1
Supporting Information:
2
The role of diversity of interaction types and space in ecological networks stability
3
Miguel Lurgi, Daniel Montoya and José M. Montoya
4
5
Appendix S1: Full description of the methods
6
Spatially explicit individual-based, bio-energetic model
7
Generation of species interactions networks
8
Once a food web was generated using the niche model (Williams & Martinez 2000) (see
9
main text), and in order to obtain a ‘hybrid’ interaction network, some links in the network,
10
among those specifying interactions between basal species and species in the first trophic
11
level (i.e. herbivore links) were selected to become mutualistic links according to the
12
corresponding MAI ratio. The MAI ratio hence corresponds to the proportion of plant
13
herbivores vs. plant mutualists in our networks. We generated networks with this
14
characteristics for MAI ratios ranging from 0 to 1.0 with steps of 0.1: [0, 0.1, 0.2, 0.3 … 1],
15
making up a total of 11 different MAI ratios, from communities with no mutualistic
16
interactions to communities with only mutualistic links between basal resources and first
17
order consumers and no herbivores. This way of choosing mutualistic links does not ensure a
18
predictable proportion of mutualistic plants versus non-mutualistic (i.e., wind dispersed)
19
plants. This fraction was however always equal or larger than the MAI ratio, because a given
20
plant species would become mutualistic (and cease to be wind dispersed) as soon as a
21
mutualistic link was assigned to it.
22
23
Our approach to create the interaction networks ensures that mutualistic partners (both animal
24
and plants) are embedded in a whole community context alongside other trophic groups such
25
as omnivores and top predators. It also ensures that a trophic pyramid is maintained within
1
26
the community, with mutualistic interactions sitting at the bottom of the pyramid between
27
basal species and primary consumers (Fig. 1 in the main text). An example of the interaction
28
networks generated in this way for one of the model communities is shown in Fig. S1. In
29
summary, this method creates a network architecture that is consistent with food webs while
30
at the same time allowing for a well-identified section composed of mutualistic interactions,
31
from which network properties that are exclusive of mutualistic networks can be calculated.
32
33
34
Figure S1. Example of a food web taken from one of the model communities. Nodes represent species
35
and are color-coded by species type: yellow = mutualistic plants, green = non-mutualistic plants,
36
blue = herbivores, purple = mutualistic animals, orange = primary predators, and red = top
37
predators. Links (lines joining nodes) are trophic or mutualistic relationships amongst them. Thicker
38
part of the line shows the end of the link. Links go from prey to predator species. The layout orders
39
species per trophic level with higher trophic levels towards the top of the diagram.
40
41
Individual-based spatially explicit dynamics
42
All individuals in the IBM are equipped with an energy storage unit. This energy is the
43
currency governing the actions performed by individuals, their lifespan, and the mean by
44
which the economy of the whole artificial ecosystem is maintained.
2
45
46
47
48
49
Thus, the basic processes that occur at the individual level are:
1. Death: Individuals possessing less than a fixed level of energy (relative to the size of
the energy storage unit) die and are removed from the system.
2. Movement: Individuals can move to one of its 8 neighbouring cells (see Fig. 2 in main
text) chosen randomly, provided that it is available.
50
3. Reproduction: Individuals can reproduce using one of two strategies if they have
51
enough resources (i.e., energy) to do so (see Table S1 for parameter values):
52
a. Sexual reproduction occurs between two individuals of the same non-basal
53
(animal) species. A reproductive event occurs if and only if two conditions are
54
met: (i) the subject individual encounters another individual of its same
55
species and with enough resources to reproduce within its neighbouring cells
56
(see Fig. 2 in main text), and (ii) there is an empty (or containing only a plant
57
individual) cell within the individual’s 4-cell radius neighbourhood. The
58
offspring (new individual) created during the reproductive event is placed in
59
this randomly chosen empty cell.
60
b. Asexual reproduction occurs in basal species (i.e., plants). The model allows
61
for two mechanisms of asexual reproduction: (1) ‘Wind dispersal’ and (2)
62
Mutualistic dispersal. In the former case, as for the sexual reproduction
63
scenario, a 4-cell radius neighbourhood around the plant individual defines the
64
spatial extent of the dispersal event. For the reproductive event to occur, the
65
plant must possess enough resources to reproduce (see Table S1 for parameter
66
value), and one of the cells (randomly chosen) in the spatial neighbourhood
67
mentioned above must be empty (or containing only an animal individual).
68
The new individual produced during the ‘wind dispersal’ event is placed in
69
that empty cell.
3
70
In the latter case (mutualistic dispersal), dispersal is done by the animal
71
partner, which means that the ‘seed’ for a new individual can travel farther
72
before it settles. The spatial extent of this dispersal event will depend on (1)
73
the movements of the animal disperser after it has visited a mutualistic plant
74
partner, (2) its efficiency in dispersing the ‘seed’ (mutualistic efficiency), and
75
(3) a cooling effect that decreases the dispersal (mutualistic) efficiency as time
76
lapses (see Table S1 for parameters governing this process and their
77
explanation). If, before this time lapses (when the dispersal efficiency
78
becomes zero), the animal partner comes across an empty cell in the
79
landscape, it ‘creates’ an offspring for the plant previously visited with a
80
probability given by its mutualistic efficiency. Although plants can reproduce
81
sexually, we only needed to simulate the ecological fact that plants can
82
reproduce without the need of physically encountering another individual of
83
the same species.
84
4. Feeding: This is perhaps the most fundamental mechanism of the model. It occurs
85
when two potentially interacting individuals, as defined in the interaction matrix, find
86
each other in space (i.e. one is occupying the cell that it was chosen by the other when
87
moving). Several outcomes are possible (see Table S1 for bioenergetic parameters
88
governing each one of these outcomes):
89
a. If both individuals are not primary producers, a predation event occurs, in
90
which the prey dies and the predator increases its energy storage unit’s level.
91
b. If one individual is a basal species and the other is a non-mutualistic primary
92
consumer (i.e., herbivore), then the herbivore takes some resources from the
93
plant and both continue living, with effects on the energy storage.
94
c. If it is a mutualistic relation (mutualistic plant and mutualistic animal), the
4
95
animal takes resources from the individual plant while at the same time ‘keeps
96
track’ of the species it belongs to until some point in the future. If, before this
97
time lapses, the animal partner comes across an empty cell in the landscape, it
98
‘creates’ an offspring for the plant previously visited with a probability given
99
by its mutualistic efficiency (see Table S1).
100
In all feeding events an efficiency or assimilation rate of resources is applied. This
101
coefficient differs between herbivore and carnivore links, since assimilation varies
102
depending on whether the food is plant material or animal material (Ings et al.
103
2009). Additionally, an omnivory trade-off is applied to the resources obtained by
104
an omnivore individual when feeding on a basal resource (plant). This models the
105
fact that omnivores are less efficient at eating plant material than herbivores. See
106
Table S1 for the specification of the parameter values governing these processes.
107
In addition to the aforementioned demographic processes, the model incorporates
108
immigration, which is simulated by assigning to each empty cell on the grid a probability (see
109
Table S1) of colonization by a new individual from a randomly chosen species from the
110
original species pool (i.e., a species from the original ecological network). Each iteration or
111
time step without an interaction event will reduce the energy storage of every individual due
112
to metabolic losses. Because reproduction demands an energy investment, this loss of energy,
113
without replenishment via feeding interactions, will make more difficult the individual’s
114
reproduction and will eventually cause the extinction of that individual. With the exception of
115
efficiency transfer coefficients, bio-energetic parameters are identical for all species in the
116
ecosystem (see Table S1 for a full list of parameters).
117
5
118
Table S1: Parameters for the model described in the text, including bio-energetic values
Parameter name
Value
Description
OCCUPIED_CELLS
0.4
MAX_RESOURCE
20
MIN_RESOURCE
3
LIVING_EXPEND
0.01
MATING_RESOURCE
0.5
MATING_ENERGY
0.2
Fraction of the grid initially occupied by individuals randomly placed on
it.
Maximum amount of resource an individual may possess at any given
time.
Death threshold: minimum amount of resource at individual may possess.
Any individual possessing less than this amount at any given iteration will
die (see text).
Fraction of resource an individual spends in living every iteration of the
model. Metabolic rate.
Fraction of MAX_RESOURCE that is required for an individual to be
able to reproduce.
Fraction of resource given to the offspring by the parent during
reproduction. Each parent gives the same fraction. The total amount
depends on how much resource the parent possesses at the time of
reproduction.
Probability that a new individual will appear in a cell of the grid each
iteration. The species this individual belongs to is randomly chosen from
the original species pool.
Fraction of resource that is autotrophically created by each individual
from the basal species every iteration. This is the only energy input to the
system.
Fraction of resource lost to herbivores by individuals belonging to a basal
species during a trophic event, i.e. a species in the first trophic level
feeding on a species in the basal level.
Fraction of resource that omnivores are effectively able to gather when
feeding on a species from the basal level (a plant).
Fraction of resource of a primary producer (basal species individual) that a
mutualistic partner obtains when an interaction of this type occurs.
Probability that a predator individual embark upon a trophic relationship
with one of its prey individuals when it encounters it.
Fraction of the resource the prey that is assimilated by the predator in a
carnivorous interaction, i.e. trophic interaction not involving individuals
from the basal species.
Fraction of the resource of the prey assimilated by the herbivore in an
herbivorous interaction.
Efficiency of an individual mutualist when dispersing a plant partner. In
other words, the probability with which a mutualistic individual will
facilitate the creation of a new individual of the last species of plant it
visited when it is positioned on an empty cell immediately after it
interacted with a mutualistic plant partner.
Cooling factor for the mutualistic efficiency of plant dispersers
(mutualists). This is the fraction of mutualistic efficiency that remains
after each iteration.
Reproduction rate of non-mutualistic plant species. Probability with which
an individual belonging to a plant species that does not possess mutualistic
partners for dispersal will create an offspring in any given iteration of the
simulation run.
IMMIGRATION
0.005
SYNTHESIS_ABILITY
0.1
HERB_FRACTION
0.7
OMNI_TRADEOFF
0.4
MUT_FRACTION
0.25
CAPTURE_PROB
0.4
EFFICIENCY_TRANSF
0.2
HERB_EFFICIENCY
0.8
MUT_EFFICIENCY
0.8
MUT_COOLING
0.9
REPROD_RATE
0.01
119
6
120
121
122
Analysis of community properties and stability metrics
Community diversity
The Shannon diversity index is defined as:
123
𝑆
124
′
𝐻 = − ∑ 𝑝𝑖 ∙ 𝑙𝑛 𝑝𝑖
𝑖=1
125
126
where S is the number of species and pi is the proportion of individuals of species i in the
127
community. When the index is calculated within functional groups the proportion of
128
individuals is taken only considering the species within the group. Shannon evenness index
129
on the other hand is calculated as:
130
𝐽′ =
𝐻′
𝑙𝑛 𝑆
131
132
133
Community stability
The centroid of each species population was calculated as the average of the positions
134
of all individuals belonging to that species in the 2D grid. It is thus a measure composed of
135
two values: x and y coordinates for the average of the locations. Moran’s I and Geary’s C are
136
metrics commonly used for the quantification of spatial correlation or ‘aggregation’ in
137
spatially explicit data (Fortin & Dale 2005). Moran’s I for a given species is defined as:
138
139
140
141
7
142
where N is the number of spatial units indexed by i and j, cells in the 2D grid in our simulated
143
landscape; X is the variable of interest, which takes the value of 1 if an individual of the
144
̅ is the
species for which the index is being calculated is present in that cell and 0 otherwise; X
145
mean of X; and wij is an element of a matrix of spatial weights. In our case, the weights of
146
this matrix are always 1 because all the cells in the grid are equally important for the
147
calculation of the index. Geary’s C is defined as:
148
149
150
151
where in addition to the values defined for the Moran’s I, we also have W, which is the sum
152
of all wij; in our case, the number of cells in the grid. These two indices are used to quantify
153
spatial aggregation at the global (Moran’s I) and at the local (Geary’s C) scales, and are
154
therefore complementary ways of quantifying spatial aggregation.
155
156
Sensitivity/Robustness analyses
157
To assess the robustness of our results to differences in species richness, landscape lattice
158
size, and number of generated communities, we performed a series of sensitivity analyses.
159
These analyses revealed that model simulation outcomes are affected to some extent by these
160
factors, but only until some identifiable threshold, after which results are qualitatively and
161
quantitatively similar. We chose the values that allowed us to obtain results consistent with
162
larger community sizes. Simulations were performed with lattice sizes of 50x50, 150x150,
163
200x200 (the value used for the simulations reported in the manuscript), and 250x250, which
164
demonstrated that for lattice sizes above 200x200 results are qualitatively and quantitatively
165
similar. For smaller sizes (i.e., 50x50 and 150x150) some of the patterns observed varied
8
166
quantitatively. Different values for the number of species in our communities were
167
considered: 40, 60, 80, and 100. In this case, again, significant differences in community
168
patterns were not observed between species richness values of 60 and greater. Experiments
169
were performed considering 25, 50, and 100 replicates of communities for a subset of the
170
MAI ratios studied. We found no statistically significant differences between results obtained
171
for these three values. Because trade-offs needed to be considered between computation costs
172
and size of the experiments, we chose the smaller lattice size, number of species, and number
173
of replicates that better allowed us to obtain the most accurate results in order to save
174
computation time.
175
Similarly, we tested the robustness of our results by conducting sensitivity analyses of the
176
different parameters used in the model. We have modified individually every parameter of
177
the model (i.e. those described in table S1). In doing so, we have changed the original value
178
reported in the table by ± 10% and used GLMs to test whether significant changes on the
179
relationship between MAI ratios and network properties/stability metrics analysed exist.
180
Results are summarized in tables S2 and S3 (GLMs, d.f., a significant difference is
181
considered for p<0.05). In general, no significant differences were observed for any of the
182
parameters modified.
183
184
185
186
187
188
189
9
190
Table S2. Sensitivity analyses for the parameter values included in the model reported in
191
table S1 - 10% of their value. P-values for the GLMs are reported, indicating whether
192
significant differences exist on the relationship between the MAI ratio and the network
193
property analyzed. Red values indicate significant differences (p<0.05).
properties
IMMIGRATION REPROD_RATE LIVING_EXP MATING_ENERGY MATING_RESOURCE MAX_RESOURCE MIN_RESOURCE SYNTHESIS_ABILITY
L
0,620
0,069
0,243
0,572
0,636
0,466
0,641
0,117
L/S
0,623
0,071
0,240
0,569
0,638
0,468
0,637
0,117
C
0,620
0,074
0,240
0,569
0,636
0,466
0,638
0,116
GenSD
0,642
0,440
0,316
0,776
0,326
0,747
0,243
0,569
VulSD
0,165
0,721
0,119
0,349
0,893
0,898
0,311
0,412
MxSim
0,396
0,336
0,160
0,757
0,093
0,965
0,483
0,117
MeanFoodChainLength
0,797
0,607
0,278
0,666
0,027
0,532
0,105
0,599
ChnSD
0,401
0,964
0,105
0,971
0,001
0,730
0,239
0,230
ChnNo
0,772
0,312
0,620
0,552
0,127
0,762
0,160
0,606
Dynamic Complexity (<i> SC^1/2)
0,982
0,332
0,685
0,459
0,644
0,263
0,873
0,102
Clustering Coefficient
0,911
0,576
0,739
0,962
0,173
0,673
0,927
0,379
Compartmentalisation
0,289
0,982
0,687
0,389
0,724
0,468
0,315
0,296
Mean Trophic Position
0,647
0,733
0,463
0,935
0,190
0,530
0,133
0,563
Standard Deviation Trophic Position
0,926
0,740
0,412
0,960
0,075
0,369
0,090
0,697
NODF (Nestedness)
0,575
0,880
0,080
0,740
0,730
0,204
0,810
0,934
Gq (Quantitative Generality)
0,229
0,661
0,289
0,870
0,921
0,126
0,145
0,619
Vq (Quantitative Vulnerability)
0,721
0,213
0,745
0,217
0,588
0,715
0,966
0,323
H2' (Mutualistic Specialisation)
0,527
0,389
0,451
0,473
0,837
0,510
0,282
0,746
Total Community Abundance
0,806
0,567
0,860
0,846
0,279
0,014
0,886
0,701
Shannon's Diversity Index
0,974
0,806
0,547
0,984
0,077
0,082
0,803
0,832
194
properties
CAPTURE_PROB HERB_EFFICIENCY
HERB_FRACTION
MUT_COOLING
L
0,178
0,489
0,039
0,653
L/S
0,179
0,490
0,039
0,650
C
0,194
0,488
0,038
0,650
GenSD
0,763
0,044
0,685
0,427
VulSD
0,739
0,537
0,298
0,490
MxSim
0,308
0,530
0,035
0,766
MeanFoodChainLength
0,888
0,358
0,917
0,383
ChnSD
0,826
0,938
0,619
0,243
ChnNo
0,933
0,192
0,330
0,423
Dynamic Complexity (<i> SC^1/2)
0,695
0,651
0,316
0,708
Clustering Coefficient
0,142
0,624
0,461
0,682
Compartmentalisation
0,534
0,978
0,794
0,325
Mean Trophic Position
0,715
0,404
0,873
0,505
Standard Deviation Trophic Position
0,755
0,706
0,937
0,501
NODF (Nestedness)
0,438
0,484
0,769
0,985
Gq (Quantitative Generality)
0,592
0,655
0,182
0,976
Vq (Quantitative Vulnerability)
0,547
0,201
0,478
0,301
H2' (Mutualistic Specialisation)
0,591
0,606
0,806
0,802
Total Community Abundance
0,929
0,836
0,294
0,964
Shannon's Diversity Index
0,584
0,456
0,448
0,994
195
196
197
198
199
200
201
202
203
204
205
206
10
MUT_EFFICIENCY
0,193
0,194
0,196
0,331
0,153
0,059
0,153
0,039
0,616
0,740
0,576
0,595
0,277
0,124
0,943
0,413
0,265
0,513
0,958
0,764
MUT_FRACTION OMNI_TRADEOFF EFFICIENCY_TRANSF
0,190
0,188
0,181
0,671
0,410
0,460
0,549
0,845
0,339
0,162
0,934
0,323
0,197
0,296
0,899
0,043
0,062
0,904
0,238
0,430
0,776
0,779
0,777
0,313
0,234
0,966
0,072
0,028
0,124
0,451
0,555
0,365
0,102
0,058
0,625
0,463
0,132
0,674
0,797
0,750
0,702
0,704
0,702
0,048
0,084
0,472
0,447
0,853
0,242
0,840
0,020
0,556
0,612
0,511
0,320
0,339
0,196
0,765
0,768
0,222
207
Table S3. Sensitivity analyses for the parameter values included in the model reported in
208
table S1 + 10% of their value. P-values for the GLMs are reported, indicating whether
209
significant differences exist on the relationship between the MAI ratio and the network
210
property analyzed. Red values indicate significant differences (p<0.05).
211
properties
IMMIGRATION
L
0,539
L/S
0,525
C
0,507
GenSD
0,874
VulSD
0,189
MxSim
0,417
MeanFoodChainLength
0,046
ChnSD
0,081
ChnNo
0,088
Dynamic Complexity (<i> SC^1/2)
0,898
Clustering Coefficient
0,114
Compartmentalisation
0,415
Mean Trophic Position
0,152
Standard Deviation Trophic Position
0,041
NODF (Nestedness)
0,333
Gq (Quantitative Generality)
0,345
Vq (Quantitative Vulnerability)
0,838
H2' (Mutualistic Specialisation)
0,063
Total Community Abundance
0,971
Shannon's Diversity Index
0,739
212
213
REPROD_RATE
0,308
0,309
0,307
0,611
0,744
0,729
0,252
0,037
0,381
0,866
0,500
0,692
0,144
0,205
0,737
0,786
0,961
0,619
0,403
0,422
LIVING_EXP
0,344
0,341
0,341
0,328
0,696
0,198
0,527
0,833
0,592
0,639
0,061
0,032
0,245
0,338
0,374
0,035
0,711
0,429
0,687
0,816
MATING_ENERGY MATING_RESOURCE MAX_RESOURCE MIN_RESOURCE
0,219
0,220
0,226
0,963
0,890
0,568
0,952
0,894
0,968
0,311
0,819
0,794
0,874
0,929
0,251
0,825
0,822
0,182
0,862
0,441
properties
CAPTURE_PROB HERB_EFFICIENCY HERB_FRACTION MUT_COOLING
L
0,536
0,187
0,295
0,309
L/S
0,533
0,194
0,296
0,369
C
0,533
0,203
0,294
0,444
GenSD
0,850
0,033
0,639
0,982
VulSD
0,759
0,732
0,589
0,728
MxSim
0,942
0,631
0,894
0,553
MeanFoodChainLength
0,927
0,917
0,323
0,875
ChnSD
0,740
0,576
0,316
0,963
ChnNo
0,600
0,806
0,301
0,947
Dynamic Complexity (<i> SC^1/2)
0,180
0,802
0,366
0,472
Clustering Coefficient
0,922
0,681
0,741
0,528
Compartmentalisation
0,496
0,020
0,667
0,761
Mean Trophic Position
0,247
0,414
0,608
0,206
Standard Deviation Trophic Position
0,497
0,880
0,550
0,922
NODF (Nestedness)
0,321
0,310
0,918
0,175
Gq (Quantitative Generality)
0,447
0,488
0,254
0,325
Vq (Quantitative Vulnerability)
0,866
0,017
0,900
0,337
H2' (Mutualistic Specialisation)
0,915
0,865
0,448
0,752
Total Community Abundance
0,425
0,521
0,544
0,237
Shannon's Diversity Index
0,358
0,588
0,863
0,712
11
0,429
0,426
0,395
0,888
0,848
0,972
0,727
0,418
0,918
0,287
0,720
0,502
1,000
0,739
0,226
0,428
0,541
0,955
0,779
0,928
MUT_EFFICIENCY
0,953
0,924
0,919
0,601
0,158
0,235
0,433
0,375
0,693
0,609
0,142
0,162
0,787
0,625
0,501
0,948
0,261
0,817
0,816
0,706
0,008
0,008
0,008
0,863
0,333
0,516
0,258
0,931
0,032
0,293
0,221
0,475
0,182
0,123
0,795
0,112
0,412
0,247
0,447
0,657
0,065
0,070
0,076
0,587
0,722
0,853
0,736
0,807
0,973
0,267
0,555
0,227
0,844
0,727
0,658
0,007
0,367
0,943
0,414
0,859
SYNTHESIS_ABILITY
0,330
0,327
0,327
0,464
0,620
0,005
0,411
0,200
0,479
0,481
0,856
0,985
0,248
0,189
0,744
0,643
0,500
0,258
0,379
0,367
MUT_FRACTION OMNI_TRADEOFF EFFICIENCY_TRANSF
0,608
0,610
0,608
0,648
0,140
0,644
0,398
0,978
0,295
0,536
0,838
0,066
0,332
0,541
0,879
0,155
0,680
0,658
0,164
0,244
0,123
0,123
0,121
0,587
0,774
0,224
0,890
0,725
0,856
0,079
0,113
0,691
0,584
0,323
0,370
0,040
0,735
0,411
0,221
0,569
0,135
0,133
0,128
0,447
0,862
0,813
0,384
0,404
0,157
0,226
1,000
0,173
0,133
0,464
0,135
0,979
0,712
0,502
0,565
0,861
214
Appendix S2: Results
215
Community structure
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Figure S2. Rank-abundance distribution plots of 10 sample networks with different MAI ratios and
218
which are representative of the set of communities studied. Lines represent a fit of the data to a
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lognormal distribution, typically observed in empirical rank-abundance distributions (Magurran
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2004).
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Figure S3. Moran’s I and Geary’s C (at the whole community level, calculated as the mean of each
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index across all the species in the community), plotted against MAI ratio, in model communities.
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Points show index values for each replicate. Line and shadow represent the fit of a linear model to the
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data and the standard error of the mean respectively (p-value < 0.001 for linear models fits to each
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13
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MAI ratio
0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1
MAI ratio
230
Figure S4. Geary’s C spatial aggregation index per trophic level, plotted against MAI ratio, in model
231
communities. Points show index values for each replicate. Line and shadow represent the fit of a
232
linear model to the data and the standard error of the mean respectively. * and *** represent p-value
233
< 0.05 and 0.001 for linear models fits to each data set respectively.
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14
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References
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Fortin, M.-J. & Dale, M. (2005). Spatial analysis: A guide for ecologists. Cambridge University Press,
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Cambridge, UK.
Ings, T.C., Montoya, J.M., Bascompte, J., Blüthgen, N., Brown, L.E., Dormann, C.F., et al. (2009).
Ecological networks - beyond food webs. JAE, 78, 253–269.
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Magurran, A.E. (2004). Measuring biological diversity. Blackwell Publishing, Oxford, UK.
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Williams, R.J. & Martinez, N.D. (2000). Simple rules yield complex food webs. Nature, 404, 180–
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183.
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15