Cross-scale interactions, nonlinearities, and forecasting catastrophic events.
ABSTRACT Catastrophic events share characteristic nonlinear behaviors that are often generated by cross-scale interactions and feedbacks among system elements. These events result in surprises that cannot easily be predicted based on information obtained at a single scale. Progress on catastrophic events has focused on one of the following two areas: nonlinear dynamics through time without an explicit consideration of spatial connectivity [Holling, C. S. (1992) Ecol. Monogr. 62, 447-502] or spatial connectivity and the spread of contagious processes without a consideration of cross-scale interactions and feedbacks [Zeng, N., Neeling, J. D., Lau, L. M. & Tucker, C. J. (1999) Science 286, 1537-1540]. These approaches rarely have ventured beyond traditional disciplinary boundaries. We provide an interdisciplinary, conceptual, and general mathematical framework for understanding and forecasting nonlinear dynamics through time and across space. We illustrate the generality and usefulness of our approach by using new data and recasting published data from ecology (wildfires and desertification), epidemiology (infectious diseases), and engineering (structural failures). We show that decisions that minimize the likelihood of catastrophic events must be based on cross-scale interactions, and such decisions will often be counterintuitive. Given the continuing challenges associated with global change, approaches that cross disciplinary boundaries to include interactions and feedbacks at multiple scales are needed to increase our ability to predict catastrophic events and develop strategies for minimizing their occurrence and impacts. Our framework is an important step in developing predictive tools and designing experiments to examine cross-scale interactions.
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ABSTRACT: Fire seasonality, an important characteristic of fire regimes, commonly is delineated using seasons based on single weather variables (rainfall or temperature). We used nonparametric cluster analyses of a 17-year (1993-2009) data set of weather variables that influence likelihoods and spread of fires (relative humidity, air temperature, solar radiation, wind speed, soil moisture) to explore seasonality of fire in pine savanna-grassland landscapes at the Avon Park Air Force Range in southern Florida. A four-variable, three-season model explained more variation within fire weather variables than models with more seasons. The three-season model also delineated intra-annual timing of fire more accurately than a conventional rainfall-based two-season model. Two seasons coincided roughly with dry and wet seasons based on rainfall. The third season, which we labeled the fire season, occurred between dry and wet seasons and was characterized by fire-promoting conditions present annually: drought, intense solar radiation, low humidity, and warm air temperatures. Fine fuels consisting of variable combinations of pyrogenic pine needles, abundant C4 grasses, and flammable shrubs, coupled with low soil moisture, and lightning ignitions early in the fire season facilitate natural landscape-scale wildfires that burn uplands and across wetlands. We related our three season model to fires with different ignition sources (lightning, military missions, and prescribed fires) over a 13-year period with fire records (1997-2009). Largest wildfires originate from lightning and military ignitions that occur within the early fire season substantially prior to the peak of lightning strikes in the wet season. Prescribed ignitions, in contrast, largely occur outside the fire season. Our delineation of a pronounced fire season provides insight into the extent to which different human-derived fire regimes mimic lightning fire regimes. Delineation of a fire season associated with timing of natural lightning ignitions should be useful as a basis for ecological fire management of humid savanna-grassland landscapes worldwide.PLoS ONE 01/2015; 10(1):e0116952. · 3.53 Impact Factor
Cross-scale interactions, nonlinearities, and
forecasting catastrophic events
Debra P. C. Peters*†, Roger A. Pielke, Sr.‡, Brandon T. Bestelmeyer*, Craig D. Allen§, Stuart Munson-McGee¶,
and Kris M. Havstad*
*U.S. Department of Agriculture Agricultural Research Service, Jornada Experimental Range, Las Cruces, NM 88003;‡Department of Atmospheric Sciences,
Colorado State University, Fort Collins, CO 80523;§U.S. Geological Survey, Fort Collins Ecological Science Center, Jemez Mountains Field Station, Los Alamos,
NM 87544; and¶Department of Chemical Engineering, New Mexico State University, Las Cruces, NM 88003
Edited by Harold A. Mooney, Stanford University, Stanford, CA, and approved September 9, 2004 (received for review May 28, 2004)
Catastrophic events share characteristic nonlinear behaviors that
are often generated by cross-scale interactions and feedbacks
among system elements. These events result in surprises that
cannot easily be predicted based on information obtained at a
single scale. Progress on catastrophic events has focused on one of
an explicit consideration of spatial connectivity [Holling, C. S.
(1992) Ecol. Monogr. 62, 447–502] or spatial connectivity and the
spread of contagious processes without a consideration of cross-
scale interactions and feedbacks [Zeng, N., Neeling, J. D., Lau, L. M.
& Tucker, C. J. (1999) Science 286, 1537–1540]. These approaches
rarely have ventured beyond traditional disciplinary boundaries.
We provide an interdisciplinary, conceptual, and general mathe-
matical framework for understanding and forecasting nonlinear
dynamics through time and across space. We illustrate the gener-
ality and usefulness of our approach by using new data and
recasting published data from ecology (wildfires and desertifica-
tion), epidemiology (infectious diseases), and engineering (struc-
tural failures). We show that decisions that minimize the likelihood
of catastrophic events must be based on cross-scale interactions,
and such decisions will often be counterintuitive. Given the con-
tinuing challenges associated with global change, approaches that
cross disciplinary boundaries to include interactions and feedbacks
at multiple scales are needed to increase our ability to predict
catastrophic events and develop strategies for minimizing their
occurrence and impacts. Our framework is an important step in
developing predictive tools and designing experiments to examine
ical, physical, and materials systems (1–3). These spatial non-
linearities and emergent behaviors challenge the ability of
scientists and engineers to understand and predict system be-
havior at one scale based on information obtained at finer or
broader scales (3, 4). Cross-scale interactions often result in
‘‘surprises’’ with severe consequences for the environment and
human welfare (5). For example, wildfire initiated by a single
lightning strike can spread nonlinearly across large forested
landscapes as a result of positive feedbacks between weather, fire
behavior, and vegetation pattern, with significant impacts on
ecosystem function, local and regional economies, and human
health (6). Similarly, the devastating impact of a relatively small
piece of foam (?0.3 m2) initiated a series of reactions that
cascaded very rapidly and nonlinearly to result in the break up
of the Columbia space shuttle within minutes after the initial
temperature increase (7).
In this article, we introduce a general framework for under-
that cross scales in space and time (Fig. 1). Our goal is to identify
the conditions leading to catastrophic events to minimize the
impacts of these events on ecosystem services, atmospheric
conditions, and human welfare. The significance of thresholds
and feedbacks is gaining recognition in various disciplines (3, 8,
their associated thresholds are common features of biolog-
9). However, the key to understanding threshold behavior
through time necessitates the incorporation of processes across
spatial scales that cross traditional disciplinary boundaries. For
example, in the U.S. in the 1930s, soil that eroded from farm
lands in the Great Plains resulted in ‘‘black blizzards,’’ which
were large dust clouds that descended on cities that were
hundreds of kilometers away (10). These conditions, constituting
what is known as the ‘‘Dust Bowl,’’ were not forecast based on
local or regional processes alone; rather, crop failures resulting
in unprotected soil had occurred previously at local scales, and
the atmospheric conditions of strong winds, low humidity, and
low precipitation were within the realm of previous experience
both ecological and atmospheric processes and their interactions
and feedbacks across spatial scales that we can explain this
nonlinear spatial amplification of soil erosion. Given the eco-
logical and social effects of erosion processes today in large
portions of Africa, Asia, and the Middle East (13), it is imper-
ative that we develop a framework for confronting nonlinear
We draw on emerging ideas in several disciplines to develop
a theoretical framework as a first step in describing cross-scale,
feedback-driven interactions as general phenomena in the Earth
System. This framework expands on previous approaches that
extrapolate information across spatial scales by including inter-
actions and feedbacks among fine- and broad-scale processes,
with an emphasis on connectivity among fine-scale units. We
illustrate our framework with diverse biological examples to
show its generality, and we then briefly present parallels with
known, we synthesize the information and data in a way that
focuses on dynamics both through time and across space. We
then show how our framework provides insights into the con-
ditions under which fine-scale processes propagate nonlinearly
to have broad-scale impacts and, conversely, the conditions at
which broad-scale drivers overwhelm fine-scale processes.
Cross-Scale Interactions and Feedbacks
Spatial nonlinearities describe the propagation of changes in
states from Y1to Y2through time across an area of fixed extent.
The amount or proportion of the total area that exists in state
Y changes nonlinearly through time and space as a result of
interactions among patterns and processes with different
characteristic spatial scales (Fig. 1). These interactions result
in qualitatively different kinds and rates of change than were
involved in the initial interaction (14). The propagation of
This paper was submitted directly (Track II) to the PNAS office.
Freely available online through the PNAS open access option.
Abbreviation: ha, hectare.
†To whom correspondence should be addressed at: U.S. Department of Agriculture Agri-
State University, Las Cruces, NM 88003-0003. E-mail: email@example.com.
© 2004 by The National Academy of Sciences of the USA
October 19, 2004 ?
vol. 101 ?
no. 42 www.pnas.org?cgi?doi?10.1073?pnas.0403822101
fine-scale changes to broader spatial scales can either be rapid
(responses are amplified) or slow (responses are buffered).
Rates depend on the spatial configuration, connectivity, and
flows within and among fine-scale units, the interaction of
these patterns with broad-scale forcing functions, and feed-
backs among these elements across scales. In our framework,
there are four major stages in the change in Y that are
characterized by different processes and distinguished by one
of three thresholds (Fig. 1). Other forms of nonlinear expan-
sion that do not include thresholds and feedbacks can be
explained with simpler formulations.
The points in time at which the rate of a process and the
resultant changes in Y accelerate or decelerate discontinuously
are thresholds. The thresholds indicate that distinct exogenous
processes or endogenous positive feedbacks are governing rates
of change (domains of scale, cf. ref. 15). The concept of spatial
nonlinearity complements antecedent concepts about the rate
connectivity within a specified extent to forcing functions and
feedbacks emerging from pattern at broader extents.
Conceptually, spatial nonlinearities can be illustrated mathe-
matically as follows: dY?dt ? g(Ig, Eg) ? f(Y, Ef) ? D(Y, ED) ?
c(Y, Ec), where each term of the equation represents one of the
four stages and refers to different processes that are important
at different values of Y. g(Ig, Eg) refers to a process that initiates
the presence of Y within the extent of interest and depends on
internal (Ig) and external (Eg) factors (e.g., weather); f(Y,Ef)
refers to changes within a locally homogeneous isotropic pop-
ulation or patch of Y as a function of both the current state of
Y and external factors (Ef). A simplified version of this term,
when combined with the dispersion term below, has been used
spread of gene frequency (16). The additional factor accounts
for variation in connectivity among units of Y that does not
depend on the amount or aggregation of Y. D(Y, ED) refers to
processes emerging from connections among the isotropic pop-
ulations or patches described by each f(Y, Ef) when acted on by
external variables ED. The importance of template heterogeneity
(ED) has been addressed in neutral, percolation-based landscape
models to predict when landscapes become fragmented (17).
Various formulations of this dispersion term have been devel-
oped for different applications, such as in describing spread of
invasive species across homogeneous landscapes and the spread
assume that dispersion depends only on D and the amount of Y
(for exceptions, see refs. 19 and 20). The final term, c(Y, Ec),
refers to broad-scale processes or forcing functions (Ec) that
influence or interact with Y to generate positive feedbacks to the
dynamics of Y (Fig. 1).
There are three critical features of our framework that are
driver of dynamics at intermediate scales that propagate change
by means of novel effects to broader scales and, in turn, feedback
to affect fine-scale dynamics. It is the failure to account for the
effects of connectivity, thresholds, and feedbacks on the prop-
agation or spread of Y that often results in surprises. Second, the
relative importance of each term in the equation depends on
both the spatial arrangement and amount of Y that change
through time and space. As the rate of change in Y increases
through time, it is increasingly governed by terms toward the
right because the amount, connectivity, and spatial extent of Y
increases. Thus, not all terms need to be included for every
application. Furthermore, the errors in commission that are
associated with including nonsignificant terms argue for includ-
ing the optimum number of terms required to capture the key
processes that determine dynamics (21). Third, nonlinear in-
creases in Y result in two major kinds of threshold values for
many systems (and also a third kind in some cases). In addition,
all terms except g depend, at least in part, on properties of Y, thus
providing the potential for feedbacks to occur.
View of Old Problems Through the Lens of Our Framework
We illustrate our framework by using examples that represent
different subdisciplines of biology and engineering. Each exam-
ple has fine-scale dynamics that propagate nonlinearly to broad-
scale dynamics with regional to global consequences.
Wildfires. Wildfires have been studied extensively at specific
spatial scales at which it is recognized that short- and long-term
weather conditions interact with the amount, moisture content,
and spatial distribution of fuels to affect fire extent, rate of
spread, and severity (22, 23). However, the factors that deter-
up’’ and create catastrophic conditions have not been quantified.
Thus, the explosive spread of wildfires across landscapes creates
‘‘ecological surprises’’ that are not easily forecast based either on
fine-scale fire behavior or broad-scale atmospheric conditions.
Our cross-scale framework may be a particularly useful tool in
explaining complex fire dynamics.
We illustrate our framework by using data from two recent
fires in Colorado with similar behavior yet different spatial
extents. Similar spatiotemporal processes and patterns in fire
spread have been documented for other major fires. Four major
stages of wildfire behavior are associated with the three thresh-
olds in our general framework (Fig. 2). The initiation of a fire or
several spot fires (stage 1) has a probability of spread that ranges
from 0 (fire goes out) to 1.0 (fire spreads). If within-patch fuel
load and connectivity are sufficient, then the fire crosses a
threshold (T1) and spreads (stage 2). The rate and extent of fire
spread from one tree to additional trees within a patch depends
on local processes, such as the leaf distribution and chemistry of
each tree, fuel amount and connectivity within the patch, and the
local weather conditions (24). As a fire increases in extent, it
burns from one patch to another at varying rates (stage 3). Fire
has a low probability of spreading beyond patches that are poorly
connected to other patches by low fuel amounts, thus restricting
time: the four stages (stages 1–4, shown from left to right) in the change in Y
through time and across space, the processes influencing each stage, and the
three thresholds (T1–T3) in dynamics that occur between stages. Stage 1,
initiation of an event: occurrence, timing, and location are often stochastic
processes that cannot be predicted based on initial conditions of Y. Stage 2,
within-patch expansion: changes in Y as a function of both its current state
and external factors that includes all local factors. Stage 3, spatial spread
among patches: depends on the spatial distribution and connectivity of Y
among patches, and template heterogeneity as well as broad-scale forcing
functions, such as weather. Stage 4, broad-scale changes: depend on forcing
functions that influence or interact with Y to generate positive feedbacks to
the dynamics of Y. These processes do not depend on the connectivity within
or among populations or patches of Y.
General framework of spatial nonlinearities and thresholds through
Peters et al. PNAS ?
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fire activity to the initial patch. However, patches that are highly
connected through the canopy or understory will cross a second
threshold (T2) at which fire spreads among patches. Rate of
spread depends on the amount and spatial distribution of fuel
among patches as well as interactions with local weather con-
ditions. Fire spreads more slowly and less completely in parts of
the landscape with low fuel connectivity than in areas with high
fuel loads and connectivity (25). For both Colorado fires, the
initial spread among patches was slow as low-connectivity sur-
generally continuous fuels were encountered with little variation
in structure and composition. Thus, the fires began to spread
more rapidly among patches.
As a fire continues to increase in extent and intensity, a third
threshold may be crossed (T3) that depends on interactions
between the fire and the atmosphere (stage 4). Fire tends to
generate its own winds as the convective rise of heated gases
leaves low air pressure that draws air into the heart of the fire.
Surface winds are created that drive fire behavior and provide
more oxygen to the burning fire front, thereby accelerating fire
intensity and rate of spread. The resulting positive feedbacks
between increased fire activity and convection-driven wind
circulations can develop rapidly in response to the heat from a
fire, causing highly dangerous blow up of fire behavior, with
preheating of fuels and ‘‘spotting’’ of burning materials ahead of
the flaming front (26). For example, the Hayman fire (Pike–San
Isabel National Forest, Colorado) spread to ?24,282 hectares
(ha) within several hours; pyrocumulus clouds developed to an
estimated 6.4 km in height, and winds gusted to 82 km/hr (23).
When initiated, the strong linkage between fire behavior and fire
weather can overwhelm finer-scale processes such that all parts
of the landscape burn (and often at hotter temperatures) re-
gardless of fuel load or connectivity. During this time period for
the Hayman fire, fuel modifications, including previous pre-
scribed and natural burns as well as thinning of trees, had little
effect on the rate and extent of fire advancement (23).
Thus, a consideration of cross-scale interactions provided by
our framework can improve fire-fighting strategy, increase fire
safety, and increase effectiveness of preemptive treatments.
For example, the Storm King Mountain fire (Glenwood
Springs, CO) resulted in 14 deaths to fire-fighters after a
sudden wind shift that was generated by the fire interacting
nonlinearly with highly connected fuels (27). In this case,
broad-scale fire–atmosphere feedbacks overwhelmed the ef-
fects of fine-scale variation in fuel load and connectivity on fire
spread that formed the basis for fire-fighting strategy. These
catastrophic events can be minimized by recognizing the points
in time when cross-scale interactions lead to the nonlinear
propagation of fire across landscapes.
Desertification. Desertification, or the encroachment by woody
plants into perennial grasslands and associated land degrada-
tion, has occurred throughout arid and semiarid regions of the
world for at least the past several centuries with local to global
consequences (28). Although it is widely recognized that mul-
tiple interacting processes and threshold behavior are involved
(29, 30), we still lack a clear consensus as to how the dominant
processes produce a variety of responses under apparently
similar conditions (31).
We propose that cross-scale linkages among local soil and
grass degradation, landscape connectivity of erosion processes,
and land cover–weather feedbacks can explain desertification
dynamics. Desertification follows the same general patterns
through time and across space as wildfires (Fig. 1), although with
distinct processes (Fig. 4). After introduction of woody plant
seeds into a grass-dominated system (stage 1), local spread often
to a dead tree (Left foreground), a live tree (Right background), or to the herbaceous understory (Right foreground). In some cases, such as lightning strikes to
isolated fuels, the fire cannot spread and burns itself out. Stage 2, in other cases, a threshold (T1) is crossed, and the fire spreads locally to surrounding trees and
herbaceous vegetation within a patch. Stage 3, as the number of individuals on fire increases through time, a second threshold (T2) can be crossed at which time
the rate of spread dramatically changes and fire spreads between patches (foreground). Isolated patches do not burn beyond the initial patch (Right
background). Stage 4, in areas of high connectivity and fuel loads, a third threshold (T3) can be crossed when the number, total area, and energy output of
burning patches of vegetation become sufficiently large for interactions with the atmosphere to become operative with feedbacks to the fire.
Major stages and thresholds in the spread of wildfires. Stage 1, initiation of a wildfire begins with ignition of an individual tree to start a canopy fire
www.pnas.org?cgi?doi?10.1073?pnas.0403822101Peters et al.
occurs as a result of feedback mechanisms between plants and
soil properties interacting with wind and water erosion to
produce fertile plant islands surrounded by bare areas (stage 2).
This rate of spread may be slower than other stages as a result
of interactions between plant life history characteristics that
occur infrequently, such as recruitment, and the low precipita-
tion and high temperatures that characterize dry regions. As the
size and density of woody plants increase through time, conta-
gious processes among patches (primarily wind and water ero-
governing the rate of desertification and the nonlinear increase
in woody plant cover (stage 3). Through time, sufficient land
area can be converted from grasslands (low bare area and low
albedo) to woodlands (high bare area and high albedo) for
regional atmospheric conditions (in particular, wind speed,
temperature, and precipitation) to be affected. At this point,
land–atmosphere interactions with feedbacks to the vegetation
control system dynamics (33), as documented for the Sahara
region of Africa (34).
observations and models based on particular scales (31). For
example, the inability of livestock management to halt or reduce
stage of the system (stage 3) at which contagious erosion
processes dominate vegetation dynamics regardless of the effects
of livestock on plant competition and seed availability.
Infectious Diseases and Insect Outbreaks. This problem remains
critical for ecological and human systems despite aggressive
identification, control, and eradication measures. Complex in-
teractions among the hosts, pathogens carrying the disease
agent, the environmental template, and weather conditions are
involved in the geographic spread of diseases (35, 36). Although
networks that include local clusters and global contacts can
account for some fine- and broad-scale patterns in the spread of
diseases (e.g., ref. 37), another approach is required to account
for catastrophic outbreaks.
The absolute rate and extent of the spread of infectious
diseases are host-, agent-, and event-specific, yet there are
fundamental features that can be captured by our general
framework. Similar to wildfires and desertification, initiation of
a disease (stage 1) is often followed by spread within the family
and community (stage 2) that is primarily a function of popu-
lation density, organism susceptibility, and local clustering (38).
If the disease spreads, then a second threshold (T2) at which the
disease spreads to other populations (stage 3) can be crossed.
Positive and negative feedbacks among hosts may be important
regulators of spread for some diseases (e.g., Lyme disease; ref.
35). For communities with low connectivity among organisms,
the rate of spread can decelerate at stage 3 and the disease
eventually dies out (e.g., smallpox; ref. 39). By contrast, systems
with high connectivity can experience widespread dispersal
among communities, as evidenced by pandemics and the rapid
international transport of severe acute respiratory syndrome
(SARS) among humans (40). In these cases, feedbacks to
broad-scale forcing functions may be insignificant.
However, interactions with broad-scale weather conditions
can be significant for insect outbreaks, such as bark beetle
infestations (41). Weather-induced stress can turn forests into
nearly continuous expanses of vulnerable host trees, triggering
regional outbreaks of bark beetles and associated widespread
forest diebacks, such as observed in the southwestern United
States during the drought of the 1950s and currently (42). Large
areas of dead trees with reduced shade and transpiration lead to
susceptibility to insect attacks and fire. Thus, the dynamics of
pest and disease agents and hosts can be understood only by
considering their spatial patterns (connectivity and density) at
several scales and, in some cases, feedbacks between these
patterns and atmospheric conditions.
Engineering Failures. Engineering failures provide numerous ex-
amples that illustrate how a small, unanticipated change can
result in unexpected consequences over a larger extent, which in
many cases, is accompanied with loss of human life. We focus on
2002, which was the largest fire (?55,800 ha) in modern history of that state
King Mountain in 1994, 856 ha were burned after 74 h (30).
Total area (ha) burned through time (h). (a) For the Hayman fire in
grama grasslands to mesquite shrublands in the Chihuahuan Desert of south-
ern New Mexico (total area, 942 ha). Field surveys (1915 and 1928–1929; ref.
32), black-and-white (1948) and color (1986) infrared photos, as well as
were scanned at 1,200 dots per inch and corrected geometrically to the
ARCGIS was used to obtain area occupied by each class through time. Aerial
photographs from grassland (Left Inset) and shrubland (Right Inset) were
scanned at 1,500 dots per inch and corrected geometrically to the 2003
QuickBird satellite image.
Desertification dynamics are shown by using a transition from black
Peters et al. PNAS ?
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vol. 101 ?
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the collapse of a bridge in Washington State to illustrate how
quantification of cross-scale interactions can allow the identifi-
cation of appropriate management responses.
Before the Tacoma Narrows Bridge collapsed in 1940, it would
‘‘bounce’’ vertically under some wind conditions (www.lib.washing-
ton.edu?specialcoll?tnb?page2.html). Although it is commonly be-
lieved that this vertical motion caused the bridge collapse, this was
not the case; ?45 min before its collapse, the bridge also began a
twisting motion that was initiated after a cable band slipped (stage
1). The motion was an inherently unstable deformation that grew
nonlinearly in magnitude (stages 2 and 3) until stresses in the
supporting cables exceeded their strength and the bridge collapsed
(stage 4). At this final stage, transient external forces (i.e., wind)
failure increased nonlinearly through time.
Although the bridge collapse was surprising at the time, it is
now mathematically understood that these external forces arise
from the aerodynamic flutter of a bluff body that cascade
nonlinearly to result in the collapse of the structure (43). Our
framework explains why structural supports that were added to
the bridge before its collapse (stage 3) were ineffective: they
failed to account for interactions with wind. Preemptive design
measures were needed to minimize the propagation of fine-scale
failures to the spatial extent of the bridge and to account for the
effects of broad-scale drivers. Our current understanding of this
can be used to understand the dynamics of ecological systems.
Insights and Perspectives
Scale and Dominant Processes. In both biological and physical
change with time and across spatial extents (stages 1–4). This
recognition can be used to forecast system behavior by focusing on
the most important set of processes and scales (21). For example,
broad-scale drivers (i.e., drought) often have limited ability to
explain initial woody plant invasion at fine scales at which estab-
lishment and survival are controlled by local processes (e.g., soil
resources). Conversely, as the spatial extent of a fire increases, it
becomes necessary to increase the extent of observations because
broad-scale processes and feedbacks become operative; fire-
fighting efforts focused on controlling spread within patches in the
early stages of a fire will likely miss the key conditions leading to
land–atmosphere feedbacks that result in a blow up of the fire (27).
Threshold Phenomena. We view threshold behavior as emerging
from interactions among fine- and broad-scale processes that
eventually overwhelm fine-scale processes. Recognizing the
potential for threshold behavior is the first step in preparing for
nonlinear changes before they occur. For example, management
practices (e.g., prescribed burns) can be used to reduce the
the potential for interactions with broad-scale processes. How-
ever, not all parts of a landscape need to be treated; reducing
connectivity at fine scales by thinning trees in small patches can
be used to reduce connectivity at landscape scales and limit the
potential for large fires to occur (44). Similarly, vaccinations are
diseases. For highly connected systems, these treatments need to
be in place before the threshold is reached at which the rate of
spread accelerates with increased connectivity and the process
becomes very difficult to control (45). Understanding when
broad-scale processes overwhelm fine-scale processes can also
help explain instances of management failure. For example, in
the surrounding area has significant effects on dynamics within
the Everglades as a result of mesoclimatic changes, thus limiting
the effectiveness of treatments within the park (46).
Identification of thresholds before they occur is a critical
challenge. Although statistical methods can be used to distin-
guish threshold behavior from natural heterogeneity, our frame-
work suggests that experiments that focus on changes in the
dominant process as a catastrophic event propagates through
time and space are needed. These experiments would need to
include detailed information on spatial patterns and rates of
change to detect the preconditions leading to threshold events.
Fallacy of Linear Extrapolations. Threshold phenomena also indi-
cate the limitations of linear extrapolation. Linear extrapola-
tion may be most appropriate within each stage of spread (Fig.
1). However, extrapolating information between stages will
likely be inaccurate unless cross-scale interactions are consid-
ered. For example, linear extrapolation based on broad-scale
rates of spread, such as after the blow up of a fire (stage 4), to
finer scales (stage 1) will overestimate these rates, particularly
for systems with low connectivity. This overestimate, for
instance, could result in unjustified extreme fire-fighting mea-
sures undertaken when fuel loads are disconnected with a
small probability of spread. Down-scaling to explain fine-scale
dynamics is often as problematic as up-scaling fine-scale
processes to broader scales (47).
Dangers of Over-Connected Systems. Highly connected systems
have the largest probability of exhibiting nonlinear spatial dy-
namics. In some cases, increasing connectivity of preferred
system elements is desirable and can hasten ecosystem recovery
efforts. Connectivity can also lead to undesirable events. For
example, whole-planet dust storms on Mars emerge from local-
ized storms that spread and likely interact with the atmosphere
to generate feedbacks to the movement of dust that rapidly
engulfs the entire planet within 2 weeks. These huge dust storms
are possible because the Martian landscape lacks spatial heter-
ogeneity in surface features to limit the connectivity of wind and
blowing dust. The human population on Earth is showing similar
signs of high connectivity; the ‘‘small-world effect’’ suggests that
the threat of pandemics is increasing as a result of local, regional,
and global connections among humans (48). Severe acute respi-
ratory syndrome (SARS) is a recent example in which extremely
fast rates of spread were possible globally because of increased
to exhibit stability and resilience and that knowledge about
connectivity can be used to limit the extent and rate of spread of
contagious events as well as to prevent or initiate new events.
Forecasting Complex Systems. Our ability to forecast future system
dynamics is severely constrained unless we can account for
spatial nonlinearities, threshold behavior, and cascading effects
(49). Even then, skillful predictions may not be possible (9, 50),
although we can identify vulnerabilities of systems to thresholds.
Research will need to adopt approaches that cross traditional
disciplinary boundaries to address system dynamics. Collabora-
tive efforts among ecosystem ecologists and atmospheric scien-
tists have made considerable progress in explaining broad-scale
patterns and dynamics in the Earth System (e.g., ref. 9), and
human behaviors are increasingly being recognized by ecologists
as integral to explaining system dynamics (e.g., ref. 28). More
intensive cross-disciplinary studies that identify the pervasive
role of cross-scale interactions are essential to understanding
and forecasting changes in the various components of the Earth
System. Our framework represents an initial step in seeking
generalities among disciplines.
Research Directions. Our framework suggests a general strategy
for characterizing and predicting thresholds in biological sys-
tems. Both manipulative experiments and simulation modeling
can be used to (i) examine the preconditions and fine-scale
www.pnas.org?cgi?doi?10.1073?pnas.0403822101Peters et al.
spatial patterns that lead to catastrophic events, (ii) document
and test the role of broad-scale feedbacks in the acceleration of
Lchange in spatial pattern, and (iii) provide tools to decision
makers for recognizing the critical spatial patterns and temporal
trends and identifying the stochastic and deterministic elements
of the key processes. Thus, our approach provides an integrating
framework for this level of research.
We thank S. Collins, W. Schlesinger, J. Brown, J. Herrick, S. Tartowski,
M. Havstad, and D. Carey for comments on the manuscript; D. Staley
for assistance in manuscript preparation; and A. Laliberte for assistance
in image analyses. Graphics design was provided by InFinite Designs
(Las Cruces, NM). This work was supported by National Science
Foundation Grants DEB 0080412 (to D.P.C.P., B.T.B., and K.M.H.),
DEB 0080529 (to D.P.C.P.), DEB 0087289 (to D.P.C.P. and B.T.B.), and
DEB 0217631 (R.A.P.) and by the U.S. Long-Term Ecological Research
Network Office. This work is a contribution of the Global Change in
Terrestrial Ecosystems (GCTE), the Land Project, Biologic Aspects of
the Hydrologic Cycle (BAHC), and Integrated Land Ecosystem–
Atmosphere Processes Study (iLEAPS) projects of the International
Geosphere–Biosphere Program (IGBP). This article is Sevilleta publi-
cation no. 305.
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