SSC), C̃(z, B, t) = C0 when
dh/dt > 0 (C0 being the prescribed SSC in the channel), whereas C̃(z, B, t) = 0 when dh/dt < 0.
Note that ws C(t)/ρb and αBβ C(t)/ρb are the instantaneous settling and trapping fluxes,
respectively, which, by integration over all tidal cycles in 1 year yield the yearly mean total
inorganic flux, Qs (t), from the water column to the surface [20]. We assume ws = 0.2 mm s−1 for
the settling velocity (typical of a sediment size of 20 µm [36]), whereas α and β synthetically
account for suspended sediment trapping by the canopy (see [20] for details). ρb = ρs · (1 − p) is
the bulk density of the sediment after compaction has taken place, with ρs = 2650 kg m−3 and
porosity p = 0.5.
Erosion is assumed to be zero when vegetation is present [30,31], otherwise it is E(B, MPB, t) ∝
(τ (z, t) − τc (MPB)), where τc (MPB) is the threshold shear stress above which erosion starts
(E(B, MPB, t) = 0 if τ (z, t) < τc (MPB)), and τ (z, t) is the wave-induced bottom shear stress (a
function of the local wind ‘climate’, and of time-varying water depths). τc (MPB) is a function
of the possible presence of microphytobenthos.
Vegetation may be assumed to respond over time scales (1 to a few years) which are shorter
than the characteristic response times of geomorphological processes (usually of the order of
several to tens of years [20]). Hence, rather than introducing a full dynamical equation for the
time evolution of biomass, we assume it to be in equilibrium with the current landform elevation,
i.e. we simply prescribe it as B(z). From observations in Mediterranean marshes, where several
vegetation species compete for marsh space, one can assume that there a biomass production
which is an increasing function of elevation (for marsh platform elevations between MSL (z = 0)
and z = H). We call this the multi-species case: B(z) = B0 · 2 · z/(H + z) (considering B0 ∼
= 1 kg m−2 ,
B(z) = B0 if z > H and B(z) = 0 if z < 0 [3]). Field observations in Spartina-dominated areas [4,37]
give, on the other hand, a biomass productivity which decreases with z in the range 0 < z < H:
B(z) = B0 · 2 · (H − z)/(2H − z); B(z) = 0 if z > H or z < 0. In either case the organic contribution to
accretion is expressed as Qo (z) = γ · B(z), where γ = 2.5 × 10−6 m−3 yr−1 g−1 incorporates typical
vegetation characteristics and the density (after compaction and partial decomposition) of the
organic soil produced [20,29].
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3. A point model: large-scale tidal patterns
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Because temporal variations in the bottom profile occur over time scales typical of
morphological changes, i.e. time scales of one to several years, equation (2.1) can be regarded as
the annually averaged mass balance equation. Qs (B, x, t), Qo (B, x, t) and E(B, MPB, x, t), however,
must be obtained by first resolving the relevant biotic and abiotic processes at the time scale of a
single tidal cycle and by then integrating the corresponding sediment fluxes to the yearly scale.
Steady-state solutions to equation (2.1) will be described below for a point model, in which the
landform being analysed is ‘collapsed’ into a single point and z can be interpreted as its mean
elevation, and for a one-dimensional representation, which allows the description of the smaller
scale morphological properties.
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10
sub-tidal platform
tidal
flat
4
marsh
dz/dt (mm yr–1)
−5
−10
−3
−2
−1
z (m above MSL)
0
1
Figure 1. Phase portrait of the biogeomorphic evolution of a point tidal landform. This example represents the Spartinadominated case, with C0 = 20 g m−3 , H = 0.74 m, R = 8.5 mm yr−1 . Three competing equilibria exist for this particular choice
of the forcings: (i) a subtidal platform, with elevation lower than the minimum tidal level (Hmin = −0.74 m in this case),
(ii) a tidal flat, with elevation between Hmin and MSL and (iii) a vegetated marsh, with elevation above MSL. (Online version
in colour.)
(b) Stable equilibria and corresponding biogeomorphic patterns
Use of the above mathematical representations of the controlling physical processes makes the
deposition and erosion fluxes on the right-hand side of equation (2.1) dependent just on z, such
that one obtains an equation describing a one-dimensional dynamical system: dz/dt = F(z) − R.
R plays the role of an external forcing and F(z) is a nonlinear function as a result of trade-offs
between biotic and abiotic geomorphic processes. The nonlinearity of F(z) in itself implies that the
system will display a set of stable and unstable fixed points, corresponding to an equal number
of competing equilibrium states [38]. Hence, from these considerations alone, we can infer the
presence of a path to pattern formation: only specific and recurring landforms may exist within
a tidal environment, corresponding to the available equilibrium states as mediated by the type
of vegetation (Spartina-dominated or multi-species) and by space-varying forcings (e.g. RSLR and
available SSC).
If the rate of RSLR is constant in space and time, it is possible to easily identify the accessible
equilibria by geometrically identifying the values of z corresponding to the condition dz/dt = 0.
Stable fixed points are defined by d/dz(dz/dt) < 0, such that, depending on R and C0 , up to
three stable equilibria may occur on the three descending branches of F(z) illustrated in figure 1
(for the Spartina-dominated case). Such stable equilibria correspond to the basic tidal landforms:
(i) subtidal platforms, with elevation lower than the minimum tidal level (Hmin = −0.74 m in this
case); (ii) tidal flats, with elevation between Hmin and MSL; and (iii) vegetated marshes (colonized
by Spartina in the case of figure 1), located above MSL. Note that changes in R, determined
by spatial (e.g. owing to heterogeneous subsidence rates) or long-term (induced by climatic
changes) variations of RSLR rates, vertically shift F(z) and may vary the number and types of
accessible equilibria, possibly triggering abrupt biogeomorphic regime shifts [3,20]. In particular,
it is seen that thresholds exist above which the marsh and the tidal flat equilibria no longer
exist (R > 9.7 mm yr−1 and R > 15.5 mm yr−1 , respectively, for the values of H and C0 assumed
in figure 1). These thresholds signal transitions to regimes characterized by a reduced number of
possible biogeomorphic patterns and mark limits to the resilience of tidal systems as we currently
know them [2,20,39]. In the multi-species case (not reproduced here for brevity), because organic
soil production increases monotonically with elevation, no stable marsh equilibrium is possible
for C0 = 20 g m3 and R = 8.5 mm yr−1 , and only subtidal platform and tidal flat equilibria exist
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Vegetation zonation is possibly the most recognizable characteristic feature of tidal marshes
over scales between a few centimetres and about 100 m. It is traditionally described as
a spatial vegetation pattern in which patches composed of single species or of mixtures
of species with approximately fixed composition display sharp transitions in the seeming
absence of appreciable topographic gradients [8–10]. We show here that zonation is, in fact, a
biogeomorphological pattern (as illustrated in figure 2), rather than simply a biological one, and
that it is the visible symptom of the underlying feedbacks between biomass productivity and
soil accretion.
(a) Modelling framework
We summarize here the approach introduced in Marani et al. [22] to study the time evolution of
a one-dimensional marsh transect perpendicular to the channel feeding the marsh with water
and suspended sediment (figure 2)). This configuration may represent both a fringing marsh,
delimited landward by a natural/artificial levee, or half of a marsh delimited by channels/creeks
on both sides. The system is governed by the one-dimensional version of equation (2.1) (where
the erosion of the surface is neglected as waves are effectively dampened by marsh vegetation),
complemented by a hydrodynamic/sediment transport formulation and by a description of the
biomass contribution to soil accretion.
The local inorganic sediment deposition rate is computed by integrating the instantaneous
settling flux, ws · C(x, t)/ρb (ws and ρb having the same value assumed in the point model case)
over a whole year, to obtain the mean annual settling rate Qs (x, t). The trapping flux is here
neglected, following Marani et al. [3] and Mudd et al. [27], who showed that particle settling
largely dominates inorganic sediment deposition for flow velocities commonly observed in tidal
marshes, i.e. less than 0.05 m s−1 in microtidal environments (i.e. areas where the tidal range is
less than 2 m). The instantaneous SSC, C(x, t), needed for the evaluation of Qs (x, t), is obtained by
assuming an advective–dispersive sediment transport process (solved at time scales shorter than
a tidal cycle) of the form
∂
∂C
∂(DC)
+
uCD − kd D
= −ws C,
(4.1)
∂t
∂x
∂x
where D(x, t) = h(t) − z(x, t) is the local instantaneous water depth; h(t) is the spatially uniform
instantaneous water elevation obtained by assuming a semidiurnal sinusoidal fluctuation of the
tidal level as described earlier; kd is the dispersion coefficient (assumed here to be 1.5 m2 s−1 [40]);
and u(x, t) is the local instantaneous fluid advective velocity, computed by assuming a quasi-static
propagation of tidal levels from the continuity equation ∂D/∂t + ∂(Du)/∂x = 0. The separation of
time scales between equations (4.1) and (2.1) allows us to decouple the two solutions: sediment
transport is solved through a finite volume numerical method over a tidal cycle, by setting the
following boundary conditions D(0, t) = h(t) − z(0, t) and u = 0|x=L in the continuity equation; and
∂C/∂x|x=L = 0 and C(0, t) = C0 in equation (4.1).
In order to evaluate the role of a set of competing vegetation species on marsh pattern
formation, we express the local organic accretion rate as dependent on a species-specific fitness
function: Qo (x, t) = γ Bi (x, t) = γ B0 fi (z(x)), ‘i’ being the species which happens to colonize site x at
......................................................
4. An one-dimensional model: marsh-scale biogeomorphic patterns
5
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(a marsh equilibrium can be recovered, however, either by lowering the value of R or by
increasing the value of C0 ). This further illustrates the strong control exerted by vegetation on the
tidal landscape (the interested reader will find a more detailed discussion of this point in [20–22]).
Competing stable states thus constitute the pattern formation mechanism for the large-scale
biogeomorphic features characteristic of tidal environments worldwide. It is now interesting
to explore the mechanisms that lead to the formation of well-known, smaller scale, patterns
associated with marsh vegetation species distributions.
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6
x
z
1.0
(m)
salt-marsh platform
z(x,t)
0.5
0
0 1 (m)
Figure 2. Conceptual illustration of a salt-marsh transect and of the characteristic biogeomorphic patterns identified and
analysed here. The sketch emphasizes the existence of intertwined vegetation and topographic patterns in which vegetation
patches develop at different elevations controlled by vegetation itself through organic soil production. The model is run on a
similar one-dimensional domain with length 40 m. The landscape-forming tide is assumed to be a sinusoidal tide with period
T = 12 h and amplitude H = 0.5 m. (Online version in colour.)
time t. 0 < fi (z) < 1 summarizes the degree of adaptation of species i to elevation z and represents
here the fraction of the maximum rate of biomass production, B0 , that species i is able to produce
at elevation z. Marsh species are commonly observed to be maximally productive within specific
ranges of optimal adaptation [6,7]. Following Marani et al. [22], we represent the decrease in
productivity as elevation moves away from the optimal range by way of a simple analytical
expression: fi (ζ ) = 2 · [exp(λ(ζ − ζ̄i )) + exp(−λ(ζ − ζ̄i ))]−1 , where ζ = z/H, ζ̄i = z̄i /H corresponds
to the maximum value of the fitness function for species i, and λ is a scale parameter, expressing
the rate at which the fitness function tends to zero as elevation departs from the optimal range.
A large value of λ represents specialized species, i.e. species which display a nearly maximum
biomass productivity (and reproduction rate, as we shall see later) just within a narrow range of
elevations. Conversely, a small value of λ represents species which are relatively well adapted to
a broader range of marsh elevations.
Ecological competition for space is of course an important driver of vegetation distribution.
Competition strategies in marshes, as in other biomes, involve investing energy and carbon,
e.g. into increasing reproductive rates, or into an apparatus which increases the chance of plant
survival (e.g. aerenchima, a system of conduits in the stem for ‘snorkelling’ air during tidal
flooding, see [24]). It is reasonable to assume that such competitive abilities must operate most
efficiently when a plant finds itself in the optimal environmental conditions for its species.
Hence, we assume here that competition, i.e. the mechanism by which a generic site at elevation
zk = z(xk , t + t) may be colonized by species j at time t + t, while it was previously colonized
by species i at time t (when elevation was z(xk , t)), should be driven by the value of the fitness
function fj (zk ) as compared to fi (zk ). We consider two competition mechanisms. A fittest-takesall mechanism, in which the species distribution throughout the system is updated at every
(yearly) time step by placing at the generic site xk the species i for which fi (zk ) > fj (zk ) ∀j = i.
This mechanism allows the study of system equilibria and patterns in the ideal case in which
stochastic heterogeneities (in the environment and in the organisms) affecting the outcome of
competition are absent. We will see that this approach produces sharp biogeomorphic patterns,
which serve to more clearly illustrate vegetation controls on marsh morphology. However,
actual marshes are characterized by heterogeneous soil properties, individually varying plant
responses and variable exposure to external forcings (wind, waves, insolation, nutrients, etc.),
such that they cannot be expected to behave according to a perfectly regular the fittesttakes-all rule. In order to obtain a more realistic representation of actual marsh processes,
we also consider the results of a stochastic competition mechanism, in which the ‘state’ of
each site xk is updated at each time step by randomly selecting species i with a probability
pi (zk ) = fi (zk )/ j fj (zk ).
The model simulations discussed below are started from a random distribution of individuals
from a set of three species, whose fitness functions are preassigned in such a way as to span the
elevation range between MSL and the maximum tidal level H in an approximately homogeneous
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h(t)
H
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We first consider a reference case (figure 3a) in which the platform is flooded by a tidal
oscillation of period T = 12 h and amplitude H = 0.50 m and is forced with a prescribed SSC
in the channel C0 = 20 mg l−1 and a rate of RSLR R = 3.5 mm yr−1 (e.g. consistent with the
local rate of RSLR for the twentieth century in the Venice Lagoon [41]). We consider three
vegetation species with the same degree of specialization (λ = 5), which differ for different
optimal elevation values (z̄1 = 0.45 m, z̄2 = 0.25 m and z̄3 = 0.10 m above MSL) for the ‘red’, ‘green’
and ‘blue’ species (dashed black line in figure 3c,f,i). Vegetation competition is modelled according
to the fittest-takes-all mechanism. Theoretical analyses of the fixed points of equation (2.1)
and simulations indicate that a stable steady-state configuration exists. Such configuration
exhibits sharp transitions between neighbouring biogeomorphic terrace-like structures (figure 3a),
which display very mild within-patch slopes and are colonized by a single vegetation species
(emphasized by colour-coding in figure 3a). These zonation patterns are very reminiscent of
the typical features observed in actual salt-marsh landscapes, and in particular of the sharp
boundaries between vegetation patches observed in nature [8–10].
Following Marani et al. [22], the physical and biological processes responsible for the
development of zonation patterns in salt-marsh landscapes can now be identified through the lens
of the quantitative framework described. In particular, one finds that zonation structures emerge
as a result of ecogeomorphic feedbacks involving inorganic deposition and organic accretion (via
biomass production), which lead to pairs of stable and unstable equilibrium states.
Let us consider, for the purpose of illustration, the feedbacks that produce the geomorphic
structure located at mid-elevations (species i = 2, identified by z̄2 = 0.25 m, in figure 3a, in green
in the online version), and in particular the equilibrium of the lowest site within this patch
(with coordinate, say, x̂2 ). The equilibrium condition ∂z/∂t = Qs (z) + γ · B0 f2 (z) − R = 0 gives
two solutions: a stable equilibrium above the maximum of f2 (z) (z̄2 = 0.25 m above MSL), with
(s)
elevation z2 = 0.26 m above MSL (solid circles in figure 3b,c, in green in the online version), and
(u)
an unstable equilibrium below the maximum of f2 (z), located at z2 = 0.22 m above MSL (open
circles in figure 3b,c, in green in the online version).
Note that Qs (z) is found (through numerical simulation) to be a monotonically decreasing
function of z because an increase in z, by decreasing the amount of the ‘precipitable’ suspended
sediment C · D = C · (h − z), decreases the amount of inorganic sediment deposited during a tidal
cycle. Therefore, the existence of two solutions to ∂z/∂t = 0 is not because of variations in Qs (z),
but is a consequence of the presence of a maximum in f2 (z), and hence of biological controls.
To determine the stability properties of the equilibrium state in x̂2 (figure 3a), we numerically
find the value of ∂z/∂t from equation (2.1) for a set of transect configurations obtained by
(s)
(u)
perturbing the local elevation in x̂2 within a neighbourhood of z2 and z2 (figure 3b), and we
look at the sign of the changes in the sediment fluxes brought about by the perturbation.
(s)
(u)
(s)
A positive perturbation above z2 (z1 > z > z2 , figure 3b) generates a decrease in biomass
production (figure 3c) and in the related organic accretion rate (because df2 (z)/dz < 0), together
with a decrease in the inorganic deposition rate, Qs (see discussion above). Because both Qs
and Qo decrease with respect to their equilibrium values, the accretion rate becomes negative,
∂z/∂t < 0 (figure 3b), thus acting to dissipate the initial perturbation and bringing the system back
(s)
(u)
(s)
to the original equilibrium. Similarly, a negative perturbation below z2 (i.e. z2 < z < z2 ) induces
an increase in the total (organic + inorganic) accretion rate, such that ∂z/∂t > 0 (figure 3b), again
(s)
acting against the imposed perturbation and bringing the system back to the equilibrium state. z2
is thus a stable equilibrium, a condition which can in summary be expressed as ∂(∂z/∂t)/∂z < 0.
A similar reasoning can be applied to all other sites belonging to the same vegetation patch
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(b) Stable equilibria and corresponding biogeomorphic patterns
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manner. Several tests suggest that the salient properties of the final equilibrium configuration
of the transect do not depend on the initial topographic profile, such that an initial topography
which linearly decreases away from the channel is assumed in the following.
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0.4
0.4
(c) 0.5
z1
0.4
0.3
ˆz 2
z(s)
2
z(u)
2
0.3
z2
0.2
0.2
0.2
0.1
0.1
initial condition
xˆ2
0
z3
z3(u)
0
0.5
(e) 0.5
( f ) 0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0
0
0
(g) 0.50
(h) 0.50
(i) 0.50
0.25
0.25
0.25
0
0
0
−0.25
−0.25
z (m above MSL)
(d )
z (m above MSL)
0
z(s)
3
0.1
R = 5 mm yr–1
−0.25
R = 7 mm yr–1
−0.50
0
5
10
15
20 25
space (m)
30
35
40
−0.50
−0.50
−1
0
1.0
0 0.5 1.0
−3
fitness
dz/dt (m × 10 /y)
Figure 3. (a) Equilibrium topographic profile for the reference case: C0 = 20 mg l−1 and R = 3.5 mm yr−1 ; (b) accretion rate as
a function of a local perturbation of elevation, the rest of the topographic profile remaining unperturbed. Solid circles represent
stable elevation equilibria, open circles unstable equilibria; (c) fitness functions of the species populating the transect (scale
parameter λ = 5). Dashed lines represent the maximum productivity elevations. (d,g) Same as in (a) except for R = 5 mm yr−1
and R = 7 mm yr−1 , respectively. The dashed line represents the equilibrium state for the reference case (a); (e) and (h) the
same as in (b) except for R = 5 mm yr−1 and R = 7 mm yr−1 , respectively; (f,i) the same as (c) except for R = 5 mm yr−1 and
R = 7 mm yr−1 , respectively. (Online version in colour.)
(‘green’ in the online version), to find that these are all stable equilibrium states, whose elevation
slowly increases from right to left because of a spatial increase in the inorganic deposition rate,
Qs , when approaching the channel, the source of inorganic sediment.
(u)
The second equilibrium solution, z2 , is also important, as it is involved in the pattern(u)
generating mechanism. Let us imagine site x̂2 having elevation z2 . A positive perturbation
(u)
bringing the local elevation to z > z2 enhances the organic accretion rate, Qo (figure 3c), while
reducing the inorganic deposition rate, Qs . However, one finds numerically that the decrease
in Qs is small compared with the increased organic sediment production, such that ∂z/∂t > 0
(figure 3b). This means that a positive perturbation is destined to grow, eventually leading the
(s)
(u)
local elevation to z2 . Similarly, a negative perturbation bringing the local elevation to z < z2
generates a decrease in the organic accretion rate (because df2 (z)/dz > 0) which is faster than the
increase induced in the inorganic deposition rate, Qs , thus making ∂z/∂t < 0 (figure 3b). Negative
(u)
(s)
perturbations to z2 are also destined to grow, thus pushing the elevation to z3 , the next lowest
stable equilibrium state, associated with species i = 3 (solid circle in figure 3b, in blue in the online
version).
8
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z1(s)
z1(u)
(b) 0.5
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z (m above MSL)
(a)
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We can now analyse the changes induced in biogeomorphic patterns when the marsh is subjected
to increasing rates of RSLR. We explore here R = 5 mm yr−1 and R = 7 mm yr−1 for comparisons
with the reference value R = 3.5 mm yr−1 , representative of many tidal environments under
current conditions. In the case of R = 5 mm yr−1 , the biogeomorphic features of the equilibrium
patterns are qualitatively similar to those of the reference case, although some differences emerge
(figure 3d). The comparison of the two equilibrium profiles shows that as the rate of RSLR
increases, the equilibrium marsh elevations tend to decrease, in agreement with the results of
other modelling approaches [2–4,21,39]. It should be noted that, under increased RSLR conditions,
the lowest sites within the two highest geomorphic structures, colonized by species i = 1 and
i = 2 (figure 3d, in red and green respectively in the online version), lie on the branch of the
fitness functions where dfi /dz > 0 and hence dQo /dz > 0 (solid circles in figure 3f ), very close
to the maximum value (dashed black lines in figure 3f ). Because ∂(∂z/∂t)/∂z < 0 (solid circles in
figure 3e), this suggests that the stability of the equilibria are here governed by the inorganic
deposition flux, i.e. ∂Qs /∂z > ∂Qo /∂z. On the contrary, the lowest equilibrium elevation in the
lowest patch, colonized by species i = 3, belongs to the branch of the fitness curve where dQo /dz <
0 (solid circle in figure 3f, in blue in the online version), and is therefore governed by the organic
flux, as in the reference case (i.e. ∂Qo /∂z > ∂Qs /∂z). Interestingly, the general lowering of the
equilibrium elevations induced by an increased rate of RSLR causes a sort of cascade effect, in
which areas that, for a lower value of R, belonged to higher elevation structures now abrupt
transition to lower equilibrium states. As a result, the area colonized by the species adapted to
the lowest elevations is larger under increased R, with a corresponding reduction in the extent of
the higher vegetation patches. A further increase in the rate of RSLR, however, may drastically
change marsh patterns, owing to the lowest sites in the lowest patch becoming unstable and
switching abruptly to a yet lower (unvegetated) equilibrium state (figure 3g,h,i). Part of the
marsh has transitioned to a tidal flat equilibrium state, a transition suggested also by the point
model. It is important to note, however, that the rates of RSLR suggested by the two models are
rather different and refer to different phenomena. The point model, in fact, can only deal with
equilibria of a tidal platform as a whole. The one-dimensional model in this case shows that for
relatively low threshold rates of RSLR (approx. 6 mm yr−1 versus 7.2 mm yr−1 obtained from the
point model, considering H = 0.50 m and C0 = 20 mg l−1 ) part of a marsh can collapse to a tidal
flat equilibrium with the consequent loss of the associated biodiversity and ecosystem services.
Interestingly, this finding points to the existence of a maximum sustainable length of vegetated
marshes dependent on the RSLR (and, similarly, on the local SSC in the feeding channel, not
explored here).
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(c) The role of the rate of relative sea-level rise
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The pattern-generating mechanism is thus clarified: step-like structures emerge because
unstable states, controlled by the shape of the fitness functions, act as repellers, pushing
topographic elevation either to the immediately higher stable equilibrium or to the next lowest
one. This picture, highlighting the role of vegetation species in tuning soil elevation within ranges
in which they are preferentially adapted, is valid when soil organic production is more sensitive
to variations in z than the inorganic sediment deposition, such that |∂Qs /∂z| |∂Qo /∂z|. In this
case, ∂Qo /∂z controls the stability of the equilibria.
On the contrary, when |∂Qs /∂z| ≈ |∂Qo /∂z|, equilibrium stability is jointly determined by
inorganic deposition and biomass production. This is for example the case for the part of the
transect adjacent the channel in figure 3a. One may, in fact, note that the equilibrium state
in the upper marsh zone (solid circle in figure 3b,c, in red in the online version) lies on the
branch of the fitness function located below the maximum (z̄1 , dashed black line in figure 3c),
which would seem to imply an unstable equilibrium because df1 (z)/dz > 0. However, because
|∂Qs /∂z| > |∂Qo /∂z|, then ∂/∂z(∂z/∂t) < 0, implying a stable equilibrium controlled by inorganic
sediment deposition.
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(b)
10
0.4
0.2
0.1
l =10
5
10
15
20
25
30
35
40
−1
(e)
(d) 0.5
0
1
0
0.5
1.0
( f)
0.4
0.3
0.2
0.1
l =2
0
5
10
15
20
25
space (m)
30
35
40
−1
0
1
0
dz/dt (m × 10−3 y–1)
0.5 1.0
fitness
Figure 4. (a) Equilibrium topographies for C0 = 20 mg l−1 , R = 3.5 mm yr−1 , λ = 10. The dashed line represents the
equilibrium topography for the reference case (λ = 5) (figure 3a) and (b) accretion rate as a function of perturbations in
the local elevation. Solid circles represent stable elevation equilibria. (c) Fitness functions of the species assumed to populate
the transect with scale parameter λ = 10. Dashed coloured lines represent the fitness functions assumed for the reference case
(λ = 5) and dashed black lines represent the maximum productivity elevations. (d–f ) the same as in (a–c), respectively, but
for λ = 2. (Online version in colour.)
(d) The role of vegetation specialization
We have shown that vegetation acts as a landscape engineer by producing particular
morphological–biological patterns in salt-marsh ecosystems. One might then wonder how
vegetation characteristics influence the biogeomorphic features of salt-marsh landscapes. To
address this question, we have considered vegetation species with different degrees of
specialization, i.e. species capable of adapting to narrower or broader ranges of marsh elevations,
as represented by different values of the scale parameter λ. In particular, we consider here
vegetation species which are both more specialized (λ = 10) and less specialized (λ = 2) than the
reference case (λ = 5). The maximum productivity elevations are in all cases the same as in the
reference case.
Model results show that when vegetation species are assumed to be more specialized
(figure 4a–c) the emerging biogeomorphic structures are more ‘focused’ within narrow elevation
ranges with smaller topographic gradients, the evidence of a stronger vegetation control on
morphodynamic processes. Such a behaviour can be illustrated by comparing the slope of the
fitness functions, dfi /dz, for the reference case and the case of more specialized vegetation species
in figure 4c. As the degree of specialization increases, the slope of the fitness function increases
(compare dashed and solid lines in figure 4c). Therefore, smaller variations in the topographic
elevation are required to determine the changes in Qo necessary to match the spatial gradients
of Qs and thus balance the rate of RSLR. When less specialized (i.e. λ = 2) vegetation species are
considered, topographies displaying smoother, and possibly more realistic, transitions between
patches are obtained (figure 4d–f ). Interestingly, because of the reduced vegetation control
on elevation, within-patch topographic gradients are greater for less specialized vegetation,
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0
z (m above MSL)
(c)
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z (m above MSL)
(a) 0.5
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frequency density (m−1)
0.4
0.3
0.2
0.1
0
5
10
15
20
25
space (m)
30
35
40
0
11
species 1
species 2
species 3
0.1 0.2 0.3 0.4 0.5 0.6
z (m above MSL)
Figure 5. (a) Multiple stable states for C0 = 20 mg l−1 , R = 3.5 mm yr−1 , for a stochastic competition mechanism. The dashed
line indicates the topography of the reference case (figure 3a) obtained with the fittest-takes-all competition mechanism and
(b) multi-modal frequency distribution of topographic elevations in which colour codes represent the most-abundant species
within each elevation interval. (Online version in colour.)
suggesting the possibility of reading the morphological signature of vegetation physiological
adaptations to environmental conditions. Our results, in fact, suggest that the slope of a vegetation
patch may be indicative of the breadth of the range of elevations to which the colonizing
vegetation species is adapted.
(e) Stochastic competition
We finally address the possible role of environmental stochasticity, e.g. associated with
heterogeneities in the soil characteristics or with the variability of individual plant properties.
More realistic and less regular patterns emerge when the outcome of competition is modelled with
a stochastic rule (figure 5a). Even though the marsh profile decreases as one moves away from the
marsh border, as in the reference case, the transect is now characterized by a somewhat irregular
trend and vegetation patches previously populated by single species are replaced by patches
of mixed vegetation species. However, the underlying biogeomorphic feedbacks, responsible
for forming the observed zonation patterns, may still be identified by studying the frequency
distribution of topographic elevation along the transect (figure 5b): the frequency density of
marsh elevations is multi-modal and each frequency maximum is robustly associated with just
one vegetation species, as emphasized by colour-coding the bars of the graph according to the
most abundant species colonizing each elevation interval. Hence, even though topographic and
vegetation patterns are not visually evident in this case, the tight relationship between peaks in
the frequency distribution and vegetation species is found to be a characteristic signature allowing
the detection of the landscape-constructing role of biota [22].
5. Discussion and conclusion
We have analysed the spatial distribution of biogeomorphic patterns characterizing tidal
environments over a wide range of scales. Our key result is that multiple competing stable
states, governed by two-way biogeomorphic feedbacks, provide a unifying framework to
explain the formation of patterns in tidal environments from the marsh to the system scale.
Furthermore, we propose that throughout such a wide range of spatial scales vegetation is not
just passively adapting to morphological features prescribed by sediment transport, but rather it
is fundamentally tuning tidal patterns according to its physiological requirements. Vegetation is
indeed the ‘Secret Gardener’ shaping the tidal landscape.
A conceptual point model, describing the time evolution of the mean elevation of a tidal
platform, shows that subtidal platforms, tidal flats and salt marshes are possible equilibrium
states resulting from the interplay of physical and biological processes. The existence of these
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vegetation fitness functions.
Funding statement. We acknowledge support by Duke University, the PhD School of Civil and Environmental
Engineering Sciences at the University of Padova and the Italian National Project ‘Eco-morfodinamica di
ambienti a marea e cambiamenti climatici’. A.D’.A. thanks the CARIPARO Project titled ‘Reading signatures
of the past to predict the future: 1000 years of stratigraphic record as a key for the future of the Venice Lagoon’.
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