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
Abstract
Ecological models give poor support for the positive diversity-stability relationship found in nature. Theory
states that complex food webs are unstable and extinction-prone while natural food webs are highly
complex and species rich and yet persistent. Can contradictions be resolved by incorporating coloured
environmental noise and spatial distribution into food web models? Our model is based on the “diamond
shaped” food web, previously used to show stabilizing effects of asynchrony and white environmental noise.
Stabilization by environmental fluctuations was enhanced by adding dispersal between subpopulations but
reduced with noise redness. Increasing environmental variance also changed the relative abundance of
species in the model, increasing the density of the species with the smallest number of individuals. Our
model can be considered as a building block for more complex food webs indicating that environmental
fluctuations coupled with spatial distribution can have major influences on stability and extinction risk in
natural food webs.
Introduction
Species rich natural food webs contain complex
interaction patterns evolved through historical
processes in dynamic environments (May 1973).
Food webs are known to endure unstable
environments for several generations (Vasseur &
Fox 2007). However, theoretical studies predict
that large complex food webs should be unstable
and extinction-prone because of high connectance,
many modes of oscillation and positive feedback
loops (May 1974; Tilman 1999; Green & Sadedin
2005; Borrvall & Ebenman 2008). Models in
ecology are not able to explain the complexity of
nature. To address this lack of knowledge, we
have investigated food web stability in a different
way than what is most commonly done in
theoretical studies. We have also added spatial
structure and coloured environmental variation in
order to decrease the gap between our model and
conditions found in natural ecosystems.
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). Addition of stabilizing
mechanisms in models has been shown to
facilitate food web complexity. For instance,
network connectivity incorporating a mix of weak
and strong links of interactions can inhibit
oscillatory subsystems limiting species richness
(Polis 1991; McCann et al. 1998). 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
2
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.
Landscapes are known to hold different biotic
and abiotic conditions giving rise to spatially
distributed populations inhabiting patches
connected with dispersal. Spatial distribution
enables re-establishment of extinct patches which
can prolong time to extinction (Engen et al. 2002;
Liebhold et al. 2004; Greenman & Benton 2005).
In order to investigate the stabilizing power of
spatial distribution on food webs, we position the
web in six patches. Individuals within species are
connected through dispersal between patches.
Incorporation of spatial distribution gives rise to
synchrony between subpopulation. Synchrony
describes how populations fluctuate in
relationship to each other. The level of synchrony
between subpopulations is
affected by
mechanisms such as dispersal and common
exogenous random factors. Synchrony can also be
measured 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 fluctuations can
improve food web stability (1/CV) by dampening
oscillations between resource and consumers
(McCann et al. 1998; Vasseur & Fox 2007). High
synchrony between species implies a lower
species extinction risk than during asynchronous
response (Borrvall & Ebenman 2008). So far, we
have added three separate parts to our
investigation; different food web stability
measurements, environmental variation and
spatial distribution with patches connected by
dispersal. The fourth and final part of our analysis
incorporates measurements of synchrony between
species.
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). By using
a mix of strong and weak links of interaction,
coloured environmental fluctuations and spatial
distribution we expect to decrease the gap
between food web theory and natural ecological
networks. We show that addition of
environmental variation can change relative
abundance of species, increasing the density of
the species with lowest population size in a
constant environment. Major effects of dispersal
on food web stability indicate that spatial
distribution can be the dominating factor enabling
food webs found in nature.
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 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).
Figure 1 The diamond shaped food web with
differential equation system (modeled after
McCann et al. 1998). 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.
3
Table 1 Parameter explanation and their values.
Environmental noise affects the two consumers’
mortality rates through an exponential filter ({{59
Gillooly,J.F. 2001; 42 Vasseur,D.A. 2007}}:
MCi (t) = MCi (0)eenvi (t)
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.
Dispersal between six subpopulations for each
species was governed by a mass-action mixing
process. There was no distance dependence
between patches. Subpopulations within species
were interconnected through a dispersal matrix.
(Caswell 2001 and Wennergren et al. 1995)
(Table 2).
Table 2 Dispersal matrix with four patches. 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 (patches)-1
and variance 0.2*(patches)-1. The distribution was
truncated by 0 and 1.2*(patches)-1.
(5)
where MCi(t) is the mortality rate at time t, MCi(0)
is the medial mortality rate, envi(t) is the
environmental noise at time t for consumer i. The
standard deviation, σenv, and consumer response
correlation, ρenv, are independent parameters
affecting the mortality rates of the consumers. The
model was integrated with environmental noise
with σenv in a range from 0 to 0.6 in steps of 0.05.
Consumer response correlation, ρenv, had values -1,
0 and 1 where -1 represented negative correlation
between species and positive correlation between
subpopulations within species, 0 represented
independent response of all subpopulations and 1
represented positive correlation between all
subpopulations.
Uncorrelated,
white,
environmental noise 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
(6)
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).
Simulations were made in MATLAB 7.5.0
(R2007b) 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 a bow the extinction
boundary until the end of the simulation. With
dispersal, populations were considered to decrease
below the extinction boundary when the sum of
4
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 cross-correlation and extinction risk
were calculated for each of the combinations of
varied parameters. Food web biomass was the
sum of all subpopulations. Stability of population
density was measured as:
1
𝜇𝑖
=
𝐶𝑉
𝜎𝑖
(7)
where CV is the coefficient of variation, σi the
standard deviation and μi the mean of population
i’s density time series.
Cross-correlation of consumer densities was
calculated through:
𝜌𝐶 =
1
𝑁𝜎𝐶1 𝜎𝐶2
𝑁
∑(𝐶1 (𝑡) − 𝜇𝐶1 ) (𝐶2 (𝑡) − 𝜇𝐶2 )(8)
𝑡=1
where N is time series length, σi standard
deviation and μi mean of consumer species i’s
time series. Cross-correlation between consumers
and environmental noise was calculated as
equation (8), when ρenv =1, in order to evaluate
the impact of environmental noise on each
consumer.
Results
Weak environmental noise, σenv ≈ 0.1 - 0.2,
enhanced food web stability whereas stronger
noise, σenv > 0.3, had a destabilising effect on the
system (Fig. 2a, d). The standard deviation of
environmental noise, σenv, generating maximum
stability, was species specific. C1 and P gained
their maximum stability from stronger noise than
C2 and R. Dispersal had minor affect during
correlated environmental response between
consumers. However, during uncorrelated
response, the stabilizing effect of weak
environmental noise was enhanced and the
destabilising effect of stronger noise was reduced
with dispersal (Fig. 2d). Reddening of the noise
decreased the stabilising effect of weak noise and
enhanced the destabilising effect of stronger
fluctuations. In addition, it lowered the σenv values
generating maximum stability (Fig. 2d).
Mean food web biomass decreased continuously
whereas its variance increased with increasing
σenv, regardless of consumer response correlation,
ρenv, (Fig. 2e, f). However, the minimum variance
did not occur during constant environment, there
was a small initial decrease in variance during
weak environmental noise. Environmental noise
changed the relative abundance of species (Fig.
2b). Mean density of the species with smallest
population in constant environment, C1, increased
and its variance was close to constant during
increasing σenv. In contrast to C1, strong noise
resulted in mean density decrease and a rapid
variance increase of the largest species in constant
environment, C2. The variance of R increased
rapidly, whereas the variance of P increased more
gradually (Fig. 2c). Dispersal coupled with
uncorrelated response reduced the effects of
increasing σenv on mean and variance of food web
biomass and species densities (Fig. 2e, f).
Reddening of the environmental noise enhanced
the effects of increasing σenv (Fig. 2e, f), making
the change in relative abundance of species occur
earlier on the σenv interval.
Subpopulation extinction risk increased with
increasing 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 increasing σenv. Reddening
of the environmental noise increased population
extinction risk where as dispersal coupled with
uncorrelated response reduced the risk of
extinction.
Both negative and positive values of consumer
response correlation, ρenv, gave rise to an increase
in consumer correlation, ρC, with increasing σenv.
Reddening of the environmental noise enhanced
consumer synchronization where as dispersal
coupled with uncorrelated response reduced the
synchronising effect.
Both consumer species was synchronized with
the noise during increasing environmental
variability, until a threshold of σenv ≈ 0.3. Results
5
Figure 2 Stability, mean and variance for species population densities and food web biomass with
environmental fluctuation strength, σenv and uncorrelated consumer response, ρenv=0. Left column;
measurements on species population density with white environmental noise 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 noise of γenv=0-0.6, without and with (crosshatch lines) dispersal.
6
Figure 3 Stability of food web biomass with environmental fluctuation strength, σenv and cross-correlation
of environmental fluctuations, ρenv. a) without dispersal b) with dispersal.
differed between consumer species for stronger
environmental fluctuations. The synchrony
between C1 and the noise continued to increase
while the synchrony between C2 and
environmental noise decreased for larger σenv.
Dispersal coupled with uncorrelated response
increased the difference between consumers while
redness of the noise increased the synchronizing
effect of increasing environmental variability.
Discussion
Stability in natural food webs is reached as a
result of complex interactions between species in
different trophic levels and between each species
and the abiotic environment. In our study, weak
environmental noise stabilized and lowered the
variance in the food web by interrupting initial
consumer asynchrony (Fig. 2a, d). The synchrony
between both consumers and their environment
were increased during increasing standard
deviation of environmental noise. Consumer
synchronisation affected resource predation
pressure which changed resource density. This
caused another consumer response which
dampened predator fluctuations. The food web,
affected by weak environmental noise, gets a
decreased variance because of dampened species
fluctuations. These results accords with findings
of Vasseur & Fox (2007). Without dispersal, the
highest level of stabilization and lowest variance
occurred during positive correlation in species
responses to noise. Similar results have been
found in a model where environmental variation
affected the growth rates of all species in the food
web (Borrvall & Ebenman 2008).
In our study, environmental noise changed the
relative abundance of species in the food web.
Food web resistance was increased by increasing
the density of the species with the smallest
population (Fig. 2b). While C1 was continuously
synchronized with noise during increasing σenv,
the synchrony between C2 and the noise decreased
after reaching a threshold value of σenv. C2 had
problems following the resource at strong
environmental noise whereas C1 was a better
tracker and was able to take advantage of
liberated resources. The dynamics are partly
caused by C2 having a lower ingestion rate than
C1. C1 was, however, still affected by high
predation pressure, limiting its increase in density.
Even though P prefers C1, it was negatively
affected by the drastic density decrease of the
originally large C2 population, causing P’s
population to drop with increasing σenv.
As expected from earlier studies (Lande 1993,
Engen et al. 2002), extinction risk for each species
in the food web increased continuously with
increasing environmental variance. Lowered mean
densities and increased variance increased the risk
of populations reaching extinction boundaries. C2
had the highest extinction risk at strong
environmental noise, despite being the largest
7
population in a constant environment. The result
was caused by poor resource tracking abilities
leading to high density variance (Fig. 2c). An
uncorrelated and negatively correlated consumer
response correlation, ρenv, gave higher mean
subpopulation extinction risks than positively
correlated response. These results are in line with
Borrvall & Ebenman 2008.
Reddening of environmental noise reduced the
stabilizing power of noise and moved the peaks of
maximum food web stability to lower σenv (Fig.
2d). Redness increased food web extinction risks
by increasing variance and decreasing mean
densities. Greenman and Benton (2005) observed
that reddened noise caused larger colour shifts and
variance differences on time series than white
noise. Cuddington and Yodzis (1999) show that
reddening of noise decreases mean persistence
time in overcompensating single population
models. The results speak against the importance
of noise as a stabilizing property of natural food
webs. However, our results showed that white and
reddened noise in particular changed the relative
abundances of species, increasing the density of
the species with the smallest population size. Ripa
& Lundberg (1996) showed that reddened noise
can increase the degree of environmental
fluctuation tracking. Reddening also caused an
increase in the synchronisation of consumers
which enhanced the power of food web dynamics
dampening species fluctuations. Weak reddened
environmental fluctuations, coupled with specific
food web dynamics, can have positive effects on
food web resistance.
Mass action mixing has no distance dependence,
which 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. Our study showed that dispersal had a
strong stabilizing effect during uncorrelated
environmental response (Fig. 2d). Dispersal
reduced negative impacts in patches with poor
environmental conditions through immigrating
individuals from patches with better conditions
(Engen et al. 2002, Liebhold et al. 2004).
Dispersal undermined the synchronization of
consumers during increasing σenv, by decreasing
destabilizing effects of noise. The food web with
dispersal affected by dark pink environmental
noise was actually more stable than the food web
without dispersal and white noise. Extinction risk
was close to zero, during the interval of
fluctuation strength, even for red noise. Larger
values of σenv generated similar effects of redness
as in the case of no dispersal. Kaitala et al. (1997)
supports our results by showing that the effect of
redness decreases as you add system complexity.
Engen et al. (2002) showed, in a single species
system, that increasing dispersal between patches
results in longer time to extinction. Our results
indicated that spatial distribution can be essential
for stabilization of complex food webs,
overshadowing stabilizing effect of weak
environmental variation. Further studies on
distance dependent dispersal withholding negative
effects, such as additional death rates on
dispersers, would further clarify the importance of
spatial distribution.
When comparing species with and without
dispersal during correlated environmental
response, the stability of the food web without
dispersal was larger than the one with dispersal.
Without dispersal, all subpopulations affected by
weak noise, 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 synchronized, preserving the variance. Time
lagged
dispersal
would
decrease
this
synchronizing effect. Food web variance and
stability were measured at patch level in this
study, when comparing food webs with and
without dispersal. It is important to consider the
differences between patch and landscape level
when measuring populations empirically. It will
have large effects on estimated extinction risks!
The commonly used stability measurement
coefficient of variation, CV, can be misleading
without additional independent studies of mean
and variance. Vasseur & Fox (2007) state that
weak white environmental noise can increase the
stability coefficient (μ/σ) of the diamond shaped
food, as did this study. However, the positive
8
effect of environmental noise can be questioned
because of the resulting decrease of 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). Also, the stabilizing effect of
environmental noise decreased when reddened
and thereby more likely to be encountered in
nature. However, environmental noise changed
the relative abundance of species. By increasing
the density of the smallest population, the food
web as a whole will be less sensitive to
demographic stochasticity and catastrophes, more
outlasting despite lowered mean food web
biomass. 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 noise (Greenman and Benton
2005).
Contradictions in food web research, regarding
the diversity-stability relationship, may be
resolved by increasing the complexity of
ecological models. McCann (2009) showed that
food web stability can be scale invariant,
stabilized by weak and strong interaction
pathways that repeat at different resolutions.
Results from this study showed that coloured
environmental noise can have a stabilizing effect
by increasing the coefficient of stability and
changing the relative abundance of species.
However, noise induced changes in density
proportions also implies that present large
population sizes are no insurance towards future
increase in temporal and regional environmental
variance. Synchrony of species environmental
response and dispersal between subpopulations
had both major influences on stability and
extinction risk of the food web. Our model can be
seen as a building block for more complex food
webs indicating that spatial structure could be one
of the dominating factors stabilizing complex
food webs found in nature.
Acknowledgements
References
Caswell, H. (2001). Matrix population models. 2:nd ed. Sinauer Associates Inc. Publishers, Sunderland,
Massachusetts.
May, R.M. (1973). Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton.
…