Nature offers us a variety of shapes and patterns of undeniable beauty. Many of these shapes are characterized by remarkable and recurrent geometrical regularities, in which the parts are similar to the whole. This is the case of fractals. A prime example of fractals is constituted by river networks. Moving in the upstream direction, the main river stem is recurrently split into smaller tributaries, giving rise to a dendritic shape.

Fractals have since long captured the interest of scientists, who have studied emergence, causes and implications of such self-similar structures. In particular for river networks, it has been found that their peculiar fractal shape bears important consequences for hydrological phenomena such as rainfall-runoff processes and sediment transport. But the shape of river networks is of pivotal importance also for freshwater biological populations. Indeed, unlike terrestrial habitats—where species can potentially move freely in a twodimensional landscape—river networks force organisms (at least those belonging to obligate water species) to follow along-stream paths. Moreover, variation in habitat size along a river network is dictated by variations in depth and width of a river, which vary predictably as a consequence of the fractal scaling of rivers.

Ecologists have been trying to reproduce dispersal mechanisms of species and the resulting patterns of metapopulations (that is, populations structured in space) via mathematical models. Needless to say, to model ecological processes in river networks, the use of an appropriate model for river networks is essential.

Naively, one could think that a simple bifurcating tree (as in Fig. 1a—i.e., starting from an initial downstream node, pairs of upstream nodes are added to each node until the desired length is reached) would be a possible candidate for a river network model. A more elaborated alternative would consist in creating a certain number of “links” of different length (which reproduce river reaches, i.e. uninterrupted segments of river between successive confluences, or between a source and a confluence), and then assemble them randomly, as in Fig. 1b.

A third river model was derived some thirty years ago by following geomorphological principles. In Nature, not all river network configurations are equally likely; indeed, natural river network configurations are those that correspond to a minimum of total energy dissipated by water as it flows downstream, which is reminiscent of the general physical principle of energy minimization. Virtual constructs that respect this rule are called Optimal Channel Networks (OCNs—Fig. 1c).

In this study, we compared these three different river models with real river networks extracted at different scales. In particular, we aimed to find which river model is able to reproduce properties of real river networks that are relevant from an ecological viewpoint. We focused our attention on two metrics, the coefficient of variation of a metapopulation and the metapopulation capacity. The former is a measure of instability of a metapopulation in the face of local perturbations, while the latter measures the ability of a metapopulation to persist in the long run, or to invade a new landscape. Interestingly, both metrics uniquely depend on landscape attributes, therefore their values are irrespective of the particular species studied. We found that only OCNs yield metapopulation metric values that are close to those derived for real river networks, while this is not the case for the other two river network models.

Moreover, we resolved a common misunderstanding in the use of purely random (i.e., non-OCN) river models in ecological studies. Algorithms to generate such random networks often depend on a so-called “branching probability” (that is, the probability that a single node is branching), which has been claimed to be a driver of metapopulation stability: the more “branching” a river is, the more the metapopulations inhabiting it would be supposed to be protected from global extinction induced by local perturbations. Actually, real river networks (and OCNs) cannot be “more” or “less” branching, as their fractal properties are universal. The different branching character of real river networks essentially depends on the scale at which these are observed: a river network made up of few, large tributaries if observed at a coarse scale can look much more branching when observed at a finer scale. Hence, branching probability is not a scale-invariant property of river networks.

Overall, our study shows that OCNs are the only appropriate model for hydrological and ecological studies in river networks. In broader terms, our results advocate a tighter integration between physical (geomorphology, hydrology) and biological (ecology) disciplines in the study of freshwater ecosystems, and particularly in the perspective of a mechanistic understanding of drivers of persistence and loss of biodiversity. 

The full article can be retrieved here.