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: 5512
No. words in the main text: 4168
No. references: 34
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 stabilise 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. A second explanation could be that the stability analysis itself may not be
sufficient. To address the question of important components, theoretical studies have started
to include factors such as density regulation, asymmetric interaction pathways, environmental
autocorrelation and dispersal (Polis 1991; McCann et al. 1998; Vasseur et al. 2005; Benincà
et al. 2009; McCann & Rooney 2009). These components are known to affect local
population dynamics and should be included when studying population dynamics and
extinction risks (Engen et al. 2002). In this study, we will take yet another important step by
investigating the stabilising role of asynchrony in the spatial dimension. As a food web
model we chose the ‘diamond shaped’ food web. This model is stabilised by consumer
asynchrony, resulting in stable oscillations during constant environment (McCann et al. 1998).
Given these oscillations in stable environments there is a potential that temporal fluctuations
in time and space may stabilize or destabilize the system. Vasseur and Fox (2007) tested the
effect of temporal fluctuations and identified the important results that weak-to-moderate
environmental variation actually stabilised the food web. The next step is to incorporate space
and we will study whether the stability induced by consumer asynchrony and weak-tomoderate environmental variation still holds, or is even more pronounced, when adding two
important components; (i) coloured (autocorrelated in time) environmental variation and (ii)
3
food web dynamics with a spatial dimension, incorporating regional dispersal between
patches. Furthermore, we address the question of sufficient stability analysis by performing a
thorough analysis of population dynamics.
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,
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
4
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
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
5
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 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
components. Vasseur and Fox (2007) showed that weak-to-moderate environmental variation
can stabilise the ‘diamond shaped’ food web. In this study, we show that redness decrease the
stabilising effect of environmental variation whereas regional dispersal, coupled with
uncorrelated response, has a strong stabilising effect. While dispersal increased stability by
increasing mean biomass and lowering the variance of densities, weak-to-moderate
environmental variation alone 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
6
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.
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):
7
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
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,
measurements in our study were taken both on local and regional scale. Patches, containing
the food web, were either isolated or connected with the other patches by dispersal. Dispersal
8
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.
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
9
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, 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 last 3000 time-steps of these time series were used in the analysis. For isolated
subpopulations, 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, 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 calculated for
each of the combinations of varied parameters. Food web biomass was the sum of all species
densities in one patch, for local scale, and in all patches, for regional scale. 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.
10
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
σenv. In addition, it lowered the σenv values generating maximum stability (Fig. 2d). Dispersal
had minor affect on local stability during correlated environmental variation (Fig. 3).
However, during uncorrelated environmental variation, the stabilising effect of weak-tomoderate σenv was enhanced and the destabilising effect of higher σenv was reduced with
dispersal (Fig. 2d, Fig. 3). Complementary measurements on food webs with high migration
rate (dij = 0.8) showed minor quantitative differences whereas low dispersal rates (dij = 0.004)
enhanced the stabilising effect of weak-to-moderate σenv. Surprisingly, the regional variance
of the food web affected by σenv < 0.6, was lower without dispersal than with dispersal,
independent on ρenv. This gave food webs with isolated patches a higher regional stability,
1/CV, than food webs with connected patches. Studies on time series of biomass and species
abundances revealed that addition of dispersal between subpopulations resulted in
11
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
12
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 > 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 McCann et al. (1998) and Vasseur and Fox (2007), 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 regional dispersal may 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) 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
13
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 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
14
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
15
effect of redness. Engen et al. (2002) showed that increasing dispersal between patches,
withholding single species, results in longer time to extinction. Gouhier et al. (2010) studied a
discrete version of the ‘diamond shaped’ food web, investigating how local, nearestneighbour, dispersal affected food web response in a spatially uniform environment. They
conclude that local dispersal may be a contributing factor determining food web response to
uniform environmental fluctuations. Results indicate that differences in dispersal rates will
have large effect on the capacity of correlated environmental variation to disrupt
compensatory dynamics. This phenomenon is not evident in our study. Results of Gouhier et
al. (2010) are dependent on one specific dispersal regime giving rise to autonomous
fluctuations on the regional scale, in constant environment. The results are difficult to
compare with results of Vasseur and Fox (2007) and ours because of differences between the
difference equation system used in Gouhier et al. (2010) and the differential equation systems
used in McCann et al. (1998), Vasseur and Fox (2007) and our current study. The minor
differences in results between migration rate 0.3 and 0.8, found in our study, depend on a
small actual difference in migrating proportion. Note that such migrating proportion can be
approximated from migration rate in analogue with the conversion of intrinsic to discrete per
capita rate of growth. Although our study is not mainly studying the effect of different
migration rates we also performed a study on very low migration rate since it ought to render
slightly other dynamics and may explain some findings of Gouhier et al. (2010). The effect
of low dispersal was as expected from general theory: the synchronising potential of
dispersal diminishes and patches are as unsynchronized as the environmental variation over a
long transient period. The length of the transient period will depend on the dispersal rate and
as the rate approaches zero the length of the transient period approaches infinity. Hence, low
dispersal coupled with weak-to-moderate environmental variation reduces the amplitude of
16
the stable limit cycles, lowering variability and increase stability. These broad and general
results are fairly close to the findings in Gouhier et al. (2010) more specific study.
We chosed to use mass action mixing in our study although it is not close to most natural
scenarios, one may for example expect distance dependence. The mass action mixing infers
similar probabilities of dispersal between all patches. However, we find it to be a good
starting point when investigating general effects of dispersal itself without confounding
effects of distance dependence and aggregational patterns etc. Our approach can be further
developed towards more specific spatio temporal systems with heterogeneous landscapes and
dispersal kernels (Lindström et al. 2008). Results from Petchey et al. (1997) showed minor
differences in population persistence when comparing landscapes with local and regional
dispersal.
The primary focus of this paper has been measurements on local scale. However, it is
important to consider differences between local and regional scale when estimating food web
resistance (Kareiva & Wennergren 1995). The choice of scale will have major effect on
estimated extinction risks. Results obtained on local scale, showing a stabilising effect of
dispersal with uncorrelated environmental variation, appear to contradict measurements on
regional scale. 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 regional scale, 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 fluctuations over
time. This gave the regional density time series an artificially low variance and hence
generated a high overall stability. With dispersal, subpopulations that initially fluctuate in
17
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 empirical 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 & Rooney 2009). Our model may be seen as a building
block for more complex food webs indicating that dispersal coupled by variability in space
18
and time can be the missing component in theory explaining the existence of large and
diverse food webs.
Acknowledgements
…
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
19
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
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 Local 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)
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dispersal. * P, C1 and C2 density stability as when ρenv=0 in Fig. 2b, c and d in Vasseur & Fox
(2007).
Figure 3 Local 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).
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