Catastrophic Thresholds: Perspectives, Definitions, and Applications

Authors


Introduction

A symposium organized and chaired by Robert A. Washington-Allen and Lucinda F. Salo for the 2006 ESA Annual Meeting at Memphis, Tennessee: “Icons and Upstarts in Ecology,” with support from the Rangeland Ecology Section of ESA.

The concept of threshold behavior of ecosystem variables and parameters in space and time received early theoretical treatment in the 1970s by the “Upstart” C. S. Holling in a landmark paper on ecological resilience (Holling 1973). Recent treatments have been eerily mirrored in recent bestsellers, Connie Willis's science fiction novel Bellwether (Willis 1998) and Malcom Gladwell's Tipping Point: How Little Things Can Make a Big Difference (Gladwell 2002), and concepts presented by Peters et al. (2004) in the paper, “Cross-scale interactions, nonlinearities, and forecasting catastrophic events.” In Willis's Bellwether the protagonist investigates the origin, spread, and end of fads, such as pet rocks, mood rings, and miniature golf. Gladwell's Tipping Point discusses the mechanisms involved in how new ideas, warnings (e.g., Paul Revere's “the British are Coming!!”), disease (HIVAIDS), and fads (Hush Puppies) are rapidly spread across society. Similarly, Peters et al. (2004) analyzed how natural phenomena such as disease, wildfire, erosion, and major dust storms spread across the environment. These publications, despite their disparate audiences, commonly discuss thresholds or tipping points and conceptual models that can be used to better understand these phenomena.

A threshold can be defined by such synonyms as a border, a regime shift, an ecotone, a discontinuity, a phase transition, a point of criticality, and the tipping point between two or more dynamic regimes or states (Mayer and Rietkerk 2004). Adjectives that have been used with “threshold” include transition (Archer 1989), critical, and catastrophic (Scheffer et al. 2001). It is a phenomenon that can be observed in both time and space, but whose observation depends on scale and criteria, e.g., ecosystem or landscape (Beisner et al. 2003).

Thresholds can be seen in settings as diverse as artwork, the economy, and in social movements/society. Bi-stability and thresholds are illustrated in M. C. Escher 's 1956 wood engraving “Swans,” where flying black swans rapidly change into white swans in a chiral Möbius strip 〈http://www.mcescher.com/〉. The musician and ecologist Marten Scheffer, author of the forthcoming book Stability: the Mechanisms of Inertia, Chaos and Collapse (Scheffer 2007) showed bi-stability in the economy with the 1929, 19 October 1987, and the 1997 stock market crashes. With reference to social systems, this ESA conference was held near the site where the “Upstart” and now “Icon” the Reverend Dr. Martin Luther King, Jr. was assassinated. His accomplishments and death were major tipping points in social justice and equality for all Americans. (Also see Gunderson and Holling [2002] and Turchin [2003] and Turchin [2006] for quantitative treatment of social and ecological phenomena in general.) Further examples of threshold behavior can be found in the Thresholds Bibliography that is archived by the Resilience Alliance (RA), a multidisciplinary research group that explores the dynamics of complex adaptive systems 〈http://www.resalliance.org〉.

Symposium talks

The speakers in this symposium consider thresholds in space and time in both terrestrial and aquatic ecosystems (Table 1). For ecological systems, R. Washington-Allen discussed thresholds in rangelands, and illustrated these with examples of changes in landscape-scale vegetation production, structure, and pattern in relation to wet and dry climatic periods, and livestock and wildlife grazing management (Washington-Allen et al. 2006, Zimmermann et al. 2006). This research used power law analysis of vegetation patch dynamics (Olsen et al. 2005) and timeseries analysis (the autocorrelation function) (Perry et at. 2000) to quantitatively detect shifts in dynamic regimes (states).

Table 1. Speakers in the symposium “The Detection of Catastrophic Thresholds: Perspectives, Definitions, and Methods” (ESA Annual Meeting, August 2006), titles of their talks, and the methods they used to detect states and thresholds.Thumbnail image of

M. Scheffer showed similar evidence for bi-stability in Saharan vegetation cover, the 1980s collapse of Caribbean coral reefs, North Sea plankton and benthos, State collapses (Sudan), and shallow aquatic systems. He went on to show other dramatic events (dinosaur extinction), but questioned whether these events really presented evidence of regimes shifts? Scheffer suggested that observing positive feedback (climate–vegetation feedbacks) is a necessary but not sufficient condition. He suggested observing system behavior using time series analysis for a sudden jump in time, multimodality, dual relationships (use of two functions instead of a single regression), divergence, triggered shifts (but this may be transient), and hysteresis (Scheffer and Carpenter 2003). These behaviors may also be necessary but not sufficient. He also suggested the use of early-warning signals for detection such as increased variance near a threshold (O'Neill et al. 1989), a “red shift” (spectral analysis), or a “critical slowdown” (broadening correlations near the threshold), but in terms of time and space, none of these patterns may be conclusive. At the end of the day there is a need for a combination of models and laboratory and field experiments to get at the whole picture.

Three speakers examined scaling laws, length scales, and increased variance using models and field data collection. C. Johnson and his colleagues are concerned with the definition of characteristic or natural length scales (CLS). Determination of CLS allows (1) the definition of the optimum scale for defining independent or complex interactions between different species, (2) monitoring, and (3) interpreting change in ecological dynamics. Determination of CLS is accomplished by measuring the changes in the magnitude of the prediction error as the scale of observation of species abundance (e.g., percent cover) changes, using two error estimates of variance spectra called the error X (a scaled error variance) and the prediction r2. Innovation is introduced by showing that the use of space-for-time substitution within a limited time series of data, rather than a more onerous time-series analysis (which requires thousands of realizations) produces the same results (Habeeb et al. 2005; C. Johnson et al., unpublished manuscript).

C. Allen observed that variance of a response variable increased near a threshold. He has observed that discontinuous structures at different spatial scales may result in discontinuities between aggregated species' body mass patterns, reflecting the scales of structure available to animal communities within a landscape. This has led to a textural discontinuity hypothesis, where it is predicted that variability in population abundance is greater in animal species near the edge of body mass aggregations than it is in species in the interior of body mass aggregations. Allen and his colleagues globally examined spatial and temporal data on fauna population abundances and confirmed this relationship. They further examined this relationship for bird populations in the Everglades, and again confirmed the relationship. This may be a predictor of invasions and nomadism.

B. Milne used power law analysis on previous studies of U.S. basins to observe that particular input and output parameters exhibited 4/7 scaling. Consider that a basin, whose length scale (L) is proportional to its Area (A) to its fractal dimension (D) divided by 2 power or L ∞ AD/2 receives solar irradiance and precipitation (PPT) as inputs. PPT is lost as runoff (Q), evaporation (E), and transpiration (T) from plants; thus the basin can be considered an exchange surface (A). Consider that renormalization (increasing pixel size, thus coarsening resolution) of the Columbia River Basin and other energy-limited areas scale as the 4/7 power of cell length, that actual ET varies with PPT to the 4/7 power in U.S. basins, and that plant cover is proportional to L to the 7/4 power. Consider that gradient percolation, an erosive force gradient from land to shore to sea, produced −4/7 scaling, suggesting a universal statistical signature in the geometry of the terrain. Can this scaling be seen in basin boundaries? B. Milne tested 29 basins and found that they supported the hypothesis of 4/7 scaling. Thus basin geometry will affect interception of solar radiation that powers evaporation and transpiration throughout the basin, or the exchange surface (A) is approximately equal to PPT × ET and to A1+4/7 = (L2)11/7 = Lπ. So where do the 4/7 and the π scaling constraints that produced this nonlinear coupling come from? The 4/7 scaling is from the gradient, and π is from the cross-sectional area of sunbeams (which are usually circular), i.e., the solar irradiance (energy for transpiration and evaporation) input as scattered hemispherical radiance fills the basin within the fractal perimeter, subject to π. This is true because the fractal dimension of the basin perimeter evolves to satisfy the equality entailed by the solar disk. B. Milne concluded that ecohydrological theory is based on a foundation of coupled fluxes of water and energy. Universal to this foundation are basin boundaries (formed by erosion) that exhibit 4/7 scaling and a fractal geometrical limit on the interception of sunbeams. Thus the sustainability of ecosystem services of food, fiber, and water can be assessed from the viewpoint of scaling theory.

Two speakers examined the application of mathematical models to field data. The Lockwoods and D. Thomas are interested in the dynamics of grasshopper and mosquito population outbreaks, respectively, and the development of nonlinear tools to provide explanation and forecasting for management strategies. The Lockwoods have used both catastrophe theory (CT) and self-organized criticality (SOC) to examine outbreaks, and Thomas has used time-scale calculus (TSC, Hilger 1990). Catastrophe theory is a mathematical framework from topology that can be used to explain both discontinuous and continuous behavior in systems (Thom 1972, Gilmore 1981). Self-organized criticality is the emergence of complex behavior from local interaction between neighbors in a system that is self-reinforcing or self-tuning, i.e., positive feedback at the local scale maintains a poised state between order and disorder. Self-organized criticality can be detected by examination of scale-invariance using either power law analysis or 1/f noise (Bak et al. 1987). Regime shifts are detected by lack of scale-invariance in a previously self-organized system (Olsen et al. 2005). The Lockwoods fit the cusp catastrophe geometry to the state variable (grasshopper population size) and two driving parameters (bimonthly temperature and precipitation) in a linear regression manner using the software GEMCAT II (Olivia et al. 1987 〈http://astro.temple.edu/~oliva/cat-theory.htm〉). They were able to temporally and spatially predict grasshopper outbreaks over a 28-year period of reconstruction. They also found self-similarity by fractal analysis and 1/f noise in the frequency distribution of areas infested by grasshoppers or scale-invariance, a feature diagnostic of a self-organized criticality (SOC). They also found, to their surprise, that CT cusp bifurcation can produce SOC dynamics. Consider that this is convergence between two different mathematical models where CT emphasizes strong exogenous controls and SOC strong endogenous processes. This brings into question whether mosquito population outbreaks and crashes are internally or externally controlled? It may be reasonable to consider that due to processes that drive evolution, the control parameters are not exogenous if they exert selective pressure on the state variables (adaptation being a feedback), as appears to be the case for El Niños (Gibbs and Grant 1987).

D. Thomas developed a simple time-scale calculus (TSC) model of mosquito, birds, and human population dynamics to successfully direct the pesticide-spraying plan for New York City to battle the transmission of West Nile encephalitis (Thomas and Urena 2001). In the spring, mosquitoes contract West Nile virus from migrant birds, transmit it to their offspring, and infect both humans and birds throughout the summer. Infected humans and birds become sick, die, or generally recover from the infection. Infected birds migrate to South America, and overwintering mosquito larvae incubate the virus in North America. These transmission dynamics are a time-varying process that presents a special mathematical challenge where both discrete (seasonal breaks or sudden jumps) and continuous (day-by-day) changes occur. Generally, discrete events are modeled using difference equations, and continuous behavior is modeled using differential equations (calculus). However, time-scale calculus is a discovery of the relationship between difference and differential equations, and thus like catastrophe theory can be used to model systems that exhibit both continuous and discrete behavior (Hilger 1990, Bohner and Peterson 2001).

T. Stringham applies the concepts of multiple stable states and regime shifts to the real-world challenge of describing vegetation dynamics on rangelands. Previously, the ecological basis of land management models assumed linear changes in plant community composition that were primarily endogenously driven. However, federal land management agencies, including the USDA's Forest Service and NRCS, and the U.S. Department of the Interior's BLM, have recently adopted the nonlinear States and Transition model that assumes states are exogenously driven by ecosystem processes. These agencies are currently preparing ecological site or dynamic regime descriptions that illustrate homogenous vegetation and soil dynamics at local management scales in the western United States. T. Stringham defines a “state” as a resilient complex of soil and vegetation that is connected by ecological processes such as hydrology. Within a state, plant community dynamics called “phases” occur. A “community pathway” is a mechanism that results in change from one phase to another. Similarly a “stressor” is a mechanism that degrades the soil–vegetation complex's ecological processes enough to result in a “transition” or change from one state to another. Sites that resist change exhibit resilience. Stringham noted that the ecological site descriptions are being developed by interview with expert stakeholders who aid in identifying reference or control ecological sites. Plot-level vegetation and soil data are collected, classified into states by cluster analysis, and inferential statistics are used to detect significant change in ecosystem processes relative to controls; this is indicative of a threshold being crossed.

Conclusion

David Briske tied the symposium together by synthesizing the concepts presented that further contributed to our growing knowledge of complex systems in ecology. The talks used nonlinear concepts such as dynamic regimes, soil–vegetation complex dynamics, scaling laws, catastrophe theory, self-organization, time-scale calculus, and cross-scale interactions, i.e., the textural discontinuity hypothesis, provided insights on constraints on basin geometry, characteristic scale length of communities, management of disease and insect outbreaks, body mass aggregations in discontinuous landscapes, and rangeland monitoring and assessment.

This symposium developed protocols for detecting states and thresholds, including the use of catastrophe theory, self-organized criticality, and time scale calculus, analysis of the increase in variance (including predicted r2 and error-X variance spectra analysis), red shifts, and critical slowdown time series analyses, multivariate and inferential statistical analyses, and power law analyses. Speakers defined characteristics of nonlinear ecology and described applications for natural resource management that have grown out of nonlinear concepts. These include state and transitions models, which grew out of alternative stable states, the concept of thresholds, which developed from regime shifts, and the idea of rangeland health, which is a practical application of ecosystem self-organization. These applications have helped natural resource managers and other decision-makers deal with “real world” problems of rangeland stewardship, management of West Nile virus, and control of grasshopper infestations.

Not surprisingly, this gathering raised even more questions. How do we proceed from here? The time scale calculus study has moved toward an increase in model complexity to account for both temperature and continued seasonal dynamics. In general, processes contributing to the observed dynamics of insect outbreaks and rangeland vegetation dynamics remain inconclusive. For example: What is the relationship of self-organized criticality to catastrophe theory? What is the relationship of time scale calculus to catastrophe theory and the cyclic regime shifts typical of insect outbreaks? Next steps must include more fully integrating concepts of nonlinear ecology, such as thresholds, into mainstream ecology and developing a more comprehensive nonlinear framework that recognizes the importance of scale in ecological systems.

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