Title: Spatially structured food webs in coloured environments
Authors: Sara Gudmundson, Frida Lögdberg, Uno Wennergren
Affiliation: IFM, Theory and Modelling, Linköpings Universitet, Linköping, Sweden
Email addresses: sargu@ifm.liu.se, friwa@ifm.liu.se, unwen@ifm.liu.se
Running title (< 45 characters including spaces): Spatially structured food webs
Keywords (< 10): Asynchrony, coloured environmental variation, dispersal, extinction risk,
food web, spatial structure, stability, subpopulation, synchrony.
Type of article: Letter
No. words in the abstract: 135
No. words in the text body: 4113
No. words in the manuscript as a whole: 5244
No. references: 28
No. figures & tables: 5
Correspondence: Assoc. Prof. Uno Wennergren, +46 13 281666, unwen@ifm.liu.se
Abstract
Ecological theory states that complex food webs are unstable and extinction-prone but yet
species rich natural food webs do exist. This study analyse how temporal and spatial
components may stabilize interactions between species. Our analyses indicate that a single
measure of stability, as CV, could be misleading. It does not reflecting significant changes in
mean densities. The ‘diamond shaped’ food web responds to an increased environmental
variance by an increase of the rare species and a decrease of the most abundant species. The
shift in relative abundances indicates an increase in food web stability with environmental
variation. The stabilisation of environmental variation decreased with increased redness while
a further increase in reality, linking environmental variation to spatially subdivided
populations, increased the stabilising effect. Spatial and temporal components, jointly
included and analysed with care, may become a new basis for explaining the existence of
large and diverse food webs.
Introduction
Species rich natural food webs contain complex interaction patterns evolved through
historical processes in dynamic environments (May 1973). Food webs are known to endure
unstable environments for several generations (Vasseur & Fox 2007). However, theoretical
studies predict that large complex food webs should be unstable and extinction-prone because
of high connectance, many modes of oscillation and positive feedback loops (May 1973;
Tilman 1999; Green & Sadedin 2005; Borrvall & Ebenman 2008). The inability of explaining
nature’s diversity implies that population dynamic theory lacks important components.
Furthermore, the stability analysis itself may not be sufficient. Density regulation,
environmental autocorrelation and dispersal are known to affect local population dynamics
and should be included in investigations regarding population dynamics and extinction risks
(Engen et al. 2002). Coupling of asymmetric interaction pathways have been shown to
facilitate food web complexity (Polis 1991; McCann et al. 1998). The diamond shaped food
web has previously been used to identify stabilizing effects of consumer asynchrony
(McCann et al. 1998) and weak-to-moderate environmental variation (Vasseur and Fox 2007).
In this study, we investigate how coloured environmental variation and spatial structure affect
the same food web by performing a thorough analysis of stability.
Food web stability is often measured as variability, which usually is calculated as the
coefficient of variation, standard deviation divided by the mean (McCann 2000). Decreased
variability implies decreased population variance which is likely to lower extinction risk
(Lande 1993; McCann 2000). However, measurements of the coefficient of variation, CV,
may not be enough for determining food webs ability to withstand stress. An increase in
stability, measured as CV, can imply an increase in mean density and a decrease in variance,
or only one of these. A population consisting of just a few individuals can misleadingly be
seen as robust to stress as long as its variance is low in comparison to its mean. To address
the risk of misinterpreting results of stability, we evaluate mean and variance one by one in
addition to standard variability measurements of food web biomass and species abundances.
The environmental variance, measured in impact and frequency of extreme weather events,
is increasing (Easterling et al. 2000). The change in climate is likely to cause increased
variability and extinction risk of ecological systems (Lande 1993; Halley & Dempster 1996;
Ripa & Lundberg 1996; Ripa & Lundberg 1996; Kaitala et al. 1997; Fontaine & Gonzalez
2005). When investigating the effect of environmental variation, it is important to consider
different magnitudes of variance. Another important property of environmental variation is its
correlation in time. Variation found in nature is considered to be the best represented by pink
1/f noise (Caswell & Cohen 1995; Halley 1996; Ripa & Lundberg 1996; Cuddington &
Yodzis 1999). It describes correlations in many different scales and does not priorities
between timescales of disturbances (Halley 1996). In order to investigate the effect of
environmental variation on food webs, we incorporate 1/f noise with different magnitudes of
variance and redness. In addition to variation in time, nature also contains variation in space.
Landscapes are known to include different biotic and abiotic conditions giving rise to
spatially separated subpopulations occupying patches. Dispersal between subpopulations
enables re-establishments which can prolong the whole population’s time to extinction
(Engen et al. 2002; Liebhold et al. 2004; Greenman & Benton 2005). Subpopulations within
the diamond shaped food web fluctuate in stable limit cycles when existing in a constant
environment (McCann et al. 1998, Vasseur and Fox 2007). These intrinsic dynamics makes it
possible to study how dispersal alone affects how subpopulations fluctuate in relation to each
other, termed (sub-) population synchrony (reviewed in Bjørnstad et al. 1999). Subpopulation
synchrony determines if stability on landscape level will be different than stability on patch
level. When modelling subpopulations in a landscape which also include variable
environmental conditions, one has to identify each subpopulation specific environmental
variation and their relation. Are all subpopulations affected by the same variation or are they
affected by different environmental variation depending on patch specific conditions? Hence
the landscape dimension includes not only the subpopulations themselves but also how
stochasticity is distributed in space. Furthermore, each patch is populated by a food web, in
our study consisting of four species. Different species situated in the same patch may respond
differently to the same environmental variation. This food web dimension implies yet another
level of synchrony of environmental stochasticities. By varying the cross-correlation of
environmental time series affecting the species and their subpopulations, we simulate
differences both in patch specific conditions and species environmental response. Crosscorrelation of environmental variation will affect population synchrony. Synchrony can be
measured both between subpopulations and between species. In this study, we will only
measure synchrony between species.
Synchrony between species has been shown to have a substantial effect on food web
stability and extinction risk. Asynchronous consumers coupled with uncorrelated
environmental variation can improve food web stability (1/CV) by dampening oscillations
between resource and consumers (McCann et al. 1998; Vasseur & Fox 2007). A positive
correlation in species environmental response implies a lower species extinction risk than
during uncorrelated response (Borrvall & Ebenman 2008). We measure synchrony between
species, according to Vasseur and Fox (2007). In addition, we measure the correlation
between each species and their environmental variation. By measuring this correlation we
aim to increase the understanding of how environmental variation affects how species
fluctuate in relation to each other.
This study addresses how the stability of food webs is affected by coloured environmental
variation and spatial structure. We simulate the same diamond shaped food web used in
McCann et al. (1998) and Vasseur and Fox (2007) in order to clarify the implications of these
additional components. Vasseur and Fox (2007) showed that weak-to-moderate
environmental variation can stabilise the diamond shaped food web. We show that redness
decrease the stabilising effect of environmental variation whereas dispersal, coupled with
uncorrelated response, has a strong stabilising effect. While dispersal increased the stability
by increasing mean biomass and lowering the variance of densities, weak-to-moderate
environmental variation actually decreased mean biomass. Single measures of stability did
not show the full picture. However, moderate environmental variation caused a change in the
relative abundance of species increasing the density of the species with the smallest
population in a constant environment. This food web would be more resistant to additional
stresses, such as demographic stochasticity and catastrophes, yet the stability, measured as CV,
is reduced in comparison to the food web with a constant environment.
Method
The diamond shaped food web contains four species. Two consumers share one resource and
have one common predator (Fig. 1). The dynamics are described by a continuous-time
differential equation system, modelled by Vasseur & Fox (2007) after McCann et al. (1998).
Resources grow logistically and consumers and predator have natural background mortality.
Consumption is limited by a type II nonlinear functional response (Yodzis & Innes 1992;
McCann et al. 1998; Vasseur & Fox 2007). The parameter values above the splitting line in
Table 1 are the same as in Vasseur and Fox (2007). The parameters are biologically plausible,
estimated from studies on species’ body mass versus metabolic and ingestion rate (Dickie et
al. 1987; Yodzis & Innes 1992; McCann et al. 1998; Vasseur & Fox 2007). Resource gain
and predator preference are set higher for C1 than for C2. C1 is the strongest resource
competitor and preferred prey of P. The competition irregularity causes intrinsic
asynchronous fluctuations of consumers. Species densities fluctuate in stable limit cycles in
constant environment.
The standard deviation, σenv, and cross-correlation, ρenv, of environmental variation are
independent parameters affecting the consumers. Environmental variation affects the two
consumers’ mortality rates through an exponential filter (Gillooly et al. 2001; Vasseur & Fox
2007):
where
is the mortality rate at time t,
is the medial mortality rate and
is the environmental variation at time t for consumer k in patch i.
Until this point, our method is the same as in Vasseur and Fox (2007). We have added an
additional index, i, to the differential equation system (Fig. 1) which represents the spatial
dimension and hence the patch number. Furthermore we have included the environmental
variation filter (eq 1). The second part of this method description incorporates our additional
components, starting off with colour in the environmental variation. At first, uncorrelated
white environmental variation was generated from a random normal distribution with zero
mean and σenv2 variance. Thereafter, we used Fourier transform to add colour to the
environmental variation. The discrete Fourier transform of the coloured environmental 1/f
noise, P(ƒ), was scaled according to:
where ƒ is frequency, X(ƒ) is the discrete Fourier transform of the previously generated white
environmental variation and the colour of P(ƒ) was determined by the value of the spectral
exponent, γenv, where γenv = 0 gives white and γenv > 0 gives red noise. After colouring the
time series, inverse Fourier transform was used on P(ƒ) to generate the coloured
environmental variation, env(t). The food web model was integrated across a range of σenv, 0
to 0.6 in steps of 0.05, and γenv, 0 to 0.6 in steps of 0.2.
In order to determine the effect of dispersal between spatially separated subpopulations,
the measurements in our study were taken from one patch in the landscape. Patches,
containing the food web, were either isolated or connected with the other patches by dispersal.
Dispersal between subpopulations was governed by a mass-action mixing process without
distance dependence. Subpopulations with dispersal were connected through a dispersal
matrix and their dynamics were described by a differential equation system (modelled after
Caswell 2001 and Wennergren et al. 1995):
where Popds,i is the density of species s in patch i when all subpopulations are connected with
dispersal. s is an element of the set {P, C1, C2, R} and i is an element of the set {1, 2,..., 6}.
dPops,i/dt is the differential equation for species s in patch i when isolated without dispersal
(Fig. 1). L is the total number of patches, L=6. di,j is the proportion of the subpopulation in
patch i that migrate to patch j in one time step. Migrating proportions, di,j, were generated
from a random normal distribution with mean 1/L and variance 0.2/ L. The distribution was
truncated by 0 and 1.2/ L.
The time series of environmental variation affecting the consumers were cross-correlated,
with ρenv = -1, 0 or 1. In addition to varying the cross-correlation of environmental variation
affecting the two consumer species, as in Vasseur & Fox 2007, we have added differences in
cross-correlation of environmental variation affecting subpopulations of the same species.
ρenv = -1 represented perfect negative cross-correlation between all pairs of time series
affecting different consumer species. All subpopulations within the same species were
affected by the same environmental time series. For ρenv = 0, both consumer species and all
their subpopulations were affected by unique independent environmental time series. ρenv = 1
represented perfect positive cross-correlation, both consumer species and all their
subpopulations were affected by the same time series of environmental variation.
Simulations were made in MATLAB 7.5.0 (R2007b, The Mathworks, Natick, MA, USA)
with 100 replicates and 3000 time-steps. Initial subpopulation densities where chosen on the
same uniform interval; 0.1 to 1.0, as in Vasseur and Fox (2007). Extinction risk was
calculated as the risk of populations decreasing below the extinction boundary 10-6 and by
how many replicates that had all subpopulations staying above the extinction boundary until
the end of the simulation. We chose to use the same extinction boundary, for isolated
subpopulations, as in Vasseur and Fox (2007). With dispersal, populations were considered to
decrease below the extinction boundary when the sum of all six subpopulations within
species decreased below 10-6. Replicates with extinctions were only analysed in respect to
extinction risk. The first quarter of the simulated time series was excluded from analysis to
avoid initial transients. Mean, variance and stability of subpopulation density, species density
and food web biomass, consumer synchrony and extinction risk were calculated for each of
the combinations of varied parameters. Food web biomass was the sum of all species
densities in one patch. Stability was measured as in Vasseur and Fox (2007), through:
where CV is the coefficient of variation, σx the standard deviation and μx the mean of
population x’s density time series. Consumer synchrony was calculated as in Vasseur and
Fox (2007), through:
where T is time series length, σk standard deviation and μk mean of consumer species k’s
time series. The cross-correlation between each consumer and its environmental variation
was calculated as equation (5), when ρenv =1, in order to evaluate the impact of environmental
variation on each consumer.
Results
The magnitude of environmental variance was of great importance for food web stability and
extinction risk. Weak-to-moderate variance lowered variability of biomass and all species
densities, except the resource, whereas higher variance destabilises the system (Fig. 2a, d, Fig.
3a). Compare results regarding the stability of C1, C2 and P (Fig. 2a) with Fig. 2b, c and d in
Vasseur and Fox (2007). The standard deviation of environmental variation, σenv, generating
maximum stability, was species specific. C1 and P gained their maximum stability from
higher σenv than C2 and R. The same pattern was found for each value of cross-correlation of
environmental variation, ρenv. Reddening of the environmental variation decreased the
stabilising effect of weak-to-moderate σenv and enhanced the destabilising effect of higher
σenv. In addition, it lowered the σenv values generating maximum stability (Fig. 2d). Dispersal
had minor affect during correlated environmental variation (Fig. 3). However, during
uncorrelated environmental variation, the stabilising effect of weak-to-moderate σenv was
enhanced and the destabilising effect of higher σenv was reduced with dispersal (Fig. 2d, Fig.
3). Studies on time series of biomass and species abundances revealed that addition of
dispersal between subpopulations resulted in maintenance of intrinsic dynamics during
moderate σenv. The stable limit cycles where not as apparent in isolated patches during the
same environmental variance (Fig. 4).
Mean food web biomass decreased and biomass variance increased with increasing σenv
(Fig. 2e, f), regardless of ρenv. However, a constant environment did not give the lowest
variance in biomass. Weak-to-moderate σenv actually resulted in a minor decrease in biomass
variance. Measurements on time series of species densities showed that the value of σenv
affected the relative abundance of species (Fig. 2b). Mean density of the species with the
smallest population in constant environment, C1, increased with increased σenv. In contrast to
C1, high σenv decreased mean density and resulted in a major increase in variance for the
largest species in constant environment, C2 (Fig. 2c). Mean density of R increased whereas
the mean of P decreased with increased σenv. Reddening of the environmental variation
enhanced the effects of increased σenv on biomass (Fig. 2e, f) and each species (results not
shown). The same change in relative species abundance occurred, but for lower values of σenv.
Dispersal coupled with uncorrelated environmental variation reduced the effects of increasing
σenv on food web biomass (Fig. 2e, f) and species densities (results not shown).
Subpopulation extinction risk increased with increased σenv, regardless of the value of ρenv.
ρenv = -1 gave the highest extinction risk whereas ρenv =1 gave the lowest. A similar pattern
was found for each species, where C2 showed the highest sensitivity to increased σenv.
Reddening of the environmental variation increased population extinction risk whereas
dispersal coupled with uncorrelated environmental variation reduced the risk of extinction.
Consumer synchrony increased with increased σenv, regardless of ρenv. Results are in line
with Vasseur and Fox (2007). Reddening of the environmental variation enhanced this effect
whereas dispersal coupled with uncorrelated environmental variation reduced the
synchronising effect of increased σenv. Both consumers become increasingly negatively
correlated with their environmental variation during weak-to moderate σenv. However, results
differed for σenv values above 0.3. The negative correlation between C1 and the environmental
variation continued to increase while the negative correlation between C2 and environmental
variation started to decrease for higher σenv. Reddening of the environmental variation
amplified the effect whereas dispersal coupled with uncorrelated environmental variation
decreased the effect of increased σenv. The pattern of differences in correlation was retained
for all different scenarios tested.
Discussion
The diamond shaped food web was first used by McCann et al. (1998) to show stabilising
effects of consumer asynchrony in constant environments. Vasseur and Fox (2007) simulated
the same food web and investigated the effects of environmental variation. By simulating the
same model, used in these two well done studies, our aim was to clarify the implications of
coloured environmental variation and spatial structure on the stability of food webs. We show
that redness decreases the stabilising effect of environmental variation whereas dispersal
between spatially subdivided populations increases the stability of the system. In addition of
using the same stability analysis as in Vasseur and Fox (2007), we also include direct analysis
of mean and variation of densities.
We initiate our investigation by confirming results of Vasseur and Fox (2007): (i) weakto-moderate environmental variation increases the system’s stability coefficient, 1/CV, by
dampening predator fluctuations, (ii) stronger environmental variation increases the
variability of densities, destabilising the system. Results from our study reveal that measuring
stability only by 1/CV can be misleading. When environmental variation increases the
stability coefficient it also causes a decrease in food web biomass (Fig. 2e). Decreased
biomass implies increased system sensitivity to demographic stochasticity and catastrophes
with increased extinction risks as a result (Lande 1993). Independent studies of mean and
variance of densities also show that variation over time can shift the relative abundance of
species in the food web, increasing the density of the smallest population, C1 (Fig. 2b). The
shift is caused by C1 having a better ability to make use of the resource than C2 during high
environmental variance. In contrast to C2, C1 was increasingly negatively correlated with the
environmental variation during the whole interval of increasing σenv. C2:s poor resource
tracking abilities resulted in high density variance. The high variance gave C2 the highest
extinction risk at high σenv, despite being the species with the largest mean density. Shifts in
relative abundance with increased σenv indicate that addition of stress factors, such as
catastrophes and demographic stochasticity, may affect the relationship between extinction
risk and environmental variance. Moderate environmental variation may decrease system
extinction risks by increasing the density of the species with the lowest population in a
constant environment. Further studies including these mechanisms may clarify the effect of
environmental variation and the importance of multiple measures when analysing food web
stability and extinction risk.
The second phase of our investigation was to add additional components to the diamond
shaped food web. Environmental variation is considered to be positively correlated in time
(Caswell & Cohen 1995; Halley 1996; Ripa & Lundberg 1996; Cuddington & Yodzis 1999).
Positively correlated, red, variation is dominated by low frequencies. This property results in
bad/good conditions being retained for several time steps. Red variation gives populations
more time to respond to differences in their environment, increasing the probability of
environmental fluctuation tracking (Ripa & Lundberg 1996). The stabilising power of weakto-moderate environmental variation was reduced and extinction risks where increased with
increased redness. These results are explained by reddened environmental variation causing
larger density variance than white environmental variation (Fig. 2f), compare with Greenman
and Benton (2005). Cuddington and Yodzis (1999) support our results by showing that
reddening of variation can decrease mean persistence time in overcompensating single
population models. Reddening of the environmental variation also amplified the shift in
relative abundances of species and increased consumer synchronisation. That redness may
increase the positive correlation between populations has also been shown in Greenman and
Benton (2005). Reduced stabilizing effects and increased extinction risks caused by redness
reduce the importance of environmental variation as an important stabilising property of food
webs. However, addition of dispersal between subdivided populations re-emphasizes the
importance of abiotic variability.
Dispersal had a strong stabilising effect during uncorrelated environmental variation (Fig.
2d, Fig. 3, Fig. 4). Individuals from patches with good conditions were able to migrate to
patches with poor conditions, compare with result of Engen et al. (2002) or Liebhold et al.
(2004). The migration undermined consumer synchronisation and evened out destabilising
effect of environmental variation. The equalising effect caused by dispersal had major
implications for food web stability and extinction risks. The food web with dispersal affected
by highly red environmental variation was more stable than the food web in an isolated patch
with white variation (Fig. 2d). Extinction risks with dispersal were actually close to zero, for
environmental variance and redness used in our study. Higher σenv values generated similar
destabilising effects of redness as in the case with isolated subpopulations. Kaitala et al.
(1997) supports our results by showing that increased system complexity can reduce the
effect of redness. Engen et al. (2002) showed that increasing dispersal between patches,
withholding single species, results in longer time to extinction. Mass action mixing, as in our
study, has no distance dependence. This infers similar probabilities of dispersal between all
patches. The assumption can be far from dispersal found in nature. However, results from
Petchey et al. (1997) showed minor differences in population persistence when comparing
landscapes with regional and local dispersal. Despite the lack of distance dependence, a
minor increase in stability was observed in some patches when adding dispersal during
correlated environmental variation. This effect can be explained by our dispersal matrixes
generation method causing a variance in dispersal rates between patches. The small dispersal
rate variance enables a rescue effect.
The presented food web stabilities and extinction risks are measured at patch level. It is
important to consider the differences between patch and landscape level when estimating
food web resistance. The choice of scale will have major effect on estimated extinction risks.
Our complementary measurements on landscape level showed that the stability, 1/CV, of the
food web affected by σenv < 0.6 was much higher without dispersal than with dispersal,
independent on ρenv. This appears to contradict our results obtained on patch level which
showed that dispersal coupled with uncorrelated environmental variation lowers the
extinction risks in the food web affected by σenv < 0.6. The differences between scales are
caused by the spatial dimension affecting subpopulation synchrony. Isolated subpopulations,
without dispersal, fluctuate in their own phase depending on initial densities and
environmental variation. When measuring densities on landscape level, we take the sum of all
subpopulation for each time step. The asynchrony between patches makes the summation of
density time series to even out the fluctuations over time. This gave the landscape density
time series a low variance and hence generated a high overall stability. With dispersal,
subpopulations that initially fluctuate in their own phase became more synchronised by
migration between patches (Bjørnstad et al. 1999). When taking the sum of such
subpopulations, the variance of the time series will be more preserved resulting in a higher
variance producing a lower stability. Our results implicate that if stability is measured by
variance, as in CV, one have to be assured that the time series are not a summation of sets of
local time series. We anticipate that this may become a problem in empirically work since it
can be cumbersome to discriminate between subpopulations.
The addition of coloured environmental variation and spatial structure had major
implications on the stability and extinction risk of the diamond shaped food web. Redness
decreased the stabilising effect of environmental variation whereas dispersal, coupled with
uncorrelated response, stabilised the system. Dispersal increased the stability by increasing
mean biomass and lowering the variance of densities. Weak-to-moderate environmental
variation actually decreased mean biomass in the same time as it increased the value of the
stability coefficient, 1/CV. Single measures of stability did not show the full picture.
Environmental variation also caused a change in the relative abundance of species increasing
the density of the species with the smallest population in a constant environment. The food
web affected by moderate environmental variation would actually be more resistant to
additional stresses, such as demographic stochasticity and catastrophes, than the same food
web situated in a constant environment. However, shifts in relative abundances of species
may have unexpected implications for species with present large population sizes. Large
populations today may not insure species against future increase in environmental variance.
Interaction pathways, exemplified in our study, have been shown to repeat at different
resolutions, making food web stability scale invariant (McCann 2009). Our model may be
seen as a building block for more complex food webs indicating that dispersal coupled by
variability in space and time can be the missing component in theory explaining the existence
of large and diverse food webs.
Acknowledgements
…
Figure and table legends
Figure 1 The diamond shaped food web with differential equation system (McCann et al.
1998; Vasseur and Fox 2007). The equation system is the same as in Fig. 1 in Vasseur and
Fox (2007) except for our additional index, i, the patch number. P is the density of the top
predator, C1 first consumer, C2 second consumer, R resource and Ωa,b, is the consumption
preference of species a for species b.
Table 1 Parameter explanation and their values. The constant parameters above the splitting
line are the same as in Table 1 in Vasseur and Fox (2007). The food web model was
integrated across a range of σenv (0 to 0.6 in steps of 0.05), γenv (0 to 0.6 in steps of 0.2) and
ρenv (-1, 0 and 1).
Figure 2 Stability, μ/σ, mean, μ, and variance, σ2, for species population densities and food
web biomass with environmental fluctuation strength, σenv, and uncorrelated environmental
variation, ρenv=0. Left column; measurements on species population densities with white
environmental variation, γenv=0, without dispersal. P is predator, C1 first consumer, C2 second
consumer and R resource. Right column; measurements on food web biomass with white and
coloured environmental variation of γenv=0-0.6, without and with (crosshatch lines) dispersal.
* P, C1 and C2 density stability as when ρenv=0 in Fig. 2b, c and d in Vasseur & Fox (2007).
Figure 3 Stability of food web biomass, μB/σB, with standard deviation of environmental
variation, σenv, and cross-correlation of environmental variation, ρenv. a) isolated patch b)
patch connected by dispersal.
Figure 4 System responses to continual synchronous point perturbations with standard
deviation of environmental variation σenv= 0.3 and uncorrelated environmental variation,
ρenv=0. The patch that is connected with the other patches by dispersal maintains the intrinsic
dynamics of the food web. * as in Vasseur & Fox (2007).
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