p.53 It is becoming increasingly clear that many complex systems
have critical thresholds - so-called tipping points - at which the system shifts abruptly from one state to another ... It
is notably hard to predict such critical transitions, because the state of the system may show little change before the tipping
point is reached. Also, models of complex systems are usually not accurate enough to predict reliably where critical thresholds
may occur. Interestingly, though, it now appears that certain generic symptoms may occur in a wide class of systems
as they approach a critical point. At first sight, it may seem surprising that disparate phenomena such as the collapse of
an overharvested population and ancient climatic transitions could be indicated by similar signals. However, as we will explain
here, the dynamics of systems near a critical point have generic properties, regardless of differences in the details of each
system9. Therefore, sharp transitions in a range of complex systems are in fact related. In models, critical thresholds for
such transitions correspond to bifurcations10. Particularly relevant are ‘catastrophic bifurcations’ (see Box
1 for an example), where, once a threshold is exceeded, a positive feedback propels the system through a phase of directional
change towards a contrasting state. Another important class of bifurcations are those that mark the transition from a stable
equilibrium to a cyclic or chaotic attractor. Fundamental shifts that occur in systems when they pass bifurcations are collectively
referred to as critical transitions11.
p.53 The most important clues that have been suggested as indicators
of whether a system is getting close to a critical threshold are related to a phenomenon known in dynamical systems theory
as ‘critical slowing down’12 ... The most straightforward implication of critical slowing down is that
the recovery rate after small experimental perturbation can be used as an indicator of how close a system is to a
bifurcation point16.
p.53 For most natural systems, it would be impractical or impossible to
monitor them by systematically testing recovery rates. However, almost all real systems are permanently subject to
natural perturbations. It can be shown that as a bifurcation is approached in such a system, certain characteristic
changes in the pattern of fluctuations are expected to occur. [JLJ - perhaps, in a game such as chess, the corresponding 'warning
signs' might be, for the side to move: increasing number of times only 1 move is playable, opponent has a number of sacrifices
that 'almost' work, opponent has a number of playable moves, future mobility of one or more pieces is restricted, ability
to constrain opponent's pieces is restricted, opponent can get a certain amount of 'critical mass' of pieces within range
of our king, future mobility of our pieces to support our king an an 'emergency' is restricted, etc.]
p.54-55 In
summary, the phenomenon of critical slowing down leads to three possible early-warning signals in the dynamics of a system
approaching a bifurcation: slower recovery from perturbations, increased autocorrelation and increased variance.
Skewness and flickering before transitions. In addition to autocorrelation and variance, the asymmetry
of fluctuations may increase before a catastrophic bifurcation28. This does
not result from critical slowing down. Instead, the explanation is that in catastrophic bifurcations such as fold bifurcations
(Box 1), an unstable equilibrium that marks the border of the basin of attraction approaches the attractor from one side (Box
1). In the vicinity of this unstable point, rates of change are lower (reflected in a less steep slope in the stability landscapes).
As a result, the system will tend to stay in the vicinity of the
unstable point relatively longer than it would on the opposite side of the stable equilibrium.
p.56 The question therefore remains of whether highly complex real systems
such as the climate or ecosystems will show the theoretically expected early-warning signals. Results from elaborate and relatively
realistic climate models including spatial dynamics and chaotic elements23 suggest that some signals might be robust in the
sense that they arise despite high complexity and noisiness. Nonetheless, it is clearly more challenging to pick up early-warning
signals in complex natural systems than in models.
p.58 A key issue when it comes to practical application
is the question of whether a signal can be detected sufficiently early for action to be taken to prevent a transition
or to prepare for one25. Understanding spatial early-warning signals better might be particularly useful in this
respect, as a spatial pattern contains much more information than does a single point in a time series, in principle allowing
shorter lead times76.
