Early-Warning Signs for Pattern-Formation in Stochastic Partial Differential Equations

There have been significant recent advances in our understanding of the potential use and limitations of early-warning signs for predicting drastic changes, so called critical transitions or tipping points, in dynamical systems. A focus of mathematical modeling and analysis has been on stochastic ordinary differential equations, where generic statistical early-warning signs can be identified near bifurcation-induced tipping points. In this paper, we outline some basic steps to extend this theory to stochastic partial differential equations with a focus on analytically characterizing basic scaling laws for linear SPDEs and comparing the results to numerical simulations of fully nonlinear problems. In particular, we study stochastic versions of the Swift-Hohenberg and Ginzburg-Landau equations. We derive a scaling law of the covariance operator in a regime where linearization is expected to be a good approximation for the local fluctuations around deterministic steady states. We compare these results to direct numerical simulation, and study the influence of noise level, noise color, distance to bifurcation and domain size on early-warning signs.Comment: Published in Communications in Nonlinear Science and Numerical Simulation (2014

All in the family: A descriptive analysis of family network change in families managing chronic illness

BackgroundSocial networks can be a source of support, including informational support, in chronic illness management, but can change over time in response to health crises. However, little is known about how families leverage their support networks to manage chronic illness over time—and how these networks may change.MethodsFor 28 families managing either diabetes or HIV, we gathered survey‐based social network data, including network size, exchanges of support, and tie strength, up to 5 times over 2 years. We then used descriptive analysis to examine changes in network size, structure and function (e.g., informational support).FindingsAlthough family networks remained stable in terms of network size and transitivity, these networks experienced regular fluctuations in both tie‐level variables (i.e., tie strength) and density of informational support exchanges.DiscussionObserved changes in these measures indicate that even family support networks are susceptible to change over time, particularly at the tie‐level, indicating a need to expand the way we think of network change beyond whole network measures when looking at small, family networks, especially examining how information exchanges fluctuate over time. Future research should explore tie‐level measures and support exchange networks to understand why networks change over time.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/163486/2/pra2247.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/163486/1/pra2247_am.pd

Coupled catastrophes: sudden shifts cascade and hop among interdependent systems

An important challenge in several disciplines is to understand how sudden changes can propagate among coupled systems. Examples include the synchronization of business cycles, population collapse in patchy ecosystems, markets shifting to a new technology platform, collapses in prices and in confidence in financial markets, and protests erupting in multiple countries. A number of mathematical models of these phenomena have multiple equilibria separated by saddle-node bifurcations. We study this behavior in its normal form as fast--slow ordinary differential equations. In our model, a system consists of multiple subsystems, such as countries in the global economy or patches of an ecosystem. Each subsystem is described by a scalar quantity, such as economic output or population, that undergoes sudden changes via saddle-node bifurcations. The subsystems are coupled via their scalar quantity (e.g., trade couples economic output; diffusion couples populations); that coupling moves the locations of their bifurcations. The model demonstrates two ways in which sudden changes can propagate: they can cascade (one causing the next), or they can hop over subsystems. The latter is absent from classic models of cascades. For an application, we study the Arab Spring protests. After connecting the model to sociological theories that have bistability, we use socioeconomic data to estimate relative proximities to tipping points and Facebook data to estimate couplings among countries. We find that although protests tend to spread locally, they also seem to "hop" over countries, like in the stylized model; this result highlights a new class of temporal motifs in longitudinal network datasets.Comment: 20 pages, 4 figures, plus a 6-page supplementary material that contains 5 figures. Accepted at Journal of the Royal Society Interfac

A Scale-Free, Relational Approach to Social Development in Late-Prehistoric Tyrrhenian Central Italy.

This dissertation examines the interplay between relational patterns and trajectories of social development within late-prehistoric Tyrrhenian Central Italy (TCI). Despite historical disagreement, TCI is now recognized as a context of radical social transformations leading to or even encompassing the formation of states and cities. Between the end of the Bronze Age and the Early Iron Age (12th-8th centuries BCE), in concert with substantial changes attested in the funerary record, hundreds of small settlements distributed evenly across the research area were replaced by a few large and formally heterogeneous centers. Based on a narrow evidentiary range, these developments tend to be viewed in terms of differential progression along a single axis of complexity, with the emerging centers categorized according to apparent scale and historical prominence. The aim of this work is not to establish TCI as a case of state formation and urbanization according to one or another definition. Following anthropological perspectives that emphasize the importance of parsing the dynamics underlying qualitative changes in the form of social collectives, I focus on tracing developmental trajectories within a systematically validated relational plane as a first step toward identifying better bases for comparison, categorization, and ultimately explanation. In practice, I derive large datasets for network reconstruction from the TCI mortuary record and apply tools developed for the study of complex networks to track the position of social collectives over time and extract summary measures of relational distance. I use these measures to evaluate three aspects of the transformations attested between the end of the Bronze Age and the Early Iron Age: 1) the degree of local and regional coordination involved in the shift from a large number of small villages to relatively few large centers; 2) the impact of shifting axes of long-distance exchange on local relational patterns; and 3) the relationship between the formal characteristics and internal substance of the new settlements. I conclude that the watershed changes attested across the research area can be understood in reference to uniform endogenous processes leading to the emergence of comparable social collectives.PhDIndependent Interdepartmental Degree ProgramUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133403/1/cangemi_1.pd

Chapter 1.2: Tipping points in the cryosphere

Drastic changes in our planet’s frozen landscapes have occurred over recent decades, from Arctic sea ice decline and thawing of permafrost soils to polar amplification, the retreat of glaciers and ice loss from the ice sheets. In this chapter, we assess multiple lines of evidence for tipping points in the cryosphere – encompassing the ice sheets on Greenland and Antarctica, sea ice, mountain glaciers and permafrost – based on recent observations, palaeorecords, numerical modelling and theoretical understanding. With about 1.2°C of global warming compared to pre-industrial levels, we are getting dangerously close to the temperature thresholds of some major tipping points for the ice sheets of Greenland and West Antarctica. Crossing these would lock in unavoidable long-term global sea level rise of up to 10 metres. There is evidence for localised and regional tipping points for glaciers and permafrost and, while evidence for global-scale tipping dynamics in sea ice, glaciers and permafrost is limited, their decline will continue with unabated global warming. Because of the long response times of these systems, some impacts of crossing potential tipping points will unfold over centuries to millennia. However, with the current trajectory of greenhouse gas (GHG) emissions and subsequent anthropogenic climate change, such largely irreversible changes might already have been triggered. These will cause far-reaching impacts for ecosystems and humans alike, threatening the livelihoods of millions of people, and will become more severe the further global warming progresses. The scientific content of this chapter is based on the following manuscript in preparation: Winkelmann et al., (in prep