Critical Transitions
in Nature and Society
Marten Scheffer
Chapter 14
How to Know if Alternative
Basins of Attraction Exist ?
14.1 Hints from Field Data
How one can tell if there are multiple attractors?
• Jumps in time-series or regime shifts
• Sharp boundaries and multimodality of frequency
distributions
• The shape of a catastrophe fold
The evidence can never be conclusive.
Discontinuities in time-series or spatial patterns can be due to
discontinuities in environmental factors.
Alternatively, the system might have a threshold response that is not
related to alternative stability domains.
14.2 Experimental Evidence
Experiments can provide evidence for multiple attractors.
• Different initial states lead to different final states
• Disturbance triggers a shift to another permanent state
• Hysteresis in response to forward and backward
changes in conditions
A hysteretic pattern may not indicate alternative attractors if the
response of the system is not fast enough relative to the rate of
change in the controle factor.
Experiments provide potentially powerful tests, but there are serious
limitations to exploring large spatial scales and long time spans.
14.3 Mechanistic Insight
Mechanistic models, formalized as graphs or equations, may explain
observed behavior and allow us to extrapolate our findings.
They allow us to predict that the system might have alternative stable
states, chaotic attractors, or other interesting stability properties, even if
those have not yet been detected in its behavior.
We want such models to become as accurate as possible and should
take their outcomes seriously, even though we have no comparable
past events to validate this aspect of their predictions.
Mapping mechanistic models to reality is the core of
science. It is what allows moving from beliefs to true
understanding of the complex world around us.
Implications of positive feedbacks
Minimal models and hypothesis testing
Realistic simulation models
To predict the consequences of the feedback, we need to take
a more quantitative approach and analyze its interaction with
other processes in the system.
Minimal, or strategic, models focus on a minimal set of
mechanisms needed to produce a certain behavior. However,
in many cases we cannot identify one isolated casual factor.
Predictive simulation models work surprisingly poorly for
ecological problems. Tuning of complex ecological models
easily leads to good results for the wrong reasons. The
assumed casual relations need not be true, and extrapolation
to new situations can lead to nonsense predictions.
Chapter 15
How to Know if a
Threshold is Near ?
15.1 The theory: Signs of Upcoming Transition
Early warning signals as predicted from theory.
• Slower recovery from perturbations
• Increasing autocorrelation
• Increasing variance
• Flickering and stochastic resonance
• Increased spatial coherence
15.2 Precursors of Transitions in Real Systems
Ancient climate shifts: an increase in autocorrelation but
no increase in variance.
Lakes shifting from turbid to clear state: increase in
variance but no rising autocorrelation.
Epileptic seizures and heart failure: changes in the
synchronization between oscillating cells corresponding
to phase locking and resonance phenomena.
15.3 Reliability of the signals
A shift due to a fast and permanent change of external
conditions cannot be detected (?)
Detecting increased autocorrelation requires long and
good time-series.
The external regime of perturbations can change over
time distorting or counteracting the expected signals (?)
The indicators will also signal a threshold that is not
related to a catastrophic bifurcation but to a strong
sensitivity of the system around a critical value.
15.3 Synthesis
We are far from being able to develop accurate models
to predict thresholds in most complex systems ranging
from cells to organism, ecosystems, or the climate.
We do not understand all relevant mechanisms and
feedbacks sufficiently well in most cases.
The generic character of early warning signals is reason
for optimism, as they occur largely independent of the
precise mechanism involved.