Scaling Theory Of Transitions Research Articles

Scaling theory of fractal complex networks

We show that fractality in complex networks arises from the geometric self-similarity of their built-in hierarchical community-like structure, which is mathematically described by the scale-invariant equation for the masses of the boxes with which we cover the network when determining its box dimension. This approach—grounded in both scaling theory of phase transitions and renormalization group theory—leads to the consistent scaling theory of fractal complex networks, which complements the collection of scaling exponents with several new ones and reveals various relationships between them. We propose the introduction of two classes of exponents: microscopic and macroscopic, characterizing the local structure of fractal complex networks and their global properties, respectively. Interestingly, exponents from both classes are related to each other and only a few of them (three out of seven) are independent, thus bridging the local self-similarity and global scale-invariance in fractal networks. We successfully verify our findings in real networks situated in various fields (information—the World Wide Web, biological—the human brain, and social—scientific collaboration networks) and in several fractal network models.

Density dependence and spatial heterogeneity limit the population growth rate of invasive pines at the landscape scale

Determining population growth across large scales is difficult because it is often impractical to collect data at large scales and over long timespans. Instead, the growth of a population is often only measured at a small, plot‐level scale and then extrapolated to derive a mean field estimate. However, this approach is prone to error since it simplifies spatial processes such as the neighbourhood effects of density and dispersal. We present a novel approach that estimates how spatial processes derived from the effects of density and dispersal affect population growth between plot scales and landscape scales. The method is based on a scale transition theory and calculates a transition term to measure the spatial scaling of population growth, which we extend to unstable, expanding populations in order to assess whether landscape‐scale population dynamics are different from those estimated at smaller spatial scales. We illustrate this approach using aerial imagery of eight locations in New Zealand experiencing non‐native pine invasions. Analyses examined the dynamics at a plot scale (1 ha) and compared this to estimates across entire landscapes (between 24 and 1600 ha), in several cases for more than one time period. We used a Bayesian spatial random effects model to examine population growth and to account for neighbourhood effects and dispersal between plots in a rapidly changing system.We found that the estimates of the scale transition term were typically 10–25% of the mean field estimates, which led to mean field estimates of population growth extrapolated from plots being considerably higher than landscape estimates. The approach we have developed will not only have applications for predicting the populations' growth of invasive species, but also for studies examining the scaling of landscape‐scale phenomena.

Topological phase transitions in glassy quantum matter

Amorphous systems have rapidly gained promise as novel platforms for topological matter. In this work we establish a scaling theory of amorphous topological phase transitions driven by the density of lattice points in two dimensions. By carrying out a finite-size scaling analysis of topological invariants averaged over discrete and continuum random geometries, we discover unique critical properties of Chern and $\mathbb{Z}_2$ glass transitions. Even for short-range hopping models the Chern glass phase may persist down to the fundamental lower bound given by the classical percolation threshold. While the topological indices accurately satisfy the postulated one-parameter scaling, they do not generally flow to the closest integer value in the thermodynamic limit. Furthermore, the value of the critical exponent describing the diverging localization length varies continuously along the phase boundary and is not fixed by the symmetry class of the Hamiltonian. We conclude that the critical behaviour of amorphous topological systems exhibit characteristic features not observed in disordered systems, motivating a wealth of new research directions.

Scale-Dependent Community Theory for Streams and Other Linear Habitats.

The maintenance of species diversity occurs at the regional scale but depends on interacting processes at the full range of lower scales. Although there is a long history of study of regional diversity as an emergent property, analyses of fully multiscale dynamics are rare. Here, we use scale transition theory for a quantitative analysis of multiscale diversity maintenance with continuous scales of dispersal and environmental variation in space and time. We develop our analysis with a model of a linear habitat, applicable to streams or coastlines, to provide a theoretical foundation for the long-standing interest in environmental variation and dispersal, including downstream drift. We find that the strength of regional coexistence is strongest when local densities and local environmental conditions are strongly correlated. Increasing dispersal and shortening environmental correlations weaken the strength of coexistence regionally and shift the dominant coexistence mechanism from fitness-density covariance to the spatial storage effect, while increasing local diversity. Analysis of the physical and biological determinants of these mechanisms improves understanding of traditional concepts of environmental filters, mass effects, and species sorting. Our results highlight the limitations of the binary distinction between local communities and a species pool and emphasize species coexistence as a problem of multiple scales in space and time.

The role of competition - colonization tradeoffs and spatial heterogeneity in promoting trematode coexistence.

Competition - colonization tradeoffs occur in many systems, and theory predicts that they can strongly promote species coexistence. However, there is little empirical evidence that observed competition- colonization tradeoffs are strong enough to maintain diversity in natural systems. This is due in part to a mismatch between theoretical assumptions and biological reality in some systems. We tested whether a competition - colonization tradeoff explains how a diverse trematode guild coexists in California horn snail populations, a system that meets the requisite criteria for the tradeoff to promote coexistence. A field experiment showed that subordinate trematode species tended to have higher colonization rates than dominant species. This tradeoff promoted coexistence in parameterized models but did not fully explain trematode diversity and abundance, suggesting a role of additional diversity maintenance mechanisms. Spatial heterogeneity is an alternative way to promote coexistence if it isolates competing species. We used scale transition theory to expand the competition - colonization tradeoff model to include spatial variation. The parameterized model showed that spatial variation in trematode prevalence did not isolate most species sufficiently to explain the overall high diversity, but could benefit some rare species. Together, the results suggest that several mechanisms combine to maintain diversity, even when a competition - colonization tradeoff occurs.

Thermal variation, thermal extremes and the physiological performance of individuals

In this review we consider how small-scale temporal and spatial variation in body temperature, and biochemical/physiological variation among individuals, affect the prediction of organisms' performance in nature. For 'normal' body temperatures - benign temperatures near the species' mean - thermal biology traditionally uses performance curves to describe how physiological capabilities vary with temperature. However, these curves, which are typically measured under static laboratory conditions, can yield incomplete or inaccurate predictions of how organisms respond to natural patterns of temperature variation. For example, scale transition theory predicts that, in a variable environment, peak average performance is lower and occurs at a lower mean temperature than the peak of statically measured performance. We also demonstrate that temporal variation in performance is minimized near this new 'optimal' temperature. These factors add complexity to predictions of the consequences of climate change. We then move beyond the performance curve approach to consider the effects of rare, extreme temperatures. A statistical procedure (the environmental bootstrap) allows for long-term simulations that capture the temporal pattern of extremes (a Poisson interval distribution), which is characterized by clusters of events interspersed with long intervals of benign conditions. The bootstrap can be combined with biophysical models to incorporate temporal, spatial and physiological variation into evolutionary models of thermal tolerance. We conclude with several challenges that must be overcome to more fully develop our understanding of thermal performance in the context of a changing climate by explicitly considering different forms of small-scale variation. These challenges highlight the need to empirically and rigorously test existing theories.

Linking patterns and processes across scales: the application of scale-transition theory to algal dynamics on rocky shores

Understanding how species and environments respond to global anthropogenic disturbances is one of the greatest challenges for contemporary ecology. The ability to integrate modeling, correlative and experimental approaches within individual research programs will be key to address large-scale, long-term environmental problems. Scale-transition theory (STT) enables this level of integration, providing a powerful framework to link ecological patterns and processes across spatial and temporal scales. STT predicts the large-scale (e.g. regional) behavior of a system on the basis of nonlinear population models describing local (e.g. patch-scale) dynamics and the interaction between these nonlinearities and spatial variation in population abundance or environmental conditions. Here we use STT to predict the dynamics of turf-forming algae on rocky shores at Capraia Island, in the northwest Mediterranean. We developed a model of algal turf dynamics based on density-dependent growth that included the effects of local interactions with canopy algae. The model was parameterized with field data and used to scale up the dynamics of algal turfs from the plot scale (20×20 cm) to the island scale (tens of km). The interaction between nonlinear growth and spatial variance in cover of turfing algae emerged as a key term to translate the local dynamics up to the island scale. The model successfully predicted short-term and long-term mean values of turf cover estimated independently from a separate experiment. These results illustrate how STT can be used to identify the relevant mechanisms that drive large-scale changes in ecological communities. We argue that STT can contribute significantly to the connection between biomechanics and ecology, a synthesis that is at the core of the emerging field of ecomechanics.

BIOPHYSICS, BIOENERGETICS AND MECHANISTIC APPROACHES TO ECOLOGY

Biophysics – the study of how the structure of individual plants and animals contributes to their function – has rarely been considered on the ecological scale. However, Mark Denny from Stanford University, USA, has a broader vision. Because the mechanisms that underpin individual performance

Scale transition theory: Its aims, motivations and predictions

Scale transition theory is an approach to understanding population and community dynamics in the presence of spatial or temporal variation in environmental factors or population densities. It focuses on changes in the equations for population dynamics as the scale enlarges. These changes are explained in terms of interactions between nonlinearities and variation on lower scales, and they predict the emergence of new properties on larger scales that are not predicted by lower scale dynamics in the absence of variation on those lower scales. These phenomena can be understood in terms of statistical inequalities arising from the process of nonlinear averaging, which translates the rules for dynamics from lower to higher scales. Nonlinearities in population dynamics are expressions of the fundamental biology of the interactions between individual organisms. Variation that interacts with these nonlinearities also involves biology fundamentally in several different ways. First, there are the aspects of biology that are sensitive to variation in space or time. These determine which aspects of a nonlinear dynamical equation are affected by variation, and whether different individuals or different species are sensitive to different extents or to different aspects of variation. Second is the nature of the variation, for example, whether it is variation in the physical environment or variation in population densities. From the interplay between variation and nonlinearities in population dynamics, scale transition theory builds a theory of changes in dynamics with changes in scale. In this article, the focus is on spatial variation, and the theory is illustrated with examples relevant to the dynamics of insect communities. In these communities, one commonly occurring nonlinear relationship is a negative exponential relationship between survival of an organism and the densities of natural enemies or competitors. This negative exponential has a biological origin in terms of independent actions of many individuals. The subsequent effects of spatial variation can be represented naturally in terms of Laplace transforms and related statistical transforms to obtain both analytical solutions and an extra level of understanding. This process allows us to analyze the meaning and effects of aggregation of insects in space. Scale transition theory more generally, however, does not aim to have fully analytical solutions but partial analytical solutions applicable for circumstances too complex for full analytical solution. These partial solutions are intended to provide a framework for understanding of numerical solutions, simulations and field studies where key quantities can be estimated from empirical data.

Using an individual-based model to quantify scale transition in demographic rate functions: Deaths in a coral reef fish

Scientifically informed population management requires quantitatively accurate demographic rate functions that apply at the spatial scale at which populations are actually managed, but practical constraints confine most field measurements of such functions to small study plots. This paper employs an individual-based population growth model to extrapolate the death rate function in a well-studied coral reef fish, the bridled goby Coryphopterus glaucofraenum, from the scale of 2 m × 2 m coral reef “cells” at which it was measured to the larger scale of an entire coral reef. Density dependence in the whole-reef function actually proves stronger than in the local function because high goby density occasionally arises in local patches with few refuges from predators, producing very high mortality there. This IBM-based approach extends the reach of scale transition theory by examining considerably more realistic models than standard analytical methods can presently handle.

Scale transition theory with special reference to species coexistence in a variable environment

Scale transition theory is a mathematical technique for understanding changes in population dynamics with changes in spatial or temporal scale. It explains the emergence of new properties on large scales from the interaction between nonlinearities and variation on small scales. It applies statistical theory for averaging nonlinear functions to understanding this interaction. The fundamental concepts are most easily illustrated with reference to spatial models where state variables on larger spatial scales are simply defined as averages of those on smaller scales. Scale transition theory also explains the conceptually difficult topic of how species coexistence arises from temporal fluctuations. In this case, averages of per capita growth rates over time define long-term population trends and outcomes, and these averages are critically affected by interactions between nonlinear dynamics and temporal variation in state variables and environmental variables. Two general mechanisms of species coexistence, the storage effect and relatively nonlinear competitive variance, emerge.

Scaling up the functional response for spatially heterogeneous systems

Scale transition theory is a framework for predicting regional population dynamics from local process functions and estimates of spatial heterogeneity. Using this framework, we estimated regional scale functional responses for a benthic predator-prey system in the Baltic Sea. Functional responses were based on laboratory experiments or field observations of stomach contents, and prey densities measured at a local scale (0.1 m(2)) or a regional scale (300 km(2)). Laboratory data overestimated consumption at high prey densities, whereas predictions based on local scale data tallied closely with consumption observed at the regional scale. The predicted regional functional response was different for increasing and decreasing prey densities, reflecting that predator and prey densities, as well as the covariance between them, exhibit oscillatory dynamics. We conclude that it is important to validate laboratory data with small-scale field observations and that scale transition is a powerful tool for scaling-up process functions in heterogeneous systems.

THE SCALE TRANSITION: SCALING UP POPULATION DYNAMICS WITH FIELD DATA

Applying the recent developments of scale transition theory, we demonstrate a systematic approach to the problem of scaling up local scale interactions to regional scale dynamics with field data. Dynamics on larger spatial scales differ from the predictions of local dynamics alone because of an interaction between nonlinearity in population dynamics at the local scale and spatial variation in density and environmental factors over the regional population. Our systematic approach to scaling up involves the following five steps. First, define a model for dynamics on the local spatial scale. Second, apply scale transition theory to identify key interactions between nonlinearity and spatial variation that translate local dynamics to the regional scale. Third, measure local-scale model parameters to determine nonlinearities at local scales. Fourth, measure spatial variation. Finally, combine nonlinearity and variation measures to obtain the scale transition. Using field data for the dynamics of grazers and periphyton in a freshwater stream, we show that scale transition terms greatly reduce the growth and equilibrium density of the periphyton population at the stream scale compared to rock scale populations, confirming the importance of spatial mechanisms to stream-scale dynamics.

Large-scale questions and small-scale data: empirical and theoretical methods for scaling up in ecology

Most experimental work is done at small spatial scales due to scientific, logistical, and financial constraints. Unfortunately, ecologists (and society) require answers to problems about dynamics that arise at larger scales than often can be studied experimentally. Thus, it is imperative to develop empirical and theoretical methods for scaling up the results of small-scale studies to predictions at larger scales. These methods should also identify the limits to scaling up: i.e., clarify when results from small-scale experiments cannot be extrapolated to larger scales. In this special topic, we identify three important hurdles associated with scaling up in population biology and methods of addressing these problems: (1) increased spatial heterogeneity with increasing spatial scale, (2) changes to species pools and species identities with changes in spatial scale, and (3) behaviors and trait-mediated indirect effects that emerge at larger scales, but are absent in small, relatively homogeneous experimental settings. Larger spatial scales will usually incorporate greater abiotic and biotic heterogeneity than small scales, including greater genetic or phenotypic variation among individuals. Simple extrapolations that assume linear scaling relationships will not provide good predictions when there are both spatial heterogeneities and nonlinear interactions among species or between species and abiotic resources. Thus it may be troubling that spatial heterogeneity is ubiquitous and many ecological interactions are known to be non-linear. Fortunately, if both the degree of nonlinearity and the amount of spatial heterogeneity at different scales were quantified, available methods may allow integrating large-scale observational studies with small-scale experiments to account for the influence of heterogeneities on the outcome of nonlinear interactions. Alternatively, models or experiments can be used to directly examine the effects of changing levels of variance and to determine the importance of different sources of heterogeneity in different systems. Three of the contributions illustrate ways to account for effects of increasing heterogeneity and non-linearities. Melbourne and Chesson use scale-transition theory to develop an expression for regional population dynamics that accounts for non-linear local density dependence and spatial heterogeneity in resources. Incorporating spatial variances and covariances, they predict the regional responses of caddisfly populations to riffle disturbance regimes. Inouye describes data collected at two nested spatial scales on the variances and covariances of distributions of competitors in a patchy and ephemeral habitat. These data are used to parameterize a model for the regional population dynamics of these competitors that includes spatial heterogeneity at both spatial scales. Helms and Hunter show that using observed spatial variation in local population growth parameters can produce regional population dynamics that are different from those obtained using average parameter values. A second hurdle in scaling up is that at larger scales, one is likely to encounter additional species or habitats within the domain of the study. In moving from studying a single species or pair of species interacting within a microcosm to studying the same species and interaction embedded in a landscape, the indirect effects of additional species may swamp whatever direct effects were measured in the microcosm. To address these problems, we need to include information on how species pools change as the spatial scales of studies increases. Srivastava addresses the relative roles of local biotic interactions and regional abiotic influences in aquatic treehole communities across a range of spatial scales and levels of diversity. Although local treehole communities are similarly structured around the world, communities Communicated by Craig W. Osenberg

Scaling up population dynamics: integrating theory and data

How to scale up from local-scale interactions to regional-scale dynamics is a critical issue in field ecology. We show how to implement a systematic approach to the problem of scaling up, using scale transition theory. Scale transition theory shows that dynamics on larger spatial scales differ from predictions based on the local dynamics alone because of an interaction between local-scale nonlinear dynamics and spatial variation in density or the environment. Based on this theory, a systematic approach to scaling up has four steps: (1) derive a model to translate the effects of local dynamics to the regional scale, and to identify key interactions between nonlinearity and spatial variation, (2) measure local-scale model parameters to determine nonlinearities at local scales, (3) measure spatial variation, and (4) combine nonlinearity and variation measures to obtain the scale transition. We illustrate the approach, with an example from benthic stream ecology of caddisflies living in riffles. By sampling from a simulated system, we show how collecting the appropriate data at local (riffle) scales to measure nonlinearities, combined with measures of spatial variation, leads to the correct inference for dynamics at the larger scale of the stream. The approach provides a way to investigate the mechanisms and consequences of changes in population dynamics with spatial scale using a relatively small amount of field data.

A scaling theory of the collapse transition in geometric cluster models of polymers and vesicles

Much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk, polygon and animal models. The crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2- alpha =1/ phi with the alpha the canonical (thermodynamic) exponent that characterizes the divergence of the specific heat. The reasons for the validity of this relation are not widely known. The authors present a scaling theory of collapse transitions in such models. The free energy and canonical partition functions have finite-length scaling forms whilst the grand partition function has a tricritical scaling form. The link between the grand and canonical ensembles leads to the above scaling relation. They then comment on the validity of current estimates of the crossover exponent for interacting self-avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values.