Environmental Variability and Dispersal affect the Stability of Food Webs in
Metacommunities: A Comment on Gouhier et al.
Uno Wennergren and Sara Gudmundson
IFM, Theory and Modelling
Linköping University
Sweden
Abstract
Gouhier et al. (2010) presented a study in The American Naturalist where they made an attempt
to study food web dynamics in a landscape setting using a set of difference equations. Their
conclusions are based on an erroneous interpretation of continuous vs discrete systems and some
important assumptions in the model lack ecological relevance. Instead we argue that this system
of diamond-shaped food webs in a landscape setting primarily ought to be analysed by systems
of differential equations instead of systems of difference equations.
Introduction
One of the large questions that theoretical ecology struggle with is the existence of highly
diverse food webs although theoretical studies predict that they should be unstable and prone to
extinction (May 1973; Tilman 1999; Green & Sadedin 2005; Borrvall & Ebenman 2008). The
theoretical issue that scientists then face is the struggle to find new components of food web
dynamics that may increase stability and favor high diversity. It is well known that spatial and
temporal dimensions are important components of extinction risk, and thereby also biodiversity,
as a result of metapopulation theory which was presented in the early 1990’s (Hanski 1994). On
the other hand, if subpopulations are completely synchronized over space the spatial component
will disappear and it will no longer reduce extinction risk or favor biodiversity.
In a recent publication in this journal, Gouhier et al. (2010) raise the important question of how
asynchrony may arise between diamond-shaped food webs in a landscape. Such an asynchrony
may then stabilize the food web on both local and regional scale by decreasing inherent
oscillations of the food web. This phenomenon may then indicate that the spatial dimension
promote biodiversity by stabilizing food webs. Asymmetric interaction pathways between
species and weak-to-moderate environmental noise have previously been shown to stabilize the
diamond-shaped food web (McCann et al. 1998, Vasseur & Fox 2007) in a non spatial setting by
reducing the amplitudes of inherent oscillations. Gouhier et al. investigated whether the stability
induced by these factors still holds, or even increase, when adding two new components: (i) food
web dynamics with a spatial dimension incorporating local dispersal between cells and (ii) color
of the environmental noise (autocorrelation in time). Their study showed that correlated
environmental fluctuations between the species in the food web can stabilize the food web by
reducing the amplitudes of species compensatory dynamics when dispersal rate is high. These
results are in line with Vasseur & Fox (2007), who investigates the same food web but without a
spatial component. However, Gouhier et al. also show some quite surprising results; firstly, when
dispersal rate is low, they conclude that asynchrony is induced over space by the dispersal
scheme itself which then promotes stability yet weak environmental fluctuations added to this
can thereby reduce food web stability by synchronising the subpopulations in the landscape. In
this comment, we show that interpretations, assumptions, in the Gouhier et al. (2010) can be
questioned and hence we argue that their results are not valid in most ecological settings.
Our comment deals with the following two issues of the modeling approach:
1. A destabilizing time lag is introduced since the model is a discretized version of the
previously analysed differential equation system of the diamond-shaped food web
(McCann et al. 1998, Vasseur & Fox 2007), fig. 1.
2. A. The authors claim that they apply asynchronous updates, yet asynchronous update is
not adopted completely on the modeling of dispersal and furthermore are the local
dynamics of the food web itself updated synchronously.
B. The assumptions that follow by the specifics of their asynchronous update are
ecologically irrelevant and smaller changes of these assumptions alter the results.
Corrections of these issues alter the conclusions and also question the applicability of any
conclusion of their modelling approach.
To test the result of Gouhier et al. we ran their simulations and replicated the results of Gouhier
et al. We made some additional modeling to test for different applications of asynchronous
updates to examine the very impact of issue 2B. Our results from these tests are shown in figs. 2,
3 and 4. The spatial context that is set up by Gouhier et al. considers a grid of 256x256 patches,
in total 65 536 patches, with periodic boundary conditions. The patches are all the same; both the
initial conditions and the environmental noise is the same all over space. Such a homogenous
landscape ought to be equally well represented with a smaller landscape given the periodic
boundaries, yet some effects will appear in the transient dynamics, fig. 3.
Background
In addition to variation in time, nature exhibits variation in space. Dispersal between spatially
separated subpopulations may enable re-establishments that can prolong the time to extinction
for entire populations (Engen et al. 2002; Liebhold et al. 2004; Greenman & Benton 2005). The
diamond-shaped food web contains four species, fig. 1. Two consumers share one resource and
have one common predator. The model was introduced as a continuous-time differential equation
system by Vasseur & Fox (2007) after McCann et al. (1998). This web is stabilized by consumer
asynchrony, which means that interactions result in stable oscillations in a constant environment
(McCann et al. 1998), and hence external temporal fluctuations in time and space, or dispersal,
may either stabilize or destabilize the system.
1. A destabilizing time lag
The dynamics of Gouhier et al.’s model is expected to be less stable than the original diamondshaped food web since the system is destabilized by the time lag that is induced by transforming
the differential equations of McCann et al. (1998) into difference equations, fig. 1. We found that
the initial conditions Gouhier et al. used in their simulations would in a non-spatial setting (not
tested by Gouhier et al.) result in a crash within a few time steps. On the other hand it will persist
with oscillations like the numerical solutions of the differential equations if initial conditions are
close to, or at, equilibrium, fig. 2. Hence, the reference model (the one cell difference equations
model) of Gouhier et al. is very different from the original differential equation model. Instead of
an equilibrium that is a global attractor with stable oscillations as of the differential equation of
Vasseur and Fox (2007) and McCann et al. (1998) the difference system of Gouhier et al. only
has a local attractor and hence will respond differently to environmental noise and most probably
also to dispersal. Both dispersal and environmental noise may disturb local dynamics either into
or away from the region of attraction to oscillatory dynamics. Gouhier et al. do not clarify that
they actually have another reference model than theVasseur and Fox (2007) and McCann et al.
(1998) model, instead they claim that the model of McCann et al. (1998) is their mean-field food
web model. The conclusion we have made, given this destabilizing time lag, is that Gouhier et al.
may study stabilizations of their system if transient analysis, initial conditions, and regions of
attractions are considered. But, we argue that it is highly inappropriate to refer back to the
dynamics of the diamond-shaped web represented by differential equations in Vasseur and Fox
(2007) and McCann et al. (1998).
The conclusion of Gouhier et al (2010) that “Low dispersal (d <0.03) decouples local food webs
and allow them to fluctuate autonomously without disrupting local compensatory dynamics.” is
misleading since dispersal does not cause any disrupting of local compensatory dynamics. The
disrupted compensatory dynamics is a result of the difference equation as shown in fig. 2. The
modeling framework of Gouhier et al. instead shows that low dispersal stabilize the dynamics by
rescue effects which transfer the solution into the oscillatory region. At medium levels of
dispersal it also synchronizes cells, as of general theory and also discussed in Gouhier et al. What
Gouhier et al. also did not consider to analyse, since they were not fully aware of the instability
of the difference equation, was the effect of high dispersal. High dispersal will turn the system
close to a single cell system. High dispersal desynchronize the dynamics and at even higher
levels also destabilizes the system of Gouhier et al., fig. 4.
Furthermore, by including environmental noise Gouhier et al. perturb their difference equations
and they may perturb it out of the region of oscillatory dynamics. Most of their interpretations of
the effect of environmental noise thereby also becomes misleading since these are not related to
this basic dynamic of the difference equation. Still, we do not perform any detailed analysis of
the effects of environmental noise since it is beyond the scoop of this comment.
The authors may claim that the asynchronous update is a methodology to transform the
simulation, when using a set of difference equations, to a trajectory close to that of a differential
equation. This could have supported their use of the differential equations as reference model.
There are two ways to find a trajectory of differential equations, either by mathematical analysis
or by using numerical methods that cope with the dynamics of the system. The Euler method,
which actually is used by Gouhier et al., is most often numerically unstable especially for stiff
equations. The method simply applies only the first derivative. The error is of course reduced by
introducing smaller time steps yet the error may still persist and this is expected for systems with
oscillatory dynamics. Today this Euler method is not a method commonly used as a numerical
method and it is completely incorrect to state that it is necessary true that an asynchronous
update equals the solution of a differential equation. Asynchronous update is a method that may
picture a process with short timesteps and in that sense it can closer relate to a continuous model
(Schönfisch and de Roos 1999). The very essence of asynchronous updates is to consider
timesteps so small that only a single event occurs (Schönfisch and de Roos 1999). Note that
asynchronous updates is not a way to transform the difference equation into a differential
equations it is simply a way to reduce the time step such that within a time step only a single
event occurs. This update of single events orginate from cellular automata models where a single
event may change the state of a cell, for example to occupied from un-occuppied. A synchronous
update in a cellular automata model will update all cells within one time step while an
asynchronous update only update one cell during a timestep. Hence the time step is reduced such
that the probability for any cell to be updated is 1/n instead of 1, given that there are n cells in the
grid. Gouhier et al have used this cellular automata approach in their modeling yet they have not
considered states of cells. Instead they used population densities and rates. They do not explain
how one shall apply asynchronous update on rates and how this reduction in length of a time step
affect the rates in the difference model. Usually a shorter time step implies a smaller rate, in their
case the dispersal and growth rates ought to be reduced in analogy with how probability of
update of a cell was reduced from 1 to 1/n in the example above.
2. Asynchronous updates of Gouhier et al. (2010)
The authors claim that they apply asynchronous updates, yet asynchronous update is not adopted
completely on the modeling of dispersal and furthermore are the dynamics of the food web itself
updated synchronously. Since the update of the local growth, that is the local food web dynamics,
is synchronously updated the local dynamics itself is perfectly pictured by the difference
equation and by a simulation in a one cell lattice, figs 1 and 2. In this case there is no reduction
of the length of the time step at all within the cell.
Gouhier et al. have used an unusual update design that one of the authors introduced in an earlier
paper (Maser et al. 2007). After growth in a local cell, individuals disperse to one of the
neighboring cells (randomly selected every time step). The proportion dispersing to the
neighboring cell depends on the dispersal rate and is weighed by the difference between the
density of the focal cell and the neighboring cell. Gouhier et al. applied the dispersal on food
web level such that all species disperse to the very same neighboring patch. This dispersal to
only one of all neighboring patches is more relevant in individual based modeling where each
individual are treated separately. Such an individual based modeling could very well represent a
single event from an ecological perspective. Yet, in the model of Gouhier et al. all dispersing
individuals within the whole food web disperse to one randomly selected neighbor and we do
question the ecological relevance of such a synchronized dispersal of both individuals and
species. The very essence of asynchronous update is to reduce the length of time steps such that
single events occur during a time step. Applying asynchronous updates at population level is
very different and hence we argue that Gouhier et al’s food web level approach of single events
is rather odd. It can be questioned from different perspectives both ecological and mathematical.
For example: Why are all dispersing to the very same cell? Why are all species, even at different
trophic levels, treated as having same frequency, time scale of events? Why are not the dispersal
and growth rates rates reduced to low levels to picture short time steps? How is the single event
defined from an ecological perspective? Schönfisch and de Roos (1999) points out that the very
specifics of an asynchronous update may alter the dynamics of cellular automata and one may
expect that this also apply on the study of Gouhier et al. We tested whether a modification
towards a more realistic version of the updates would alter the results of Gouhier et al. We
decoupled the species in the food web during dispersal. At each update we randomly selected
one cell and one species. This decoupling of species dispersal partly altered the dynamics at low
and high dispersal, fig. 4. Note that we do not at all argue that this modeling approach is
sufficient or is a true asynchronous update. For example, the model is still a difference model
with rates and not with states, the local population growth are not treated as asynchronous
updates, etc. To actually picture a dynamic that relates to previous studies on the diamondshaped food web, which are made on differential equations, it could perhaps suffices to reduce
the length of the time steps by decreasing growth and dispersal rates, given that the rules of
dispersal are set up by ecological reasoning. The aim with such an attempt would be to get rid of
the destabilizing effect of the difference equations such that the oscillations will be a global
attractor and no longer only exist for a local region. Yet, to us it is not the most efficient or
successful way to construct numerical solutions. We do think it is much more reasonable to
apply the differential equation models in a spatial setting and correctly use numerical methods
available for example in Matlab, R, or other software.
To conclude we like to point out that it is not that simple to switch between discrete and
continuous models and there are several pitfalls that are maybe not always that apparent. In this
case the authors made one mistake since they were not aware of the basic dynamics of their one
cell food web dynamics. This mistake rendered some severe misinterpretations of their results.
Furthermore, the approach of asynchronous update rendered a dispersal modeling clearly
unrealistic and not supported by ecological reasoning. On the other hand we do think that the
study stress the importance and need of careful and in-depth analysis of food web dynamics in
spatial settings.
References
Borrvall, C., and B. Ebenman. 2008. Biodiversity and persistence of ecological communities in
variable environments. Ecological Complexity 5:99-105.
Engen, S., R. Lande, and B.E. Saether. 2002. The spatial scale of population fluctuations and
quasi-extinction risk. American Naturalist 160:439-451.
Gouhier, T.C., F. Guichard, and A. Gonzalez. 2010. Synchrony and stability of food webs in
metacommunities. American Naturalist 175:E16-E34.
Green, D.G., and S. Sadedin. 2005. Interactions matter: complexity in landscapes and
ecosystems. Ecological Complexity 2:117-130.
Greenman, J.V., and T.G. Benton. 2005. The frequency spectrum of structured discrete time
population models: Its properties and their ecological implications. Oikos 110:369-389.
Hanski, I. 1994. A practical model of metapopulation dynamics. Journal of Animal Ecology 63,
1:151-162.
Hassell, M.P., and H.B. Wilson. 1998. The dynamics of spatially distributed host-parasitoid
system. Pages 75–110 in: Tilman, D., and P. Kareiva, ed. Spatial Ecology. Princeton University
Press, Princeton.
Liebhold, A., W. D. Koenig, and O. N. Bjornstad. 2004. Spatial synchrony in population
dynamics. Annual Review of Ecology, Evolution and Systematics 35:467-490.
Maser, G.L., F. Guichard, and K. McCann. 2007. Weak trophic interactions and the balance of
enriched metacommunities. Journal of Theoretical Biology 247:337-345.
May, R.M. 1973. Stability and complexity in model ecosystems. Princeton University Press,
Princeton.
McCann, K.S., A. Hastings, and G.R. Huxel. 1998. Weak trophic interactions and the balance of
nature. Nature 395:794-798.
Schönfisch B., and de Roos A. 1999. Synchronous and asynchronous updating in cellular
Automata. BioSystems 51 (1999) 123–143
Tilman, D. 1999. The ecological consequences of changes in biodiversity: A search for general
principles. Ecology 80:1455-1474.
Vasseur, D.A., and J.W. Fox. 2007. Environmental fluctuations can stabilize food web dynamics
by increasing synchrony. Ecology Letters 10:1066-1074.
Legends
Figure 1. The diamond-shaped food web. From left to right described by: a set of differential
equations (McCann et al. 1998), a graph, and a set of difference equations (Gouhier et al. 2010).
Figure 2. Simulations of Gouhier et al. (2010) food web model to test the effect of different
initial conditions on local dynamics.
Figure 3. Simulations of Gouhier et al. (2010) diamond-shaped food web model to test the effect
of different lattice size.
Figure 4. Simulation of Gouhier et al. (2010) food web model to test the effect of different
asynchronous updates of the dispersal and also to test the effect of low, medium and high
dispersal, d=[0.004 0.5 0.996].