Spatially structured food webs in a coloured environment
Sara Gudmundson, Frida Lögdberg and Uno Wennergren*
Department of Theoretical Biology, Linköping University, Sweden, *Correspondence: Email: unwen@ifm.liu.se
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 1974;
(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). In this study, we
add coloured environmental variation and spatial structure to the diamond shaped food web.
The system 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).
When investigating how coloured environmental variation and spatial structure affect the
stability of the food web we also perform a thorough analysis of the 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 have evaluated 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 inhabiting patches. Dispersal between subpopulations
enables re-establishment of extinct patches which can prolong the whole population’s time to
extinction (Engen et al. 2002; Liebhold et al. 2004; Greenman & Benton 2005). In order to
investigate stabilizing properties of dispersal between subpopulations, we position our food
web in six spatially separated patches connected with dispersal. When modelling
subpopulations in a landscape incorporating variable conditions, one also has to specify each
subpopulation specific environmental variation. Is all subpopulations affected by the same
environmental variation or are they affected by different environmental variation depending
on the patch specific conditions? The landscape dimension includes not only the
subpopulations themselves but also one have to define how stochasticity is distributed in
space. Furthermore, each patch incorporates a food web of four species. Different species
situated in the same patch may respond differently to the same environmental variation. 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. The cross-correlation of environmental variation will affect how
populations fluctuate in relation to each other. This property is called population synchrony
(reviewed in Bjørnstad et al. 1999). It can be measured both between species and between
subpopulations. In this study, we have only measured 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 have measured synchrony
between species, according to Vasseur and Fox (2007). In addition, we have measured 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 simulated 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 where as 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, 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. This food web would be more resistant to additional
stresses, such as demographic stochasticity and catastrophes, yet the stability measured as CV is
reduced compared 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 Vasseur and Fox (2007). These biologically plausible parameters are
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.
Figure 1 The diamond shaped food web with differential equation system (McCann et al.
1998; Vasseur and Fox 2007). P is the density of the top predator, C1 first consumer, C2
second consumer, R the resource species and Ωi,j, is the consumption preference of species i
for species j.
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 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):
MCi (t) = MCi (0)eenvi (t)
(2)
where MCi(t) is the mortality rate at time t, MCi(0) is the medial mortality rate, envi(t) is the
environmental variation at time t for consumer i.
Until this point, our method is the same as in Vasseur and Fox (2007). The second part
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)|2 𝑓 −𝛾
(1)
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, γ, where γ = 0 gives white and γ > 0 gives red noise. After colouring the time series,
inverse Fourier transform was used on P(ƒ) to generate the coloured environmental variation,
envi(t). The food web model was integrated across a range of σenv, 0 to 0.6 in steps of 0.05,
and γ, 0 to 0.6 in steps of 0.2.
In order to determine the effect of dispersal between spatially separated subpopulations, all
measurements in our study were taken from one patch of 6 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 continuous-time differential equation system
represented in Fig. 2 (Caswell 2001 and Wennergren et al. 1995).
Figure 2 Differential equation and dispersal matrix for each species specific subpopulation
connected with dispersal (modelled after Caswell 2001 and (Wennergren et al. 1995). dSi/dt
is the differential equation for species, S, and subpopulation, i, without dispersal (Fig. 1). P=6,
the total number of patches, and dij represents the proportion of the subpopulation in patch i
that migrates to patch j in one time step.
Migrating proportions, dij, were generated from a random normal distribution with mean 1/P
and variance 0.2/P where P=6, the total number of patches. The distribution was truncated by
0 and 1.2/P. The same dispersal matrix was used for all four species.
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 as in Vasseur and
Fox (2007). With dispersal, populations were considered to decrease below the extinction
boundary when the sum of all 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 patch 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 subpopulations. Stability was measured as
in Vasseur and Fox (2007), through:
1
𝜇𝑖
=
𝐶𝑉
𝜎𝑖
(3)
where CV is the coefficient of variation, σi the standard deviation and μi the mean of
population i’s density time series. Consumer synchrony was calculated as in Vasseur and Fox
(2007), through:
𝜌𝐶 =
1
𝑁𝜎𝐶1 𝜎𝐶2
𝑁
∑(𝐶1 (𝑡) − 𝜇𝐶1 ) (𝐶2 (𝑡) − 𝜇𝐶2 ) (4)
𝑡=1
where N is time series length, σi standard deviation and μi mean of consumer species i’s time
series. The cross-correlation between each consumer and its environmental variation was
calculated as equation (4), 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. 3a, d, Fig.
4a). The results regarding the stability of C1, C2 and P (Fig. 3a) are in line with 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. 3d). Dispersal had minor affect
during correlated environmental variation (Fig. 4). 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. 3d, Fig. 4). 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. 5).
Mean food web biomass decreased and biomass variance increased with increasing σ env (Fig.
3e, 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. 3b). Mean density of the species with the smallest
population in constant environment, C1, increased with increased σenv (Fig. 3c). 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. Mean density of R increased where as the mean
of P decreased with increased σenv. Reddening of the environmental variation enhanced the
effects of increased σenv on biomass (Fig. 3e, f) and each species. 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. 3e, f)
and species densities.
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 where as
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
where as 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 where as 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.
Figure 3 Stability, mean and variance 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 density with white environmental
variation of γenv=0, without dispersal. P is predator, C1 first consumer, C2 second consumer
and R resource. Right column; measurements on food web biomass with coloured
environmental variation of γenv=0-0.6, without and with (crosshatch lines) dispersal. * as in
Vasseur & Fox (2007).
Figure 4 Stability of food web biomass with standard deviation of environmental variation,
σenv and cross-correlation of environmental variation, ρenv. a) isolated patch b) patch
connected by dispersal.
Figure 5 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 by dispersal with the other patches maintains the intrinsic
dynamics of the food web. * as in Vasseur & Fox (2007).
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) 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. 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. 3b). 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 found in nature 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
environmental 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 weak-to-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. 3f). The same results have been found by 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.
3d, Fig. 4, Fig. 5). Individuals from patches with good conditions were able to migrate to
patches with poor conditions, compare with for example the 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 actually more stable than the food web in
an isolated patch with white variation. Extinction risks with dispersal were actually close to
zero, for environmental variance and redness used in our study. However, 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.
Food web stability and extinction risk were measured at patch level in this study. 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.
Results from our study showed that the stability of the food web affected by positively
correlated environmental variation was much higher without dispersal than with dispersal
when measured on landscape level. This contradicts our results obtained on patch level which
show that dispersal lowers extinction risks. Without dispersal, subpopulations will fluctuate
in their own phase, depending on initial densities and environmental variation. When taking
the sum of all subpopulations, the asynchrony will be even out differences resulting in a
reduced variance and increased landscape stability. With dispersal, subpopulations that
initially fluctuate in their own phase may become synchronised by the migration between
patches. When taking the sum of all subpopulations, the variance will be preserved resulting
in a higher variance and lower stability than without dispersal. Time lagged dispersal, more
close to dispersal found in nature would decrease this synchronising effect. However, it is
still important to consider these different scales both when investigating model food webs
and when measuring populations empirically.
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 decrease the
stabilising effect of environmental variation where as dispersal, coupled with uncorrelated
response, stabilise 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. This food web would 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.