A prevailing paradigm suggests that species richness increases with area in a decelerating way. This ubiquitous power law scaling, the species–area relationship, has formed the foundation of many conservation strategies. In spatially complex ecosystems, however, the area may not be the sole dimension to scale biodiversity patterns because the scale-invariant complexity of fractal ecosystem structure may drive ecological dynamics in space. Here, we use theory and analysis of extensive fish community data from two distinct geographic regions to show that riverine biodiversity follows a robust scaling law along the two orthogonal dimensions of ecosystem size and complexity (i.e., the dual scaling law). In river networks, the recurrent merging of various tributaries forms fractal branching systems, where the prevalence of branching (ecosystem complexity) represents a macroscale control of the ecosystem’s habitat heterogeneity. In the meantime, ecosystem size dictates metacommunity size and total habitat diversity, two factors regulating biodiversity in nature. Our theory predicted that, regardless of simulated species’ traits, larger and more branched “complex” networks support greater species richness due to increased space and environmental heterogeneity. The relationships were linear on logarithmic axes, indicating power law scaling by ecosystem size and complexity. In support of this theoretical prediction, the power laws have consistently emerged in riverine fish communities across the study regions (Hokkaido Island in Japan and the midwestern United States) despite hosting different fauna with distinct evolutionary histories. The emergence of dual scaling law may be a pervasive property of branching networks with important implications for biodiversity conservation.
Scale invariance, which refers to the preservation of geometric properties regardless of observation scale, is a prevalent phenomenon in ecological systems. This concept is closely associated with fractals, and river networks serve as prime examples of fractal systems. Quantifying river network complexity is crucial for unveiling the role of river fractals in riverine ecological dynamics, and researchers have used a metric of “branching probability” to do so. Previous studies showed that this metric reflects the fractal nature of river networks. However, a recent article by Carraro and Altermatt (2022) contradicted this classical observation and concluded that branching probability is “scale dependent.” I dispute this claim and argue that their major conclusion is derived merely from their misconception of scale invariance. Their analysis in the original article (fig. 3a) provided evidence that branching probability is scale‐invariant (i.e., branching probability exhibits a power‐law scaling), although the authors erroneously interpreted this result as a sign of scale dependence. In this article, I re‐introduce the definition of scale invariance and show that branching probability meets this definition. This provided an opportunity to address the divergent use of “scale invariance” and “scaling” between fractal theory and ecology.
more » « less- Award ID(s):
- 2015634
- NSF-PAR ID:
- 10439603
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Population Ecology
- Volume:
- 66
- Issue:
- 2
- ISSN:
- 1438-3896
- Format(s):
- Medium: X Size: p. 108-114
- Size(s):
- ["p. 108-114"]
- Sponsoring Org:
- National Science Foundation
More Like this
-
more » « less
-
more » « less
Abstract Discharge from multiple wastewater treatment plants (WWTPs) distributed in urbanized river basins contributes to impairments of river water‐quality and aquatic ecosystem integrity, with size and location of WWTPs determined by population distribution within a river basin. Here we used geo‐referenced data for WWTPs in Germany to investigate the spatial organization of three attributes of interest in this study: population, population equivalents (the aggregated population served by each WWTP), and the number/sizes of WWTPs. To this end, we selected as case studies three large urbanized river basins (Weser, Elbe, and Rhine), home to about 70% of the population in Germany. We employed fractal river networks as structural platforms to examine the spatial patterns from two perspectives: spatial hierarchy (stream order) and patterns along longitudinal flow paths (width function). Moreover, we proposed three dimensionless scaling indices to quantify (1) human settlement preferences by stream order, (2) non‐sanitary flow contribution to total wastewater treated at WWTPs, and (3) degree of centralization in WWTPs locations. Across the three river basins, we found scale‐invariant distributions for each of the three attributes with stream order, quantified using extended Horton scaling ratios. We found a weak downstream clustering of population in the three basins. Variations in population equivalent clustering among different class‐sizes of WWTPs reflected the size, number, and locations of urban agglomerations in these river basins. We discussed the applicability of this approach to other large urbanized basins to analyze spatial organization of population and WWTPs.
-
Today in the age of advanced ceramic civilization, there are a variety of applications for modern ceramics materials with specific properties. Our up-to date research recognizes that ceramics have a fractal configuration nature on the basis of different phenomena. The key property of fractals is their scale-independence. The practical value is that the fractal objects’ interaction and energy is possible at any reasonable scale of magnitude, including the nanoscale and may be even below. This is a consequence of fractal scale independence. This brings us to the conclusion that properties of fractals are valid on any scale (macro, micro, or nano). We also analyzed these questions with experimental results obtained from a comet, here 67P, and also from ceramic grain and pore morphologies on the microstructure level. Fractality, as a scale-independent morphology, provides significant variety of opportunities, for example for energy storage. From the viewpoint of scaling, the relation between large and small in fractal analysis is very important. An ideal fractal can be magnified endlessly but natural morphologies cannot, what is the new light in materials sciences and space.more » « less
-
Similarity search is the basis for many data analytics techniques, including k-nearest neighbor classification and outlier detection. Similarity search over large data sets relies on i) a distance metric learned from input examples and ii) an index to speed up search based on the learned distance metric. In interactive systems, input to guide the learning of the distance metric may be provided over time. As this new input changes the learned distance metric, a naive approach would adopt the costly process of re-indexing all items after each metric change. In this paper, we propose the first solution, called OASIS, to instantaneously adapt the index to conform to a changing distance metric without this prohibitive re-indexing process. To achieve this, we prove that locality-sensitive hashing (LSH) provides an invariance property, meaning that an LSH index built on the original distance metric is equally effective at supporting similarity search using an updated distance metric as long as the transform matrix learned for the new distance metric satisfies certain properties. This observation allows OASIS to avoid recomputing the index from scratch in most cases. Further, for the rare cases when an adaption of the LSH index is shown to be necessary, we design an efficient incremental LSH update strategy that re-hashes only a small subset of the items in the index. In addition, we develop an efficient distance metric learning strategy that incrementally learns the new metric as inputs are received. Our experimental study using real world public datasets confirms the effectiveness of OASIS at improving the accuracy of various similarity search-based data analytics tasks by instantaneously adapting the distance metric and its associated index in tandem, while achieving an up to 3 orders of magnitude speedup over the state-of-art techniques.more » « less
-
more » « less
Abstract Most terrestrial allochthonous organic matter enters river networks through headwater streams during high flow events. In headwaters, allochthonous inputs are substantial and variable, but become less important in streams and rivers with larger watersheds. As allochthonous dissolved organic matter (DOM) moves downstream, the proportion of less aromatic organic matter with autochthonous characteristics increases. How environmental factors converge to control this transformation of DOM at a continental scale is less certain. We hypothesized that the amount of time water has spent travelling through surface waters of inland systems (streams, rivers, lakes, and reservoirs) is correlated to DOM composition. To test this hypothesis, we used established river network scaling relationships to predict relative river network flow‐weighted travel time (FWTT) of water for 60 stream and river sites across the contiguous United States (3090 discrete samples over 10 water years). We estimated lentic contribution to travel times with upstream in‐network lake and reservoir volume. DOM composition was quantified using ultraviolet and visible absorption and fluorescence spectroscopy. A combination of FWTT and lake and reservoir volume was the best overall predictor of DOM composition among models that also incorporated discharge, specific discharge, watershed area, and upstream channel length. DOM spectral slope ratio (R2 = 0.77) and Freshness Index (R2 = 0.78) increased and specific ultraviolet absorbance at 254 nm (R2 = 0.68) and Humification Index (R2 = 0.44) decreased across sites as a function of FWTT and upstream lake volume. This indicates autochthonous‐like DOM becomes continually more dominant in waters with greater FWTT. We assert that river FWTT can be used as a metric of the continuum of DOM composition from headwaters to rivers. The nature of the changes to DOM composition detected suggest this continuum is driven by a combination of photo‐oxidation, biological processes, hydrologically varying terrestrial subsidies, and aged groundwater inputs.
