Title: Spatially structured food webs in coloured environments
Authors: Sara Gudmundson, Frida Lögdberg, Uno Wennergren
Affiliation: IFM, Theory and Modelling, Linköping University, Linköping, Sweden
Email addresses: sargu@ifm.liu.se, friwa@ifm.liu.se, unwen@ifm.liu.se
Running title: Spatially structured food webs
Keywords: Asynchrony, coloured environmental variation, dispersal, extinction risk, food
web, stability, subpopulation, synchrony.
Type of article: Letter
No. words in the abstract: 147
No. words in the manuscript as a whole: 5483
No. words in the main text: 4141
No. references: 33
No. figures: 4
No. tables: 1
Correspondence: Assoc. Prof. Uno Wennergren, +46 13 281666, unwen@ifm.liu.se
1
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 1/CV, could be misleading. It does not reflect 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 basis for explaining the existence of large
and diverse food webs.
Keywords: Asynchrony, coloured environmental variation, dispersal, extinction risk, food
web, stability, subpopulation, synchrony.
2
Introduction
Natural food webs contain complex interaction patterns evolved through historical processes
in dynamic environments (May 1973). Diverse communities are expected to have less
temporal variability in biomass than species-poor counterparts (Odum 1971). However,
theoretical studies predict that large complex food webs should be unstable and extinctionprone 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; Vasseur et al. 2005;
Benincà et al. 2009; McCann & Rooney 2009). 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 1/CV, can imply an increase in mean density and a decrease in variance,
3
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; 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 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
(Kareiva & Wennergren 1995; 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
4
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. One has to identify each subpopulation specific
environmental variation and their relation when modelling subpopulations in a landscape. 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
5
understanding of how environmental variation affects how species fluctuate in relation to
each other.
This study addresses how the stability of food webs are 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. A food web in such moderate environment would be
more resistant to additional stresses, such as demographic stochasticity and catastrophes, yet
the stability, measured as 1/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.
6
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):
i
M Ci k (t )  M Ck (0)e envk (t )
(1)
where MiCk(t) is the mortality rate at time t, MCk(0) is the medial mortality rate and envik(t) 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. The spatial dimension is also included in the
environmental variation filter (eqn 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
7
to add colour to the environmental variation. The discrete Fourier transform of the coloured
environmental 1/f noise, P(ƒ), was scaled according to:
P( f )  X ( f ) f  en v
2
(2)
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):

L
dPopsd,i dPops ,i

  dij Popsd, j  d ji Popsd,i
dt
dt
j i

(3)
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). The migration rate of subpopulation j to patch i, dij, was set to 0.3/(L-1). L is the
total number of patches, L=6.
8
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).
Initial densities for undisturbed isolated food webs where drawn from the same uniform
interval; 0.1 to 1.0, as in Vasseur and Fox (2007). These systems were integrated for 3000
time steps. The last 500 time steps, with stable limit cycles, were used for generating initial
densities for simulations incorporating environmental variation and dispersal. Simulations
with initial densities from the stable limit cycles were run for 6000 time steps with 50
replicates. The first 3000 time-steps were excluded in order to avoid initial transients.
Extinction risk was calculated as the risk of any population decreasing below the extinction
boundary 10-6 during the time interval 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. Mean, variance and stability of subpopulation
density, species density and food web biomass, consumer synchrony and extinction risk were
9
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:
1

 x
CV  x
(4)
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:
C 
1
N C1 C 2
 C (t )   C (t )   
T
t 1
1
C1
2
(5)
C2
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 in eqn 5, with 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 P, C1 and C2 (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
10
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
11
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
12
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) weak-tomoderate 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.
13
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
14
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 and can be further
developed with heterogeneous landscapes and dispersal kernels (Lindström et al. 2008).
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 (Kareiva & Wennergren 1995). 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
15
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 an artificially 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 1/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
16
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 & Rooney 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
…
17
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).
Parameter
Description
Value
r
Resource intrinsic rate of growth
1.0
K
Resource carrying capacity
1.0
JC 1
Consumer (C1) ingestion rate
0.8036
JC 2
Consumer (C2) ingestion rate
0.7
JP
Predator ingestion rate
0.4
MC1(0)
Medial consumer (C1) mortality rate
0.4
MC2(0)
Medial consumer (C2) mortality rate
0.2
MP
Predator mortality rate
0.08
R01
Half saturation constant
0.16129
R02
Half saturation constant
0.9
C0
Half saturation constant
0.5
ΩPC1
Preference coefficient
0.92
ΩC1R
Preference coefficient
1.0
ΩC2R
Preference coefficient
0.98
σenv
Standard deviation of environmental variation
0 - 0.6
γenv
Colour of environmental variation
0 - 0.6
ρenv
Cross-correlation of environmental variation
-1, 0, 1
18
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.
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. σenv=0.3 gave the highest predator stability when patches were
connected. * as in Vasseur & Fox (2007).
19
References
Benincà, E., Jöhnk, K. D., Heerkloss, R. & Huisman, J. (2009). Coupled predator–prey
oscillations in a chaotic food web. Ecol. Lett., 12, 1367-1378.
Bjørnstad, O. N., Ims, R. A. & Lambin, X. (1999). Spatial population dynamics:
Analyzing patterns and processes of population synchrony. Trends Ecol. Evol., 14, 427432.
Borrvall, C. & Ebenman, B. (2008). Biodiversity and persistence of ecological
communities in variable environments. Ecol. Comp., 5, 99-105.
Caswell, H. & Cohen, J. (1995). Red, white and blue: Environmental variance spectra
and coexistence in metapopulations. J.Theor.Biol., 176, 301-316.
Caswell, H. (2001). Matrix population models. 2:nd ed. Sinauer Associates Inc.
Publishers, Sunderland, Massachusetts.
Cuddington, K. M. & Yodzis, P. (1999). Black noise and population persistence.
Proc.Biol.Sci., 266, 969-973.
Dickie, L. M., Kerr, S. R. & Boudreau, P. R. (1987). Size-dependent processes
underlying regularities in ecosystem structure. Ecol.Monogr., 57, 233-250.
Easterling, D. R., Meehl, G. A., Parmesan, C., Changnon, S. A., Karl, T. R. & Mearns, L.
O. (2000). Climate extremes: Observations, modeling, and impacts. Science, 289, 20682074.
Engen, S., Lande, R. & Saether, B. E. (2002). The spatial scale of population
fluctuations and quasi-extinction risk. Am.Nat., 160, 439-451.
Fontaine, C. & Gonzalez, A. (2005). Population synchrony induced by resource
fluctuations and dispersal in an aquatic microcosm. Ecology, 86, 1463-1471.
Gillooly, J. F., Brown, J. H., West, G. B., Savage, V. M. & Charnov, E. L. (2001).
Effects of size and temperature on metabolic rate. Science, 293, 2248-2251.
Green, D. G. & Sadedin, S. (2005). Interactions matter - complexity in landscapes and
ecosystems. Ecological Complexity, 2, 117-130.
Greenman, J. V. & Benton, T. G. (2005). The frequency spectrum of structured discrete
time population models: Its properties and their ecological implications. Oikos, 110,
369-389.
Halley, J. M. (1996). Ecology, evolution and 1/f-noise. Trends Ecol. Evol., 11, 33-37.
Halley, J. M. & Dempster, J. P. (1996). The spatial population dynamics of insects
exploiting a patchy food resource: A model study of local persistence. J. Appl. Ecol., 33,
439-454.
20
Kaitala, V., Ylikarjula, J., Ranta, E. & Lundberg, P. (1997). Population dynamics and the
colour of environmental noise. Proc.Biol.Sci., 264, 943-948.
Kareiva, P. & Wennergren, U. (1995). Connecting landscape patterns to ecosystem and
population processes. Nature, 373, 299-302.
Lande, R. (1993). Risks of population extinction from demographic and environmental
stochasticity and random catastrophes. Am.Nat., 142, 911-927.
Liebhold, A., Koenig, W. D. & Bjornstad, O. N. (2004). Spatial synchrony in population
dynamics. Ann. Rev. Ecol. Evol. Syst., 35, 467-490.
Lindström, T., Håkansson, N., Westerberg, L., Wennergren, U. (2008). Splitting the tail
of the displacement kernel shows the unimportance of kurtosis. Ecology, 89, 1784–1790.
May, R. M. (1973). Stability and Complexity in Model Ecosystems. Princeton
University Press, Princeton.
McCann, K. S. (2000). The diversity–stability debate. Nature, 405, 228-233.
McCann, K. S. & Rooney, N. (2009). The more food webs change the more they stay the
same. Phil. Trans. R. Soc. B., 364, 1789-1801.
McCann, K. S., Hastings, A. & Huxel, G. R. (1998). Weak trophic interactions and the
balance of nature. Nature, 395, 794-798.
Odum, E. P. (1971). Fundamentals of ecology, 3:rd ed. W. B. Saunders Company, USA.
Petchey, O. L., Gonzalez, A. & Wilson, H. B. (1997). Effects on population persistence:
The interaction between environmental noise colour, intraspecific competition and space.
Proc.Biol.Sci., 264, 1841-1847.
Polis, G. A. (1991). Complex trophic interactions in deserts: An empirical critique of
food-web theory. Am.Nat., 138, 123-155.
Ripa, J. & Lundberg, P. (1996). Noise colour and the risk of population extinctions.
Proc.Biol.Sci., 263, 1751-1753.
Tilman, D. (1999). The ecological consequences of changes in biodiversity: A search for
general principles. Ecology, 80, 1455-1474.
Vasseur, D. A. & Fox, J. W. (2007). Environmental fluctuations can stabilize food web
dynamics by increasing synchrony. Ecol. Lett., 10, 1066-1074.
Vasseur, D. A., Gaedke, U. & McCann, K. S. (2005). A seasonal alternation of coherent
and compensatory dynamics occurs in phytoplankton. Oikos, 110, 507-514.
Wennergren, U., Ruckelshaus, M. & Kareiva, P. (1995). The promise and limitations of
spatial models in conservation biology. Oikos, 74, 349-356.
21
Yodzis, P. & Innes, S. (1992). Body size and consumer-resource dynamics. Am.Nat.,
139, 1151-1175.
22