## 1. Introduction

In the last few decades, many studies have analyzed the morphological evolution of alluvial watercourses affected by anthropogenic disturbances, highlighting that changes occur not only along the stream, but also across it, influencing both bed morphology and riparian vegetation at the local scale [

1,

2,

3]. Following these observations, various works argue that hydrology, morphology, and riparian plants represent three mutually inter-dependent components of the riverine environment [

4], thus requiring an interdisciplinary approach at various scales, which is yet to be comprehensively established.

Indeed, these kinds of studies should handle many issues related to the spatial and temporal scale of the problem. On the one hand, generally, hydro-ecological studies are very site-specific and based on particular observations at a small detailed scale. On the other hand, fluvial modelling should consider many interactions, which have, in principle, 3D [

5,

6] or, at least, 2D spatial features [

7,

8], besides the temporal variability embedded in natural systems. Therefore, as a result, numerical models can be quite hard to manage under the engineering and computing points of view, especially in the description of the behavior of large rivers [

9] or when accounting for a detailed description of all the possible components involved. To overcome these limitations, simplified 1D physically-based models can efficiently be applied, simulating the long-term interactions between hydrology, morphology and biodynamics at the watershed scale, considering spatial variability by the use of appropriate coefficients, thus involving a reduced computational effort and giving valuable results [

4].

Various studies have proven that riparian vegetation is influenced by the local fluvial hydrological regime through the control exerted by water discharges, water table elevation, flooding events, and hyporheic fluxes [

10,

11,

12]. However, at the same time, plants interact with the flow in that they influence the width, height, and stability of the watercourse surface, as well as other fluvial processes [

4,

13]. Similarly, other studies have shown that the spatial pattern of riparian vegetation can be considered to be a good indicator of this strong influence: sparse plants are generally associated with high discharge variability, while uniform vegetation is typical of more regulated streams or lower discharges [

14], reflecting how vegetation patterns and morphology dynamics are closely interconnected. A few (site-specific) field works have tried to describe the existing relationships between riparian vegetation patterns and fluvial landforms [

15,

16], while some others identified the geomorphological characteristics of locations in which seedlings are most likely to germinate and survive [

17]. As showed by Bendix and Hupp in 2000 [

18], and confirmed by other research [

19,

20,

21,

22,

23], generally the species are sorted along the river banks in relation to the water table gradient and the flow velocity, since the height of the water table and the magnitude of the flow are two of the main controls of the riparian vegetation growth. In fact, plants benefit significantly from high flow conditions that supply moisture, seeds, and nutrients [

2,

24], albeit these conditions are often related to negative effects and associated with physical damages [

25] like bank erosion, uprooting and sediment removal [

26], anoxia [

27,

28], and burial [

20,

29], depending on the species considered.

Under a modelling point of view, in the past, conceptual qualitative models [

30,

31,

32] and regression analyses between vegetation growth and flow duration [

19,

33] or flood magnitude [

34] were developed. These approaches have the limitation of not completely addressing the riparian vegetation dynamics in a physically based manner, which appears, instead, crucial in the light of the model here proposed. Numerical simulations [

35,

36,

37] and laboratory experiments [

38] were performed to evaluate the feedbacks of hydro-morphological constraints on riparian vegetation, but the results were locally limited and, therefore, issues were raised on the extrapolation to large-scale and long-term analyses. Recently, a new approach was developed accounting for the stochastic of the dynamics involved [

39,

40,

41,

42].

In the present paper, the effects of the water flow on riparian vegetation are investigated, starting from the approach recently proposed by Nones and Di Silvio [

4], with the aim to evaluate the importance of the parameters involved in simulating the growth of riparian vegetation. Different from the approach proposed in [

4], the present version pays particular attention to plant growth rate and initial population density, trying to give additional insights into the changing water-plants relationships, and to suggest a range for the parameters analyzed, which is still missing in the pre-existing literature. After a description of the main features of logistic curves and carrying capacity, frequently adopted to simulate the growth of biological populations, the main forcing terms are presented, focusing on the damages that different flow conditions can cause. Adopting data retrieved from remote sensing imagery and covering large and small watercourses, the results of the sensitivity analysis show the dependency of the proposed model on the parameters selected. In fact, these are frequently site-specific and, therefore, difficult to estimate due to a lack of data in previously existing scientific literature, and the importance of the vegetation growth rate rather than the initial plant density. In the conclusion section, the validity and limitations of the model are highlighted, and possible future developments and open questions are proposed for scholars and researchers.

## 3. Results and Discussion

The sensitivity analysis reported here has been performed by changing the values of initial population

P_{0} and growth rate

r of the riparian vegetation, in accordance with available evidence in the established literature [

59,

60], which is, unfortunately, not exhaustive in addressing these parameters to date. The other parameters, like extremal damages and morphological coefficients, are kept constant and equal to the values obtained by Nones and Di Silvio in their analysis [

4], with the aims of reducing the variables used in the present analysis. A further development of the present model, however, would require a site-specific calibration of these parameters too, depending on the river location.

The simulations have been performed assuming a temporal horizon of one hydrological year (i.e., submergence time t spans from 0 to 1), schematized by means of the long-term maximum, minimum, and median discharges and, therefore, the adopted flow duration curve can represent quite well the typical discharge rating curve of the studied rivers. To better describe the influence that the parameters analyzed have with respect to the watercourse sizes, large rivers are also evaluated separately, assuming a total river width B_{tot} > 1000 m as the threshold. Because the main aim of the model is to reproduce the active width of alluvial watercourses, the results report the comparison between measured and computed active river widths, derived from the above reported model. As one can observe, such values are strictly related to the water discharges and the presence of riparian vegetation along the banks. In fact, in adopting the regime equation, one obtains that the lower the discharges, the lower the active width. Similarly, from Equation (4), the vegetational carrying capacity is directly proportional to the water discharge.

In the future, a wider study is yet necessary to assess the relative importance of each parameter and the possibility to rearrange the formulation used, involving other contributions to the total carrying capacity, either increasing the number of species considered or choosing different approaches to describe the population growth, depending on the location of the watercourses under study.

#### 3.1. Initial Population

The solution of Equation (4) requires, as an initial condition, an initial population

P_{0} (dimensionless), representing the density of riparian vegetation per unit area at the initial time

t = 0. In the literature, very little evidence is available regarding the magnitude of this parameter and the wide range of rivers analyzed here does not permit the selection of a significant value a priori. To overcome these limitations, a sensitivity analysis is performed (

Table 1), evaluating the capability of the model in representing the measured active widths

B(

t) for different values of the initial population

P_{0}, which cover some orders of magnitude. The comparison is made in terms of mean relative error

E (Equation (12)) and coefficients of determination

R^{2}, and the latter is computed for the entire river dataset and only for large rivers.

The other parameters are assumed constant: the extremal damages are the same adopted in [

4]: Δ

_{A} = 0.19, Δ

_{W} = 0.17, Δ

_{E} = 0.13, Δ

_{B} = 0.31; while the growth rate is imposed equal to 0.025 [

59] and the morphological parameters are derived from [

65].

In

Figure 3 the results of the six tests are shown, highlighting the two groups of rivers analyzed: small Italian watercourses exhibit measured widths of maximum one kilometer, while large rivers can reach values significantly higher. This figure compares the transport (active) widths measured by remote imagery with the ones derived from the vegetation model, starting from the measure of the total width

B_{tot} and computing the vegetated width

B_{v} by means of the above reported equations.

As one can observe, the relative error

E decreases by increasing the initial population density

P_{0}, in particular thanks to the presence of small streams. In fact, regarding the coefficient of determination, on the one hand, for small rivers the model reproduces quite well the active widths, indicating that the impact of the riparian vegetation is relatively small in defining this parameter. On the other hand, in the case of large rivers, the active width is reproduced better for higher values of the initial population

P_{0} (

Table 1). Considering all the rivers, for an initial population density greater than 1000 there are no significant changes in the performance of the model against measured widths, indicating a threshold value for this parameter. Intermediate widths (1.5–2.5 km) are not well represented due to the present model structure and possibly because of the data chosen for the calibration: for such reasons, future research is still necessary, considering also the revision of the structure itself and the analyses of more case studies.

Furthermore, for some rivers the present discharge is lower than the bankfull discharge, yielding to inaccuracy in representing active widths by the model. As visible from

Figure 3, the computed widths remain quite constant even though the measured ones vary from 1 to 2 km. This inaccuracy can be related to the lower discharges measured during the study and used as inputs for the model. Analyzing the constitutive equations, in fact, it can be noticed that higher flow rates can yield to higher predicted active widths.

The results obtained may lead to the conclusion that the initial population is not a major limiting factor. However, initial values in the range of 100–200 individuals allow for a relatively better evolution of the river in accordance with the model structure, giving rise to computed widths closer to the measured ones, both in the case of small watercourses and for large rivers. As can be observed, the model results are quite insensitive to the initial population and, therefore, additional analyses on a larger database are necessary to confirm these preliminary results, pointing out possible biases related to the parameters adopted. On the other hand, this application could give new insights into the importance of the initial density in simulating the long-term growth of riparian vegetation.

#### 3.2. Vegetation Growth Rate

Regardless of the initial density of the plants, the validity of the model is strictly related to the value of the vegetation growth rate

r chosen during the simulations, as observable in

Figure 4. In the established literature there are specific studies focused on the relative growth rate of various species [

58,

59,

60] and, therefore, for the present study these values are assumed as a basis for the sensitivity analysis. Moreover, values of

r outside of the range proposed by the previous studies seem to not have any physical meaning.

In

Table 2 the non-dimensional growth rates adopted, the mean relative errors

E, and the coefficients of determination

R^{2} for all watercourses and large rivers are reported. In this case, the analysis is performed assuming an initial population

P_{0} = 100, while the other parameters like extremal damages were the ones reported in [

4,

65] and adopted in the previous analysis. As observed in the previous section, the initial population cannot be considered to be a limiting factor and, consequently, similar results are expected regardless the value of

P_{0} adopted.

In this case, the lower the growth rate, the better the reproduction of the active river width by the model (i.e., the lower the relative errors), especially due to the good representation for large rivers, as highlighted in the following. Similar to the previous analysis, in this case active widths between 1.5 and 2.5 km are also not well represented.

As can be seen from

Table 2 and

Figure 4, in fact, higher growth rates lead to better representation of the measured river widths, evaluating small and large rivers at the same time. Despite this perception, however, by observing the results it is possible to notice that, for higher growth rates, small Italian rivers are not well represented, while the results for large rivers computed properly. Therefore, intermediate values, like the ones proposed by Gleeson and Tilman [

59], allow, in general, for a better assessment of the measured width for the entire database.

From this sensitivity analysis, one can argue that a unique vegetation growth rate is not feasible for reproducing the behavior of two very different fluvial environments (small vs. large rivers) and, hence, the use of calibrated and site-specific vegetation growth rates is suggested for future applications.

## 4. Conclusions

The present work points out the importance of two biological parameters in defining the active river width by means of a simplified mathematical model. Indeed, the sensitivity analysis of vegetation growth rate and initial density reported here highlights the feedbacks of such parameters in computing the active width, frequently assumed only as dependent of the water discharge (i.e., regime equation). For this first calibration, the model is applied to a relatively small number of cross-sections of small and large rivers, showing the validity of the hypotheses assumed, though highlighting some limitations related to the parameters adopted during the simulations and possible biases in reproducing the measured widths due to many constraints (e.g., river discharges, morphological and biological parameters, etc.). The present model structure is capable of reproducing small and large river widths, but needs improvements to simulate a larger variability of active widths.

As argued in past research [

4], the application of the model to other case studies would require some changes of the calibration parameters (extremal damages, vegetal growth rate, initial population, morphological coefficients). Perhaps a revision of the model structure could be required, introducing other contributions to the aggregate carrying capacity, related, for example, to salinity, ice, bottom capillarity, local climate, presence of sediments, groundwater dependency [

66], depending on the river location and the local climate.

In spite of the relatively small number of cross-sections studied, the sensitivity analysis showed that the vegetation growth rate plays a major role in defining the vegetated width and, consequently, the active one. For this reason, in evaluating large rivers lower rates are suggested (0.010–0.015), while, for small rivers like the ones studied here, growth rates in the order of 0.03–0.04 can be adopted during the simulations, as suggested by evidence in the existing literature, probably because of the different species involved. The initial population, on the other hand, seems to have a lower influence on the final outcomes of the model. In this case, in fact, the preliminary analysis performed and the related coefficients of determination suggest that a lower initial population is better suitable for small rivers, while a higher initial population can contribute to an improved simulation of the behavior of large rivers. This could be due to the inadequacy of the model in reproducing, at the temporal scale adopted, a fast growth of plants despite the space available.

Notwithstanding the several simplifications adopted in the model and the small database used, the outcomes of this preliminary sensitivity analysis suggest that the basic concepts utilized in the present approach might have a general validity and, therefore, the model could be applied to future case studies aimed at confirming its potential for modelling different environmental conditions. Further research is also necessary: (i) in the estimation of the growth rate, which creates site-specific effects and, therefore, requires a careful evaluation to limit errors in simulating the riparian vegetation growth; (ii) in the definition of the long-term evolution of riparian vegetation, assuming a temporal horizon characterized by a series of consecutive hydrological years; (iii) in the implementation of the model structure, to simulate a larger variability of river widths; (iv) in the integration of this 2D description of the cross-sections with 1D and even 0D models to simulate the very long-term (geological) evolution of rivers.