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Environmental Variability and Dispersal affect the Stability of Food Webs in
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Metacommunities: A Comment on Gouhier et al.
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Uno Wennergren and Sara Gudmundson
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IFM, Theory and Modelling
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Linköping University
Sweden
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unwen@ifm.liu.se
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sargu@ifm.liu.se
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Figures in colour: Figs 2, 3 and 4
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Abstract
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Gouhier et al. (2010) presented a study in The American Naturalist where they made an attempt
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to study food web dynamics in a landscape setting using a set of difference equations. Their
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conclusions are based on an erroneous interpretation of continuous vs discrete systems and some
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important assumptions in the model lack ecological relevance. Instead we argue that this system
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of diamond-shaped food webs in a landscape setting primarily ought to be analysed by systems
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of differential equations instead of systems of difference equations.
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Introduction
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One of the large questions that theoretical ecology struggle with is the existence of highly
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diverse food webs although theoretical studies predict that they should be unstable and prone to
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extinction (May 1973; Tilman 1999; Green & Sadedin 2005; Borrvall & Ebenman 2008). The
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theoretical issue that scientists then face is the struggle to find new components of food web
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dynamics that may increase stability and favor high diversity. It is well known that spatial and
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temporal dimensions are important components of extinction risk, and thereby also biodiversity,
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as a result of metapopulation theory which was presented in the early 1990’s (Hanski 1994). On
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the other hand, if subpopulations are completely synchronized over space the spatial component
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will disappear and it will no longer reduce extinction risk or favor biodiversity.
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In a recent publication in this journal, Gouhier et al. (2010) raise the important question of how
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asynchrony may arise between diamond-shaped food webs in a landscape. Such an asynchrony
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may then stabilize the food web on both local and regional scale by decreasing inherent
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oscillations of the food web. This phenomenon may then indicate that the spatial dimension
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promote biodiversity by stabilizing food webs. Asymmetric interaction pathways between
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species and weak-to-moderate environmental noise have previously been shown to stabilize the
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diamond-shaped food web (McCann et al. 1998, Vasseur & Fox 2007) in a non spatial setting by
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reducing the amplitudes of inherent oscillations. Gouhier et al. investigated whether the stability
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induced by these factors still holds, or even increase, when adding two new components: (i) food
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web dynamics with a spatial dimension incorporating local dispersal between cells and (ii) color
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of the environmental noise (autocorrelation in time). Their study showed that correlated
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environmental fluctuations between the species in the food web can stabilize the food web by
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reducing the amplitudes of species compensatory dynamics when dispersal rate is high. These
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results are in line with Vasseur & Fox (2007), who investigates the same food web but without a
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spatial component. However, Gouhier et al. also show some quite surprising results; firstly, when
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dispersal rate is low, they conclude that asynchrony is induced over space by the dispersal
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scheme itself which then promotes stability yet weak environmental fluctuations added to this
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can thereby reduce food web stability by synchronising the subpopulations in the landscape. In
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this comment, we show that interpretations, assumptions, in the Gouhier et al. (2010) can be
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questioned and hence we argue that their results are not valid in most ecological settings.
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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
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previously analysed differential equation system of the diamond-shaped food web
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(McCann et al. 1998, Vasseur & Fox 2007), fig. 1.
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2. A. The authors claim that they apply asynchronous updates, yet asynchronous update is
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not adopted completely on the modeling of dispersal and furthermore are the local
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dynamics of the food web itself updated synchronously.
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B. The assumptions that follow by the specifics of their asynchronous update are
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ecologically irrelevant and smaller changes of these assumptions alter the results.
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Corrections of these issues alter the conclusions and also question the applicability of any
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conclusion of their modelling approach.
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To test the result of Gouhier et al. we ran their simulations and replicated the results of Gouhier
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et al. We made some additional modeling to test for different applications of asynchronous
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updates to examine the very impact of issue 2B. Our results from these tests are shown in figs. 2,
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3 and 4. The spatial context that is set up by Gouhier et al. considers a grid of 256x256 patches,
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in total 65 536 patches, with periodic boundary conditions. The patches are all the same; both the
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initial conditions and the environmental noise is the same all over space. Such a homogenous
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landscape ought to be equally well represented with a smaller landscape given the periodic
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boundaries, yet some effects will appear in the transient dynamics, fig. 3.
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Background
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In addition to variation in time, nature exhibits variation in space. Dispersal between spatially
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separated subpopulations may enable re-establishments that can prolong the time to extinction
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for entire populations (Engen et al. 2002; Liebhold et al. 2004; Greenman & Benton 2005). The
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diamond-shaped food web contains four species, fig. 1. Two consumers share one resource and
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have one common predator. The model was introduced as a continuous-time differential equation
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system by Vasseur & Fox (2007) after McCann et al. (1998). This web is stabilized by consumer
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asynchrony, which means that interactions result in stable oscillations in a constant environment
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(McCann et al. 1998), and hence external temporal fluctuations in time and space, or dispersal,
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may either stabilize or destabilize the system.
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1. A destabilizing time lag
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The dynamics of Gouhier et al.’s model is expected to be less stable than the original diamond-
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shaped food web since the system is destabilized by the time lag that is induced by transforming
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the differential equations of McCann et al. (1998) into difference equations, fig. 1. We found that
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the initial conditions Gouhier et al. used in their simulations would in a non-spatial setting (not
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tested by Gouhier et al.) result in a crash within a few time steps. On the other hand it will persist
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with oscillations like the numerical solutions of the differential equations if initial conditions are
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close to, or at, equilibrium, fig. 2. Hence, the reference model (the one cell difference equations
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model) of Gouhier et al. is very different from the original differential equation model. Instead of
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an equilibrium that is a global attractor with stable oscillations as of the differential equation of
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Vasseur and Fox (2007) and McCann et al. (1998) the difference system of Gouhier et al. only
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has a local attractor and hence will respond differently to environmental noise and most probably
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also to dispersal. Both dispersal and environmental noise may disturb local dynamics either into
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or away from the region of attraction to oscillatory dynamics. Gouhier et al. do not clarify that
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they actually have another reference model than theVasseur and Fox (2007) and McCann et al.
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(1998) model, instead they claim that the model of McCann et al. (1998) is their mean-field food
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web model. The conclusion we have made, given this destabilizing time lag, is that Gouhier et al.
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may study stabilizations of their system if transient analysis, initial conditions, and regions of
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attractions are considered. But, we argue that it is highly inappropriate to refer back to the
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dynamics of the diamond-shaped web represented by differential equations in Vasseur and Fox
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(2007) and McCann et al. (1998).
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The conclusion of Gouhier et al (2010) that “Low dispersal (d <0.03) decouples local food webs
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and allow them to fluctuate autonomously without disrupting local compensatory dynamics.” is
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misleading since dispersal does not cause any disrupting of local compensatory dynamics. The
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disrupted compensatory dynamics is a result of the difference equation as shown in fig. 2. The
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modeling framework of Gouhier et al. instead shows that low dispersal stabilize the dynamics by
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rescue effects which transfer the solution into the oscillatory region. At medium levels of
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dispersal it also synchronizes cells, as of general theory and also discussed in Gouhier et al. What
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Gouhier et al. also did not consider to analyse, since they were not fully aware of the instability
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of the difference equation, was the effect of high dispersal. High dispersal will turn the system
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close to a single cell system. High dispersal desynchronize the dynamics and at even higher
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levels also destabilizes the system of Gouhier et al., fig. 4.
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Furthermore, by including environmental noise Gouhier et al. perturb their difference equations
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and they may perturb it from the region of oscillatory dynamics. Most of their interpretations of
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the effect of environmental noise becomes misleading since these are not related to the basic
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dynamic of the difference equation. Still, we do not perform any detailed analysis of the effects
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of environmental noise since it is beyond the scoop of this comment.
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The authors may claim that the asynchronous update is a methodology to transform the
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simulation, when using a set of difference equations, to a trajectory close to that of a differential
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equation. This could have supported their use of the differential equations as reference model.
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There are two ways to find a trajectory of differential equations, either by mathematical analysis
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or by using numerical methods that cope with the dynamics of the system. The Euler method,
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which actually is used by Gouhier et al., is most often numerically unstable especially for stiff
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equations. The method simply applies only the first derivative. The error is of course reduced by
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introducing smaller time steps yet the error may still persist and this is expected for systems with
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oscillatory dynamics. Today this Euler method is not a method commonly used as a numerical
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method and it is completely incorrect to state that it is necessary true that an asynchronous
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update equals the solution of a differential equation. Asynchronous update is a method that may
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picture a process with short timesteps and in that sense it can closer relate to a continuous model
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(Schönfisch and de Roos 1999). The very essence of asynchronous updates is to consider
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timesteps so small that only a single event occurs (Schönfisch and de Roos 1999). Note that
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asynchronous updates is not a way to transform the difference equation into a differential
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equations it is simply a way to reduce the time step such that within a time step only a single
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event occurs. This update of single events orginate from cellular automata models where a single
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event may change the state of a cell, for example to occupied from un-occuppied. A synchronous
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update in a cellular automata model will update all cells within one time step while an
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asynchronous update only update one cell during a timestep. Hence the time step is reduced such
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that the probability for any cell to be updated is 1/n instead of 1, given that there are n cells in the
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grid. Gouhier et al have used this cellular automata approach in their modeling yet they have not
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considered states of cells. Instead they used population densities and rates. They do not explain
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how one shall apply asynchronous update on rates and how this reduction in length of a time step
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affect the rates in the difference model. Usually a shorter time step implies a smaller rate, in their
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case the dispersal and growth rates ought to be reduced in analogy with how probability of
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update of a cell was reduced from 1 to 1/n in the example above.
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2. Asynchronous updates of Gouhier et al. (2010)
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The authors claim that they apply asynchronous updates, yet asynchronous update is not adopted
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completely on the modeling of dispersal and furthermore are the dynamics of the food web itself
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updated synchronously. Since the update of the local growth, that is the local food web dynamics,
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is synchronously updated the local dynamics itself is perfectly pictured by the difference
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equation and by a simulation in a one cell lattice, figs 1 and 2. In this case there is no reduction
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of the length of the time step at all within the cell.
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Gouhier et al. have used an unusual update design that one of the authors introduced in an earlier
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paper (Maser et al. 2007). After growth in a local cell, individuals disperse to one of the
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neighboring cells (randomly selected every time step). The proportion dispersing to the
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neighboring cell depends on the dispersal rate and is weighed by the difference between the
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density of the focal cell and the neighboring cell. Gouhier et al. applied the dispersal on food
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web level such that all species disperse to the very same neighboring patch. This dispersal to
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only one of all neighboring patches is more relevant in individual based modeling where each
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individual are treated separately. Such an individual based modeling could very well represent a
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single event from an ecological perspective. Yet, in the model of Gouhier et al. all dispersing
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individuals within the whole food web disperse to one randomly selected neighbor and we do
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question the ecological relevance of such a synchronized dispersal of both individuals and
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species. The very essence of asynchronous update is to reduce the length of time steps such that
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single events occur during a time step. Applying asynchronous updates at population level is
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very different and hence we argue that Gouhier et al’s food web level approach of single events
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is rather odd. It can be questioned from different perspectives both ecological and mathematical.
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For example: Why are all dispersing to the very same cell? Why are all species, even at different
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trophic levels, treated as having same frequency, time scale of events? Why are not the dispersal
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and growth rates rates reduced to low levels to picture short time steps? How is the single event
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defined from an ecological perspective? Schönfisch and de Roos (1999) points out that the very
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specifics of an asynchronous update may alter the dynamics of cellular automata and one may
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expect that this also apply on the study of Gouhier et al. We tested whether a modification
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towards a more realistic version of the updates would alter the results of Gouhier et al. We
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decoupled the species in the food web during dispersal. At each update we randomly selected
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one cell and one species. This decoupling of species dispersal partly altered the dynamics at low
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and high dispersal, fig. 4. Note that we do not at all argue that this modeling approach is
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sufficient or is a true asynchronous update. For example, the model is still a difference model
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with rates and not with states, the local population growth are not treated as asynchronous
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updates, etc. To actually picture a dynamic that relates to previous studies on the diamond-
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shaped food web, which are made on differential equations, it could perhaps suffices to reduce
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the length of the time steps by decreasing growth and dispersal rates, given that the rules of
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dispersal are set up by ecological reasoning. The aim with such an attempt would be to get rid of
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the destabilizing effect of the difference equations such that the oscillations will be a global
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attractor and no longer only exist for a local region. Yet, to us it is not the most efficient or
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successful way to construct numerical solutions. We do think it is much more reasonable to
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apply the differential equation models in a spatial setting and correctly use numerical methods
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available for example in Matlab, R, or other software.
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To conclude we like to point out that it is not that simple to switch between discrete and
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continuous models and there are several pitfalls that are maybe not always that apparent. In this
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case the authors made one mistake since they were not fully aware of the basic dynamics of their
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one cell food web dynamics. This mistake rendered some severe misinterpretations of their
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results. Furthermore, the approach of asynchronous update resulted in a dispersal model not
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supported by ecological reasoning. On the other hand we do think that the study stress the
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importance and need of careful and in-depth analysis of food web dynamics in spatial settings.
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References
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Borrvall, C., and B. Ebenman. 2008. Biodiversity and persistence of ecological communities in
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variable environments. Ecological Complexity 5:99-105.
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Engen, S., R. Lande, and B.E. Saether. 2002. The spatial scale of population fluctuations and
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quasi-extinction risk. American Naturalist 160:439-451.
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Gouhier, T.C., F. Guichard, and A. Gonzalez. 2010. Synchrony and stability of food webs in
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metacommunities. American Naturalist 175:E16-E34.
226
Green, D.G., and S. Sadedin. 2005. Interactions matter: complexity in landscapes and
227
ecosystems. Ecological Complexity 2:117-130.
228
Greenman, J.V., and T.G. Benton. 2005. The frequency spectrum of structured discrete time
229
population models: Its properties and their ecological implications. Oikos 110:369-389.
230
Hanski, I. 1994. A practical model of metapopulation dynamics. Journal of Animal Ecology 63,
231
1:151-162.
232
Hassell, M.P., and H.B. Wilson. 1998. The dynamics of spatially distributed host-parasitoid
233
system. Pages 75–110 in: Tilman, D., and P. Kareiva, ed. Spatial Ecology. Princeton University
234
Press, Princeton.
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Liebhold, A., W. D. Koenig, and O. N. Bjornstad. 2004. Spatial synchrony in population
236
dynamics. Annual Review of Ecology, Evolution and Systematics 35:467-490.
237
Maser, G.L., F. Guichard, and K. McCann. 2007. Weak trophic interactions and the balance of
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enriched metacommunities. Journal of Theoretical Biology 247:337-345.
239
May, R.M. 1973. Stability and complexity in model ecosystems. Princeton University Press,
240
Princeton.
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241
McCann, K.S., A. Hastings, and G.R. Huxel. 1998. Weak trophic interactions and the balance of
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nature. Nature 395:794-798.
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Schönfisch B., and de Roos A. 1999. Synchronous and asynchronous updating in cellular
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Automata. BioSystems 51 (1999) 123–143
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Tilman, D. 1999. The ecological consequences of changes in biodiversity: A search for general
246
principles. Ecology 80:1455-1474.
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Vasseur, D.A., and J.W. Fox. 2007. Environmental fluctuations can stabilize food web dynamics
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by increasing synchrony. Ecology Letters 10:1066-1074.
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Legends
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Figure 1. The diamond-shaped food web. From left to right described by: a set of differential
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equations (Vasseur and Fox 2007 and McCann et al. 1998), a graph, and a set of difference
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equations (Gouhier et al. 2010). P –predator, C1-Consumer 1, C2-Consumer 2, and R- Resource
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and for more details on parameters se Gouhier et al 2010 and Vasseur and Fox 2007.
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Figure 2. Simulations of Gouhier et al. (2010) food web model to test the effect of different
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initial conditions on local dynamics. Red –predator, blue-Consumer 1, and green-Consumer 2.
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Figure 3. Simulations of Gouhier et al. (2010) diamond-shaped food web model to test the effect
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of different lattice size.Cell abundance mean is calculated as mean density over all cells. Red –
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predator, blue-Consumer 1, and green-Consumer 2.
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Figure 4. Simulation of Gouhier et al. (2010) food web model to test the effect of different
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asynchronous updates of the dispersal and also to test the effect of low, medium and high
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dispersal, d=[0.004 0.5 0.996]. Cell abundance mean is calculated as mean density over all cells.
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Red –predator, blue-Consumer 1, and green-Consumer 2.
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