1,2,*
2
2
* Corresponding author: tarik.gouhier@gmail.com | Phone (541) 737-5405 | Fax: (541) 737-3360
1 Present address: Department of Zoology, 3029 Cordley Hall, Oregon State University, Corvallis,
OR 97331, USA
2 Department of Biology, McGill University, 1205 Avenue Docteur Penfield, Montréal, Québec,
H3A 1B1, Canada
Number of figures: 10
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SUMMARY OF REVIEW
Wennergren and Gudmundson claim that our recent findings regarding the effects of dispersal and environmental variability on food web stability published in The American Naturalist contradict previous work because of methodological errors and questionable assumptions. Here, we demonstrate that (1) our work supports and extends—rather than contradicts—existing theory, and
(2) that their issues with our methodology/assumptions stem from a fundamental misunderstanding over the use of asynchronous updates to simulate a continuous time process in discrete spatial ecological systems. For these reasons, we believe the comment should be declined.
REVIEW
Previous work has shown that asymmetrical trophic interactions can generate asynchronous fluctuations in the abundance of competing species (i.e. compensatory dynamics) that promote coexistence and the stability of diamond-shaped food webs (McCann et al. 1998). Recent work has shown that in diamond-shaped food webs, exposing competing species to weak-to-moderate levels of correlated environmental fluctuations can further promote food web stability by limiting the amplitude of compensatory dynamics (Vasseur and Fox 2007). In an article published in The
American Naturalist this year (Gouhier et al. 2010), we sought to extend these studies to spatial ecological systems by incorporating this food web module into a metacommunity (i.e. a set of 256 2 food web modules—or communities—connected by dispersal) in order to understand how dispersal, environmental variability and compensatory dynamics interact to govern stability. We showed that high rates of dispersal synchronized food web dynamics in space and that mild-tomoderate levels of correlated environmental fluctuations stabilized food webs by reducing the amplitude of compensatory dynamics (Fig 1). These results were qualitatively identical to those found by Vasseur and Fox (2007) in a non-spatial version of the model. Additionally, we found that in the absence of environmental variability, low rates of dispersal stabilize food webs at local and
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19 global scales by spatially desynchronizing their dynamics (Gouhier et al. 2010). However, when dispersal rates are low, the addition of spatially-correlated environmental variability reduces food web stability at local and regional scales by spatially synchronizing their dynamics via the Moran effect (Gouhier et al. 2010). In an effort to demonstrate the robustness of our findings, we also showed that autocorrelated environmental variability (i.e. reddened environmental noise) does not affect the main conclusions of our study, which were obtained with non-autocorrelated environmental variability (i.e. white noise; see Appendix B of Gouhier et al. (2010)).
In their comment, Wennergren and Gudmundson claim that our findings (1) conflict with existing work (page 2, lines 40-47) and (2) stem from questionable assumptions and flaws in our modeling framework (page 1, lines 6-12; page 2, lines 47-49). The authors claim five separate flaws in our approach: (1) discretizing the original differential equation model introduces a destabilizing time lag, (2) the study is limited to transient dynamics, (3) patches are updated asynchronously, (4) computations are incorrect because of small number representation errors and (5) dispersal is set up as an individual based model yet whole food webs are treated as individuals (page 3, lines 51-63).
Below, we summarize and address the authors’ comments in two sections: (1) our findings conflicting with existing theory and (2) the technical details and assumptions in our modeling framework.
1.
Our results conflict with existing theory (page 2, lines 39-47):
The authors claim that two of our findings conflict with existing theory.
“[…] Gouhier et al. also show some quite surprising results; firstly, when dispersal rate is low, asynchrony is induced over space by the dispersal scheme itself which then promotes biodiversity yet weak environmental fluctuations added to this can thereby reduce food web stability by synchronizing the subpopulations in the
landscape […]” (page 2, 40-43):
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The authors state that this finding conflicts with existing theory, but neither clarify how nor offer any specific citation to substantiate their claim. Because of the compound structure of the sentence, it is difficult to understand whether the authors object to our finding that (1) low dispersal induces asynchrony and thus stabilizes food webs (we assume that authors meant “stability” when referring to “biodiversity”) and/or (2) that weak but fully spatially correlated environmental fluctuations induce synchrony and destabilize food webs. We now discuss both potential sources of conflict.
First, previous work has shown that in the absence of environmental variability, low dispersal can stabilize food webs by decoupling them in space and inducing spatial asynchrony (see review by Briggs and Hoopes 2004). As summarized in our paper (page E27, “The Stabilizing Effect of Space”), this stabilizing effect of low dispersal can occur via three separate mechanisms. First, the averaging of spatially heterogeneous food web dynamics increases global stability via “statistical stabilization”. Second, nonlinear spatial averaging can stabilize (or destabilize) local food webs when spatially heterogeneous densities are combined with nonlinear responses to density. Third, low dispersal can decouple local abundance from immigration and thus increase both local and global food web stability. Indeed, this decoupling leads to a negative correlation between local abundance and immigration that dampens local fluctuations by (1) promoting immigration when abundance is low and (2) limiting immigration when local abundance is high. This mechanism can stabilize food webs at both local and global scales (Briggs and Hoopes 2004). Hence, our finding that low dispersal induces spatial asynchrony and promotes local and global stability is perfectly consistent with existing theory.
Second, it has long been established that spatially-correlated environmental noise can generate synchrony in the dynamics of spatially disjunct populations via the Moran effect (Moran
1953). By spatially synchronizing food web dynamics, the Moran effect prevents stabilization via
“the decoupling of immigration and local abundance” and “statistical stabilization” (the mechanisms
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19 described above), and thus leads to a reduction in food web stability at local and regional scales. This finding is also consistent with established theory. Perhaps the authors were contending that our results conflict with those of Vasseur and Fox (2007), who show that low-to-moderate levels of correlated environmental fluctuations stabilize food webs in non-spatial models by reducing the amplitude of compensatory dynamics. However, Vasseur and Fox (2007)’s non-spatial model is only comparable to ours when the rate of dispersal is high in our spatial model (i.e. when our metacommunity becomes a mean-field approximation). Indeed, we show that when high levels of dispersal spatially synchronize food webs, the addition of weak-to-moderate correlated environmental fluctuations stabilizes food web dynamics at the global scale by limiting the amplitude of compensatory dynamics (Fig. 3 in our original paper, reproduced here for convenience as Fig. 1).
At high rates of dispersal, our model replicates the results of Vasseur and Fox (2007) over the entire range of environmental correlation and fluctuation strength (Fig. 1). It is precisely the spatial nature of our model (i.e. at low dispersal rates) that makes our results (1) novel, (2) perfectly consistent with existing theory describing the effects of dispersal on stability (Briggs and Hoopes 2004), but (3) not directly comparable to the results generated by Vasseur and Fox (2007)’s non-spatial model.
“[…] secondly, the authors also claim that redness of environmental variation, i.e. autocorrelation over time, has no effect on this system. These two results are inconsistent and contradict previous studies in reduced systems, for example that reddening of environmental noise affect density and mean persistence time
(Cuddington & Yodzis 1999; Greenman & Benton 2005).” (page 2, lines 43-47):
We tested the robustness of our results to a number of different implementations of environmental variability and underlying food web dynamics (see Appendix B of Gouhier et al. 2010). For one of those tests, we showed that implementing environmental fluctuations as red noise (i.e. autocorrelated) instead of white noise (i.e. non-autocorrelated) did not affect our main conclusions
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2 and stated in our original paper that “[…] our findings are robust to temporally autocorrelated environmental fluctuations” (page E19, “Environmental Variability” section). However, that does not mean that red noise has “[…] no effect on this system”, as stated by the authors (page 2, line
44). Indeed, the results presented in Appendix B (Fig B4 in our original paper, reproduced here for convenience as Fig 2) clearly show that red noise reduces stability and persistence, as predicted by current theory (compare Fig 3 vs. Fig B4 in the original paper, or Fig 1 and Fig 2 here). Indeed, red noise not only reduces the stability of food webs compared to white noise, but it also reduces the parameter space over which food webs persist (compare the missing values in Fig 3 vs. Fig B4 in the original paper, or Fig 1 vs. 2 here). Red noise leads to global extinction at relatively low levels of environmental fluctuations (as early as 

=0.3) and for all levels of environmental correlation (Fig.
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B4 in original paper or Fig. 2), whereas white noise leads to global extinction at only very high levels


> 0.5) and only for strongly correlated (positively or negatively) environmental fluctuations (Fig. 3 in original paper or Fig. 2). Hence, our results are consistent with 13
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22 persistence of ecological systems (see Ruokolainen et al. 2009 for a review).
In this section, we have shown that the results and interpretations presented in our original paper are perfectly consistent with exiting theory. The inconsistencies described by the authors stem from inappropriate comparisons (i.e. comparing the results from Vasseur and Fox (2007)’s nonspatial model to our spatial model when dispersal is low) or misinterpretations of our statements regarding the robustness of our results and original figures. We now turn to the authors’ technical issues with our work.
2.
Methodological errors or erroneous assumptions
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“A destabilizing time lag is introduced since the model is a discretized version of the previous differential equation system of the diamond-shaped food web […]” (page 3,
lines 52-54; pages 4-5, lines 96-119):
We do not agree with the authors: the discretization of the original differential equation would have resulted in destabilization had we performed synchronous updates. This is why we elected to use
asynchronous updates to simulate the original continuous time differential equation model by inducing gradual changes in the system (Durrett and Levin 1994b). The use of asynchronous updates to approximate a continuous time process has a rich history in cellular automata theory (Durrett and
Levin 1994a, 1994b, Schonfisch and de Roos 1999, Nehaniv 2002, 2004, Cornforth et al. 2003, 2005,
Fates and Morvan 2005). Classical cellular automata have often been criticized for using synchronous updates, which are unrealistic since they assume that all components of the system respond to a common, global clock or trigger (Schonfisch and de Roos 1999, Cornforth et al. 2003,
2005, Fates and Morvan 2005). While real spatial ecological systems may appear synchronous at broad temporal scales (e.g. ecological processes often share a common response to seasonality, phenology), most locations typically do not undergo any form of demographic change (e.g. births, deaths, immigration or emigration) at finer temporal scales (Schonfisch and de Roos 1999). Hence, at finer temporal scales, real ecological systems can be perceived and modeled asynchronously since some locations may undergo (multiple) demographic changes while other locations exhibit no changes. In this context, asynchronous updates can be viewed as the discretization and approximation of the continuous time processes observed in natural systems (Schonfisch and de
Roos 1999).
The asynchronous update scheme clearly does a good job of approximating a continuous time process since it allowed us to reproduce the results generated by the differential equation models developed by Vasseur and Fox (2007) and McCann et al. (1998) (e.g. Figs. 1-10). As noted by
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19 the authors (page 5, lines 101-104), extremely small levels of dispersal (or no dispersal) cause our approximation of a continuous time process (i.e. asynchronous updates) to fail and the entire metacommunity to collapse within a few time steps when using the initial conditions specified in the original paper. However, this failure was noted in the original article:
“When dispersal is extremely low (d < 0.001), our stochastic approximation of the continuous-time model breaks down and leads to large changes in the mean abundance of all species (Fig. 2B). The quasi extinctions of consumer C
2
and predator P (Fig. 2B) allow the trivial disruption of compensatory dynamics (Fig. 2F).
We now focus on the broad section of the dispersal gradient that remains unaffected by this artifact (i.e., d>0.001).” (page E21, “Food Web Stability in the Absence of
Environmental Fluctuations” section).
As indicated by the last sentence, we focused on the broad range of dispersal that remained unaffected by this failure because we were investigating the effect dispersal on food web stability.
Hence, the failure of our approximation at low (to zero) levels of dispersal—although noted in our original paper—is irrelevant to the presentation or interpretation of our results.
In addition, we found that when we used the constant initial conditions cited in our paper, our results were consistent with those obtained with random and spatially heterogeneous initial conditions falling within the range (0.1, 1) used by Vasseur and Fox (2007) (see page E19 of our paper). Contrary to the authors’ assertions (page 5, lines 104-119), our model does not have a different set of global stability properties than that of Vasseur and Fox (2007) and McCann et al.
(1998). Instead, the differences in the global stability between the differential equation systems of
Vasseur and Fox (2007) and McCann et al. (1998) and our model at low dispersal (prior to the breakdown of our approximation, i.e. d > 0.001) are attributable to the documented stabilizing effect
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24 of low dispersal at local and global scales (Briggs and Hoopes 2004), as discussed in the previous section.
“The study is unintentionally limited to the transient phase of the dynamics.” (page
3, line 55; pages 5-6, lines 121-139):
We took great care to get rid of transients prior to running our analyses. These transient periods can last anywhere from 500 to 50,000 time steps depending on the rate of dispersal and the size of the lattice (Figs. 3-10), and we thus conducted our analyses after 50,000 time steps on large lattices
(256x256) in the original paper. This should have been stated explicitly in the original paper and we regret the omission. Here, we have provided supplementary simulations that show that on large lattices (256x256), the system reaches its asymptotic behavior after ~6,000 time steps and that the dynamics remain identical for the next 94,000 time steps (Figs 3-10). Hence, the analyses presented and interpreted in the original paper were indeed the final post-transient dynamics of the system and not transients. We also show that the spatial coefficient of variation of abundance for each species in the food web remains positive and quasi-constant after the transient period (i.e. for ~94,000 time steps, Fig 5), indicating that spatial heterogeneity does not decay in time as observed by Wennergren and Gudmundson in their simulations.
Instead, the complex and long transient dynamics described by Wennergren and
Gudmundson are a byproduct of their use of asynchronous updates on small lattices (5x5 to 12x12 maximum size compared to 256x256 in our simulations). Contrary to their claims (page 6, lines 134-
139), the identical initial conditions of all cells/patches and the periodic boundary conditions do not mean that the number of cells or patches (i.e. the lattice size) does not matter. Instead, theory has shown that using random sampling with replacement to conduct asynchronous updates (as we have done) in a lattice of n cells means that the expected (or mean) time between successive updates of a given cell c is
   n and its variance is
    
1

(Schonfisch and de Roos 1999). As lattice
 
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22 size n decreases, the asynchronous update mechanism breaks down because the mean and the variance of the period between successive updates for a given cell c both decrease. Decreased lattice size therefore reduces the stochasticity of the update process and allows the system to become increasingly synchronous and discrete. Hence, asynchronous updates via random sampling with replacement provide an unbiased approximation of a continuous time process, but only at large
enough lattice sizes (Schonfisch and de Roos 1999). This explains the breakdown of the asynchronous update process and the subsequent emergence of complex dynamics, long transients and system-wide extinction at small lattice sizes shown by Wennergren and Gudmundson, and by supplementary simulations of our model for small lattice sizes (Figs. 3-10).
“Patches are updated asynchronously (page 3, 56-58, page 6-7, lines 141-165):
Contrary to the authors’ claims, the update process that we have used (i.e. the random update with replacement of n cells per time step in a lattice of size n) is not “unusual” nor was it “introduced” by us (page 6, lines 143-144). It has been used repeatedly (e.g., Durrett and Levin 1994a, 1994b,
Schonfisch and de Roos 1999) and shown to be the most effective and unbiased method for approximating a continuous time process via asynchronous updates, as long as the lattice size is sufficiently large (Schonfisch and de Roos 1999). The so-called “mixing of time of update
(population growth) and the time of dispersal” (page 7, lines 150-151) described by the authors is actually the very definition of the asynchronous update process and not some artifact that we have introduced (see previous sections). With asynchronous updates in a lattice of size n, n cells are selected for update at random (with replacement) at each time step. Since the selection of cells is random with replacement, some cells may be updated twice or more per time step whereas others may remain unchanged. However, the expected time between updates is
   n for all cells, and

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2 more generally the probability P of a cell being updated k times in u updates is binomially-distributed with (Schonfisch and de Roos 1999):
  k
 
 u !
k !
u
 k


1
 n

 k 

1

1 n

 u
 k
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Even if we had separated the time of update from the dispersal process by first iterating the process, we would still be “mixing the time of update and the time of dispersal” since by virtue of the random nature of cell updates, some cells would be updated multiple times per time step whereas others would have remained unchanged.
Once again, this asynchronous update process is not ad hoc but based on established cellular automata theory (Durrett and Levin 1994a, 1994b, Schonfisch and de Roos 1999), and it allows discrete systems to reproduce the results of continuous time models (e.g. compare column 1 of Fig.
2 to Vasseur and Fox (2007)). If the “mixing of time steps” described by the authors had introduced some kind of artifact, it would have been evident in the local time series (Fig 1B or Fig. 5 in the original paper) or the global time series (Figs. 3-10).
Computations do not correct for errors due to small number representation (page 3,
lines 59-60, page 7, lines 167-195):
Wennergren and Gudmundson claim that problems with small number representation introduced artifactual spatial heterogeneity in the metacommunity, and thus allowed density-dependent dispersal to occur between patches. They argue that this phenomenon is responsible for the asynchrony between local food webs observed at low levels of dispersal (page 8, 193-195). In our original simulations, we corrected densities by setting them to a threshold value of zero if they ever became negative after the update process. To test the notion that the small number representation problem was responsible for the spatial asynchrony, we set densities that were below the threshold of 10 -7 to zero, as described by Wennergren and Gudmundson. The new results presented here show that
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19 spatial asynchrony persists for the duration of our long-term simulations (100,000 time steps), regardless of whether or not we set densities that fall below the threshold described by Wennergren and Gudmundson to zero prior to calculating the difference between densities (Figs. 3-10). The strong effect they perceived in their simulations is most likely due to the use of asynchronous updates on lattices that are too small.
The dispersal is set up as an individual based model yet a whole food web is treated as an individual. As a result all species in a focal patch disperse into one and the
same randomly selected patch (page 3, lines 61-63; page 8-9, lines 198-208):
Wennergren and Gudmundson claim that our dispersal process is not appropriate because all individuals disperse to the same randomly-selected neighboring location, “and such a modeling framework may cause very specific dynamics” (page 9, lines 202-203). They suggest that we should allow individuals from a focal location to disperse to all neighboring locations, rather than a single randomly-selected location. However, we found that our dispersal mechanism did not introduce any particular artifact in our results, as were able to reproduce the results of Vasseur and Fox (2007) at high rates of dispersal (Fig 1., column 1). Furthermore, we elected to use this dispersal mechanism because we wanted to focus on the effect of dispersal on food webs at the metacommunity level.
Allowing the different components of the food web (i.e. the resource, the consumers and the predator) to disperse to different locations would have added another layer of complexity (and spatial heterogeneity) that we were not interested in exploring at the time. The authors’ inability to generate spatial heterogeneity with a different dispersal mechanism (mass-action mixing) is not surprising given their inability to generate persistent spatial heterogeneity with our single, random neighbor dispersal process (e.g. Figs. 3-4 in their comment). The lack of spatial heterogeneity in their simulations is once again most likely attributable to the limited size of their lattice and its effect on the asynchronous update process, and not the specifics of the dispersal process.
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CONCLUSION
We have shown that the issues raised by Wennergren and Gudmundson regarding our modeling framework are all related to a misunderstanding of the nature of the asynchronous update process and its requirements (in terms of lattice size). Additionally, we have demonstrated that our results and interpretations are all perfectly consistent with existing theory regarding the effects of dispersal and reddened (and correlated) environmental fluctuations on the stability of food webs. Hence, we recommend that the comment be declined.
LITERATURE CITED
Briggs, C. J., and M. F. Hoopes. 2004. Stabilizing effects in spatial parasitoid-host and predator-prey models: a review. Theoretical Population Biology 65:299-315.
Cornforth, D., D. G. Green, D. Newth, and M. Kirley. 2003. Do artificial ants march in step? ordered asynchronous processes and modularity in biological systems. Pages 28-32 in
Artificial life VIII.
Cornforth, D., D. G. Green, and D. Newth. 2005. Ordered asynchronous processes in multi-agent systems. Physica D: Nonlinear Phenomena 204:70-82. doi: doi: DOI:
10.1016/j.physd.2005.04.005.
Durrett, R., and S. Levin. 1994a. The Importance of Being Discrete (and Spatial). Theoretical
Population Biology 46:363-394.
Durrett, R., and S. A. Levin. 1994b. Stochastic Spatial Models - a Users Guide to Ecological
Applications. Philosophical Transactions of the Royal Society of London Series B-Biological
Sciences 343:329-350.
Fates, N., and M. Morvan. 2005. An experimental study of robustness to asynchronism for elementary cellular automata. Complex Systems 16:1-27.
Gouhier, T. C., F. Guichard, and A. Gonzalez. 2010. Synchrony and Stability of Food Webs in
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Metacommunities. The American Naturalist 175:E16-E34.
McCann, K., A. Hastings, and G. R. Huxel. 1998. Weak trophic interactions and the balance of nature. Nature 395:794-798.
Moran, P. A. P. 1953. The Statistical Analysis of the Canadian Lynx Cycle .2. Synchronization and
Meteorology. Australian Journal of Zoology 1:291-298.
Nehaniv, C. 2002. Evolution in asynchronous cellular automata. Pages 65-73 in Artificial life VIII.
MIT Press.
Nehaniv, C. 2004. Asynchronous automata networks can emulate any synchronous automata network. International Journal of Algebra and Computation 14:719-739.
Ruokolainen, L., A. Lindén, V. Kaitala, and M. S. Fowler. 2009. Ecological and evolutionary dynamics under coloured environmental variation. Trends in Ecology & Evolution 24:555-
563. doi: 10.1016/j.tree.2009.04.009.
Schonfisch, B., and A. de Roos. 1999. Synchronous and asynchronous updating in cellular automata.
Biosystems 51:123-143. doi: doi: DOI: 10.1016/S0303-2647(99)00025-8.
Vasseur, D. A., and J. W. Fox. 2007. Environmental fluctuations can stabilize food web dynamics by increasing synchrony. Ecology Letters 10:1066-1074.
FIGURE LEGENDS
Figure 1: The effect of environmental fluctuations on (A, B) global stability ( 
/
 ) for predator P,
(C, D) consumer C
1
, (E, F) consumer C
2
and (G, H) global consumer correlation ( 
G
) for metacommunities with high (d=0.5, first column) and low (d=0.004, second column) dispersal. All results represent means from 10 replicate simulations. This figure is an unaltered reproduction of
Figure 3 in the Gouhier et al. (2010) AmNat article.
Figure 2: The effect of autocorrelated
  
0.8

environmental fluctuations on (A, B) global stability ( 
/
 ) for predator P, (C, D) consumer C
1
, (E, F) consumer C
2
and (G, H) global consumer

14


correlation ( 
G
) for metacommunities with high (d=0.5, first column) and low (d=0.004, second column) dispersal. All results represent means from 10 replicate simulations. This figure is an unaltered reproduction of Figure B4 in Appendix B of the Gouhier et al. (2010) AmNat
article.
Figure 3: The effects of lattice size and density threshold on the global abundance time series
(100,000 time steps) of consumer C
1
(blue), consumer C
2
(green) and predator P (red) in metacommunities with low dispersal (d=0.004).
Figure 4: The effects of lattice size and density threshold on the global abundance time series
(100,000 time steps) of consumer C
1
(blue), consumer C
2
(green) and predator P (red) in metacommunities with high dispersal (d=0.5).
Figure 5: The effects of lattice size and density threshold on the time series of the spatial coefficient of variation of abundance (100,000 time steps) for consumer C
1
(blue), consumer C
2
(green) and predator P (red) in metacommunities with low dispersal (d=0.004).
Figure 6: The effects of lattice size and density threshold on the time series of the spatial coefficient of variation (100,000 time steps) of abundance for consumer C
1
(blue), consumer C
2
(green) and predator P (red) in metacommunities with high dispersal (d=0.5).
Figure 7: The effects of lattice size and density threshold on the first 6,000 time steps of the global abundance time series of consumer C
1
(blue), consumer C
2
(green) and predator P (red) in metacommunities with low dispersal (d=0.004).
Figure 8: The effects of lattice size and density threshold on the first 6,000 time steps of the global abundance time series of consumer C
1
(blue), consumer C
2
(green) and predator P (red) in metacommunities with high dispersal (d=0.5).
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Figure 9: The effects of lattice size and density threshold on the first 6,000 time steps of the time series of the spatial coefficient of variation of abundance for consumer C
1
(blue), consumer C
2
(green) and predator P (red) in metacommunities with low dispersal (d=0.004).
Figure 10: The effects of lattice size and density threshold on the first 6,000 time steps of the time series of the spatial coefficient of variation of abundance for consumer C
1
(blue), consumer C
2
(green) and predator P (red) in metacommunities with high dispersal (d=0.5).
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FIGURES
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Figure 10
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