New Results

Generation and application of river network analogues for use in ecology and evolution

Luca Carraro, Enrico Bertuzzo, Emanuel A. Fronhofer, Reinhard Furrer, Isabelle Gounand, Andrea Rinaldo, Florian Altermatt
doi: https://doi.org/10.1101/2020.02.17.948851

Abstract

  1. Several key processes in freshwater ecology and evolution are governed by the connectivity inherent to dendritic river networks. These networks have extensively been analyzed from a geomorphological and hydrological viewpoint, yet network structures classically used in modelling have only been partially representative of the structure of real river basins, and have often failed to capture well known scaling features of real river networks. Pioneering work has identified optimal channel networks (OCNs) as spanning trees that reproduce all scaling features characteristic of real, natural stream networks worldwide. While these networks have been used to generate landscapes for studies on metapopulations, biodiversity and epidemiology, their generation has not been generally accessible.

  2. Given the increasing interest in dendritic riverine networks by ecologists and evolutionary biologists, we here present a method to generate OCNs and, to facilitate its application, we also provide the R-package OCNet. Owing to the random search process that generates OCNs, multiple network replicas spanning the same surface can be built, allowing one to perform computational experiments whose results do not depend on the particular shape of a single river network. The OCN construct also enables the generation of elevational gradients derived from the optimal network configuration, which can constitute three-dimensional landscapes for spatial studies in both terrestrial and freshwater realms. Moreover, the OCNet package provides functions that aggregate the OCN into an arbitrary number of nodes, calculate several metrics and descriptors of river networks, and draw relevant features of the network.

  3. We describe the main functionalities of the package and present how it can be integrated into other R-packages commonly used in spatial ecology. Moreover, we exemplify the generation of OCNs and discuss an application to a metapopulation model for an invasive riverine species.

  4. In conclusion, OCNet provides a powerful tool to generate and use realistic river network analogues for various applications. It thereby allows the design of spatially realistic studies in increasingly impacted ecosystems, and enhances our knowledge on spatial processes in freshwater ecology in general.

1 Introduction

The central goal of ecology is to causally understand patterns and processes in ecological systems, such as species coexistence, biodiversity patterns, or the unfolding of species invasions [Gause, 1934; Elton, 1958]. Much of ecological theory and empirical work has either focused on local patterns and dynamics or has taken a spatially implicit perspective. However, virtually all natural ecosystems are spatially structured, and the relevance of the spatial dimension on ecological systems can hardly be overestimated [Levin, 1992; Hanski and Gaggiotti, 2004; Holyoak et al., 2005]. Consequently, over the last decades, ecologists have started to account for spatial processes on population and community dynamics as well as biodiversity. Theoretical, comparative and experimental studies have increasingly been done in a spatially explicit perspective (e.g., Hanski and Ovaskainen [2000]; Cadotte and Fukami [2005]; Holyoak et al. [2005]; Altermatt et al. [2011]; Gilarranz and Bascompte [2012]; Dale and Fortin [2014]), especially promoted by theories on metapopulation and metacommunity dynamics.

A direct consequence of this spatial approach to ecology is the need to describe and understand the spatial structure and layout of natural ecosystems. While initial models of spatial dynamics assumed spatially implicit networks of populations or communities [Levins, 1970], all natural ecosystems follow spatially explicit structures. These structures, such as those typically found in coral reefs and atolls, mountainous landscapes and their elevational gradients, or tidal pools, are shaped by general geophysical processes resulting in characteristic landscape structures. Arguably the most characteristic (but also among the most widespread) landscape structure is found in riverine networks [Leopold et al., 1964; Rodriguez-Iturbe and Rinaldo, 2001]: erosional forces balancing uplift create dendritic networks of rivers and streams following universal patterns. These networks are characterized by their fractal, scale-free structure, as well as by universally applicable laws regarding many geomorphological and hydrological variables of direct relevance to ecology, such as catchment area, river bed width and depth, or variation in discharge [Horton, 1945; Leopold and Maddock, 1953; Rodriguez-Iturbe and Rinaldo, 2001]. In contrast to these specific features of natural landscape structures, much of ecological and evolutionary theory and experiments, but also much of the species-distribution modelling still assumes either random networks or simply structured linear, circular or Cartesian networks, in which local patches are connected to their 2, 4 or 8 nearest neighbours [e.g. Bascompte and Solé, 1996; Bell and Gonzalez, 2011; Holland and Hastings, 2008]. This oversimplification of spatial network structures may limit the plausibility and relevance of the findings. An application to more realistic network structures has, however, often been hindered by the lack of formalized, spatially correct and generalizable network structures as well as easily accessible tools generating them.

Riverine ecosystems are not only of high interest to ecologists due to their universal network structure, but also due to the considerable biodiversity inhabiting them [Balian et al., 2008; Altermatt, 2013]. River networks cover less than 1% of the landmasses, but contain up to 10% of all species. However, this high biodiversity, as well as the associated ecosystem functions, are threatened by various anthropogenically induced causes, including pollution, biological invasions, or damming and modification of the network structure [Vörösmarty et al., 2010; Darwall et al., 2018]. An understanding of many of these processes requires a spatially explicit approach, such as how pollution and chemicals are transported in riverine networks [Helton et al., 2018], how organisms spread along rivers and invade riverine ecosystems [Mari et al., 2014; Giometto et al., 2017], or how the modification of network structures across drainage basins affects local diversity [Leuven et al., 2009]. Consequently, there has been a rapid increase in ecological and evolutionary studies considering the effect of river-like network structures on ecological dynamics over the last two decades [Fagan, 2002; Campbell Grant et al., 2007], paralleled by an increase in methodological tools to analyse such spatial datasets [Muneepeerakul et al., 2008; Rodriguez-Iturbe et al., 2009; Peterson et al., 2013; Paz-Vinas and Blanchet, 2015; Welty et al., 2015; Duarte et al., 2019; Rinaldo et al., 2020].

While all of these works acknowledge the importance of studying rivers in a spatially explicit perspective, a large part of them is built on networks that do not factually capture many of the inherent characteristics of true riverine networks. Notable examples range from the River Continuum Concept [Vannote et al., 1980], which describes rivers as a single, linear array of patches, to slightly more complex bifurcation networks or alterations thereof [Fagan, 2002; Morrissey and De Kerckhove, 2009; Chaput-Bardy et al., 2009; Brown and Swan, 2010; Yeakel et al., 2014; Seymour and Altermatt, 2014; Seymour et al., 2015; Paz-Vinas et al., 2015; Anderson and Hayes, 2018]. All of these studies use networks that may at first sight look like “river” networks, but do not satisfy the necessary constraint posed by draining a given surface (Fig. 1). Furthermore, these constructs do not adequately represent the connectivity and several geometric properties (like the distributions of upstream and downstream lengths, and of total contributing area at a point) inherent to natural river networks, and lack the space-filling attribute of small to smallest streams not only incrementally flowing into larger streams, but also the common direct inflow of very small streams into large streams. As such, all of this work has been ignoring the extensive and long-lasting knowledge from geomorphology that has appropriately acknowledged and formalized the spatial unfolding of dendritic river networks.

Figure 1:
Figure 1:

Examples of river network analogues with increasing level of resemblance with real river networks. Line width increases towards the downstream direction. a) Linear array of patches [Vannote et al., 1980]. b) Binary-fission-like tree [Fagan, 2002; Paz-Vinas et al., 2015]. c) An OCN spanning a 50×100 lattice, aggregated with a threshold area equal to 25 pixels. d) A real river (Thur, Switzerland), spanning an area of 760 km2, extracted from a digital elevation model with a threshold area equal to 4 km2.

In particular, the fractal character of river networks, epitomized by Horton’s laws [Horton, 1945] on bifurcation and length ratios, was observed with regards to several morphological and hydrological characteristics of river basins and expressed by means of a number of power-law relationships, which are the signatures of fractal behaviour [Mandelbrot, 1983; Maritan et al., 1996]. Notable examples are Hack’s law [Hack, 1957] L ∼ Ah, linking the maximum upstream channelized length L at any location in the river with the corresponding drainage area A; the slope-area relationship s ∼ Aγ−1, where s is the channel slope [Tar-boton et al., 1989]; the scaling of the probability distribution of drainage areas P(A ≥ a) ∼ a−β [Rodriguez-Iturbe et al., 1992a]. Typical values observed in real rivers for the scaling exponents are h ≈ 0.57, γ ≈ 0.5, β ≈ 0.43 [Rinaldo et al., 2014]. From a hydraulic geometry viewpoint, Leopold’s relationships [Leopold and Maddock, 1953] express how mean river depth, width and velocity change, both at-a-station and along the river’s course, as power-law functions of discharge, which in turn scales linearly with drainage area (strictly speaking, this applies to landscape-forming discharges [Rodriguez-Iturbe and Rinaldo, 2001]).

Such scale-invariant properties of river networks prompted the development of a model of idealized stream networks: optimal channel networks (OCNs). OCNs are ‘optimal’ inasmuch as their configuration corresponds to a minimum of total energy expenditure and reproduces all scaling features of real rivers [Rodriguez-Iturbe et al., 1992b; Rinaldo et al., 1992; Maritan et al., 1996; Rinaldo et al., 2014]. Importantly, OCNs are exact stationary solutions of the general equation describing landscape evolution [Banavar et al., 2001]. The OCN construct allows the generation of an unlimited number of different network replicas spanning the same drainage domain, therefore enabling one to run computational experiments and derive results that are independent of the shape of a single river network, which would not be the case if real rivers were used as landscapes. Moreover, OCNs enable the investigation of spatial processes occurring not only in dendritic river networks, but also along the elevational gradients of fluvial landscapes [Bertuzzo et al., 2016; Giezendanner et al., 2019]. To this regard, it is worthwhile to note that the elevational landscape generated by an OCN is such that the graph obtained by following the steepest descent directions reproduces the OCN structure [Balister et al., 2018].

OCNs have been used to investigate a number of ecological issues, ranging from metapopulation structure in riverine [Mari et al., 2014; Bertuzzo et al., 2015; Fronhofer and Altermatt, 2017] and terrestrial landscapes [Bertuzzo et al., 2016; Giezendanner et al., 2019]; habitat fragmentation [Sarker et al., 2019]; spreading of human [Bertuzzo et al., 2010; Gatto et al., 2013; Mari et al., 2019] and animal [Carraro et al., 2018] waterborne pathogens; ecosystem processes, such as carbon [Bertuzzo et al., 2017; Koenig et al., 2019] and nitrogen cycling [Helton et al., 2018]; migration fronts of human populations [Campos et al., 2006]; cross-ecosystem subsidies [Harvey et al., 2019]; riverine biodiversity patterns from a theoretical viewpoint [Muneepeerakul et al., 2019] or by means of mesocosm experiments [Carrara et al., 2012, 2014; Harvey et al., 2018].

Despite the longstanding establishment of the OCN concept, its application especially in ecology and evolutionary biology has been lagging behind, likely because easily accessible code or appropriate tools have been lacking. This is particularly regrettable considering the recent bloom of tools allowing the statistical analysis of data from real dendritic networks (e.g. the R-package SSN [Ver Hoef et al., 2014]). However, such tools are specifically designed for real river networks, while their applicability to virtually generated networks is limited. To fill this gap, we here describe the methodological and mathematical frameworks that underlies OCNet, an R-package for the generation and analysis of optimal channel networks, and provide guidelines and examples to facilitate the use of this tool.

2 The OCNet package

2.1 Theoretical background

The OCN concept is based on the assumption that river network configurations occurring in nature minimize a functional describing total energy dissipation [Rodriguez-Iturbe et al., 1992b; Rinaldo et al., 1992]. This assumption, and the subsequent construction of OCNs, is well supported by a comparison with river networks globally. Let us consider a regular lattice made up of N cells, where each cell represents the generic node i of the network. Each node i is connected via a link to one of its nearest neighbours. The energy dissipation across the ith network link is proportional to QiΔhi, where Qi is the landscape-forming discharge in the link [Rinaldo et al., 2014], and Δhi = siLi the corresponding elevation drop, with si identifying slope and Li link length. By assuming Qi ∼ Ai [Rodriguez-Iturbe and Rinaldo, 2001], where Ai is the area drained by link i, and the slope-area relationship Embedded Image [Tarboton et al., 1989], the functional representing total energy expenditure across a landscape formed by N cells reads Embedded Image

Note that link lengths do not appear in the above formula, as they can be considered constant with no loss of generality. The OCN configuration is defined by an adjacency matrix W whose corresponding set of drainage areas A = [A1, …, AN] yields a local, dynamically accessible minimum of (1). Note that the correspondence between A and the adjacency matrix W of a tree is subsumed by the relationship (IN − WT)A = 1, where IN is the identity matrix of order N, and 1 a N-dimensional vector of ones [Bertuzzo et al., 2015].

Minimization of (1) is operated by means of a simulated annealing technique: starting from a feasible initial flow configuration (i.e., a spanning tree, see Fig. 2a, e), a link at a time is rewired to one of its nearest neighbours; if the obtained configuration is a spanning tree, H is computed; the new configuration is accepted if it lowers total energy expenditure; if this is not the case, the new configuration can still be accepted with a probability controlled by the cooling schedule of the simulated annealing algorithm. Such myopic search, which only explores close configurations, actually mimics the type of optimization that nature performs, at least in fluvial landscapes [Rinaldo et al., 2014]. Notably, restricting the search of a network yielding a minimum of Eq. (1) to spanning, loopless configurations entails no approximation, because every spanning tree is a local minimum of total energy dissipation [Banavar et al., 2000]. The shape of the so-obtained OCNs retains the heritage of the initial flow configuration, although the extent to which this occurs is partly controlled by the cooling schedule adopted (Fig. 2). This underpins the concept of feasible optimality, i.e., the search for dynamically accessible configurations.

Figure 2:
Figure 2:

Effect of initial network state (rows) and cooling schedule (columns) on the final OCN configuration. b, c, d) OCNs on 250×250 lattices generated from the initial state shown in panel a. f, g, h) As above but with initial state as shown in panel e. i) Dynamics of total energy expenditure H (Eq. (1)) and temperature T (i.e., cooling schedule of the simulated annealing algorithm) for the 6 OCNs displayed above. Values of H and T are normalized by the energy of the initial network state. Note that, for graphical reasons, the initial states shown in panels a, e refer to 25×25 lattices. A script generating this figure can be found in the Supporting Information.

2.2 Overall setup of the package

The OCNet package consists of a series of functions that allow constructing river-analogue networks as well as calculating a number of metrics and descriptors commonly used in spatial ecology. The networks constructed by the package are built at several levels of aggregation. At each level, they are generally defined by a number of nodes, an adjacency matrix, a vector of contributing areas and two vectors with longitudinal and latitudinal coordinates of the nodes. The functions constituting the OCNet package are intended to be applied in sequential order, and the respective output can be directly used to visualize the created networks and linked to other commonly used R-packages.

The first function, create_OCN, only requires the longitudinal and latitudinal dimensions of the lattice as necessary inputs, while several other parameters can be optionally tuned to obtain customized results. Some examples are provided in the following section; extensive further information is given in the package documentation. The output of create_OCN is a list containing a sub-list termed FD that, in turn, encloses key information on the topology of the network, among which the adjacency matrix (written in sparse form via the spam format [Furrer and Sain, 2010]) and a vector of contributing areas. The subsequent functions landscape_OCN (generation of the three-dimensional landscape derived from the network configuration), aggregate_OCN (aggregation of the OCN at various levels – see Aggregation levels), paths OCN (evaluation of paths among network nodes, and lengths thereof), rivergeometry_OCN (hydraulic geometry of the river network, following Leopold and Maddock [1953]) require as necessary input the output list produced by the previous function, in the aforementioned order (except rivergeometry_OCN, which can be executed after aggregate_OCN). The output of these functions is a list where all objects of the input list are copied, and to which new objects are added. Note that output lists contain all input values, to avoid inconsistencies in the sequential application of functions. A group of functions (identified by the prefix “draw_”, see examples in Fig. 3) are devoted to graphical representations of the OCN.

Figure 3:
Figure 3:

Examples of outputs from OCNet’s graphical functions. a) Representation of an OCN generated on a 250×250 lattice (draw_simple_OCN). Note that the network spans the whole lattice; for graphical reasons, the portion of network exceeding a given AT is plotted in blue. b) Planar representation of the elevational landscape generated by the OCN of panel a (draw_elev2D_OCN). c) Partitioning of the lattice into subcatchments for the OCN of panel a (draw subcatchments OCN); blue dots indicate locations of the nodes at the AG level. d) Representation of an OCN generated on a 400×400 lattice, with all perimetral pixels as outlets (draw_contour_OCN); black solid lines display partitioning among catchments; the grey background identifies the largest catchment. e) 3D representation of the largest catchment within the OCN of panel d (draw elev3Drgl OCN). f) Strahler stream order values for the largest catchment within the OCN of panel d (draw_thematic_OCN). g) Representation of an OCN generated on a 300×300 lattice, with 4 outlets (shown by black squares) and periodic boundaries (draw_contour_OCN). h) Perspective 3D representation of the OCN of panel g (draw_elev3D_OCN). i) Real-shaped representation of the OCN of panel g (draw_contour_OCN). A script generating this figure can be found in the Supporting Information.

2.3 Aggregation levels

Before moving to the illustration of some possible applications of the package, we here clarify some concepts and terminology with respect to the aggregation of OCNs. Additional details are provided in the package documentation. Networks produced by the OCN algorithm can be used in a variety of fashions (see Table 1 for a review) by exploiting different connectivity metrics that are embedded in the OCN construct. At a first, non-aggregated level, each cell of the lattice (also termed as pixel) constitutes a node of the network (see Fig. 4), and the connectivity among nodes is ruled by the flow direction pattern (represented by the adjacency matrix) obeying the OCN principle. This is here referred to as the flow direction level (FD).

View this table:
Table 1:

Types of OCN aggregation schemes used in previous studies.

Figure 4:
Figure 4:

Representation of the different aggregation levels. Top row: example from a single-outlet 8×8 lattice describing how the aggregation procedure operated by aggregate_OCN works. Letter ‘O’ identifies the outlet pixel. Arrows on the other pixels identify flow directions; note that the these are not representative of an OCN, but are here presented only for illustrative purposes. Numbers represent the cumulative drainage area (in number of pixels). At the FD level, all 64 pixels belong to the network. To obtain the RN level, a threshold area AT = 5 pixels is applied to distinguish pixels belonging to the river network. Bottom row: the same procedure is applied to a single-outlet, 20×20 OCN (obtained via the script presented in Generation of an OCN). Aggregation is performed with AT = 5 pixels. Note that river width is proportional to the square root of drainage area [Leopold and Maddock, 1953]. A script generating this figure can be found in the Supporting Information.

As customary in hydrology when extracting a river network based on digital elevation models of the terrain [O’Callaghan and Mark, 1984], a threshold AT on drainage area can be imposed to identify those pixels of the lattice that constitute nodes of the river network (RN, second level – see Terui et al. [2018] for an example of an ecological application of non-OCN synthetic networks akin to OCNs at the RN level). In a third, aggregated level (AG), nodes correspond to sources, confluences and outlet(s) of the river network identified at the RN level. The whole lattice is then partitioned into areas that directly drain into the nodes at the AG level, or the edges departing from them, thereby constituting the fourth, subcatchment level (SC).

A fifth level (catchment, CM) partitions the lattice into regions that are drained by different outlets, when the multiple-outlet option in create OCN is enabled (see following section). Finally, in an optional, zero-level spatial structure, all lattice pixels are treated as nodes, but connectivity follows the Von Neu-mann (4 nearest neighbours, level N4) or Moore (8 nearest neighbours, level N8) neighbourhoods, as in the green network described by Altermatt [2013]. In this case, the OCN was used to generate a realistic elevation gradient governed by fluvial erosion, on which, for instance, the structure of terrestrial metapopulations can be studied [Bertuzzo et al., 2016; Giezendanner et al., 2019].

The final output of the OCNet functions is a list of lists, each of which named after the corresponding aggregation level (N4/N8, FD, RN, AG, SC, CM) and containing relevant topological and morphological information for that level. Variables may vary in number, type and definition among the sub-lists, although the adjacency matrix is defined for all levels, while the drainage area vector is defined for all levels but N4/N8.

3 Overview of package features

3.1 Number of outlets, boundary types, and elevational gradients

Although the OCN principle is primarily intended to be applied to networks spanning the whole drainage domain (where the area drained by the outlet is equal to the lattice area, see an example in Fig. 3a–c), the generalization to the case of multiple networks—each of which subsumed by a different outlet—within the same lattice is straightforward. Indeed, the very same mathematical formulation presented in Theoretical background holds when multiple outlet pixels are imposed. Technically, this is done by preventing these pixels to drain into their neighbouring pixels. In this case, the sum of the areas drained by all outlet pixels is equal to the lattice area. In the package, the multiple-outlet option can be activated by means of the optional input nOutlet of function create OCN. In the limiting case, all pixels at the lattice boundary can be treated as outlets [Sun et al., 1994; Bertuzzo et al., 2017]. This is done by setting nOutlet = “All” in create_OCN. A graphical representation of an OCN obtained for the latter case is shown in Fig. 3d.

When a pixel’s flow direction is rewired during the search for an optimal network configuration, possible directions are generally those towards the eight neighbouring pixels. This is not the case for the outlet pixels (which cannot be rewired) and the pixels at the lattice boundaries, which can be rewired to either three (corner pixels) or five (side pixels) neighbouring cells. This latter assumption can be relaxed by allowing pixels at the boundary to drain into eight neighbours, by also considering pixels at the opposite sides as feasible directions. In OCNet, periodic boundaries can be enabled via the optional input periodicBoundaries of create OCN. An example is shown in Fig. 3g–i. Such option can be useful when the OCN lattice is to be considered as the periodic unit of an infinite landscape [Bertuzzo et al., 2016; Giezendanner et al., 2019], or when one aims at building OCNs spanning domains that are not lattice-shaped (see Fig. 3f, i).

Once an OCN has been created by the simulated annealing algorithm, the iterative application of the slope-area relationship starting from the outlet node and moving in the upstream directions enables the derivation of the elevation field subsumed by the OCN (up to two constants, e.g., the elevation and slope of the outlet pixel). Some examples of elevational landscapes built on OCNs are provided in Fig. 3b, e, h. Importantly, the slope-area relationship only holds for the channeled portion of the domain, which implies, strictly speaking, that the OCN must not be aggregated if one aims at making use of a three-dimensional landscape generated by an OCN. Moreover, the slope-area relationship is actually multiscaling [Tarboton et al., 1989], therefore the simple recursive application of Embedded Image (as performed by the function landscape_OCN) to yield an elevational landscape is to be considered as a first approximation, suitable for ecological applications. Methods to account for the scaling of the variance of the slope-area relationship exist [Grimaldi et al., 2005], but are beyond the scope of this work.

3.2 Relationship between threshold drainage area and number of nodes

Owing to the somewhat heuristic procedure for the definition of an aggregated network based on a threshold drainage area value AT, it is not possible to establish a priori how many nodes at the AG level correspond to a given AT. This in fact depends on the configuration of the OCN at the FD level, which is the result of a stochastic process. This issue is particularly relevant when OCNs are used in experiments where practical reasons enforce a limitation on the number of nodes that can be handled, or when several OCN replicas with the same number of aggregated nodes are required.

To help overcome this issue, OCNet includes the function find area threshold OCN, which requires as input a non-aggregated OCN (produced by landscape_OCN) and evaluates the number of nodes resulting from the aggregation procedure for different values of AT. Such a function can therefore be used prior to aggregate_OCN to assess which threshold has to be used to obtain a network with the desired aggregation structure. Additionally, find_ area_threshold_OCN also evaluates other variables that help characterize the network structure from a hydrological perspective, such as maximum stream order and drainage density. Maximum stream order can be of interest in some studies, when patch sizes need to be related to the structure of the underlying network but only few discrete dimensions are available, so that it is convenient to employ different patch sizes for different stream order values of the corresponding nodes (e.g., Harvey et al. [2018]). Drainage density is relevant because it allows the assessment of hydrological characteristics of the aggregated OCN for a given metric resolution (i.e., the length in meters attributed to a pixel length—corresponding to the optional input cellsize of create_OCN), such as aridity and timing of the hydrologic response [Pallard et al., 2009].

Fig. 5 shows results from the application of find_area_threshold_OCN to several OCNs built on large lattices. When AT is lower than 2% of the lattice size, the number of nodes scales fairly well as a power law of the normalized threshold area (Fig. 5a). This relationship allows qualitatively assessing the relevant range of AT corresponding to a sought number of nodes at the AG level, which can be used as input in find_area_threshold_OCN to speed up its execution, especially for large networks. Scaling relationships with AT are also found for maximum Strahler stream order (Fig. 5b) and drainage density (Fig. 5c). To provide an example, if a threshold AT = 20 pixels is applied to a 200×200 OCN, the expected number of nodes at the AG level is 1052.4, the expected maximum stream order is 5.56, and the expected drainage density is Dd = 0.1454 inverse planar units, which corresponds to a relatively wet catchment (Dd = 2.91 km−1) of area 100 km2 (when 1 planar unit represents 50 m), or to a rather arid catchment (Dd = 1.46 km-1) of area 400 km2 (if 1 planar unit is equal to 100 m). Notably, the relationship between drainage density and threshold area AT mirrors the scaling behaviour of drainage areas (Fig. 5d), which is characterized by an exponent β in the range [0.42; 0.45] (see Introduction and Rinaldo et al. [2014]). Indeed, drainage density for a given AT is roughly (i.e., if differences in lengths between vertical/horizontal and diagonal flow directions are neglected) equal to the number of pixels whose area is greater than or equal to AT. Fig. 5d also represents how the number of nodes at the RN level (NRN) scales with varying AT: to this end, it suffices to replace a with AT and P(A ≥ a) with NRN /N.

Figure 5:
Figure 5:

Effect of choice of threshold area AT on OCN configuration. Aggregation of four large OCNs is performed ∀AT = {1, …, N} via function find_area_threshold_OCN. For all panels, vertical lines indicate the cutoff value AT = 0.02 N; only points corresponding to threshold area values below the cutoff are used to estimate the (dashed) regression lines, whose equations and R2 values are reported. a) Number of nodes at the AG level scales as a power-law function of the normalized threshold area aT = AT /N. b) Maximum Stahler stream order value as a function of normalized threshold area aT. c) Drainage density scales as a power-law function of threshold area AT. d) Scaling behaviour of OCNs: probability P(A ≥ a) of randomly sampling a pixel within the lattice whose drainage area A is not greater than a given value a. A script generating this figure can be found in the Supporting Information.

The scaling behaviour of OCNs displayed in Fig. 5 can also provide useful information with respect to the choice of values of relevant parameters N and AT that allow generating an OCN of adequate size for practical applications. To this extent, it is worthwhile to note that the OCN construct is invariant under coarse graining [Rodriguez-Iturbe and Rinaldo, 2001; Rinaldo et al., 2014], which means that the choice of the lattice dimension N does not affect the scaling of drainage areas. As in the example above, such choice should rather be based on geomorphological arguments, that is, the answers to the questions: What is the area that the OCN is supposed to drain? What are the expected values of maximum stream order and drainage density on this area? However, as a rule-of-thumb indication, we suggest to perform aggregation with a threshold not greater than AT = 0.02 · N, such that the obtained configuration is not affected by the finite-size scaling effect; this corresponds to an expected NAG ≥ 30 (see Fig. 5a).

3.3 Compatibility with existing R-packages

Specific functions of OCNet enable transformation of OCNs into objects that can be used by other commonly used R-packages in spatial ecology. In particular, compatibility with igraph [Csardi and Nepusz, 2006], a package for network analysis and visualization, is provided by function OCN_to_igraph. More-over, function OCN_to_SSN transforms an OCN at a desired aggregation level into an object that can be read by SSN, a package on spatial statistical modeling and prediction for data on stream networks [Ver Hoef et al., 2014]. Examples for these functions are shown in Fig. 6. Finally, output from OCNet can be used in combination with R-packages for geostatistical modelling such as gstat [Pebesma, 2004], based on the coordinates of nodes of an OCN given at any aggregation level. Remarkably, adjacency matrices and other information can easily be extracted as base R objects, which guarantees compatibility with virtually every R-package and even other programming languages.

Figure 6:
Figure 6:

Compatibility of OCNet with packages igraph and SSN. Examples are built on the OCN obtained in Generation of an OCN. a) The OCN, aggregated at the AG level, is transformed into an igraph object (via OCN_to_igraph), and plotted via the function plot.igraph of igraph. b) The same OCN is plotted via draw_thematic_OCN. c) The same OCN is transformed into an SSN object (via OCN_to_SSN) and plotted via the function plot.SpatialStreamNetwork of SSN. Here, 50 observation points have been sampled along the network by means of a binomial design, and their distance from the outlet is displayed. A script generating this figure can be found in the Supporting Information.

4 Application example: a metapopulation model

In order to show a possible application of the OCNet package, we now apply a simple metapopulation model for an invasive riverine species to an OCN.

4.1 Generation of an OCN

Let us build an OCN with the following assumptions: it spans a 20×20 lattice, with a single outlet located close to the southwestern corner of the lattice, and each pixel represents a square of side 500 m (total size of the catchment is therefore 100 km2); the elevation, slope and channel width of the outlet node are 0 m a.s.l., 0.01, and 5 m, respectively; the threshold area used to aggregate the network is equal to 1.25 km2. The code lines to build such network are the following: set.seed(1) # use fixed random number generator OCN <- create_OCN(20, 20, outletPos = 3, cellsize = 500) OCN <- landscape_OCN(OCN, slope0 = 0.01) OCN <- aggregate_OCN(OCN, thrA = 1.25e6) OCN <- paths_OCN(OCN, pathsRN = TRUE) OCN <- rivergeometry_OCN(OCN, widthMax = 5)

The resulting OCN is shown in Fig. 4.

4.2 Metapopulation model

Let us build a discrete-time, deterministic metapopulation model on the previously built OCN, according to the following assumptions: (i) the model is run on the OCN aggregated at the RN level (consisting of Nn nodes); (ii) population growth at each node follows the Beverton-Holt model [Beverton and Holt, 1957], with baseline fecundity rate r = 1.05 constant for all nodes, and carrying capacity Ki = 10 · Wi, where Wi is the river width of the network node i; (iii) at each time step t, the number of individuals moving from node i is equal to gPi(t), where g = 0.1 is a mobility rate constant for all nodes, and Pi(t) is the (expected) population size at node i and time t; (iv) at each time step, individuals at node i can only move to a node that is directly connected to i, either downstream or upstream; (v) pd and pu = 1 − pd identify the probability to move downstream or upstream, respectively; (vi) if the indegree of node i is larger than 1 (namely the node has multiple upstream connections), individuals moving upstream are split among the possible destination nodes into fractions ϒi proportional to their drainage areas; (vii) as initial condition, all network nodes are uninhabited barring the node f that is farthest from the outlet (where Pf ,1 = 1). The model equation hence reads Embedded Image where wij is a generic entry of the adjacency matrix W expressing OCN connectivity at the RN level; Di (Ui) is equal to one if there is a downstream (upstream) connection available from node i and is null otherwise. Weights ϒi are defined as Embedded Image where i′ is such that wii′ = 1, and Ai is the drainage area at node i. A code script performing model simulations is reported in the Supporting Information.

We performed two model simulations to investigate the effect of parameters pd, pu on the time elapsed until the system reaches a steady state; in a first (default) run, no preferential direction of movement was assumed (pd = pu = 0.5); in the second run, a preference for downstream movement (pd = 0.7, pu = 0.3) was hypothesized. Results are shown in Fig. 7. When pd = 0.5, the invading species rapidly reaches the equilibrium in the initially occupied node, while colonization of the downstream patches is delayed. When a preference for downstream movement is attributed (pd = 0.7), local population growth in the onset (green) node is slower, whereas invasion of the outlet node occurs faster, both in terms of initial growth and establishment of the equilibrium (see coloured vertical lines in Fig. 7a). Colonization of the headwater that is farthest from the onset node is also delayed with respect to the default case. As a result, when pd = 0.7, the metapopulation size initially grows faster than when pd = 0.5, due to fast invasion of the downstream nodes and growth of the local populations therein (see Fig. 7b); in a second phase, the growth rate of the metapopulation is reduced, because invasion of the upstream nodes is hampered by the low pu value, and the establishment of the equilibrium is delayed. As for the spatial spread of the metapopulation, when a preference for downstream movement is adopted, local population sizes at equilibrium tend to increase in the downstream nodes and decrease in the upstream nodes with respect to the default case (Fig. 7a), resulting in a slightly lower overall metapopulation size at equilibrium (Fig. 7b).

Figure 7:
Figure 7: Results from the application of a metapopulaton model on an OCN.

a) Dynamics of local population size in three nodes: green represents the headwater that is invaded at the beginning of the simulation; red, the outlet; blue, the headwater that is farthest from the green node (see circles in the top-left corner of panel c). Coloured vertical lines represent the time steps when the respective local population has reached the equilibrium (arbitrarily imposed as 99% of the population size at t = 800). Black solid lines identify the time steps used in panel c. b) Dynamics of the overall metapopulation size. Line styles as in panel a. c) Snapshots (obtained via draw_thematic_OCN) of spatial spread of the metapopulation at 3 different time steps. A script generating this figure can be found in the Supporting Information.

5 Conclusions

The importance of adequately representing spatial processes in ecological and evolutionary studies cannot be overstated. In the realm of freshwater ecology in particular, it is essential to consider how geomorphology shapes the structure of dendritic river networks and the ensuing connectivity configuration, which in turn control the variability of physical habitats and environmental variables, the dispersal of species and pathogens, and the spatial patterns of biodiversity and ecosystem processes. To this end, we presented OCNet, an R-package that enables the generation of optimal channel networks, spanning trees that reproduce all scaling features of real river networks throughout the globe. These can be used as realistic riverine landscape analogues for a number of ecological, epidemiological, ecohydrological and evolutionary studies. We reviewed the theoretical background of the OCN concept and the existing applications on problems of ecological relevance, provided an overview of the main functionalities of the package, and proposed an example of application in the context of an invasive riverine species. We believe that this tool will allow a leap forward in the way spatial processes in river networks are investigated.

Authors contributions

LC wrote the first draft of the manuscript, and developed the package based on preliminary scripts from EB with inputs and contributions from EAF, RF, and IG. EB, AR, and FA conceived the study. All authors contributed to the final version of the manuscript.

Data accessibility

The code of the OCNet package is accessible on both CRAN (https://CRAN.R-project.org/package=OCNet) and Github (http://doi.org/10.5281/zenodo.3669873 - development version).

Acknowledgements

FA acknowledges funding from the Swiss National Science Foundation Grants No PP00P3 179089 and 31003A 173074 and the University of Zurich Research Priority Programme “URPP Global Change and Biodiversity”. This is publication ISEM-YYYY-XXX of the Institut des Sciences de l’Evolution – Montpellier.

References

  1. ↵
    Altermatt, F. (2013). Diversity in riverine metacommunities: A network perspective. Aquatic Ecology, 47(3):365–377.
    OpenUrl
  2. ↵
    Altermatt, F., Schreiber, S., and Holyoak, M. (2011). Interactive effects of disturbance and dispersal directionality on species richness and composition in metacommunities. Ecology, 92(4):859–870.
    OpenUrlCrossRefPubMedWeb of Science
  3. ↵
    Anderson, K. E. and Hayes, S. M. (2018). The effects of dispersal and river spatial structure on asynchrony in consumer–resource metacommunities. Freshwater Biology, 63(1):100–113.
    OpenUrl
  4. ↵
    Balian, E. V., Segers, H., Lévèque, C., and Martens, K. (2008). The freshwater animal diversity assessment: An overview of the results. Hydrobiologia, 595(1):627–637.
    OpenUrlCrossRefWeb of Science
  5. ↵
    Balister, P., Balogh, J., Bertuzzo, E., Bollobás, B., Caldarelli, G., Maritan, A., Mastrandrea, R., Morris, R., and Rinaldo, A. (2018). River landscapes and optimal channel networks. Proceedings of the National Academy of Sciences of the United States of America, 115(26):6548–6553.
    OpenUrlAbstract/FREE Full Text
  6. ↵
    Banavar, J. R., Colaiori, F., Flammini, A., Maritan, A., and Rinaldo, A. (2000). Topology of the fittest transportation networks. Physical Review Letters, 84(20):4745–4748.
    OpenUrlCrossRefPubMedWeb of Science
  7. ↵
    Banavar, J. R., Colaiori, F., Flammini, A., Maritan, A., and Rinaldo, A. (2001). Scaling, optimality, and landscape evolution. Journal of Statistical Physics, 104(1–2):1–48.
    OpenUrl
  8. ↵
    Bascompte, J. and Solé, R. V. (1996). Habitat fragmentation and extinction thresholds in spatially explicit models. Journal of Animal Ecology, 65(4):465–473.
    OpenUrlCrossRef
  9. ↵
    Bell, G. and Gonzalez, A. (2011). Adaptation and evolutionary rescue in metapopulations experiencing environmental deterioration. Science, 332(6035):1327–1330.
    OpenUrlAbstract/FREE Full Text
  10. ↵
    Bertuzzo, E., Carrara, F., Mari, L., Altermatt, F., Rodriguez-Iturbe, I., and Rinaldo, A. (2016). Geomorphic controls on elevational gradients of species richness. Proceedings of the National Academy of Sciences of the United States of America, 113(7):1737–1742.
    OpenUrlAbstract/FREE Full Text
  11. ↵
    Bertuzzo, E., Casagrandi, R., Gatto, M., Rodriguez-Iturbe, I., and Rinaldo, A. (2010). On spatially explicit models of cholera epidemics. Journal of the Royal Society Interface, 7(43):321–333.
    OpenUrl
  12. ↵
    Bertuzzo, E., Helton, A. M., Hall, R. O. J., and Battin, T. J. (2017). Scaling of dissolved organic carbon removal in river networks. Advances in Water Resources, 110:136–146.
    OpenUrl
  13. ↵
    Bertuzzo, E., Rodriguez-Iturbe, I., and Rinaldo, A. (2015). Metapopulation capacity of evolving fluvial landscapes. Water Resources Research, 51(4):2696–2706.
    OpenUrlGeoRef
  14. ↵
    Beverton, R. J. H. and Holt, S. J. (1957). On the dynamics of exploited fish populations. Chapman & Hall, London.
  15. ↵
    Brown, B. L. and Swan, C. M. (2010). Dendritic network structure constrains metacommunity properties in riverine ecosystems. Journal of Animal Ecology, 79(3):571–580.
    OpenUrl
  16. ↵
    Cadotte, M. W. and Fukami, T. (2005). Dispersal, spatial scale, and species diversity in a hierarchically structured experimental landscape. Ecology Letters, 8(5):548–557.
    OpenUrlCrossRefPubMed
  17. ↵
    Campbell Grant, E. H., Lowe, W. H., and Fagan, W. F. (2007). Living in the branches: Population dynamics and ecological processes in dendritic networks. Ecology Letters, 10(2):165–175.
    OpenUrlCrossRefPubMedWeb of Science
  18. ↵
    Campos, D., Fort, J., and Méndez, V. (2006). Transport on fractal river networks: Application to migration fronts. Theoretical population biology, 69(1):88–93.
    OpenUrlCrossRefPubMedWeb of Science
  19. ↵
    Carrara, F., Altermatt, F., Rodriguez-Iturbe, I., and Rinaldo, A. (2012). Dendritic connectivity controls biodiversity patterns in experimental metacommunities. Proceedings of the National Academy of Sciences of the United States of America, 109(15):5761–5766.
    OpenUrlAbstract/FREE Full Text
  20. ↵
    Carrara, F., Rinaldo, A., Giometto, A., and Altermatt, F. (2014). Complex interaction of dendritic connectivity and hierarchical patch size on biodiversity in river-like landscapes. American Naturalist, 183(1):13–25.
    OpenUrlCrossRefPubMedWeb of Science
  21. ↵
    Carraro, L., Mari, L., Gatto, M., Rinaldo, A., and Bertuzzo, E. (2018). Spread of proliferative kidney disease in fish along stream networks: A spatial metacommunity framework. Freshwater Biology, 63(1):114–127.
    OpenUrl
  22. ↵
    Chaput-Bardy, A., Fleurant, C., Lemaire, C., and Secondi, J. (2009). Modelling the effect of in-stream and overland dispersal on gene flow in river networks. Ecological Modelling, 220(24):3589–3598.
    OpenUrl
  23. ↵
    Csardi, G. and Nepusz, T. (2006). The igraph software package for complex network research. InterJournal, Complex Systems:1695.
  24. ↵
    Dale, M. R. and Fortin, M.-J. (2014). Spatial analysis: a guide for ecologists. Cambridge University Press.
  25. ↵
    Darwall, W., Bremerich, V., De Wever, A., Dell, A. I., Freyhof, J., Gessner, M. O., Grossart, H.., Harrison, I., Irvine, K., Jähnig, S. C., Jeschke, J. M., Lee, J. J., Lu, C., Lewandowska, A. M., Monaghan, M. T., Nejstgaard, J. C., Patricio, H., Schmidt-Kloiber, A., Stuart, S. N., Thieme, M., Tockner, K., Turak, E., and Weyl, O. (2018). The alliance for freshwater life: A global call to unite efforts for freshwater biodiversity science and conservation. Aquatic Conservation: Marine and Freshwater Ecosystems, 28(4):1015–1022.
    OpenUrl
  26. ↵
    Duarte, G., Segurado, P., Oliveira, T., Haidvogl, G., Pont, D., Ferreira, M. T., and Branco, P. (2019). The river network toolkit –RivTool. Ecography, 42(3).
  27. ↵
    Elton, C. S. (1958). The ecology of invasions by animals and plants. John Wiley.
  28. ↵
    Fagan, W. F. (2002). Connectivity, fragmentation, and extinction risk in dendritic metapopulations. Ecology, 83(12):3243–3249.
    OpenUrlCrossRefWeb of Science
  29. ↵
    Fronhofer, E. A. and Altermatt, F. (2017). Classical metapopulation dynamics and eco-evolutionary feed-backs in dendritic networks. Ecography, 40(12):1455–1466.
    OpenUrl
  30. ↵
    Furrer, R. and Sain, S. R. (2010). spam: A sparse matrix R package with emphasis on MCMC methods for Gaussian Markov random fields. Journal of Statistical Software, 36(10):1–25.
    OpenUrl
  31. ↵
    Gatto, M., Mari, L., Bertuzzo, E., Casagrandi, R., Righetto, L., Rodriguez-Iturbe, I., and Rinaldo, A. (2013). Spatially explicit conditions for waterborne pathogen invasion. American Naturalist, 182(3):328–346.
    OpenUrlCrossRefPubMedWeb of Science
  32. ↵
    Gause, G. (1934). The Struggle for Existence. Dover Publications, New York, US.
  33. ↵
    Giezendanner, J., Bertuzzo, E., Pasetto, D., Guisan, A., and Rinaldo, A. (2019). A minimalist model of extinction and range dynamics of virtual mountain species driven by warming temperatures. PLoS ONE, 14(3).
  34. ↵
    Gilarranz, L. J. and Bascompte, J. (2012). Spatial network structure and metapopulation persistence. Journal of Theoretical Biology, 297:11–16.
    OpenUrlCrossRefPubMedWeb of Science
  35. ↵
    Giometto, A., Altermatt, F., and Rinaldo, A. (2017). Demographic stochasticity and resource autocorrelation control biological invasions in heterogeneous landscapes. Oikos, 126(11):1554–1563.
    OpenUrl
  36. ↵
    Grimaldi, S., Teles, V., and Bras, R. L. (2005). Preserving first and second moments of the slope area relationship during the interpolation of digital elevation models. Advances in Water Resources, 28(6):583–588.
    OpenUrl
  37. ↵
    Hack, J. T. (1957). Studies of longitudinal stream profiles in Virginia and Maryland, volume 294. US Government Printing Office.
  38. ↵
    Hanski, I. and Ovaskainen, O. (2000). The metapopulation capacity of a fragmented landscape. Nature, 404(6779):755–758.
    OpenUrlCrossRefPubMedWeb of Science
  39. ↵
    Hanski, I. A. and Gaggiotti, O. E. (2004). Ecology, genetics and evolution of metapopulations. Academic Press.
  40. ↵
    Harvey, E., Gounand, I., Fronhofer, E. A., and Altermatt, F. (2018). Disturbance reverses classic biodiversity predictions in river-like landscapes. Proceedings of the Royal Society B: Biological Sciences, 285(1893).
  41. ↵
    Harvey, E., Gounand, I., Fronhofer, E. A., and Altermatt, F. (2019). Metaecosystem dynamics drive community composition in experimental, multi-layered spatial networks. Oikos.
  42. ↵
    Helton, A. M., Hall, R. O., J., and Bertuzzo, E. (2018). How network structure can affect nitrogen removal by streams. Freshwater Biology, 63(1):128–140.
    OpenUrl
  43. ↵
    Holland, M. D. and Hastings, A. (2008). Strong effect of dispersal network structure on ecological dynamics. Nature, 456(7223):792–795.
    OpenUrlCrossRefPubMedWeb of Science
  44. ↵
    Holyoak, M., Leibold, M. A., and Holt, R. D. (2005). Metacommunities: spatial dynamics and ecological communities. University of Chicago Press.
  45. ↵
    Horton, R. E. (1945). Erosional development of streams and their drainage basins; hydrophysical approach to quantitative morphology. Geological society of America bulletin, 56(3):275–370.
    OpenUrlAbstract/FREE Full Text
  46. ↵
    Koenig, L. E., Helton, A. M., Savoy, P., Bertuzzo, E., Heffernan, J. B., Hall Jr, R. O., and Bernhardt, E. S. (2019). Emergent productivity regimes of river networks. Limnology and Oceanography Letters, 4(5):173–181.
    OpenUrl
  47. ↵
    Leopold, L., Wolman, M., and Miller, J. (1964). Fluvial Processes in Geomorphology. Dover Publications Inc., New York, US.
  48. ↵
    Leopold, L. B. and Maddock, T. (1953). The hydraulic geometry of stream channels and some physiographic implications, volume 252. US Government Printing Office.
  49. ↵
    Leuven, R. S. E. W., van der Velde, G., Baijens, I., Snijders, J., van der Zwart, C., Lenders, H. J. R., and bij de Vaate, A. (2009). The river Rhine: A global highway for dispersal of aquatic invasive species. Biological Invasions, 11(9):1989–2008.
    OpenUrl
  50. ↵
    Levin, S. A. (1992). The problem of pattern and scale in ecology. Ecology, 73(6):1943–1967.
    OpenUrlCrossRefWeb of Science
  51. ↵
    1. Gerstenhaber, M.
    Levins, R. (1970). Extinction. In Gerstenhaber, M., editor, Some Mathematical Problems in Biology, pages 77–107. American Mathematical Society, Providence.
  52. ↵
    Mandelbrot, B. B. (1983). The Fractal Geometry of Nature, volume 173. WH Freeman New York.
  53. ↵
    Mari, L., Casagrandi, R., Bertuzzo, E., Rinaldo, A., and Gatto, M. (2014). Metapopulation persistence and species spread in river networks. Ecology Letters, 17(4):426–434.
    OpenUrl
  54. ↵
    Mari, L., Casagrandi, R., Bertuzzo, E., Rinaldo, A., and Gatto, M. (2019). Conditions for transient epidemics of waterborne disease in spatially explicit systems. Royal Society Open Science, 6(5).
  55. ↵
    Maritan, A., Rinaldo, A., Rigon, R., Giacometti, A., and Rodríguez-Iturbe, I. (1996). Scaling laws for river networks. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, 53(2):1510–1515.
    OpenUrl
  56. ↵
    Morrissey, M. B. and De Kerckhove, D. T. (2009). The maintenance of genetic variation due to asymmetric gene flow in dendritic metapopulations. American Naturalist, 174(6):875–889.
    OpenUrlCrossRefPubMedWeb of Science
  57. ↵
    Muneepeerakul, R., Bertuzzo, E., Lynch, H. J., Fagan, W. F., Rinaldo, A., and Rodriguez-Iturbe, I. (2008). Neutral metacommunity models predict fish diversity patterns in mississippi-missouri basin. Nature, 453(7192):220–222.
    OpenUrlCrossRefPubMedWeb of Science
  58. ↵
    Muneepeerakul, R., Bertuzzo, E., Rinaldo, A., and Rodriguez-Iturbe, I. (2019). Evolving biodiversity patterns in changing river networks. Journal of Theoretical Biology, 462:418–424.
    OpenUrl
  59. ↵
    O’Callaghan, J. F. and Mark, D. M. (1984). The extraction of drainage networks from digital elevation data. Computer Vision, Graphics, and Image Processing, 28(3):323–344.
    OpenUrlCrossRef
  60. ↵
    Pallard, B., Castellarin, A., and Montanari, A. (2009). A look at the links between drainage density and flood statistics. Hydrology and Earth System Sciences, 13(7):1019–1029.
    OpenUrl
  61. ↵
    Paz-Vinas, I. and Blanchet, S. (2015). Dendritic connectivity shapes spatial patterns of genetic diversity: A simulation-based study. Journal of Evolutionary Biology, 28(4):986–994.
    OpenUrlCrossRef
  62. ↵
    Paz-Vinas, I., Loot, G., Stevens, V. M., and Blanchet, S. (2015). Evolutionary processes driving spatial patterns of intraspecific genetic diversity in river ecosystems. Molecular Ecology, 24(18):4586–4604.
    OpenUrlCrossRef
  63. ↵
    Pebesma, E. J. (2004). Multivariable geostatistics in S: the gstat package. Computers & Geosciences, 30(7):683–691.
    OpenUrl
  64. ↵
    Peterson, E. E., Ver Hoef, J. M., Isaak, D. J., Falke, J. A., Fortin, M.., Jordan, C. E., McNyset, K., Monestiez, P., Ruesch, A. S., Sengupta, A., Som, N., Steel, E. A., Theobald, D. M., Torgersen, C. E., and Wenger, S. J. (2013). Modelling dendritic ecological networks in space: An integrated network perspective. Ecology Letters, 16(5):707–719.
    OpenUrlCrossRefPubMed
  65. ↵
    Rinaldo, A., Gatto, M., and Rodriguez-Iturbe, I. (2020). River networks as ecological corridors. Species, populations, pathogens. Cambridge University Press, New York, US.
  66. ↵
    Rinaldo, A., Rigon, R., Banavar, J. R., Maritan, A., and Rodriguez-Iturbe, I. (2014). Evolution and selection of river networks: Statics, dynamics, and complexity. Proceedings of the National Academy of Sciences of the United States of America, 111(7):2417–2424.
    OpenUrlAbstract/FREE Full Text
  67. ↵
    Rinaldo, A., Rodriguez-Iturbe, I., Rigon, R., Bras, R. L., Ijjasz-Vasquez, E., and Marani, A. (1992). Minimum energy and fractal structures of drainage networks. Water Resources Research, 28(9):2183–2195.
    OpenUrlCrossRefGeoRefWeb of Science
  68. ↵
    Rodriguez-Iturbe, I., Ijjász-Vásquez, E. J., Bras, R. L., and Tarboton, D. G. (1992a). Power law distributions of discharge mass and energy in river basins. Water Resources Research, 28(4):1089–1093.
    OpenUrlCrossRefWeb of Science
  69. ↵
    Rodriguez-Iturbe, I., Muneepeerakul, R., Bertuzzo, E., Levin, S. A., and Rinaldo, A. (2009). River networks as ecological corridors: A complex systems perspective for integrating hydrologic, geomorphologic, and ecologic dynamics. Water Resources Research, 45(1). W01413.
    OpenUrl
  70. ↵
    Rodriguez-Iturbe, I. and Rinaldo, A. (2001). Fractal River Basins. Chance and self-organization. Cambridge University Press, New York, US.
  71. ↵
    Rodriguez-Iturbe, I., Rinaldo, A., Rigon, R., Bras, R. L., Marani, A., and Ijjasz-Vasquez, E. (1992b). Energy dissipation, runoff production, and the three-dimensional structure of river basins. Water Resources Research, 28(4):1095–1103.
    OpenUrlCrossRefWeb of Science
  72. ↵
    Sarker, S., Veremyev, A., Boginski, V., and Singh, A. (2019). Critical nodes in river networks. Scientific Reports, 9(1):11178.
    OpenUrl
  73. ↵
    Seymour, M. and Altermatt, F. (2014). Active colonization dynamics and diversity patterns are influenced by dendritic network connectivity and species interactions. Ecology and Evolution, 4(8):1243–1254.
    OpenUrl
  74. ↵
    Seymour, M., Fronhofer, E. A., and Altermatt, F. (2015). Dendritic network structure and dispersal affect temporal dynamics of diversity and species persistence. Oikos, 124(7):908–916.
    OpenUrlCrossRef
  75. ↵
    Sun, T., Meakin, P., and Jøssang, T. (1994). The topography of optimal drainage basins. Water Resources Research, 30(9):2599–2610.
    OpenUrlCrossRefGeoRefWeb of Science
  76. ↵
    Tarboton, D. G., Bras, R. L., and Rodriguez-Iturbe, I. (1989). Scaling and elevation in river networks. Water Resources Research, 25(9):2037–2051.
    OpenUrlGeoRefWeb of Science
  77. ↵
    Terui, A., Ishiyama, N., Urabe, H., Ono, S., Finlay, J. C., and Nakamura, F. (2018). Metapopulation stability in branching river networks. Proceedings of the National Academy of Sciences of the United States of America, 115(26):E5963–E5969.
    OpenUrlAbstract/FREE Full Text
  78. ↵
    Vannote, R. L., Minshall, G. W., Cummins, K. W., Sedell, J. R., and Cushing, C. E. (1980). The river continuum concept. Canadian Journal of Fisheries and Aquatic Sciences, 37(1):130–137.
    OpenUrlCrossRefWeb of Science
  79. ↵
    Ver Hoef, J. M., Peterson, E. E., Cliord, D., and Shah, R. (2014). SSN: An R package for spatial statistical modeling on stream networks. Journal of Statistical Software, 56(3):1–45.
    OpenUrl
  80. ↵
    Vörösmarty, C. J., McIntyre, P. B., Gessner, M. O., Dudgeon, D., Prusevich, A., Green, P., Glidden, S., Bunn, S. E., Sullivan, C. A., Liermann, C. R., and Davies, P. M. (2010). Global threats to human water security and river biodiversity. Nature, 467(7315):555–561.
    OpenUrlCrossRefPubMedWeb of Science
  81. ↵
    Welty, E. Z., Torgersen, C. E., Brenkman, S. J., Duda, J. J., and Armstrong, J. B. (2015). Multiscale analysis of river networks using the R package linbin. North American Journal of Fisheries Management, 35(4):802–809.
    OpenUrl
  82. ↵
    Yeakel, J. D., Moore, J. W., Guimarães, P. R., and de Aguiar, M. A. M. (2014). Synchronisation and stability in river metapopulation networks. Ecology Letters, 17(3):273–283.
    OpenUrl
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Generation and application of river network analogues for use in ecology and evolution
Luca Carraro, Enrico Bertuzzo, Emanuel A. Fronhofer, Reinhard Furrer, Isabelle Gounand, Andrea Rinaldo, Florian Altermatt
bioRxiv 2020.02.17.948851; doi: https://doi.org/10.1101/2020.02.17.948851
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