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).
We investigate how coloured environmental variation and spatial structure affect the stability
of the food web by performing a more thorough analysis concerning the mean and variance
of densities.
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 able to withstand stresses. An increase in
stability, measured as CV, can either imply an increase in mean density or a decrease in
variance. A population consisting of just a few individuals can misleadingly be seen as robust
to stresses 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 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 holds variation in space.
Landscapes are known to hold different biotic and abiotic conditions giving rise to spatially
separated subpopulations inhabiting patches. Dispersal between subpopulations enables reestablishment 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
variable environment, one also has to specify the correlation between each subpopulations
environmental response. Do all subpopulations respond the same to the environmental
variation or do they have different responses? We have investigated this phenomenon by
varying the cross-correlation of environmental time series affecting the subpopulations. By
doing so, we simulate differences in environmental response. However, our study
incorporates a food web of four species in each patch. For this reason, we have decided to
add differences in environmental response both between species and between subpopulations
of the same species. Differences in environmental response will affect how populations
fluctuate in relation to each other. This property is called synchrony.
Synchrony between species has been shown to have a substantial effect on food web
stability and extinction risk. Asynchronous consumers coupled with uncorrelated
environmental fluctuations 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 asynchronous response (Borrvall & Ebenman 2008). We have measured synchrony
between species, according to (Vasseur & 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 really
affects how our species fluctuate in relation to each other.
This study addresses the stability of food webs 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 our
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, than the same food web situated in
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 biologically plausible parameter values have
previously been used by Vasseur & Fox (2007) and McCann et al. (1998) (Table 1). The
values 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 (modeled after
McCann et al. (1998) and 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, represent the trophic
interaction strength between the species.
Table 1 Parameter explanation and their values. The parameter values are the same as in
Vasseur and Fox (2007) except our addition; colour of environmental variation.
The standard deviation, σenv, colour, γ, and cross-correlation, ρenv, of environmental variation
are independent parameters affecting the mortality rates of the consumers. Uncorrelated,
white, environmental variation was generated from a random normal distribution with zero
mean and σenv2 variance. Fourier transform was used to generate coloured 1/f noise. The
discrete Fourier transform of the coloured 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
noise 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 noise, 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.
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.
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 (Caswell 2001 and Wennergren et al. 1995) (Table 2).
Table 2 Dispersal matrix with six patches. dij represents the proportion of the subpopulation
in patch i that migrates to patch j in one time step.
0
d21 d31 d41 d51 d61
d12
0
d32
d42
d52
d62
d13
d23
0
d43
d53
d63
d14
d24
d34
0
d54
d64
d15
d25
d35
d45
0
d65
d16
d26
d36
d46
d56
0
Migrating proportions, dij, were generated from a random normal distribution with mean
(patches)-1 and variance 0.2*(patches)-1. The distribution was truncated by 0 and
1.2*(patches)-1. The same dispersal matrix was used for all four species in this study.
The time series of environmental variation affecting the consumers were cross-correlated,
with ρenv = -1, 0 or 1. ρenv = -1 represented perfect negative cross-correlation between all pairs
of time series affecting subpopulations of different consumer species. All subpopulations
within the same species were affected by the same environmental time series. For ρenv = 0, all
subpopulations was affected by unique independent environmental time series. ρenv = 1
represented perfect positive cross-correlation, all 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
uniform interval; 0.1 to 1.0. 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. 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 density variability:
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 (Vasseur & Fox 2007).
Consumer synchrony was calculated 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 (Vasseur & Fox 2007). 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. 2a, d, Fig.
3a). 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-tomoderate σenv and enhanced the destabilising effect of higher σenv. In addition, it lowered the
σenv values generating maximum stability (Fig. 2d). Dispersal had minor affect during
correlated environmental variation (Fig. 3). However, during uncorrelated environmental
variation, the stabilising effect of weak-to-moderate σenv was enhanced and the destabilising
effect of higher σenv was reduced with dispersal (Fig. 2d, Fig. 3). Studies on time series of
biomass and species abundances revealed that addition of dispersal between subpopulations
resulted in maintenance of intrinsic dynamics during moderate σenv. The stable limit cycles
where not as apparent in isolated patches during the same environmental variance (Fig. 4).
Mean food web biomass decreased and biomass variance increased with increasing σ env (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 (Fig. 2c). 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. 2e, 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. 2e, f)
and species densities.
Figure 2 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.
Figure 3 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 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 by dispersal with the other patches maintains the intrinsic
dynamics of the food web. * as in Vasseur & Fox (2007).
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.
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.
Consumer synchrony increased with increased σenv, regardless of ρenv. Reddening of the
environmental variation enhanced the effect where as dispersal coupled with uncorrelated
environmental variation reduced the synchronising effect of increased σenv.
Discussion
We have added coloured environmental variation and spatial structure to the diamond shaped
food web. The model 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. The aim of our study was to
identify how coloured environmental variation and spatial structure affects the stability of
food webs. We chose to simulate the same food web used in the two well done studies by
McCann et al. (1998) and by Vasseur and Fox (2007) in order to clarify the implications of
coloured environments and spatial structure. In addition of using the same stability analysis
as in Vasseur and Fox (2007), we have done a more comprehensive analysis of stability
concerning mean and variation of densities.
Vasseur and Fox (2007) show that weak-to-moderate environmental variation stabilise the
diamond shaped food web by interrupting initial consumer asynchrony. Stronger
environmental variation destabilise the system by increasing the variability of species
densities. The stabilisation by weak-to-moderate environmental variation is caused by the
systems intrinsic dynamics. Consumer synchronisation causes a shift in total resource
predation pressure which affects resource density. The shift in resource density causes
another quick consumer response dampening predator fluctuations. Fluctuation dampening
decreases the variance in the system, increasing the stability coefficient (1/CV). The highest
stability was found when consumers were affected by positively correlated environmental
variation, ρenv. (Vasseur and Fox 2007)
We first confirm the results of Vasseur and Fox (2007) in order to clarify effects caused by
the components we add to the model. Our results show that coloured environmental variation
and spatial structure had major implications for the stability of the diamond shaped food web.
Redness decreased the stabilising effect of environmental variation whereas dispersal
between spatially subdivided populations increased the stability of the system. The mixing of
individuals from subpopulations in different environments diluted the destabilizing effect of
variation in time. According to our results, variation in space had a much stronger stabilizing
effect than variation in time. However, measuring food web stability only by variability
(1/CV) can be misleading. Results from our study shows that independent studies of mean
and variance of densities is needed in order to evaluate the actual stability of the food web.
The stabilising effect of weak-to-moderate environmental variation, found by Vasseur and
Fox (2007), can be questioned because of the resulting decrease in mean food web biomass
and increased extinction risk. A decreased mean has negative effects on population
persistence, such as increased effects of demographic stochasticity and catastrophes (Lande
1993). However, our study reviled that variation in time can shift the relative abundance of
species. The species with the smallest population in a constant environment actually gained a
larger density during environmental variation (Fig. 2b).
The shift in relative abundance of species was caused by the intrinsic dynamics of the food
web. It is important to have in mind that food web structure and choice of model parameter
will affect the degree of sensitivity to different kinds of environmental variation (Greenman
and Benton 2005). In the diamond shaped food web, C2 has a lower ingestion rate than C1.
This means that C1 has a better ability to take advantage of its resource than C2 during high
environmental variance. Despite the increase in available R, C1 was still affected by a high
predation pressure from P limiting its density increase. Even though P prefers C1, it was
negatively affected by the drastic density decrease of the originally large C2 population. This
caused P:s population to decrease with increased σenv. C1:s superior ability to take advantage
of R was apparent in the correlation between each consumer and the environmental variation.
The negative correlation between C1 and the environmental variation increased continuously
with increased σenv where as the negative correlation between C2 and the environmental
variation decreased after reaching a σenv threshold. A density increase of the smallest
population, C1, will have positive effect on the persistence of the food web. Food webs
withholding species with large populations will have a smaller overall risk of suffering from
catastrophes and demographic stochasticity than food webs withholding small populations.
In addition to measures of stability, we have investigated how different magnitudes of
environmental variance affect the extinction risks in our model food web. As expected from
earlier studies (Lande 1993, Engen et al. 2002), extinction risk for each species in the food
web increased with increased σenv. Lowered mean densities and increased variance increased
the risk of populations reaching extinction boundaries. C2:s poor resource tracking abilities
gave C2 the highest extinction risk at high σenv, despite being the largest population in a
constant environment. The result was caused by C2:s high density variance (Fig. 2c). Isolated
subpopulations with uncorrelated and negatively correlated environmental variation had
higher mean extinction risks than isolated subpopulations withholding a positively correlated
environmental variation. Results may be explained by decreased variance in species densities
with increased correlation (Borrvall & Ebenman 2008). Results of 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 the risk of
extinction by increasing the density of the species with the lowest population in a constant
environment. Further studies including these mechanisms may further clarify the effect of
environmental variation and the importance of multiple measures when analysing food web
stability and extinction risk.
Environmental variation found in nature is considered to be positively correlated in time
(Caswell & Cohen 1995; Halley 1996; Ripa & Lundberg 1996; Cuddington & Yodzis 1999).
Red variation is dominated by low frequencies which result 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. 2f). 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. Redness
increasing the positive correlation between populations has also been shown in Greenman
and Benton (2005). Reduced stabilizing effects and increased extinction risks caused by
redness speak against the importance of environmental variation as an important stabilising
property of food webs. However, addition of dispersal between subdivided populations
clarifies 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 (Engen et al. 2002, 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 dark red environmental
variation was actually more stable than the food web in an isolated patch affected by white
variation. Extinction risks with dispersal were close to zero, during the interval of
environmental variance and redness. 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 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
global 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 the variance in dispersal rate
between patches caused by our dispersal matrixes generation method. The small variance in
dispersal rates makes it possible for individuals in patches with larger dispersal rates to save
other patches with lower dispersal rates at the cost of their original subpopulation density.
Further studies on distance dependent dispersal withholding negative effects, such as
additional death rates on dispersers, would further clarify the importance of dispersal between
subdivided populations.
Food web stability and extinction risk were measured at subpopulation level in this study. It
is important to consider the differences between patch and landscape level when estimating
food web resistance. The choice will have large effects on estimated extinction risks! When
comparing species with and without dispersal during correlated environmental variation, the
stability of the food web in the landscape without dispersal was much higher than the one
with dispersal. Without dispersal, all subpopulations affected by weak variation, will
fluctuate in their own phase, depending on initial densities. This asynchrony minimises the
variance of the sum of all subpopulations which leads to larger landscape stability. With
dispersal, patches that originally fluctuate in their own phase will eventually be more
synchronised, preserving the variance in the sum of all subpopulations. Time lagged
dispersal, more close to dispersal found in nature would decrease this synchronising effect.
However, it is still important to think of 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 for 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, an important
implication of the shift in relative abundances is that present large population sizes may not
give species insurance towards 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 resistance of diverse
food webs.