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# Parametric transitions between bare and vegetated states in water-driven patterns

Edited by Andrea Rinaldo, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, and approved June 26, 2018 (received for review December 14, 2017)

## Significance

Since the appearance of land plants in Devonian time, vegetation has played a key role in the coevolution of life and landscapes as a result of mutual orchestrated processes between vegetation characteristics, environmental disturbances, and soil allometry. We mathematically frame the interactions between these three processes into a single parameter that discriminates between vegetated and bare states. In agreement with theories linking ecosystem development to hydrosphere and lithosphere connectivity, this theory suggests that the vegetation biodiversity of river sediment deposits occurs as a selection process in and among the biomechanical characteristics of species. Verified against field observations, this theory allows for water management applications and has important implications to understand natural and man-induced changes in biogeomorphological styles.

## Abstract

Conditions for vegetation spreading and pattern formation are mathematically framed through an analysis encompassing three fundamental processes: flow stochasticity, vegetation dynamics, and sediment transport. Flow unsteadiness is included through Poisson stochastic processes whereby vegetation dynamics appears as a secondary instability, which is addressed by Floquet theory. Results show that the model captures the physical conditions heralding the transition between bare and vegetated fluvial states where the nonlinear formation and growth of finite alternate bars are accounted for by Center Manifold Projection. This paves the way to understand changes in biogeomorphological styles induced by man in the Anthropocene and of natural origin since the Paleozoic (Devonian plant hypothesis).

Mankind has lived by and controlled water systems (e.g., catchments, rivers, lakes, and sea environments) for millennia. Present day, about 80% of the world’s population lives close to and depends on freshwaters and related ecosystems. As a consequence, 48% of all rivers worldwide are hydrologically altered by human activities (1), with a biodiversity threat exceeding the 75th percentile (2). In the future, changes in precipitation-continental patterns due to global temperature rise over the 21st century (3) are expected to have tremendous impacts on freshwater biology [20,000 species are at risk (4)] by perturbing the ecogeomorphological equilibrium of fluvial landscapes and related water infrastructures. Although the important implications, which range from a better understanding of man-induced changes to studying the Devonian plant hypothesis (e.g., refs. 5 and 6), how and which physical conditions determine vegetation encroachment and pattern formation have never been fully analytically framed. This gap also limits our quantitative understanding of the impact of human activities and changing climatic scenarios on water-driven morphological patterns.

In a more physical parlance, surface-water bodies, such as streams, marsh lands, or coastal zones, behave as nonlinear dissipative dynamical systems characterized by stochastic processes, threshold mechanisms, and pattern formation induced by parametric transitions from different states. Such transitions can be smooth or critical [i.e., inducing catastrophic shifts, when small changes in certain parameters of a nonlinear system suddenly cause equilibria to appear or disappear (7⇓–9)].

Due to such a confluence of complexity, a quantitative physically based understanding of the relation between pattern formation and vegetation dynamics remains elusive, as in river systems for example. The reason for this failure is connected to our limited knowledge of the mutual triad interaction among three fundamental processes: (*i*) flow variability, (*ii*) sediment transport, and (*iii*) vegetation dynamics. These three cornerstones (henceforth referred to as ecomorphological triad) are common to many aquatic and aeolian morphogenic processes, and understanding their interactions is the mission of the emerging discipline of ecomorphodynamics (10). In practice, this goal has been addressed through field observations (11), experiments (12), or numerical simulations (13, 14). A limitation of these approaches is that they are partially able to shed light on the fundamental aspects of the involved dynamics and need thus to be complemented with analytical models. To this end, advances have been achieved by considering just two issues at a time, such as flow stochasticity and vegetation (15), vegetation and sediment transport (16).

In this article, a fully analytical physically based approach is formulated to link all of the three above mentioned aspects within the same theory and to address some open issues concerning vegetation inception and morphological pattern formation (17, 18). The rationale of the following theory is general, and it can be broadened to different kinds of spatially extended vegetated systems forced by time-dependent disturbances, such as salt marshes, wetlands, mangrove ecosystems, meandering rivers, aeolian dunes covered by biogenic crusts, or coastal foredunes (19⇓⇓–22) (see Fig. 1 *A*–*C*). Indeed, all these systems share very similar processes embedded into the above mentioned triad. Other remarkable examples of transitions—not considered here—are narrowing–widening and braiding–meandering (23). Instead, we apply our theory first to tackle a yet open puzzling question of physical geography: Why can two single-thread fluvial systems, having nearly the same hydrogeomorphologic features (sediment size d, slope S, and mean annual discharge *D* and *E*)? Physically speaking, the answer is that the phase space describing the dynamical system exhibits two different equilibrium points in the basin of attraction, thus making parametric transitions possible from one state to the other. We present an analytical theory that accounts for the aforementioned triad interaction and describes the vegetated–unvegetated transition in morphologically active rivers. Implications range from unraveling certain mechanisms underpinning effects of plants on rivers and landscape evolution through Earth history to understanding the role of environmental and anthropogenic disturbances in water-driven patterns under changing scenarios.

## Ecomorphological Triad

The main result of our theory states that vegetation encroachment on bare sediment undergoing occasional inundation is a secondary instability problem whose asymptotic behavior can be summarized into a single parameter, α—given by Eq. **6**—which determines whether and how vegetation patterns develop. Thus, the spreading of plants is possible only if their biomechanical characteristics (e.g., growth rate and rooting efficiency) can cope with the removal action by stochastic floods in that particular sedimentary environment. The links among hydrology, biomechanics, and morphodynamics are all contained in the parameter α and determine its sign, which controls the transition from one state to another. Eventually, these processes select suitable plant characteristics in and among species and thus contribute to determine riparian vegetation biodiversity. We obtained this result by mathematically framing the processes of the ecomorphological triad as follows.

### Flow Variability.

Modeling flow variability using a minimalist approach is a challenging task in river science, because of the nontrivial features of the discharge time series *A* (blue line). At steady state, the PDF of the CPP is Gamma-distributed (Fig. 2 *A*, *Right*) and the upcrossing time **1** will be used later on in the triad interaction.

### Morphodynamics.

The second element of the triad interaction—sediment transport—is a threshold process that activates when the flow rate overcomes a lower critical value and becomes morphologically effective when an upper statistically rare flow rate *A*). This so-called formative discharge, an open issue in geomorphology, resets the riverbed by erasing any previous pattern and uprooting all vegetation. The bare flat state so created is the starting point of our analysis.

In a straight river with nonerodible banks, when *D* and *E*), which were theoretically studied through linear and weakly nonlinear analyses (27, 28). Most previous studies neglected the role of flow variability, with the notable exceptions of refs. 29 and 30. Starting from infinitesimal perturbations, the bar amplitude *C*):

Let us introduce the following scaling quantities: the river half-width B for the horizontal lengths, the depth *D*). The linear analysis also provides the bar wavelength (fastest growing mode). Further details are provided in *Methods* and *SI Appendix*, *Section S2*.

The weakly nonlinear analysis through CMP leads to Eq. **2**, from which one obtains **1**, such condition leads to

### Vegetation Dynamics.

After the formative event is extinguished, the flow decreases to ordinary values, reducing the water level and letting the crests of the recently formed bars emerge. At this point, vegetation may grow on the dried areas depending on flow variability. Vegetation is assumed to develop by following a logistic Holling type III equation, wherein the carrying capacity, *SI Appendix*, *Section S4*); **4** are space- and time-dependent. During ordinary flows, it is likely that sediment transport vanishes in most of the submerged sites, so that the bars can be approximately regarded as stable, until the occurrence of the next formative event. Moreover, the different time scales for bar saturation, order of days, and vegetation growth, order of years, allow us to separate mathematically the process and treat it analytically.

In this description, vegetation develops as a secondary instability (36) over the finite-amplitude bed topography. Encompassing bar topography in space and flow variability in time, a secondary instability is therefore performed by linearizing Eq. **4** about a new basic state (labeled with tilde), namely**5**, the first term r.h.s. refers to the uniform solution, which is dependent on time through the CPP (*Methods* for more details), in which s and n are the longitudinal and transversal coordinates and *A*): (*i*) The duration of the event equals the average interval between subsequent shots, *ii*) the discharge mean value between formative events is preserved; and (*iii*) the peak value is such that the coefficient of variation **4** becomes periodic and Floquet’s theorem can be applied. Thus, linearizing Eq. **4** around Eq. **5**, the solution for the vegetation is **6** is particularly cumbersome. Although the above theory has been described for river systems, it is quite general, and the same steps could be repeated for other ecogeomorphologic systems driven by unsteady flows.

## Results and Discussion

### Parametric Transitions Driven by Flow Variability.

Once the triad interaction is set up (i.e., hydrodynamics, sediment, and vegetation parameters are provided), the present mathematical framework can evaluate the surface area of alternate bars where plants endure flow variability (*A*. Let us define the AVI as the ratio of the vegetated area over the emerged area at the minimum discharge, *B* shows the dependence of AVI on flow variability **3**. The two cases exhibit analogies and discrepancies. In fact, both rivers show that higher flow variability (increasing **3**—to a discharge time series that is more correlated in the former case. This leads to longer submergence periods in the sand river and to plant uprooting for a fatigue stress. In the gravel river, the periods of bar submergence are instead shorter, but the more frequent events are equally able to remove mature vegetation. Another discrepancy concerns the two values of

The value of *SI Appendix*, *Section S5*. However, as the uprooting mechanism is much faster than plant growth, it is possible to simplify

Fig. 3c shows that the above approximation is asymptotically exact for very fast uprooting (i.e., **7**, the behavior of *D*). In the region where bedload transport occurs and alternate bars develop,

### Test Cases.

The computation of AVI is here shown for five actual fluvial cases of widespread interest to the scientific community (a summary of river features and validation results is provided in Table 1). The Isère (France) and the Alpine Rhine (border Austria–Switzerland) share almost identical hydrodynamic and sediment parameters (Fig. 1 *SI Appendix*, Fig. S5). Finally, the Arc river (France) has undergone a width reduction due to the construction of a highway in the 1990s (40). This triggered a parametric transition between a fully vegetated condition to a weakly vegetated bed (see *SI Appendix*, Fig. S6), where plants are periodically uprooted by annual flood events.

As also reported in Fig. 4, the present theory quantitatively captures the underlying physics providing a good matching between theoretical predictions and measurements. A partial exception is the actual vegetation cover of the Arc river, which is not fully captured. This might be due to (*i*) the very strong anthropic influences upstream of the study area (dams and sediment mining) and (*ii*) the fact that the Arc river has not yet reached the ecomorphological equilibrium and thus we might expect bars to slowly vegetate in the future.

These encouraging results may be useful to depict changes in biogeomorphological styles induced by man in the Anthropocene and of natural origin since the Paleozoic. For example, the model is able to capture the underlying physics for the river environments tested here and provide insights about the physical conditions that must subsist to allow for the colonization and the spreading of plants. Thus, the recent Devonian plant hypothesis, which suggests how plant root evolution might have contributed to the spreading of vascular plants (6), is supported by this theory. In fact, more robust root systems lead to a decrease in the uprooting coefficient *C*). However, further implications concerning how established plants have in turn affected landscape evolution are not contained in the present theory. This requires introducing the feedback of vegetation cover on morphodynamics and would explain why our river bar model underestimates the vegetated bar length (Table 1) with respect to the linear theory. Similarly, other processes not accounted for in this analytical theory concern sediment mining, alteration of sediment supply because of hydropower regulation, different root morphologies, and conditions for seed dispersal.

## Conclusions

We presented a methodology to couple the three main processes controlling ecomorphodynamic pattern formation under a single analytical model. Results show that flow variability discourages vegetation growth, up to the point that above a certain threshold (**7**) plant spread is completely inhibited. Such transition from vegetated to bare state depends parametrically on flow stochasticity, morphology, and plant biomechanical characteristics. In particular,

## Methods

Dimensionless shallow water equations for straight rivers (*SI Appendix*, *Section S1*. By perturbing the system around the basic uniform solution as **13** in the nonlinear system (Eq. **12**) and taking the internal product with the adjoint eigenfunctions, a Galerkin system of nonlinear differential equations for the amplitudes **2** is eventually achieved. Further details on the derivation of the Landau Eq. **2** may be found in *SI Appendix*, *Section S3*, together with the analytical expressions of its coefficients Ω and σ.

## Acknowledgments

We thank Guido Zolezzi for his useful advice on the validation and Alistair Borthwick for the editing.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. Email: matteo.bertagni{at}polito.it.

Author contributions: M.B.B., P.P., and C.C. designed research, performed research, and wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1721765115/-/DCSupplemental.

Published under the PNAS license.

## References

- ↵.
- Grill G, et al.

- ↵
- ↵.
- Chrinstensen O,
- Chrinstensen J

- ↵.
- Strayer DL

- ↵.
- Algeo TJ,
- Scheckler SE

- ↵
- ↵.
- Thom R

- ↵.
- Arnold V

- ↵.
- Meron E

- ↵.
- D’Alpaos A,
- Toffolon M,
- Camporeale C

- ↵.
- Bertoldi W, et al.

- ↵.
- Gran K,
- Paola C

- ↵.
- Bertoldi W, et al.

- ↵.
- Siviglia A,
- Crosato A

- ↵.
- Camporeale C,
- Ridolfi L

- ↵.
- Bärenbold F,
- Crouzy B,
- Perona P

- ↵.
- Camporeale C,
- Perucca E,
- Ridolfi L,
- Gurnell A

- ↵
- ↵.
- Kinast S,
- Meron E,
- Yizhaq H,
- Ashkenazy Y

- ↵.
- Hurst TA,
- Pope AJ,
- Quinn GP

- ↵.
- Goldstein E,
- Moore LJ

- ↵.
- Marani M,
- D’Alpaos A,
- Lanzoni S,
- Carniello L,
- Rinaldo A

- ↵
- ↵.
- Botter G,
- Basso S,
- Rodriguez-Iturbe I,
- Rinaldo A

- ↵.
- Laio F,
- Porporato A,
- Ridolfi L,
- Rodriguez-Iturbe I

- ↵.
- Abramowitz M,
- Stegun I, et al.

- ↵.
- Blondeaux P,
- Seminara G

- ↵.
- Colombini M,
- Seminara G,
- Tubino M

- ↵.
- Tubino M

- ↵.
- Hall P

- ↵.
- Cheng M,
- Chang H

- ↵.
- Bertagni M,
- Camporeale C

- ↵.
- Muneepeerakul R,
- Rinaldo A,
- Rodríguez-Iturbe I

- ↵
- ↵.
- Perona P,
- Crouzy B,
- McLelland S,
- Molnar P,
- Camporeale C

- ↵.
- Schmid P,
- Henningson D

- ↵.
- Serlet A, et al.

- ↵.
- Adami L,
- Bertoldi W,
- Zolezzi G

- ↵.
- Ferguson R,
- Bloomer D,
- Church M

- ↵.
- Jaballah M,
- Benoît C,
- Lionel P,
- André P

- ↵
- ↵.
- Wohl E,
- Lane S,
- Wilcox AC

- ↵.
- Guckenheimer J,
- Holmes P

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