theunstab le poi nt rel ati vel y lon ger tha n it wou ld on the opp osi te sid e of the stable equilibrium. The skewness of the distribution of states is expected to increase not only if the system approaches a catastrophic bif urc ati on, but als o if the sys tem is dri ven clo ser to the bas in bou nd- ary by an increasing amplitude of perturbation

28

. Another phenomenon that can be seen in the vicinity of a cata- strophic bifurcation point is flickering. This happens if stochastic forcing is strong enough to move the system back and forth between the basins of attraction of two alternative attractors as the system enters the bistable region before the bifurcation

26,29

. Such behaviour is also considered an early warning, because the system may shift permanently to the alternative state if the underlying slow change in conditions persists, moving it eventually to a situation with only one stable state. Flickering has been shown in models of lake eutro- phication

24

and trophic cascades

30

, for instance. Also, as discussed below, data suggest that certain climatic shifts and epileptic seizures may be pre sag ed by fli cke rin g. Sta tis tic all y, fli cke rin g can be obs erv ed in the frequency distribution of states as increased variance and ske wne ss as wel l as bim oda lit y (re fle cti ng the two alt ern ati ve regimes)

24

.

Ind ica tor s in cyc lic andchaoti c sys tem s

. Th e pri nci ple s dis cus sed so far apply to systems that may be stochastically forced but have an underlying attractor that corresponds to a stable point (for example theclass ic fol d cat ast rop he ill ust rat ed in Box1). Cri tic al tra nsi tio ns in cyc lic a nd cha oti c sys tem s are les s wel l stu die d fro m the po int of view

Box 3

j

The relation between critical slowing down, increased autocorrelation and increased variance

Critical slowing down will tend to lead to an increase in the autocorrelation and variance of the fluctuations in a stochastically forced system approaching a bifurcation at a threshold value of a control parameter. The example described here illustrates why this is so. We assume that there is a repeated disturbance of the state variable after each period

D

t

(that is, additive noise). Between disturbances, the return to equilibrium is approximately exponential with a certain recovery speed,

l

. In a simple autoregressive model this can be described as follows:

x

n

z

1

{



x x

~

e

l

D

t

(

x

n

{



x x

)

z

se

n

y

n

z

1

~

e

l

D

t

y

n

z

se

n

Here

y

n

is thedevia tio n of thestat e var iab le

x

from the equil ibriu m,

e

n

is a random number from a standard normal distribution and

s

is the standard deviation. If

l

and

D

t

are independent of

y

n

, this model can also be written as a first-order autoregressive (AR(1)) process:

y

n

z

1

~

a

y

n

z

se

n

The autocorrelatio n

a

;

e

l

D

t

iszerofor wh it e no is e an d cl os e toone fo r red (autocorrelated) noise. The expectation of an AR(1) process

y

n

z

1

~

c

z

a

y

n

z

se

n

is

18

E

(

y

n

z

1

)

~

E

(

c

)

z

a

E

(

y

n

)

z

E

(

se

n

)

[

m

~

c

z

am

z

0

[

m

~

c

1

{

a

For c

5

0, the mean equals zero and the variance is found to be Var

(

y

n

z

1

)

~

E

(

y

2

n

)

{

m

2

~

s

2

1

{

a

2

Close to the critical point, the return speed to equilibrium decreases, imply ing that

l

appro ache s zeroand the auto corre lati on

a

ten ds toone. Thus , the varia nce tend s to infi nity.These early -war ning signa ls are the result of critical slowing down near the threshold value of the control parameter.

Box 2

j

Critical slowing down: an example

To see why the rate of recovery rate after a small perturbation will be reduced, and will approach zero when a system moves towards a cata strop hic bifu rcati on point , consi der the foll owin g simpl e dyna mical syste m, wher e

c

is a pos iti ve sca lin g fac to r and

a

and

b

are param eter s: d

x

d

t

~

c

(

x

{

a

)(

x

{

b

)

ð

1

Þ

It can easily be seen that this model has two equilibria,



x x

1

5

a

and



x x

2

5

b

, of which one is stable and the other is unstable. If the value of parameter

a

equals that of

b

, the equilibria collide and exchange stability (in a transcritical bifurcation). Assuming that



x x

1

is the stable equilibrium, we can now study what happens if the state of the equilibrium is perturbed slightly (

x

5



x x

1

1

e

): d

(



x x

1

z

e

)

d

t

~

f

(



x x

1

z

e

)

Here

f

(

x

) is th e rig ht han d sid e of equ ati on (1) . Lin ear izi ng thi s equ ati on using a first-order Taylor expansion yields d

(



x x

1

z

e

)

d

t

~

f

(



x x

1

z

e

)

<

f

(



x x

1

)

z

L

f

L

x

   



x x

1

e

which simplifies to

f

(



x x

1

)

z

d

e

d

t

~

f

(



x x

1

)

z

L

f

L

x

   



x x

1

e

[

d

e

d

t

~

l

1

e

ð

2

Þ

With eigenvalues

l

1

and

l

2

in this case, we have

l

1

~

L

f

L

x

   

a

~{

c

(

b

{

a

)

ð

3

Þ

and, for the other equilibrium

l

2

~

L

f

L

x

   

b

~

c

(

b

{

a

)

ð

4

Þ

If

b

.

a

then the first equilibrium has a negative eigenvalue,

l

1

, and is thus stable (as the perturbation goes exponentially to zero; see equa tion(2)). It is easy to see from equat ions (3) and (4) that at the bifurcation (

b

5

a

) the recovery rates

l

1

and

l

2

are both zero and perturbations will not recover. Farther away from the bifurcation, the rec ove ry rat e in thi s mod el is lin ea rlydepen den t on th e siz e of thebasin of attraction (

b

2

a

). For more realistic models, this is not necessarily true but the relation is still monotonic and is often nearly linear

16

.

A R ( 1 ) c o e f f . s . d . R e s i d u a l

a b c d

0 2 4 6 0 0.10 0.12 0.65 0.75

B i o m a s s ( a . u . )

F

2

F

1

Increasing harvest rate over time

Figure 2

|

Early warning signals for a critical transition in a time series generated by a mo del of a harvested population

77

driven slowly across a bifur catio n. a

, Biom ass time serie s.

b

,

c

,

d

, Analys is of the filter ed time serie s (

b

) shows that the catastrophic transition is preceded by an increase both in theamplit udeof flu ctu ati on,expre sse d as s.d . (

c

), andin slo wne ss,estim ate d as thelag-1 aut ore gre ssi on (AR (1) ) coe ffi cie nt (

d

), as pre dict ed fro m the ory . The grey band in

a

identifies the transition phase. The horizontal dashed arr owshowsthe wid th of themovin g win dowusedto com put e theindic ato rs sho wn in

c

and

d

, an d th e re d li neis th e tr en d us edforfilt er in g (s eeref.22 fo r the methods used). The dashed curve and the points

F

1

and

F

2

represent the equil ibriu m curve and bifurc ation poin ts as in Box 1 Figurec, d. a.u., arbitrary units.

NATURE

j

Vol 461

j

3 September 2009

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55

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