Rising variance: a leading indicator of ecological transition

Abstract

Regime shifts are substantial, long-lasting reorganizations of complex systems, such as ecosystems. Large ecosystem changes such as eutrophication, shifts among vegetation types, degradation of coral reefs and regional climate change often come as surprises because we lack leading indicators for regime shifts. Increases in variability of ecosystems have been suggested to foreshadow ecological regime shifts. However, it may be difficult to discern variability due to impending regime shift from that of exogenous drivers that affect the ecosystem. We addressed this problem using a model of lake eutrophication. Lakes are subject to fluctuations in recycling associated with regime shifts, as well as fluctuating nutrient inputs. Despite the complications of noisy inputs, increasing variability of lake-water phosphorus was discernible prior to the shift to eutrophic conditions. Simulations show that rising standard deviation (SD) could signal impending shifts about a decade in advance. The rising SD was detected by studying variability around predictions of a simple time-series model, and did not depend on detailed knowledge of the actual ecosystem dynamics.

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LETTER
Rising variance: a leading indicator of ecological
transition
S. R. Carpenter1* and
W. A. Brock2
1Center for Limnology,
University of Wisconsin,
Madison, WI, USA
2Department of Economics,
University of Wisconsin,
Madison, WI, USA
*Correspondence: E-mail:
srcarpen@wisc.edu
Abstract
Regime shifts are substantial, long-lasting reorganizations of complex systems, such as
ecosystems. Large ecosystem changes such as eutrophication, shifts among vegetation
types, degradation of coral reefs and regional climate change often come as surprises
because we lack leading indicators for regime shifts. Increases in variability of ecosystems
have been suggested to foreshadow ecological regime shifts. However, it may be difficult
to discern variability due to impending regime shift from that of exogenous drivers that
affect the ecosystem. We addressed this problem using a model of lake eutrophication.
Lakes are subject to fluctuations in recycling associated with regime shifts, as well as
fluctuating nutrient inputs. Despite the complications of noisy inputs, increasing
variability of lake-water phosphorus was discernible prior to the shift to eutrophic
conditions. Simulations show that rising standard deviation (SD) could signal impending
shifts about a decade in advance. The rising SD was detected by studying variability
around predictions of a simple time-series model, and did not depend on detailed
knowledge of the actual ecosystem dynamics.
Keywords
Alternate stable states, dynamic linear model, eutrophication, indicator, lake, regime shift,
stationary distribution, stochastic differential equation, variance.
Ecology Letters (2006) 9: 311–318
INTRODUCTION
Regime shifts are substantial, long-term reorganizations of
complex systems such as societies, ecosystems or climate
(Steele 1998; Scheffer et al. 2001; National Research Council
2002; Carpenter 2003; Foley et al. 2003; Scheffer & van Nes
2004; Brock et al. 2006; Folke et al. 2005; Hsieh et al. 2005;
Brock 2006). In ecology, some of the well-studied cases
include eutrophication of lakes and coastal oceans, shifts
among grassy and woodland cover types in rangelands,
degradation of coral reefs and regional climate change
(Scheffer et al. 2001; Folke et al. 2005). Feedbacks that
control key system processes are different after a regime
shift (Holling 1973). For example, in oligotrophic (clear
water) lakes the concentrations of nutrients that affect
primary production are controlled primarily by inputs from
the watershed, whereas in eutrophic (turbid) lakes the
concentrations of nutrients are driven by recycling within
the lake as well as inputs (Carpenter 2003). Other kinds of
feedback changes are known from regime shifts in other
types of ecosystems (Scheffer et al. 2001; Carpenter 2003;
Foley et al. 2003; Walker & Meyers 2004; Folke et al. 2005).
Brock (2006) reviews social systems in which tipping points
are found.
Regime shifts are difficult to study (Carpenter 2003;
Scheffer & Carpenter 2003). They occur in large, spatially
heterogeneous systems, and usually involve processes at
more than one spatial scale. From the perspective of a
human lifetime, regime shifts are infrequent events that may
play out over many years, even though the change is rapid in
comparison with routine ecological change. Regime shifts
have multiple causes, so studies must track multiple
variables simultaneously for long periods of time. Inference
requires several lines of evidence, such as long-term
observations or paleo-ecological data, comparisons of
ecosystems across gradients of key drivers, models of
varioustypes and appropriately
(Carpenter 2003). It takes considerable effort to build
understanding of regime shifts.
Numerous models can potentially explain any particular
regime shift, and it may be very difficult to discriminate
among these models, particularly in early stages of an
scaled experiments
Ecology Letters, (2006) 9: 311–318doi: 10.1111/j.1461-0248.2005.00877.x
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investigation (Carpenter 2003). The kinds of evidence
that are needed to understand a regime shift depend on
the set of models that are believed to describe the regime
shift. The appropriate evidence cannot be gathered without
knowing the models, and the models cannot be assessed
without the evidence. This circularity poses a challenge
to researchers. It is important to design measurement
programmes that could provide relevant evidence for a wide
range of models, often over long periods of time and
extensive spatial ranges.
Recently, however, researchers have noticed that the
variability of system behaviour changes in advance of a
regime shift (Kleinen et al. 2003; van Nes & Scheffer
2003; Brock et al. 2006; Oborny et al. 2005). These
changes in variance may be discernible without detailed
knowledge of the underlying ecosystem dynamics. Kleinen
et al. showed that the variance spectra of an ocean–
atmosphere model shifted to lower frequencies as the
system approached a regime shift in the circulation of the
North Atlantic ocean. van Nes & Scheffer (2003) showed
that variability of an individual-based model of an aquatic
macrophytes increased near thresholds between alternate
attractors. Using a generalized model of spatial dynamics,
Oborny et al. (2005) showed that spatial variance of
terrestrial mosaics increased near the critical threshold for
percolation. Brock et al. (2006) showed that rising variance
near thresholds should be expected in a wide range of
social and ecological systems with multiple attractors,
including scientific paradigms, elections, monetary policy,
lake eutrophication, fisheries collapse, ocean dynamics and
climate change.
In general, the variance of temporal fluctuations in certain
state variables increases and the variance spectra shifts
towards longer wavelengths (lower frequencies) just before
the regime shift occurs (Kleinen et al. 2003; Brock et al.
2006). The reasons for this behaviour seem general, and
should apply to a wide range of plausible models of regime
shifts (Horsthemke & Lefever 1984; Berglund & Gentz
2002a,b; Kleinen et al. 2003; Brock et al. 2006; Appen-
dix S2). Therefore, increased variance may be an important
clue of regime shifts even in cases where the appropriate
model is unknown. Furthermore, increased variance may
provide a leading indicator of regime shifts that can be used
in ecosystem management. Regime shifts often have large
ecological and economical consequences, e.g. losses of water
quality, declining productivity of fish stocks or rangelands,
or breakdowns of dryland agriculture. It would be useful to
have advance warning of these changes so that managers
have the opportunity to avoid them. Eutrophication, the
regime shift studied in this paper, is a syndrome of aquatic
ecosystems leading to blooms of nuisance (often toxic)
algae, anoxic events, fish mortality and substantial economic
losses (Carpenter 2003).
In practice, changes in variance due to impending regime
shifts may be difficult to distinguish from other drivers of
variance such as exogenous noise that affects the ecosystem.
In lakes, for example, variance because of fluctuations in
recycling (which may indicate a regime shift) may be difficult
to discriminate from variance because of noisy nutrient
inputs. Furthermore, the true model for ecosystem dynam-
ics is unknown and inferences about changing variance must
be drawn from approximate models. We addressed this
problem using a model of lake eutrophication that includes
both input noise and recycling noise. In addition to the
complication introduced by input noise, the model has three
state variables with distinctly different turnover times
(Carpenter 2005). Thus the model is more realistic and
complex than the one-dimensional systems examined in
some previous work. Despite these complications, the
regime shift to eutrophication could be detected years in
advance by studying the standard deviation (SD) of
phosphorus concentration in the water during summer
stratification. Change in the SD could be detected by
analysing time series of phosphorus in lake water using a
simple empirical model that did not require detailed
knowledge of the ecosystem dynamics.
METHODS
The model, which was modified slightly from previous
papers (Carpenter et al. 1999; Carpenter 2003, 2005; Ludwig
et al. 2003), is a system of stochastic differential equations
(SDE) for phosphorus density (g m)2) in soil (U ), lake
water (X ) and surface sediment (M ) (Fig. 1).
dU
dt
¼ F ? cUH;
ð1Þ
Figure 1 Phosphorus flows in the model.
312 S. R. Carpenter and W. A. Brock
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dX
dt
¼ cUH ? ðs þ hÞX þ rMRðXÞ þ rMRðXÞdW
dt
;
ð2Þ
dM
dt
¼ sX ? bM ? rMRðXÞ ? rMRðXÞdW
dt
;
ð3Þ
where F is the input rate of phosphorus to soil (e.g. from
fertilizer use, dust deposition or weathering), c is a coeffi-
cient for transfer of soil phosphorus to the lake, H is noise
for input to the lake (see below), s is sedimentation loss, h is
hydrologic loss (outflow), r is a recycling coefficient, r is the
SD of recycling noise and b is the permanent burial rate of
phosphorus in sediments; dW is a white noise process with
mean zero and variance dt. The recycling function R(X ) is
Xq
mqþ Xq;
where m is the value of X at which recycling is half the
maximum rate and the exponent q determines the slope of
R(X ) near m (Carpenter et al. 1999). The annual load dis-
turbance H is calculated within each year as the solution of
the Ito SDE
RðXÞ ¼
ð4Þ
dH
H
with initial condition H0¼ 1; dZ is a white noise process
with mean zero and variance dt (independent of dW ). Using
standard methods for Ito SDEs (Malliaris & Brock 1982),
the solution of eqn 5 is obtained as
?
Our numerical analyses use parameters estimated for
Lake Mendota, Wisconsin (Table S1; Carpenter 2005). The
model is solved for successive summer stratified seasons.
We assume that the annual nutrient input enters prior to
summer stratification. During the summer stratified season,
recycling from sediments occurs in a series of stochastic
events driven by wind (Soranno et al. 1997). Frequent
shocks to recycling because of wind events within the
summer season are represented by rMR(X )dW/dt, where
dW is a white noise process with mean zero and variance dt
(Horsthemke & Lefever 1984). We set the scale parameter r
to 0.01, which produced variability in recycling similar to
that observed by Soranno et al. (1997).
According to our hypothesis, the SD of water phospho-
rus (X ) should increase prior to a regime shift. We used two
approaches to estimate the SD. First, we computed
stationary distributions of X (Horsthemke & Lefever
1984). Over a long period of time, the state variables
approach a stationary distribution, just as a deterministic
model would approach a stable point. While stationary
distributions expose the causes of changes in the SD of X,
¼ kdZ
ð5Þ
Ht¼ exp kZt?tk2
2
?
:
ð6Þ
stationary distributions may never be observed in ecosys-
tems because the rate of convergence to the stationary
distribution is not fast enough compared with the changes
in slow variables such as soil or sediment phosphorus. To
examine changes in the stationary distribution of X under
conditions that might be observed in field studies, we
simulated changes in a lake undergoing eutrophication,
corresponding to changes that have been observed in many
lakes (Carpenter 2003).
We computedthe mean
distributions of water phosphorus by two methods: (i) the
stationary probability densities obtained from Fokker–
Planck equations presented by Horsthemke & Lefever
(1984); and (ii) Monte Carlo simulation. Appendix S1 shows
that these two methods give similar results. Because the
Fokker–Planck expressions are not always computable due
to overflow errors, we present Monte Carlo results here. For
each estimate of the mean and SD, we simulated eqns 1–3
from each of 1000 different initial conditions for 1080 time
steps (300 years with 36 time steps per year) using Ito
calculus and the Euler method. The 1000 final values of X
(one from each of the 1000 initial points) were used to
calculate the mean and SD.
In addition to studying the stationary distributions, we
examined changes in the SD during dynamic simulations.
These come closer to the situation studied in the field,
where ecosystems may be far from the stationary distribu-
tion. We used dynamic simulations to examine changes in
the within-year SD of water phosphorus associated with
transitions to eutrophy under non-equilibrium conditions.
Simulations were computed with various schedules of
phosphorus input to soil (F, eqn 1) for 300 years with 36
time steps per year, using Ito calculus and the Euler method.
In our simulations, as in ecosystems, changes in state
variables are affected by proximity to the threshold as well
as other factors such as input disturbances which may
complicate the detection of threshold effects. Furthermore,
the true processes generating the observed dynamics of the
ecosystem will be unknown. Therefore, the analyst will work
with an approximate model of the ecosystem, and attempt
to use this model to discriminate the effects of input
disturbances from other sources of variability that affect
ecosystem dynamics. To analyse the lake simulations, we
derived a simplified approximate model. It is reasonable to
think that such a simplified model might be used by an
analyst to estimate variability of lake phosphorus from
observed time series.
Suppose the analyst believes that the true deterministic
model for the lake can be approximated over a short period
of time as
and SD for stationary
dX
dt
¼ a0þ L ? a1X;
ð7Þ
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where a0and a1are functions of the actual but unknown
parameters of the true but unknown ecosystem processes,
and the time series of input L and water phosphorus X are
measured. Equation 7 is a minimal empirical representation
of phosphorus dynamics that depend on input (L), losses
(a1) and recycling (a0) in a situation where X is the only state
variable that is monitored. L is assumed constant over one
annual time step. This situation is similar to many lake
monitoring programmes that estimate annual values for L
and X, where the annual mean estimate of X is based on
many observations in the course of the year.
Over one time step the solution of eqn 7 from initial
condition X ¼ X0at t ¼ 0 is
X1¼ X0e?a1þ1 ? e?a1
a1
L þa0ð1 ? e?a1Þ
a1
:
ð8Þ
Dynamic linear models (DLMs) are among the statistical
methods available for analysing time-series processes such
as eqn 8 (Pole et al. 1994). In DLMs, the parameters can
change slowly over time as new data are observed. Thus
DLMs can adapt to gradual changes in the underlying
ecosystem dynamics. In this situation, where ecosystem
dynamics depend on slow change in unobserved variables
(soil and sediment phosphorus), the DLM approach is
reasonable (Pole et al. 1994; Cottingham et al. 2000;
Carpenter 2003). Equation 8 converts to the following
DLM:
2
Xt¼ ½b0
bP
bL?t
1
Xt?1
Lt
4
3
5þ xt;
bL?t?1þ mt:
ð9Þ
½b0
bP
bL?t¼ ½b0
bP
ð10Þ
In eqns 9 and 10, x and m are independent normally
distributed observation and process errors respectively. The
parameters b0, bPand bLcorrespond to a0(1 ) exp()a1))/
a1, exp()a1) and (1 ) exp()a1))/a1respectively. At each
time step, parameter estimates are updated using measure-
ments of L and X (Pole et al. 1994). Because of the regular
updating driven by data, the parameters do not follow a
random walk but instead move with trends in the data. The
updating equations are well known (Pole et al. 1994) and will
not be repeated here.
RESULTS
To illustrate the stationary distributions, we present an
example in which stationary distributions were computed
across a gradient of the loading coefficient, c (eqns 1 and 2).
Manipulations of c are sometimes used as a control
parameter by lake managers. Restorations of riparian
vegetation to reduce c may mitigate eutrophication by
decreasing phosphorus inputs, for example. We focus on
the stationary distribution of water phosphorus (X ) for a
lake that has been oligotrophic for a long time. Monte Carlo
simulations of the stationary distribution were initiated at
the deterministic steady-state values of soil phosphorus (U )
and sediment phosphorus (M ) by setting the load distur-
bance k to 0. We used the oligotrophic steady state for M if
the system was bistable (otherwise the sole steady state of M
was used). Simulations were initiated over a wide range of
values of X. We present only the final values of X
(Appendix S1).
Equilibria of phosphorus in the water vs. the loading
coefficient, c, indicates alternate stable states, as known from
previous studies of this model (Fig. 2a). At low c, there is
only one equilibrium, an oligotrophic attractor. As c rises
above roughly 0.00025, a eutrophic attractor appears,
separated from the lower attractor by a repelling threshold.
As c rises above roughly 0.0025, the lower oligotrophic
attractor disappears. Similar behaviour occurs if the equil-
ibria are plotted against other parameters, such as the
recycling coefficient r or the more slowly changing variables,
Figure 2 (a) Equilibria of water P vs. input coefficient (c). Stable
equilibria are solid lines, unstable equilibria are the dotted line, and
open circles are mean values of the stationary distribution from
Monte Carlo simulation. Vertical solid line shows the relationship
between the eutrophication threshold and the standard deviation.
(b) Standard deviation of the stationary distribution versus input
coefficient (c). Note log y-axis.
314 S. R. Carpenter and W. A. Brock
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soil and sediment phosphorus. Appendix S1 shows that
similar effects occur across a gradient of sediment phos-
phorus.
Mean values of the stationary distribution closely track
the oligotrophic attractor for c less than c. 0.0023 (Fig. 2a).
At higher levels of c, the mean values lie between the two
attractors. The mean values gradually approach the upper
eutrophic attractor as c rises to 0.003.
The SD of the stationary distribution of water phospho-
rus rises steadily as c passes through the shift to alternate
stable states at c ¼ 0.00025, until c reaches c. 0.0021
(Fig. 2b). Just below the c value where the lower oligo-
trophic attractor disappears, the SD increases sharply. The
rise in SD between about c ¼ 0.0021 and the transition
point at c ¼ 0.0025 could provide an advance warning that
the system was approaching a regime shift.
Frequency distributions from Monte Carlo simulation
illustrate the changes that occur near the transition point. At
c ¼ 0.0021, frequency is sharply clustered near the oligo-
trophic attractor, with a long tail to the right extending
towards the eutrophic attractor (Fig. 3d). Near the transition
point, frequency is spread broadly across the range of water
phosphorus, with a peak slightly above the oligotrophic
attractor (Fig. 3c). There is further spreading of the
frequency distribution as c rises above the transition point
(Fig. 3b). Finally, with c ¼ 0.0030, frequencies seem to
approach a bell-like distribution centred just below the
eutrophic attractor (Fig. 3a).
Although the stable distribution is useful for exposing the
underlying causes of the change in variability, it is not useful
in practice because all of the dynamic variables (soil,
sediment and water phosphorus) are changing at the same
time. We used dynamic simulations to explore changes in
variability of water phosphorus near the transition point. An
example is presented in Fig. 4. We focus on the time period
near the transition to eutrophy. The full 300-year simulation
is presented in Fig. S1.
The upper, eutrophic attractor appears in year 101, and
the lower oligotrophic attractor disappears in year 139
(Fig. 4). The lake shifts to the eutrophic state c. years 160–
162.
Often, a field scientist will observe time series of a fast
variable (such as water phosphorus), but will not know the
slowly changing variables (in this case soil and sediment
phosphorus) or the true ecosystem dynamics (eqns 1–6). In
this situation, the analyst will often approximate ecosystem
dynamics using a simplified model such as the one we
derived in eqns 7–10. We calculated two measures of SD of
water phosphorus. One is the within-year SD around the
annual mean (within-year SD), and the other is the within-
year SD around the prediction of the DLM (DLM SD).
Uncertainty around both measures of SD is small. With 36
observations per year, the 95% CI around each SD ranges
from 0.81 to 1.29 times the estimated SD, a range that is too
narrow to be plotted on Fig. 4.
The within-year SD rises, with fluctuations, over the
course of the simulation (Fig. 4). The within-year SD
represents the combined effects of varying phosphorus
inputs to the lake and endogenous variance related to the
regime shift. Variance of phosphorus inputs is rising
through the simulation, because of rising soil phosphorus
(Fig. S1) which increases mean input rate (cUH, eqn 2) as
well as its variance (eqn 6). Despite the complicating effects
of variable phosphorus inputs, there is some indication of
regime shift effects in the within-year SD. The within-year
SD shows unusual peaks in years 147, 152, 157 and 159,
which could perhaps be taken as indicators of an impending
transition.
The DLM SD provides a clearer indication of impending
transition. The DLM accounts statistically for the fluctua-
Figure 3 Stationary distributions of water phosphorus from Monte
Carlo simulation at four values of the input coefficient (c) near the
threshold for eutrophication at c ¼ 0.0025.
Variance and ecological transition 315
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